an introduction to knowledge representation and nonmonotonic...
TRANSCRIPT
![Page 1: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/1.jpg)
An Introduction toKnowledge Representation
and Nonmonotonic Reasoning
Pedro Cabalar
Depto. ComputaciónUniversity of Corunna, SPAIN
June 29, 2010
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 1 / 123
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Outline
1 Motivation and goals
2 Survey of NMR formalismsCircumscriptionDefault LogicAutoepistemic Logic
3 Answer Set ProgrammingAnswer Set ProgrammingDiagnosisEquilibrium Logic
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 2 / 123
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Motivation and goals
A bit of history
What is Artificial Intelligence?“It is the science and engineering of making intelligentmachines, especially intelligent computer programs”[John McCarthy]
Programs with Commonsense [McCarthy59] introduces the first AIsystem: the Advice Taker.
Keypoint: make an explicit representation of the domain usinglogical formulae. In McCarthy’s words:
“In order for a program to be capable of learningsomething it must first be capable of being told it”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 3 / 123
![Page 4: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/4.jpg)
Motivation and goals
A bit of history
What is Artificial Intelligence?“It is the science and engineering of making intelligentmachines, especially intelligent computer programs”[John McCarthy]
Programs with Commonsense [McCarthy59] introduces the first AIsystem: the Advice Taker.
Keypoint: make an explicit representation of the domain usinglogical formulae. In McCarthy’s words:
“In order for a program to be capable of learningsomething it must first be capable of being told it”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 3 / 123
![Page 5: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/5.jpg)
Motivation and goals
A bit of history
What is Artificial Intelligence?“It is the science and engineering of making intelligentmachines, especially intelligent computer programs”[John McCarthy]
Programs with Commonsense [McCarthy59] introduces the first AIsystem: the Advice Taker.
Keypoint: make an explicit representation of the domain usinglogical formulae.
In McCarthy’s words:“In order for a program to be capable of learningsomething it must first be capable of being told it”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 3 / 123
![Page 6: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/6.jpg)
Motivation and goals
A bit of history
What is Artificial Intelligence?“It is the science and engineering of making intelligentmachines, especially intelligent computer programs”[John McCarthy]
Programs with Commonsense [McCarthy59] introduces the first AIsystem: the Advice Taker.
Keypoint: make an explicit representation of the domain usinglogical formulae. In McCarthy’s words:
“In order for a program to be capable of learningsomething it must first be capable of being told it”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 3 / 123
![Page 7: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/7.jpg)
Motivation and goals
Reasoning about Actions and Change (RAC)
Thus, under this focusing, Knowledge Representation (KR) playsa central role.
Historically, one of the first challenging areas for KR has beenReasoning about Actions and Change.
Some philosophical problems from the standpoint of ArtificialIntelligence [McCarthy & Hayes 69]
They introduce Situation Calculus = First Order Logic + 3 sorts:1 Fluents: system properties whose values may vary along time.
These values configure the system state.2 Actions: possible operations that allow a state transition.3 Situations: terms that identify a given instant. They have the form:
S0 - initial situationResult(a, s) - situation after perfoming action a at situation s.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 4 / 123
![Page 8: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/8.jpg)
Motivation and goals
Reasoning about Actions and Change (RAC)
Thus, under this focusing, Knowledge Representation (KR) playsa central role.
Historically, one of the first challenging areas for KR has beenReasoning about Actions and Change.
Some philosophical problems from the standpoint of ArtificialIntelligence [McCarthy & Hayes 69]
They introduce Situation Calculus = First Order Logic + 3 sorts:1 Fluents: system properties whose values may vary along time.
These values configure the system state.2 Actions: possible operations that allow a state transition.3 Situations: terms that identify a given instant. They have the form:
S0 - initial situationResult(a, s) - situation after perfoming action a at situation s.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 4 / 123
![Page 9: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/9.jpg)
Motivation and goals
Reasoning about Actions and Change (RAC)
Thus, under this focusing, Knowledge Representation (KR) playsa central role.
Historically, one of the first challenging areas for KR has beenReasoning about Actions and Change.
Some philosophical problems from the standpoint of ArtificialIntelligence [McCarthy & Hayes 69]
They introduce Situation Calculus = First Order Logic + 3 sorts:1 Fluents: system properties whose values may vary along time.
These values configure the system state.
2 Actions: possible operations that allow a state transition.3 Situations: terms that identify a given instant. They have the form:
S0 - initial situationResult(a, s) - situation after perfoming action a at situation s.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 4 / 123
![Page 10: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/10.jpg)
Motivation and goals
Reasoning about Actions and Change (RAC)
Thus, under this focusing, Knowledge Representation (KR) playsa central role.
Historically, one of the first challenging areas for KR has beenReasoning about Actions and Change.
Some philosophical problems from the standpoint of ArtificialIntelligence [McCarthy & Hayes 69]
They introduce Situation Calculus = First Order Logic + 3 sorts:1 Fluents: system properties whose values may vary along time.
These values configure the system state.2 Actions: possible operations that allow a state transition.
3 Situations: terms that identify a given instant. They have the form:S0 - initial situationResult(a, s) - situation after perfoming action a at situation s.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 4 / 123
![Page 11: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/11.jpg)
Motivation and goals
Reasoning about Actions and Change (RAC)
Thus, under this focusing, Knowledge Representation (KR) playsa central role.
Historically, one of the first challenging areas for KR has beenReasoning about Actions and Change.
Some philosophical problems from the standpoint of ArtificialIntelligence [McCarthy & Hayes 69]
They introduce Situation Calculus = First Order Logic + 3 sorts:1 Fluents: system properties whose values may vary along time.
These values configure the system state.2 Actions: possible operations that allow a state transition.3 Situations: terms that identify a given instant. They have the form:
S0 - initial situationResult(a, s) - situation after perfoming action a at situation s.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 4 / 123
![Page 12: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/12.jpg)
Motivation and goals
RAC scenarios
Typically, (discrete) dynamic systems: state transitions.
A simple scenario: a lamp in a corridor with 3 switches.
Fluents: up1,up2,up3, light (Boolean).
Actions: toggle1, toggle2, toggle3.
State: a possible configuration of fluent values. Example:{up1,up2,up3, light}.Situation: a moment in time. We can just use 0,1,2, . . .
up1 up2 up3 up1 up2 up3 up1 up2 up3
light light light
toggle1 toggle3
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 5 / 123
![Page 13: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/13.jpg)
Motivation and goals
RAC scenarios
Typically, (discrete) dynamic systems: state transitions.
A simple scenario: a lamp in a corridor with 3 switches.
Fluents: up1,up2,up3, light (Boolean).
Actions: toggle1, toggle2, toggle3.
State: a possible configuration of fluent values. Example:{up1,up2,up3, light}.Situation: a moment in time. We can just use 0,1,2, . . .
up1 up2 up3 up1 up2 up3 up1 up2 up3
light light light
toggle1 toggle3
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 5 / 123
![Page 14: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/14.jpg)
Motivation and goals
RAC goals
We want to solve some typical reasoning problems.
Here is a (non-exhaustive) list of the most usual ones:1 Prediction, simulation or temporal projection2 Postdiction or temporal explanation3 Planning4 Diagnosis5 Checking properties: system properties; representation properties
For instance, in our example . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 6 / 123
![Page 15: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/15.jpg)
Motivation and goals
RAC goals
We want to solve some typical reasoning problems.
Here is a (non-exhaustive) list of the most usual ones:1 Prediction, simulation or temporal projection2 Postdiction or temporal explanation3 Planning4 Diagnosis5 Checking properties: system properties; representation properties
For instance, in our example . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 6 / 123
![Page 16: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/16.jpg)
Motivation and goals
RAC goals
We want to solve some typical reasoning problems.
Here is a (non-exhaustive) list of the most usual ones:1 Prediction, simulation or temporal projection2 Postdiction or temporal explanation3 Planning4 Diagnosis5 Checking properties: system properties; representation properties
For instance, in our example . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 6 / 123
![Page 17: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/17.jpg)
Motivation and goals
Prediction (simulation, or temporal explanation)
Knowing: initial state + sequence of actions
Find out: final state (alternatively sequence of intermediatestates)
up1 up2 up3
light
toggle1 toggle3
? ?
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 7 / 123
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Motivation and goals
Postdiction (or temporal explanation)
Knowing: partial observations of states and performed actions
Find out: complete information on states and performed actions
up3 up1 up3 up1 up2 up3
light light light
toggle3
?
?
??
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 8 / 123
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Motivation and goals
Planning
Knowing: initial state + goal (partial description of final state)
Find out: plan (sequence of actions) that guarantees reaching thegoal
up1 up2 up3 up1 up2 up3
light light
? ??
? ? ?
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 9 / 123
![Page 20: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/20.jpg)
Motivation and goals
Planning vs Postdiction
Note that planning seems a type of postdiction. For deterministicsystems, this is true, but . . .
Nondeterministic transition system: fixing current state +performed action −→ several possible successor states.
For instance, switch 1 when moved up may fail to turn the lighton...
up1 up2 up3
up1 up2 up3
light
light
toggle1
up1 up2 up3
light
toggle1
Switch 1 "failed"
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 10 / 123
![Page 21: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/21.jpg)
Motivation and goals
Planning vs Postdiction
up1 up2 up3
light light
?
? ? ?
For postdiction, one valid explanation is: we performed toggle1,and it succeeded to turn the light on.
For planning, toggle1 is not a valid plan: it does not guaranteereaching the goal light . Possible plans are toggle2 or toggle3.
Planning in nondeterministic systems is more related to abduction.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 11 / 123
![Page 22: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/22.jpg)
Motivation and goals
Planning vs Postdiction
up1 up2 up3
light light
?
? ? ?
For postdiction, one valid explanation is: we performed toggle1,and it succeeded to turn the light on.
For planning, toggle1 is not a valid plan: it does not guaranteereaching the goal light .
Possible plans are toggle2 or toggle3.
Planning in nondeterministic systems is more related to abduction.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 11 / 123
![Page 23: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/23.jpg)
Motivation and goals
Planning vs Postdiction
up1 up2 up3
light light
?
? ? ?
For postdiction, one valid explanation is: we performed toggle1,and it succeeded to turn the light on.
For planning, toggle1 is not a valid plan: it does not guaranteereaching the goal light . Possible plans are toggle2 or toggle3.
Planning in nondeterministic systems is more related to abduction.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 11 / 123
![Page 24: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/24.jpg)
Motivation and goals
Planning vs Postdiction
up1 up2 up3
light light
?
? ? ?
For postdiction, one valid explanation is: we performed toggle1,and it succeeded to turn the light on.
For planning, toggle1 is not a valid plan: it does not guaranteereaching the goal light . Possible plans are toggle2 or toggle3.
Planning in nondeterministic systems is more related to abduction.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 11 / 123
![Page 25: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/25.jpg)
Motivation and goals
Diagnosis
Knowing: a model distinguishing between normal and abnormaltransitions + a partial set of observations (usually implyingabnormal behavior).
Find out: the minimal set of abnormal transitions that explains theobservations.
Similar to postdiction, but we are additionally interested inminimality of explanations.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 12 / 123
![Page 26: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/26.jpg)
Motivation and goals
Diagnosis
Knowing: a model distinguishing between normal and abnormaltransitions + a partial set of observations (usually implyingabnormal behavior).
Find out: the minimal set of abnormal transitions that explains theobservations.
Similar to postdiction, but we are additionally interested inminimality of explanations.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 12 / 123
![Page 27: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/27.jpg)
Motivation and goals
Checking properties
Some elaborated problems are related to checking properties(less common in RAC).
One may be interested in system properties, as in modelchecking:
1 Safety: a given (unsafe) state or condition is never reached.“Something bad never happens”
2 Liveness: after some condition, something will be eventuallyreached. “Something good will eventually happen”.Example: any request is eventually attendedLiveness can only be violated with infinite traces.
3 Fairness: fair resolution of nondeterminism.Example: avoid starvation. A process turn cannot be infinitelydenied.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 13 / 123
![Page 28: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/28.jpg)
Motivation and goals
Checking properties
Some elaborated problems are related to checking properties(less common in RAC).
One may be interested in system properties, as in modelchecking:
1 Safety: a given (unsafe) state or condition is never reached.“Something bad never happens”
2 Liveness: after some condition, something will be eventuallyreached. “Something good will eventually happen”.Example: any request is eventually attendedLiveness can only be violated with infinite traces.
3 Fairness: fair resolution of nondeterminism.Example: avoid starvation. A process turn cannot be infinitelydenied.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 13 / 123
![Page 29: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/29.jpg)
Motivation and goals
Checking properties
Some elaborated problems are related to checking properties(less common in RAC).
One may be interested in system properties, as in modelchecking:
1 Safety: a given (unsafe) state or condition is never reached.“Something bad never happens”
2 Liveness: after some condition, something will be eventuallyreached. “Something good will eventually happen”.
Example: any request is eventually attendedLiveness can only be violated with infinite traces.
3 Fairness: fair resolution of nondeterminism.Example: avoid starvation. A process turn cannot be infinitelydenied.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 13 / 123
![Page 30: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/30.jpg)
Motivation and goals
Checking properties
Some elaborated problems are related to checking properties(less common in RAC).
One may be interested in system properties, as in modelchecking:
1 Safety: a given (unsafe) state or condition is never reached.“Something bad never happens”
2 Liveness: after some condition, something will be eventuallyreached. “Something good will eventually happen”.Example: any request is eventually attended
Liveness can only be violated with infinite traces.3 Fairness: fair resolution of nondeterminism.
Example: avoid starvation. A process turn cannot be infinitelydenied.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 13 / 123
![Page 31: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/31.jpg)
Motivation and goals
Checking properties
Some elaborated problems are related to checking properties(less common in RAC).
One may be interested in system properties, as in modelchecking:
1 Safety: a given (unsafe) state or condition is never reached.“Something bad never happens”
2 Liveness: after some condition, something will be eventuallyreached. “Something good will eventually happen”.Example: any request is eventually attendedLiveness can only be violated with infinite traces.
3 Fairness: fair resolution of nondeterminism.Example: avoid starvation. A process turn cannot be infinitelydenied.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 13 / 123
![Page 32: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/32.jpg)
Motivation and goals
Checking properties
Some elaborated problems are related to checking properties(less common in RAC).
One may be interested in system properties, as in modelchecking:
1 Safety: a given (unsafe) state or condition is never reached.“Something bad never happens”
2 Liveness: after some condition, something will be eventuallyreached. “Something good will eventually happen”.Example: any request is eventually attendedLiveness can only be violated with infinite traces.
3 Fairness: fair resolution of nondeterminism.Example: avoid starvation. A process turn cannot be infinitelydenied.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 13 / 123
![Page 33: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/33.jpg)
Motivation and goals
Checking properties
Or we may be interested in representation properties.
Are two different representations equivalent? That is, do theygenerate the same state transition system?
We will see one stronger concept: are two differentrepresentations strongly equivalent? That is, do they generate thesame system even when included in a common larger description(or context)?
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 14 / 123
![Page 34: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/34.jpg)
Motivation and goals
Checking properties
Or we may be interested in representation properties.
Are two different representations equivalent? That is, do theygenerate the same state transition system?
We will see one stronger concept: are two differentrepresentations strongly equivalent? That is, do they generate thesame system even when included in a common larger description(or context)?
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 14 / 123
![Page 35: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/35.jpg)
Motivation and goals
Checking properties
Or we may be interested in representation properties.
Are two different representations equivalent? That is, do theygenerate the same state transition system?
We will see one stronger concept: are two differentrepresentations strongly equivalent?
That is, do they generate thesame system even when included in a common larger description(or context)?
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 14 / 123
![Page 36: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/36.jpg)
Motivation and goals
Checking properties
Or we may be interested in representation properties.
Are two different representations equivalent? That is, do theygenerate the same state transition system?
We will see one stronger concept: are two differentrepresentations strongly equivalent? That is, do they generate thesame system even when included in a common larger description(or context)?
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 14 / 123
![Page 37: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/37.jpg)
Motivation and goals
Example-based methodology
Paraphrasing McCarthy’s comment in a workshop:AI researchers start from examples and then try to generalize.
Philosophers start from the most general case, and never useexamples unless they are forced to.
Advantage: focus on features under study using a synthetic,limited scenario (games, puzzles, etc)
Real problems usually contain complex factors that happen to beirrelevant for the property under study.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 15 / 123
![Page 38: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/38.jpg)
Motivation and goals
Example-based methodology
Paraphrasing McCarthy’s comment in a workshop:AI researchers start from examples and then try to generalize.Philosophers start from the most general case, and never useexamples unless they are forced to.
Advantage: focus on features under study using a synthetic,limited scenario (games, puzzles, etc)
Real problems usually contain complex factors that happen to beirrelevant for the property under study.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 15 / 123
![Page 39: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/39.jpg)
Motivation and goals
Example-based methodology
Paraphrasing McCarthy’s comment in a workshop:AI researchers start from examples and then try to generalize.Philosophers start from the most general case, and never useexamples unless they are forced to.
Advantage: focus on features under study using a synthetic,limited scenario (games, puzzles, etc)
Real problems usually contain complex factors that happen to beirrelevant for the property under study.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 15 / 123
![Page 40: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/40.jpg)
Motivation and goals
Example-based methodology
Paraphrasing McCarthy’s comment in a workshop:AI researchers start from examples and then try to generalize.Philosophers start from the most general case, and never useexamples unless they are forced to.
Advantage: focus on features under study using a synthetic,limited scenario (games, puzzles, etc)
Real problems usually contain complex factors that happen to beirrelevant for the property under study.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 15 / 123
![Page 41: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/41.jpg)
Motivation and goals
Example-based methodology
A classical (planning) example: the N-puzzle.
4 1 37 2 58 6
1 2 34 5 67 6
? ??
Well known: the 8-puzzle has 181440 sates, the 15-puzzle morethan 1013.
Complexity: NP-complete.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 16 / 123
![Page 42: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/42.jpg)
Motivation and goals
Example-based methodology
A classical (planning) example: the N-puzzle.
4 1 37 2 58 6
1 2 34 5 67 6
? ??
Well known: the 8-puzzle has 181440 sates, the 15-puzzle morethan 1013.
Complexity: NP-complete.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 16 / 123
![Page 43: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/43.jpg)
Motivation and goals
Example-based methodology
A classical (planning) example: the N-puzzle.
4 1 37 2 58 6
1 2 34 5 67 6
? ??
Well known: the 8-puzzle has 181440 sates, the 15-puzzle morethan 1013.
Complexity: NP-complete.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 16 / 123
![Page 44: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/44.jpg)
Motivation and goals
Example-based methodology
An interesting analogy: “Chess is the Drosophila of AI” [A.Kronrod 65]. That is, games for AI can play the same role as fruitflies for Genetics.
Warning: avoid too much focus on the toy problem. Remember wemust be capable of generalizing the obtained results.
Back to the chess example:“Unfortunately, the competitive and commercial aspectsof making computers play chess have taken precedenceover using chess as a scientific domain. It is as if thegeneticists after 1910 had organized fruit fly races andconcentrated their efforts on breeding fruit flies that couldwin these races.” [McCarthy]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 17 / 123
![Page 45: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/45.jpg)
Motivation and goals
Example-based methodology
An interesting analogy: “Chess is the Drosophila of AI” [A.Kronrod 65]. That is, games for AI can play the same role as fruitflies for Genetics.
Warning: avoid too much focus on the toy problem. Remember wemust be capable of generalizing the obtained results.
Back to the chess example:“Unfortunately, the competitive and commercial aspectsof making computers play chess have taken precedenceover using chess as a scientific domain. It is as if thegeneticists after 1910 had organized fruit fly races andconcentrated their efforts on breeding fruit flies that couldwin these races.” [McCarthy]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 17 / 123
![Page 46: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/46.jpg)
Motivation and goals
Example-based methodology
An interesting analogy: “Chess is the Drosophila of AI” [A.Kronrod 65]. That is, games for AI can play the same role as fruitflies for Genetics.
Warning: avoid too much focus on the toy problem. Remember wemust be capable of generalizing the obtained results.
Back to the chess example:“Unfortunately, the competitive and commercial aspectsof making computers play chess have taken precedenceover using chess as a scientific domain. It is as if thegeneticists after 1910 had organized fruit fly races andconcentrated their efforts on breeding fruit flies that couldwin these races.” [McCarthy]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 17 / 123
![Page 47: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/47.jpg)
Motivation and goals
Example-based methodology
Take the 8-puzzle example. Which is our main goal?
Making avery fast solver for 8-puzzle?
But what can we learn from that? Which is the application to otherscenarios?
We should perhaps wonder which other scenarios. Originally, AIgoal was any scenario (General Problem Solver) but was tooambitious.
It could perhaps suffice with similar scenarios. Small variations orelaborations.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 18 / 123
![Page 48: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/48.jpg)
Motivation and goals
Example-based methodology
Take the 8-puzzle example. Which is our main goal? Making avery fast solver for 8-puzzle?
But what can we learn from that? Which is the application to otherscenarios?
We should perhaps wonder which other scenarios. Originally, AIgoal was any scenario (General Problem Solver) but was tooambitious.
It could perhaps suffice with similar scenarios. Small variations orelaborations.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 18 / 123
![Page 49: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/49.jpg)
Motivation and goals
Example-based methodology
Take the 8-puzzle example. Which is our main goal? Making avery fast solver for 8-puzzle?
But what can we learn from that? Which is the application to otherscenarios?
We should perhaps wonder which other scenarios. Originally, AIgoal was any scenario (General Problem Solver) but was tooambitious.
It could perhaps suffice with similar scenarios. Small variations orelaborations.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 18 / 123
![Page 50: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/50.jpg)
Motivation and goals
Example-based methodology
Take the 8-puzzle example. Which is our main goal? Making avery fast solver for 8-puzzle?
But what can we learn from that? Which is the application to otherscenarios?
We should perhaps wonder which other scenarios. Originally, AIgoal was any scenario (General Problem Solver) but was tooambitious.
It could perhaps suffice with similar scenarios. Small variations orelaborations.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 18 / 123
![Page 51: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/51.jpg)
Motivation and goals
Example-based methodology
Take the 8-puzzle example. Which is our main goal? Making avery fast solver for 8-puzzle?
But what can we learn from that? Which is the application to otherscenarios?
We should perhaps wonder which other scenarios. Originally, AIgoal was any scenario (General Problem Solver) but was tooambitious.
It could perhaps suffice with similar scenarios. Small variations orelaborations.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 18 / 123
![Page 52: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/52.jpg)
Motivation and goals
Elaboration
Example: assume we may allow now two holes.
1 25 7 34 6
? ?? 1 25 7
34 6
Less steps to solve. We can even allow simultaneous movements.Can we easily adapt our solver to this elaboration?
Think about an optimized heuristic search algorithm programmedin C, for instance.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 19 / 123
![Page 53: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/53.jpg)
Motivation and goals
Elaboration
Example: assume we may allow now two holes.
1 25 7 34 6
? ?? 1 25 7
34 6
Less steps to solve. We can even allow simultaneous movements.
Can we easily adapt our solver to this elaboration?
Think about an optimized heuristic search algorithm programmedin C, for instance.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 19 / 123
![Page 54: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/54.jpg)
Motivation and goals
Elaboration
Example: assume we may allow now two holes.
1 25 7 34 6
? ?? 1 25 7
34 6
Less steps to solve. We can even allow simultaneous movements.Can we easily adapt our solver to this elaboration?
Think about an optimized heuristic search algorithm programmedin C, for instance.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 19 / 123
![Page 55: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/55.jpg)
Motivation and goals
Keypoint: representation
A much more flexible solution:add a description of the scenario as an input to our solver.
In this way, variations of the scenario would mean changing theproblem description . . . Knowledge Representation (KR) is crucial!
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 20 / 123
![Page 56: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/56.jpg)
Motivation and goals
Keypoint: representation
A much more flexible solution:add a description of the scenario as an input to our solver.
In this way, variations of the scenario would mean changing theproblem description . . .
Knowledge Representation (KR) is crucial!
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 20 / 123
![Page 57: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/57.jpg)
Motivation and goals
Keypoint: representation
A much more flexible solution:add a description of the scenario as an input to our solver.
In this way, variations of the scenario would mean changing theproblem description . . . Knowledge Representation (KR) is crucial!
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 20 / 123
![Page 58: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/58.jpg)
Motivation and goals
Keypoint: representation
Which are the desirable properties of a good KR?1 Simplicity
2 Natural understanding: clear semantics3 Allows automated reasoning methods that:
are efficientor at least, their complexity can be assessed
4 Elaboration tolerance [McCarthy98]
“A formalism is elaboration tolerant to the extent that it isconvenient to modify a set of facts expressed in theformalism to take into account new phenomena orchanged circumstances.” [McCarthy98]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 21 / 123
![Page 59: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/59.jpg)
Motivation and goals
Keypoint: representation
Which are the desirable properties of a good KR?1 Simplicity2 Natural understanding: clear semantics
3 Allows automated reasoning methods that:are efficientor at least, their complexity can be assessed
4 Elaboration tolerance [McCarthy98]
“A formalism is elaboration tolerant to the extent that it isconvenient to modify a set of facts expressed in theformalism to take into account new phenomena orchanged circumstances.” [McCarthy98]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 21 / 123
![Page 60: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/60.jpg)
Motivation and goals
Keypoint: representation
Which are the desirable properties of a good KR?1 Simplicity2 Natural understanding: clear semantics3 Allows automated reasoning methods that:
are efficientor at least, their complexity can be assessed
4 Elaboration tolerance [McCarthy98]
“A formalism is elaboration tolerant to the extent that it isconvenient to modify a set of facts expressed in theformalism to take into account new phenomena orchanged circumstances.” [McCarthy98]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 21 / 123
![Page 61: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/61.jpg)
Motivation and goals
Keypoint: representation
Which are the desirable properties of a good KR?1 Simplicity2 Natural understanding: clear semantics3 Allows automated reasoning methods that:
are efficientor at least, their complexity can be assessed
4 Elaboration tolerance [McCarthy98]
“A formalism is elaboration tolerant to the extent that it isconvenient to modify a set of facts expressed in theformalism to take into account new phenomena orchanged circumstances.” [McCarthy98]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 21 / 123
![Page 62: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/62.jpg)
Motivation and goals
Keypoint: representation
Example of representation: an automaton is simple, and has aclear semantics . . .
But fails in elaboration tolerance! A small change (say, adding newswitches or lamps) means a complete rebuilding
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 22 / 123
![Page 63: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/63.jpg)
Motivation and goals
Keypoint: representation
Example of representation: an automaton is simple, and has aclear semantics . . .But fails in elaboration tolerance! A small change (say, adding newswitches or lamps) means a complete rebuilding
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 22 / 123
![Page 64: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/64.jpg)
Motivation and goals
Keypoint: representation
A practical alternative: use rules to describe the local effects ofeach performed action.
For each switch X ∈ {1,2,3}
Action precondition ⇒ effect(s)
toggle(X ) : up(X ) ⇒ up(X )
toggle(X ) : up(X ) ⇒ up(X )
toggle(X ) : light ⇒ light
toggle(X ) : light ⇒ light
This language is similar to STRIPS [Fikes & Nilsson 71] still usedin planning systems.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 23 / 123
![Page 65: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/65.jpg)
Motivation and goals
Keypoint: representation
A practical alternative: use rules to describe the local effects ofeach performed action.
For each switch X ∈ {1,2,3}
Action precondition ⇒ effect(s)
toggle(X ) : up(X ) ⇒ up(X )
toggle(X ) : up(X ) ⇒ up(X )
toggle(X ) : light ⇒ light
toggle(X ) : light ⇒ light
This language is similar to STRIPS [Fikes & Nilsson 71] still usedin planning systems.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 23 / 123
![Page 66: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/66.jpg)
Motivation and goals
Keypoint: representation
A practical alternative: use rules to describe the local effects ofeach performed action.
For each switch X ∈ {1,2,3}
Action precondition ⇒ effect(s)
toggle(X ) : up(X ) ⇒ up(X )
toggle(X ) : up(X ) ⇒ up(X )
toggle(X ) : light ⇒ light
toggle(X ) : light ⇒ light
This language is similar to STRIPS [Fikes & Nilsson 71] still usedin planning systems.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 23 / 123
![Page 67: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/67.jpg)
Motivation and goals
Logical Knowledge Representation
Can we just use classical logic instead?
toggle(X ) : up(X ) ⇒ up(X )
toggle(X , I) ∧ up(X , true, I) → up(X , false, I + 1)
where we include as new arguments, the temporal indices I, I + 1plus the fluent values true, false.
Problem: when toggle(1), what can we conclude about up(2) andup(3)?They should remain unchanged! However, our logical theoryprovides no information (we also have models where their valuechange).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 24 / 123
![Page 68: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/68.jpg)
Motivation and goals
Logical Knowledge Representation
Can we just use classical logic instead?
toggle(X ) : up(X ) ⇒ up(X )
toggle(X , I) ∧ up(X , true, I) → up(X , false, I + 1)
where we include as new arguments, the temporal indices I, I + 1plus the fluent values true, false.Problem: when toggle(1), what can we conclude about up(2) andup(3)?
They should remain unchanged! However, our logical theoryprovides no information (we also have models where their valuechange).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 24 / 123
![Page 69: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/69.jpg)
Motivation and goals
Logical Knowledge Representation
Can we just use classical logic instead?
toggle(X ) : up(X ) ⇒ up(X )
toggle(X , I) ∧ up(X , true, I) → up(X , false, I + 1)
where we include as new arguments, the temporal indices I, I + 1plus the fluent values true, false.Problem: when toggle(1), what can we conclude about up(2) andup(3)?They should remain unchanged! However, our logical theoryprovides no information (we also have models where their valuechange).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 24 / 123
![Page 70: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/70.jpg)
Motivation and goals
Logical Knowledge Representation
We would need much more formulae
toggle(1, I) ∧ up(2, true, I) → up(2, true, I + 1)
toggle(1, I) ∧ up(2, false, I) → up(2, false, I + 1)
toggle(1, I) ∧ up(3, true, I) → up(3, true, I + 1)
toggle(1, I) ∧ up(3, false, I) → up(3, false, I + 1)
...
and so on, for any fluent and value that are unrelated to toggle(1).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 25 / 123
![Page 71: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/71.jpg)
Motivation and goals
Logical Knowledge Representation
We would need much more formulae
toggle(1, I) ∧ up(2, true, I) → up(2, true, I + 1)
toggle(1, I) ∧ up(2, false, I) → up(2, false, I + 1)
toggle(1, I) ∧ up(3, true, I) → up(3, true, I + 1)
toggle(1, I) ∧ up(3, false, I) → up(3, false, I + 1)
...
and so on, for any fluent and value that are unrelated to toggle(1).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 25 / 123
![Page 72: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/72.jpg)
Motivation and goals
Default reasoning
Frame problem: adding a simple fluent or action meansreformulating all these formulae! [McCarthy & Hayes 69]
We need a kind of default reasoning.Inertia rule: fluents remain unchanged by default
“By default” = when no evidence on the contrary is available. Wemust extract conclusions from absence of information.
Unfortunately, Classical Logic is not well suited for this purposebecause
Γ ` α implies Γ ∪∆ ` α
This is called monotonic consequence relation.
But Γ ` α by default could mean that adding ∆, Γ ∪∆ 6` α.We need Nonmonotonic Reasoning (NMR).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 26 / 123
![Page 73: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/73.jpg)
Motivation and goals
Default reasoning
Frame problem: adding a simple fluent or action meansreformulating all these formulae! [McCarthy & Hayes 69]
We need a kind of default reasoning.Inertia rule: fluents remain unchanged by default
“By default” = when no evidence on the contrary is available. Wemust extract conclusions from absence of information.
Unfortunately, Classical Logic is not well suited for this purposebecause
Γ ` α implies Γ ∪∆ ` α
This is called monotonic consequence relation.
But Γ ` α by default could mean that adding ∆, Γ ∪∆ 6` α.We need Nonmonotonic Reasoning (NMR).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 26 / 123
![Page 74: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/74.jpg)
Motivation and goals
Default reasoning
Frame problem: adding a simple fluent or action meansreformulating all these formulae! [McCarthy & Hayes 69]
We need a kind of default reasoning.Inertia rule: fluents remain unchanged by default
“By default” = when no evidence on the contrary is available. Wemust extract conclusions from absence of information.
Unfortunately, Classical Logic is not well suited for this purposebecause
Γ ` α implies Γ ∪∆ ` α
This is called monotonic consequence relation.
But Γ ` α by default could mean that adding ∆, Γ ∪∆ 6` α.We need Nonmonotonic Reasoning (NMR).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 26 / 123
![Page 75: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/75.jpg)
Motivation and goals
Default reasoning
Frame problem: adding a simple fluent or action meansreformulating all these formulae! [McCarthy & Hayes 69]
We need a kind of default reasoning.Inertia rule: fluents remain unchanged by default
“By default” = when no evidence on the contrary is available. Wemust extract conclusions from absence of information.
Unfortunately, Classical Logic is not well suited for this purposebecause
Γ ` α implies Γ ∪∆ ` α
This is called monotonic consequence relation.
But Γ ` α by default could mean that adding ∆, Γ ∪∆ 6` α.We need Nonmonotonic Reasoning (NMR).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 26 / 123
![Page 76: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/76.jpg)
Motivation and goals
Default reasoning
Frame problem: adding a simple fluent or action meansreformulating all these formulae! [McCarthy & Hayes 69]
We need a kind of default reasoning.Inertia rule: fluents remain unchanged by default
“By default” = when no evidence on the contrary is available. Wemust extract conclusions from absence of information.
Unfortunately, Classical Logic is not well suited for this purposebecause
Γ ` α implies Γ ∪∆ ` α
This is called monotonic consequence relation.
But Γ ` α by default could mean that adding ∆, Γ ∪∆ 6` α.We need Nonmonotonic Reasoning (NMR).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 26 / 123
![Page 77: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/77.jpg)
Motivation and goals
Default reasoning
Frame problem: adding a simple fluent or action meansreformulating all these formulae! [McCarthy & Hayes 69]
We need a kind of default reasoning.Inertia rule: fluents remain unchanged by default
“By default” = when no evidence on the contrary is available. Wemust extract conclusions from absence of information.
Unfortunately, Classical Logic is not well suited for this purposebecause
Γ ` α implies Γ ∪∆ ` α
This is called monotonic consequence relation.
But Γ ` α by default could mean that adding ∆, Γ ∪∆ 6` α.
We need Nonmonotonic Reasoning (NMR).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 26 / 123
![Page 78: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/78.jpg)
Motivation and goals
Default reasoning
Frame problem: adding a simple fluent or action meansreformulating all these formulae! [McCarthy & Hayes 69]
We need a kind of default reasoning.Inertia rule: fluents remain unchanged by default
“By default” = when no evidence on the contrary is available. Wemust extract conclusions from absence of information.
Unfortunately, Classical Logic is not well suited for this purposebecause
Γ ` α implies Γ ∪∆ ` α
This is called monotonic consequence relation.
But Γ ` α by default could mean that adding ∆, Γ ∪∆ 6` α.We need Nonmonotonic Reasoning (NMR).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 26 / 123
![Page 79: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/79.jpg)
Motivation and goals
Default reasoning
An example: suppose up(2, true,0) and we perform toggle(1,0).Inertia should allow us to conclude that switch 2 is unaffected:
Γ ` up(2, true,1)
Elaboration: we are said now that toggle(1) affects up(2) in thefollowing way:
toggle(1, I) ∧ up(2, true, I)→ up(2, false, I + 1) (1)
We will need retract our previous conclusion
Γ ∪ (1) 6` up(2, true,1)
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 27 / 123
![Page 80: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/80.jpg)
Motivation and goals
Default reasoning
An example: suppose up(2, true,0) and we perform toggle(1,0).Inertia should allow us to conclude that switch 2 is unaffected:
Γ ` up(2, true,1)
Elaboration: we are said now that toggle(1) affects up(2) in thefollowing way:
toggle(1, I) ∧ up(2, true, I)→ up(2, false, I + 1) (1)
We will need retract our previous conclusion
Γ ∪ (1) 6` up(2, true,1)
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 27 / 123
![Page 81: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/81.jpg)
Motivation and goals
Other typical representational problems
Qualification problem: preconditions are affected by conditionsthat qualify an action.
Example: when can we toggle the switch? Elaborations: switch isnot broken, switch has not been stuck, we must be close enough,etc.
The explicit addition of any imaginable “disqualification” isunfeasible. Again: by default, toggle works when nothing preventsit.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 28 / 123
![Page 82: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/82.jpg)
Motivation and goals
Other typical representational problems
Qualification problem: preconditions are affected by conditionsthat qualify an action.
Example: when can we toggle the switch?
Elaborations: switch isnot broken, switch has not been stuck, we must be close enough,etc.
The explicit addition of any imaginable “disqualification” isunfeasible. Again: by default, toggle works when nothing preventsit.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 28 / 123
![Page 83: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/83.jpg)
Motivation and goals
Other typical representational problems
Qualification problem: preconditions are affected by conditionsthat qualify an action.
Example: when can we toggle the switch? Elaborations: switch isnot broken, switch has not been stuck, we must be close enough,etc.
The explicit addition of any imaginable “disqualification” isunfeasible. Again: by default, toggle works when nothing preventsit.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 28 / 123
![Page 84: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/84.jpg)
Motivation and goals
Other typical representational problems
Qualification problem: preconditions are affected by conditionsthat qualify an action.
Example: when can we toggle the switch? Elaborations: switch isnot broken, switch has not been stuck, we must be close enough,etc.
The explicit addition of any imaginable “disqualification” isunfeasible.
Again: by default, toggle works when nothing preventsit.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 28 / 123
![Page 85: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/85.jpg)
Motivation and goals
Other typical representational problems
Qualification problem: preconditions are affected by conditionsthat qualify an action.
Example: when can we toggle the switch? Elaborations: switch isnot broken, switch has not been stuck, we must be close enough,etc.
The explicit addition of any imaginable “disqualification” isunfeasible. Again: by default, toggle works when nothing preventsit.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 28 / 123
![Page 86: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/86.jpg)
Motivation and goals
Other typical representational problems
Elaboration: there is a light sensor that activates an alarm, if thelatter is connected.
The alarm causes locking the door.
In STRIPS, this means relating indirect effects alarm to eachpossible action toggle(X ).
Action precondition ⇒ effect(s)
toggle(X ) : light , connected ⇒ alarm
toggle(X ) : light , connected ⇒ lock
Problem: there may be other new ways to turn on a light, or toactivate the alarm. We will be forced to relate lock to theperformed actions!
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 29 / 123
![Page 87: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/87.jpg)
Motivation and goals
Other typical representational problems
Elaboration: there is a light sensor that activates an alarm, if thelatter is connected. The alarm causes locking the door.
In STRIPS, this means relating indirect effects alarm to eachpossible action toggle(X ).
Action precondition ⇒ effect(s)
toggle(X ) : light , connected ⇒ alarm
toggle(X ) : light , connected ⇒ lock
Problem: there may be other new ways to turn on a light, or toactivate the alarm. We will be forced to relate lock to theperformed actions!
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 29 / 123
![Page 88: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/88.jpg)
Motivation and goals
Other typical representational problems
Elaboration: there is a light sensor that activates an alarm, if thelatter is connected. The alarm causes locking the door.
In STRIPS, this means relating indirect effects alarm to eachpossible action toggle(X ).
Action precondition ⇒ effect(s)
toggle(X ) : light , connected ⇒ alarm
toggle(X ) : light , connected ⇒ lock
Problem: there may be other new ways to turn on a light, or toactivate the alarm. We will be forced to relate lock to theperformed actions!
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 29 / 123
![Page 89: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/89.jpg)
Motivation and goals
Other typical representational problems
This is called ramification problem: postconditions are affected byinteractions due to indirect effects.
lock is an indirect effect of toggling a switch(toggle 7→ light 7→ alarm 7→ lock ).We would need something like:
light(true, I) ∧ connected(true, I) → alarm(true, I)alarm(true, I) → lock(true, I)
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 30 / 123
![Page 90: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/90.jpg)
Motivation and goals
Other typical representational problems
This is called ramification problem: postconditions are affected byinteractions due to indirect effects.
lock is an indirect effect of toggling a switch(toggle 7→ light 7→ alarm 7→ lock ).We would need something like:
light(true, I) ∧ connected(true, I) → alarm(true, I)alarm(true, I) → lock(true, I)
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 30 / 123
![Page 91: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/91.jpg)
Survey of NMR formalisms
NMR formalisms
Starting from a monotonic framework (Classical Logic, forinstance) nonmonotonicity can be obtained in different ways.
Semantic way: establishing a models selection criterion.Sel(M) ⊆ M.
Γ |= α means Models(Γ) ⊆ Models(α).
Adding ∆, we must have Γ ∪∆ |= α. Proof:
Models(Γ∪∆) = Models(Γ)∩Models(∆) ⊆ Models(Γ) ⊆ Models(α).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 31 / 123
![Page 92: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/92.jpg)
Survey of NMR formalisms
NMR formalisms
Starting from a monotonic framework (Classical Logic, forinstance) nonmonotonicity can be obtained in different ways.
Semantic way: establishing a models selection criterion.Sel(M) ⊆ M.
Γ |= α means Models(Γ) ⊆ Models(α).
Adding ∆, we must have Γ ∪∆ |= α. Proof:
Models(Γ∪∆) = Models(Γ)∩Models(∆) ⊆ Models(Γ) ⊆ Models(α).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 31 / 123
![Page 93: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/93.jpg)
Survey of NMR formalisms
NMR formalisms
Starting from a monotonic framework (Classical Logic, forinstance) nonmonotonicity can be obtained in different ways.
Semantic way: establishing a models selection criterion.Sel(M) ⊆ M.
Γ |= α means Models(Γ) ⊆ Models(α).
Adding ∆, we must have Γ ∪∆ |= α. Proof:
Models(Γ∪∆) = Models(Γ)∩Models(∆) ⊆ Models(Γ) ⊆ Models(α).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 31 / 123
![Page 94: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/94.jpg)
Survey of NMR formalisms
NMR formalisms
Starting from a monotonic framework (Classical Logic, forinstance) nonmonotonicity can be obtained in different ways.
Semantic way: establishing a models selection criterion.Sel(M) ⊆ M.
Γ |= α means Models(Γ) ⊆ Models(α).
Adding ∆, we must have Γ ∪∆ |= α. Proof:
Models(Γ∪∆) = Models(Γ)∩Models(∆) ⊆ Models(Γ) ⊆ Models(α).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 31 / 123
![Page 95: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/95.jpg)
Survey of NMR formalisms
NMR formalisms
Starting from a monotonic framework (Classical Logic, forinstance) nonmonotonicity can be obtained in different ways.
Semantic way: establishing a models selection criterion.Sel(M) ⊆ M.
Γ |= α means Models(Γ) ⊆ Models(α).
Adding ∆, we must have Γ ∪∆ |= α. Proof:
Models(Γ∪∆) = Models(Γ)∩Models(∆) ⊆ Models(Γ) ⊆ Models(α).
!
"
#
! U #
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 31 / 123
![Page 96: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/96.jpg)
Survey of NMR formalisms
NMR formalisms
When selecting models Γ |∼ α meansSel(Models(Γ)) ⊆ Models(α).
Now, it may be the case that:Sel(Models(Γ)) ⊆ Models(α), that is, Γ |∼ αSel(Models(Γ ∪∆)) 6⊆ Models(α), that is, Γ ∪∆ |∼/ α.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 32 / 123
![Page 97: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/97.jpg)
Survey of NMR formalisms
NMR formalisms
When selecting models Γ |∼ α meansSel(Models(Γ)) ⊆ Models(α).
Now, it may be the case that:Sel(Models(Γ)) ⊆ Models(α), that is, Γ |∼ αSel(Models(Γ ∪∆)) 6⊆ Models(α), that is, Γ ∪∆ |∼/ α.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 32 / 123
![Page 98: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/98.jpg)
Survey of NMR formalisms
NMR formalisms
When selecting models Γ |∼ α meansSel(Models(Γ)) ⊆ Models(α).
Now, it may be the case that:Sel(Models(Γ)) ⊆ Models(α), that is, Γ |∼ αSel(Models(Γ ∪∆)) 6⊆ Models(α), that is, Γ ∪∆ |∼/ α.
!
"
#
! U #
Sel(!)
Sel(!U#)
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 32 / 123
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Survey of NMR formalisms
NMR formalisms
Syntactic way: we have Γ and want to complete its consequencesin an extended theory E .
We typically use a fixpoint condition of the form
Γ ∪ Assmp(E) ` E
where Assmp(E) is a set of assumptions (formulae) we fix usingE itself.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 33 / 123
![Page 100: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/100.jpg)
Survey of NMR formalisms
NMR formalisms
Syntactic way: we have Γ and want to complete its consequencesin an extended theory E .
We typically use a fixpoint condition of the form
Γ ∪ Assmp(E) ` E
where Assmp(E) is a set of assumptions (formulae) we fix usingE itself.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 33 / 123
![Page 101: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/101.jpg)
Survey of NMR formalisms
NMR formalisms
To show its nonmonotonicity, suppose our theory is Γ = {p → q}and we can always assume p, that is:Assmp(E)
def= {p} if p ∈ E ; ∅ otherwise.
E = Cn({p,q}) is an extension because Γ ∪ {p} ` {p,q}.
If we add the information ∆ = {¬q}Cn({p,q}) is not an extension becauseΓ ∪∆ ∪ {p} ` ⊥ 6= Cn({p,q}).Cn({¬p,¬q}) is an extension because Γ ∪∆ ∪ ∅ ` {¬p,¬q}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 34 / 123
![Page 102: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/102.jpg)
Survey of NMR formalisms
NMR formalisms
To show its nonmonotonicity, suppose our theory is Γ = {p → q}and we can always assume p, that is:Assmp(E)
def= {p} if p ∈ E ; ∅ otherwise.
E = Cn({p,q}) is an extension because Γ ∪ {p} ` {p,q}.
If we add the information ∆ = {¬q}Cn({p,q}) is not an extension becauseΓ ∪∆ ∪ {p} ` ⊥ 6= Cn({p,q}).Cn({¬p,¬q}) is an extension because Γ ∪∆ ∪ ∅ ` {¬p,¬q}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 34 / 123
![Page 103: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/103.jpg)
Survey of NMR formalisms
NMR formalisms
To show its nonmonotonicity, suppose our theory is Γ = {p → q}and we can always assume p, that is:Assmp(E)
def= {p} if p ∈ E ; ∅ otherwise.
E = Cn({p,q}) is an extension because Γ ∪ {p} ` {p,q}.
If we add the information ∆ = {¬q}Cn({p,q}) is not an extension becauseΓ ∪∆ ∪ {p} ` ⊥ 6= Cn({p,q}).
Cn({¬p,¬q}) is an extension because Γ ∪∆ ∪ ∅ ` {¬p,¬q}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 34 / 123
![Page 104: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/104.jpg)
Survey of NMR formalisms
NMR formalisms
To show its nonmonotonicity, suppose our theory is Γ = {p → q}and we can always assume p, that is:Assmp(E)
def= {p} if p ∈ E ; ∅ otherwise.
E = Cn({p,q}) is an extension because Γ ∪ {p} ` {p,q}.
If we add the information ∆ = {¬q}Cn({p,q}) is not an extension becauseΓ ∪∆ ∪ {p} ` ⊥ 6= Cn({p,q}).Cn({¬p,¬q}) is an extension because Γ ∪∆ ∪ ∅ ` {¬p,¬q}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 34 / 123
![Page 105: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/105.jpg)
Survey of NMR formalisms Circumscription
Outline
1 Motivation and goals
2 Survey of NMR formalismsCircumscriptionDefault LogicAutoepistemic Logic
3 Answer Set ProgrammingAnswer Set ProgrammingDiagnosisEquilibrium Logic
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 35 / 123
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Survey of NMR formalisms Circumscription
Circumscription
Circumscription [McCarthy 80] is one of the most popular NMRformalisms. Many variants exist.
Keypoint: try to minimize the extent of some predicate P.
Let M1 and M2 be two first order models. We define M1 ≤P M2when:
1 M1 and M2 coincide in their universes and valuations for everythingexcepting P and
2 M1[P] ⊆ M2[P].
We want our selected models to be the ≤P minimal models.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 36 / 123
![Page 107: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/107.jpg)
Survey of NMR formalisms Circumscription
Circumscription
Circumscription [McCarthy 80] is one of the most popular NMRformalisms. Many variants exist.
Keypoint: try to minimize the extent of some predicate P.
Let M1 and M2 be two first order models. We define M1 ≤P M2when:
1 M1 and M2 coincide in their universes and valuations for everythingexcepting P and
2 M1[P] ⊆ M2[P].
We want our selected models to be the ≤P minimal models.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 36 / 123
![Page 108: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/108.jpg)
Survey of NMR formalisms Circumscription
Circumscription
Circumscription [McCarthy 80] is one of the most popular NMRformalisms. Many variants exist.
Keypoint: try to minimize the extent of some predicate P.
Let M1 and M2 be two first order models. We define M1 ≤P M2when:
1 M1 and M2 coincide in their universes and valuations for everythingexcepting P and
2 M1[P] ⊆ M2[P].
We want our selected models to be the ≤P minimal models.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 36 / 123
![Page 109: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/109.jpg)
Survey of NMR formalisms Circumscription
Circumscription
Circumscription [McCarthy 80] is one of the most popular NMRformalisms. Many variants exist.
Keypoint: try to minimize the extent of some predicate P.
Let M1 and M2 be two first order models. We define M1 ≤P M2when:
1 M1 and M2 coincide in their universes and valuations for everythingexcepting P and
2 M1[P] ⊆ M2[P].
We want our selected models to be the ≤P minimal models.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 36 / 123
![Page 110: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/110.jpg)
Survey of NMR formalisms Circumscription
Circumscription
This semantic characterisation is equivalent to a Second OrderLogic definition.
Let us introduce the notation:
p ≤ P def= ∀x (p(x)→ P(x))
p = P def= ∀x (p(x)↔ P(x))
p < P def= (p ≤ P) ∧ ¬(p = P)
Then, we define:
CIRC[Γ; P]def= Γ(P) ∧ ¬∃p (Γ(p) ∧ p < P)
Theorem: models of CIRC[Γ; P] are the ≤P-minimal models of Γ[Lifschitz93]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 37 / 123
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Survey of NMR formalisms Circumscription
Circumscription
This semantic characterisation is equivalent to a Second OrderLogic definition.
Let us introduce the notation:
p ≤ P def= ∀x (p(x)→ P(x))
p = P def= ∀x (p(x)↔ P(x))
p < P def= (p ≤ P) ∧ ¬(p = P)
Then, we define:
CIRC[Γ; P]def= Γ(P) ∧ ¬∃p (Γ(p) ∧ p < P)
Theorem: models of CIRC[Γ; P] are the ≤P-minimal models of Γ[Lifschitz93]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 37 / 123
![Page 112: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/112.jpg)
Survey of NMR formalisms Circumscription
Circumscription
This semantic characterisation is equivalent to a Second OrderLogic definition.
Let us introduce the notation:
p ≤ P def= ∀x (p(x)→ P(x))
p = P def= ∀x (p(x)↔ P(x))
p < P def= (p ≤ P) ∧ ¬(p = P)
Then, we define:
CIRC[Γ; P]def= Γ(P) ∧ ¬∃p (Γ(p) ∧ p < P)
Theorem: models of CIRC[Γ; P] are the ≤P-minimal models of Γ[Lifschitz93]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 37 / 123
![Page 113: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/113.jpg)
Survey of NMR formalisms Circumscription
Circumscription
This semantic characterisation is equivalent to a Second OrderLogic definition.
Let us introduce the notation:
p ≤ P def= ∀x (p(x)→ P(x))
p = P def= ∀x (p(x)↔ P(x))
p < P def= (p ≤ P) ∧ ¬(p = P)
Then, we define:
CIRC[Γ; P]def= Γ(P) ∧ ¬∃p (Γ(p) ∧ p < P)
Theorem: models of CIRC[Γ; P] are the ≤P-minimal models of Γ[Lifschitz93]
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 37 / 123
![Page 114: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/114.jpg)
Survey of NMR formalisms Circumscription
Circumscription
How does it work? Freely decide all constants and then minimizethe extent of P.
For default reasoning: the purpose is minimizing someabnormality predicate Ab. But fixed circumscription has someproblems . . .
Example of default: Germans normally drink beer
∀x (German(x) ∧ ¬Ab(x)→ Drinks(x ,Beer))
German(Peter)
We get two ≤Ab minimal models:{¬Ab(Peter),Drinks(Peter ,Beer)} but also{¬Drinks(Peter ,Beer),Ab(Peter)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 38 / 123
![Page 115: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/115.jpg)
Survey of NMR formalisms Circumscription
Circumscription
How does it work? Freely decide all constants and then minimizethe extent of P.
For default reasoning: the purpose is minimizing someabnormality predicate Ab.
But fixed circumscription has someproblems . . .
Example of default: Germans normally drink beer
∀x (German(x) ∧ ¬Ab(x)→ Drinks(x ,Beer))
German(Peter)
We get two ≤Ab minimal models:{¬Ab(Peter),Drinks(Peter ,Beer)} but also{¬Drinks(Peter ,Beer),Ab(Peter)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 38 / 123
![Page 116: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/116.jpg)
Survey of NMR formalisms Circumscription
Circumscription
How does it work? Freely decide all constants and then minimizethe extent of P.
For default reasoning: the purpose is minimizing someabnormality predicate Ab. But fixed circumscription has someproblems . . .
Example of default: Germans normally drink beer
∀x (German(x) ∧ ¬Ab(x)→ Drinks(x ,Beer))
German(Peter)
We get two ≤Ab minimal models:{¬Ab(Peter),Drinks(Peter ,Beer)} but also{¬Drinks(Peter ,Beer),Ab(Peter)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 38 / 123
![Page 117: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/117.jpg)
Survey of NMR formalisms Circumscription
Circumscription
How does it work? Freely decide all constants and then minimizethe extent of P.
For default reasoning: the purpose is minimizing someabnormality predicate Ab. But fixed circumscription has someproblems . . .
Example of default: Germans normally drink beer
∀x (German(x) ∧ ¬Ab(x)→ Drinks(x ,Beer))
German(Peter)
We get two ≤Ab minimal models:{¬Ab(Peter),Drinks(Peter ,Beer)} but also{¬Drinks(Peter ,Beer),Ab(Peter)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 38 / 123
![Page 118: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/118.jpg)
Survey of NMR formalisms Circumscription
Circumscription
How does it work? Freely decide all constants and then minimizethe extent of P.
For default reasoning: the purpose is minimizing someabnormality predicate Ab. But fixed circumscription has someproblems . . .
Example of default: Germans normally drink beer
∀x (German(x) ∧ ¬Ab(x)→ Drinks(x ,Beer))
German(Peter)
We get two ≤Ab minimal models:{¬Ab(Peter),Drinks(Peter ,Beer)} but also{¬Drinks(Peter ,Beer),Ab(Peter)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 38 / 123
![Page 119: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/119.jpg)
Survey of NMR formalisms Circumscription
Variable Circumscription
Variable circumscription: allow some tuple of predicates Z to vary
CIRC[Γ; P; Z ]def= Γ(P,Z ) ∧ ¬∃p, z (Γ(p, z) ∧ p < P)
The models minimization is now M1 ≤P,Z M2 when1 M1[Q] = M2[Q] for any predicate Q excepting P,Z ;2 M1[P] ⊆ M2[P].
In the example CIRC[Γ; Ab; Drinks]:1 Freely decide the extent of German2 Minimize the extent of Ab3 Get the consequences for Drinks from those Ab-minimal models
We only get one selected model{¬Ab(Peter),Drinks(Peter ,Beer)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 39 / 123
![Page 120: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/120.jpg)
Survey of NMR formalisms Circumscription
Variable Circumscription
Variable circumscription: allow some tuple of predicates Z to vary
CIRC[Γ; P; Z ]def= Γ(P,Z ) ∧ ¬∃p, z (Γ(p, z) ∧ p < P)
The models minimization is now M1 ≤P,Z M2 when1 M1[Q] = M2[Q] for any predicate Q excepting P,Z ;2 M1[P] ⊆ M2[P].
In the example CIRC[Γ; Ab; Drinks]:1 Freely decide the extent of German2 Minimize the extent of Ab3 Get the consequences for Drinks from those Ab-minimal models
We only get one selected model{¬Ab(Peter),Drinks(Peter ,Beer)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 39 / 123
![Page 121: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/121.jpg)
Survey of NMR formalisms Circumscription
Variable Circumscription
Variable circumscription: allow some tuple of predicates Z to vary
CIRC[Γ; P; Z ]def= Γ(P,Z ) ∧ ¬∃p, z (Γ(p, z) ∧ p < P)
The models minimization is now M1 ≤P,Z M2 when1 M1[Q] = M2[Q] for any predicate Q excepting P,Z ;2 M1[P] ⊆ M2[P].
In the example CIRC[Γ; Ab; Drinks]:1 Freely decide the extent of German2 Minimize the extent of Ab3 Get the consequences for Drinks from those Ab-minimal models
We only get one selected model{¬Ab(Peter),Drinks(Peter ,Beer)}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 39 / 123
![Page 122: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/122.jpg)
Survey of NMR formalisms Default Logic
Outline
1 Motivation and goals
2 Survey of NMR formalismsCircumscriptionDefault LogicAutoepistemic Logic
3 Answer Set ProgrammingAnswer Set ProgrammingDiagnosisEquilibrium Logic
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 40 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Default Logic (DL) [Reiter80] has been widely and successfullyused:
1 Original definition is expressive enough: no need for variations.
2 Based on inference rules: directional reasoning solves frame,ramification, qualification problems.
3 Answer Set Programming can be seen as a subset of DL.
Main idea: logics have inference rules
α
γ
meaning “when α has been already proved, the inference ruleallows proving β”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 41 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Default Logic (DL) [Reiter80] has been widely and successfullyused:
1 Original definition is expressive enough: no need for variations.2 Based on inference rules: directional reasoning solves frame,
ramification, qualification problems.
3 Answer Set Programming can be seen as a subset of DL.
Main idea: logics have inference rules
α
γ
meaning “when α has been already proved, the inference ruleallows proving β”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 41 / 123
![Page 125: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/125.jpg)
Survey of NMR formalisms Default Logic
Default Logic
Default Logic (DL) [Reiter80] has been widely and successfullyused:
1 Original definition is expressive enough: no need for variations.2 Based on inference rules: directional reasoning solves frame,
ramification, qualification problems.3 Answer Set Programming can be seen as a subset of DL.
Main idea: logics have inference rules
α
γ
meaning “when α has been already proved, the inference ruleallows proving β”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 41 / 123
![Page 126: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/126.jpg)
Survey of NMR formalisms Default Logic
Default Logic
Default Logic (DL) [Reiter80] has been widely and successfullyused:
1 Original definition is expressive enough: no need for variations.2 Based on inference rules: directional reasoning solves frame,
ramification, qualification problems.3 Answer Set Programming can be seen as a subset of DL.
Main idea: logics have inference rules
α
γ
meaning “when α has been already proved, the inference ruleallows proving β”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 41 / 123
![Page 127: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/127.jpg)
Survey of NMR formalisms Default Logic
Recall: inference methods
Inference or formal proof: we make syntactic manipulation offormulae. To do so, we use:
An initial set of formulae: axioms (usually axiom schemata).
Syntactic manipulation rules: inference rules.
As a result of applying these rules, we go obtaining new formulae:theorems
Best known proof methods:Axiomatic Method (Hilbert)ResolutionNatural Deduction / Sequent Calculus (Gentzen)
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 42 / 123
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Survey of NMR formalisms Default Logic
Recall: inference methods
Inference or formal proof: we make syntactic manipulation offormulae. To do so, we use:
An initial set of formulae: axioms (usually axiom schemata).
Syntactic manipulation rules: inference rules.
As a result of applying these rules, we go obtaining new formulae:theorems
Best known proof methods:Axiomatic Method (Hilbert)ResolutionNatural Deduction / Sequent Calculus (Gentzen)
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 42 / 123
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Survey of NMR formalisms Default Logic
Recall: inference methods
Notation: Γ ` α means that formula α can be derived or inferredfrom theory Γ.
A logic L can be defined in terms of logical axioms and logicalinference rules. “Logical” means that they induce the logic L.
Usually, logical axioms are not represented inside Γ. Thus, ` αmeans that α is a theorem (from logic L).
Given a language L, a logic L is a subset of L. It can be defined:Semantically: L = {α ∈ L | |= α}.Syntactically: L = {α ∈ L | ` α}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 43 / 123
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Survey of NMR formalisms Default Logic
Recall: inference methods
Notation: Γ ` α means that formula α can be derived or inferredfrom theory Γ.
A logic L can be defined in terms of logical axioms and logicalinference rules. “Logical” means that they induce the logic L.
Usually, logical axioms are not represented inside Γ. Thus, ` αmeans that α is a theorem (from logic L).
Given a language L, a logic L is a subset of L. It can be defined:Semantically: L = {α ∈ L | |= α}.Syntactically: L = {α ∈ L | ` α}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 43 / 123
![Page 131: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/131.jpg)
Survey of NMR formalisms Default Logic
Recall: inference methods
Notation: Γ ` α means that formula α can be derived or inferredfrom theory Γ.
A logic L can be defined in terms of logical axioms and logicalinference rules. “Logical” means that they induce the logic L.
Usually, logical axioms are not represented inside Γ. Thus, ` αmeans that α is a theorem (from logic L).
Given a language L, a logic L is a subset of L. It can be defined:Semantically: L = {α ∈ L | |= α}.Syntactically: L = {α ∈ L | ` α}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 43 / 123
![Page 132: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/132.jpg)
Survey of NMR formalisms Default Logic
Recall: inference methods
Notation: Γ ` α means that formula α can be derived or inferredfrom theory Γ.
A logic L can be defined in terms of logical axioms and logicalinference rules. “Logical” means that they induce the logic L.
Usually, logical axioms are not represented inside Γ. Thus, ` αmeans that α is a theorem (from logic L).
Given a language L, a logic L is a subset of L. It can be defined:Semantically: L = {α ∈ L | |= α}.Syntactically: L = {α ∈ L | ` α}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 43 / 123
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Survey of NMR formalisms Default Logic
Recall: inference methods
These are some well-known logical inference rules:Modus Ponens:
α, α→ β
β
Modus tollens
¬β, α→ β
¬α
Hypothetical Syllogism
α→ β, β → γ
α→ γ
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 44 / 123
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Survey of NMR formalisms Default Logic
Recall: inference methods
When we are interested in a particular theory, we usually includenon-logical axioms. Example: PA = set axioms for PeanoArithmetics.
If a logic satisfies the deduction (meta)theorem, we can transforminference with non-logical axioms into implications:
Theorem (Deduction)
Γ ∪ {α} ` β iff Γ ` α→ β.
Classical Logic satisfies the deduction theorem. Example: we cantransform PA ` β into ` PA→ β.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 45 / 123
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Survey of NMR formalisms Default Logic
Recall: inference methods
When we are interested in a particular theory, we usually includenon-logical axioms. Example: PA = set axioms for PeanoArithmetics.
If a logic satisfies the deduction (meta)theorem, we can transforminference with non-logical axioms into implications:
Theorem (Deduction)
Γ ∪ {α} ` β iff Γ ` α→ β.
Classical Logic satisfies the deduction theorem. Example: we cantransform PA ` β into ` PA→ β.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 45 / 123
![Page 136: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/136.jpg)
Survey of NMR formalisms Default Logic
Recall: inference methods
When we are interested in a particular theory, we usually includenon-logical axioms. Example: PA = set axioms for PeanoArithmetics.
If a logic satisfies the deduction (meta)theorem, we can transforminference with non-logical axioms into implications:
Theorem (Deduction)
Γ ∪ {α} ` β iff Γ ` α→ β.
Classical Logic satisfies the deduction theorem. Example: we cantransform PA ` β into ` PA→ β.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 45 / 123
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Survey of NMR formalisms Default Logic
Back to Default Logic
Default logic considers using non-logical inference rules.
light ∧ connectedalarm
Directionality of inference rules. Quite different to an implication
light ∧ connected → alarm (2)
For instance: {¬alarm, (2)} allows concluding¬(light ∧ connected) by Modus Tollens
Replacing (2) by the inference rule does not yield thatconsequence.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 46 / 123
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Survey of NMR formalisms Default Logic
Back to Default Logic
Default logic considers using non-logical inference rules.
light ∧ connectedalarm
Directionality of inference rules. Quite different to an implication
light ∧ connected → alarm (2)
For instance: {¬alarm, (2)} allows concluding¬(light ∧ connected) by Modus Tollens
Replacing (2) by the inference rule does not yield thatconsequence.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 46 / 123
![Page 139: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/139.jpg)
Survey of NMR formalisms Default Logic
Back to Default Logic
Default logic considers using non-logical inference rules.
light ∧ connectedalarm
Directionality of inference rules. Quite different to an implication
light ∧ connected → alarm (2)
For instance: {¬alarm, (2)} allows concluding¬(light ∧ connected) by Modus Tollens
Replacing (2) by the inference rule does not yield thatconsequence.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 46 / 123
![Page 140: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/140.jpg)
Survey of NMR formalisms Default Logic
Back to Default Logic
Default logic considers using non-logical inference rules.
light ∧ connectedalarm
Directionality of inference rules. Quite different to an implication
light ∧ connected → alarm (2)
For instance: {¬alarm, (2)} allows concluding¬(light ∧ connected) by Modus Tollens
Replacing (2) by the inference rule does not yield thatconsequence.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 46 / 123
![Page 141: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/141.jpg)
Survey of NMR formalisms Default Logic
Default Logic
Some definitions:A theory (set of formulas) E is closed wrt a set of inference rulesD iff:
for any α ∈ E andα
β∈ D we also have β ∈ E
E is logically closed if E is closed wrt the set of classical logicalrules (Modus Ponens, etc). This means that E is infinite. Forinstance, from an atom p we derive . . .
E = {p, p ∨ p, p ∨ ¬p, p ∨ q, q → p, . . . }
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 47 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Some definitions:A theory (set of formulas) E is closed wrt a set of inference rulesD iff:
for any α ∈ E andα
β∈ D we also have β ∈ E
E is logically closed if E is closed wrt the set of classical logicalrules (Modus Ponens, etc).
This means that E is infinite. Forinstance, from an atom p we derive . . .
E = {p, p ∨ p, p ∨ ¬p, p ∨ q, q → p, . . . }
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 47 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Some definitions:A theory (set of formulas) E is closed wrt a set of inference rulesD iff:
for any α ∈ E andα
β∈ D we also have β ∈ E
E is logically closed if E is closed wrt the set of classical logicalrules (Modus Ponens, etc). This means that E is infinite. Forinstance, from an atom p we derive . . .
E = {p, p ∨ p, p ∨ ¬p, p ∨ q, q → p, . . . }
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 47 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Some definitions:If W is a theory and D a set of rules, Cn(W ,D) stands for theconsequences of W and D:
the least set of formulas that are both logically closedand closed wrt D
We can simplify Cn(W ,D) as Cn(∅,D′) or just Cn(D′) where
D′ def= D ∪
{>α
∣∣∣∣ α ∈W}.
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Survey of NMR formalisms Default Logic
Default Logic
Some definitions:If W is a theory and D a set of rules, Cn(W ,D) stands for theconsequences of W and D:
the least set of formulas that are both logically closedand closed wrt D
We can simplify Cn(W ,D) as Cn(∅,D′) or just Cn(D′) where
D′ def= D ∪
{>α
∣∣∣∣ α ∈W}.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 48 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Cn(D) can be incrementally obtained using the directconsequences operator:
TD(E) = Cn( {
γ
∣∣∣∣ αγ ∈ D and α ∈ E} )
Start with E0 = Cn(∅)
Define Ei+1 = TD(Ei )
A fixpoint TD(Ei ) = Ei must be reached; this fixpoint is Ei = Cn(D).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 49 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Cn(D) can be incrementally obtained using the directconsequences operator:
TD(E) = Cn( {
γ
∣∣∣∣ αγ ∈ D and α ∈ E} )
Start with E0 = Cn(∅)Define Ei+1 = TD(Ei )
A fixpoint TD(Ei ) = Ei must be reached; this fixpoint is Ei = Cn(D).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 49 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Cn(D) can be incrementally obtained using the directconsequences operator:
TD(E) = Cn( {
γ
∣∣∣∣ αγ ∈ D and α ∈ E} )
Start with E0 = Cn(∅)Define Ei+1 = TD(Ei )
A fixpoint TD(Ei ) = Ei must be reached; this fixpoint is Ei = Cn(D).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 49 / 123
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Survey of NMR formalisms Default Logic
Default Logic
But how can we represent defaults?
α : β1, . . . , βn
γ
Informal meaning:
When α is proved and β1, . . . , βn are currently consistent, then wecan infer γ.
α is called the prerequisite, βi are justifications and γ theconsequent.
When α = > we omit it. A normal default has the form:: γ
γ
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Survey of NMR formalisms Default Logic
Default Logic
But how can we represent defaults?
α : β1, . . . , βn
γ
Informal meaning:
When α is proved and β1, . . . , βn are currently consistent, then wecan infer γ.
α is called the prerequisite, βi are justifications and γ theconsequent.
When α = > we omit it. A normal default has the form:: γ
γ
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 50 / 123
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Survey of NMR formalisms Default Logic
Default Logic
But how can we represent defaults?
α : β1, . . . , βn
γ
Informal meaning:
When α is proved and β1, . . . , βn are currently consistent, then wecan infer γ.
α is called the prerequisite, βi are justifications and γ theconsequent.
When α = > we omit it. A normal default has the form:: γ
γ
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 50 / 123
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Survey of NMR formalisms Default Logic
Default Logic
But how can we represent defaults?
α : β1, . . . , βn
γ
Informal meaning:
When α is proved and β1, . . . , βn are currently consistent, then wecan infer γ.
α is called the prerequisite, βi are justifications and γ theconsequent.
When α = > we omit it.
A normal default has the form:: γ
γ
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 50 / 123
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Survey of NMR formalisms Default Logic
Default Logic
But how can we represent defaults?
α : β1, . . . , βn
γ
Informal meaning:
When α is proved and β1, . . . , βn are currently consistent, then wecan infer γ.
α is called the prerequisite, βi are justifications and γ theconsequent.
When α = > we omit it. A normal default has the form:: γ
γ
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 50 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Given D and E logically closed, we define the modulo DE as
DE def=
{α
γ
∣∣∣∣ α : β1, . . . , βm
γ∈ D, and there is no ¬βj ∈ E
}
Definition (Extension)
A logically closed theory E is an extension of D iff E = Cn(DE ).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 51 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Given D and E logically closed, we define the modulo DE as
DE def=
{α
γ
∣∣∣∣ α : β1, . . . , βm
γ∈ D, and there is no ¬βj ∈ E
}
Definition (Extension)
A logically closed theory E is an extension of D iff E = Cn(DE ).
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 51 / 123
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Survey of NMR formalisms Default Logic
Default Logic
We can use variables, but they are understood as patterns(abbreviations for their ground instances).
Example. Let D be the set of rules:
German(x) ∧ ¬Ab(x)
Drinks(x ,Beer)
>German(Peter)
: ¬Ab(x)
¬Ab(x)
Check thatCn( {German(Peter),¬Ab(Peter),Drinks(Peter ,Beer)} ) is anextension.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 52 / 123
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Survey of NMR formalisms Default Logic
Default Logic
We can use variables, but they are understood as patterns(abbreviations for their ground instances).
Example. Let D be the set of rules:
German(x) ∧ ¬Ab(x)
Drinks(x ,Beer)
>German(Peter)
: ¬Ab(x)
¬Ab(x)
Check thatCn( {German(Peter),¬Ab(Peter),Drinks(Peter ,Beer)} ) is anextension.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 52 / 123
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Survey of NMR formalisms Default Logic
Default Logic
We can use variables, but they are understood as patterns(abbreviations for their ground instances).
Example. Let D be the set of rules:
German(x) ∧ ¬Ab(x)
Drinks(x ,Beer)
>German(Peter)
: ¬Ab(x)
¬Ab(x)
Check thatCn( {German(Peter),¬Ab(Peter),Drinks(Peter ,Beer)} ) is anextension.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 52 / 123
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Survey of NMR formalisms Default Logic
Default Logic
Example 2: assume we include now in D
German(x) ∧Mormon(x)
Ab(x)
>Mormon(Peter)
Check that Cn( {German(Peter),Ab(Peter)} ) is an extension.
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Survey of NMR formalisms Default Logic
Default Logic
Example 2: assume we include now in D
German(x) ∧Mormon(x)
Ab(x)
>Mormon(Peter)
Check that Cn( {German(Peter),Ab(Peter)} ) is an extension.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 53 / 123
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Survey of NMR formalisms Default Logic
Default Logic
A default theory may have several extensions:: ¬p
q: ¬q
p
has two Cn({p}) and Cn({q}).
It may also have no extension at all:: ¬p
p
This is different from having inconsistent extensions:
>p ∧ ¬p
has one extension Cn({⊥}).
D is consistent if it has at least one consistent extension.
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Survey of NMR formalisms Default Logic
Default Logic
A default theory may have several extensions:: ¬p
q: ¬q
p
has two Cn({p}) and Cn({q}).
It may also have no extension at all:: ¬p
p
This is different from having inconsistent extensions:
>p ∧ ¬p
has one extension Cn({⊥}).
D is consistent if it has at least one consistent extension.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 54 / 123
![Page 163: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/163.jpg)
Survey of NMR formalisms Default Logic
Default Logic
A default theory may have several extensions:: ¬p
q: ¬q
p
has two Cn({p}) and Cn({q}).
It may also have no extension at all:: ¬p
p
This is different from having inconsistent extensions:
>p ∧ ¬p
has one extension Cn({⊥}).
D is consistent if it has at least one consistent extension.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 54 / 123
![Page 164: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/164.jpg)
Survey of NMR formalisms Default Logic
Default Logic
A default theory may have several extensions:: ¬p
q: ¬q
p
has two Cn({p}) and Cn({q}).
It may also have no extension at all:: ¬p
p
This is different from having inconsistent extensions:
>p ∧ ¬p
has one extension Cn({⊥}).
D is consistent if it has at least one consistent extension.P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 54 / 123
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Survey of NMR formalisms Autoepistemic Logic
Outline
1 Motivation and goals
2 Survey of NMR formalismsCircumscriptionDefault LogicAutoepistemic Logic
3 Answer Set ProgrammingAnswer Set ProgrammingDiagnosisEquilibrium Logic
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 55 / 123
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Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Autoepistemic logic (AEL) [Moore 85] belongs to the family ofnonmonotonic modal logics.
Brief overview on modal logic . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 56 / 123
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Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Autoepistemic logic (AEL) [Moore 85] belongs to the family ofnonmonotonic modal logics.
Brief overview on modal logic . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 56 / 123
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Survey of NMR formalisms Autoepistemic Logic
Recall: modality
Capture some frequent aspect of knowledge in the studieddomain. Examples:
time instantsalways P(x), sometimes P(x)
an agent’s knowledge states (epistemology)A_knows_that ( B_knows_that P(X ) )
processes, program states, word or sentence interpretations(natural language), . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 57 / 123
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Survey of NMR formalisms Autoepistemic Logic
Recall: modality
Capture some frequent aspect of knowledge in the studieddomain. Examples:
time instantsalways P(x), sometimes P(x)
an agent’s knowledge states (epistemology)A_knows_that ( B_knows_that P(X ) )
processes, program states, word or sentence interpretations(natural language), . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 57 / 123
![Page 170: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/170.jpg)
Survey of NMR formalisms Autoepistemic Logic
Recall: modality
Capture some frequent aspect of knowledge in the studieddomain. Examples:
time instantsalways P(x), sometimes P(x)
an agent’s knowledge states (epistemology)A_knows_that ( B_knows_that P(X ) )
processes, program states, word or sentence interpretations(natural language), . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 57 / 123
![Page 171: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/171.jpg)
Survey of NMR formalisms Autoepistemic Logic
Modality: an example
Example extracted form Wikipedia rule of Neutral Point Of View(NPOV) about Religion:
NPOV policy means that Wikipedia editors ought to try towrite sentences like this: "Certain adherents of this faith(say which) believe X, and also believe that they havealways believed X; . . . "
http://en.wikipedia.org/wiki/Wikipedia:Neutral_point_of_view
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 58 / 123
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Survey of NMR formalisms Autoepistemic Logic
Relation to First Order Logic
Can we use First Order Logic (FOL) instead?
Yes, although with some notational “effort”. Example:“Always P(x , y)” = ∀t(Instant(t)→ P(x , y , (t))
Notice how we indexed all predicates wrt t (time has been reified)
. . . In fact, modal logics can be translated to FOL.
Then, which is their utility?Notation and deduction methods are more comfortableWe lose expressivity but we gain in restricting the reasoningmethodsUsually, propositional version is decidable
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 59 / 123
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Survey of NMR formalisms Autoepistemic Logic
Relation to First Order Logic
Can we use First Order Logic (FOL) instead?Yes, although with some notational “effort”. Example:“Always P(x , y)” = ∀t(Instant(t)→ P(x , y , (t))
Notice how we indexed all predicates wrt t (time has been reified)
. . . In fact, modal logics can be translated to FOL.
Then, which is their utility?Notation and deduction methods are more comfortableWe lose expressivity but we gain in restricting the reasoningmethodsUsually, propositional version is decidable
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 59 / 123
![Page 174: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/174.jpg)
Survey of NMR formalisms Autoepistemic Logic
Relation to First Order Logic
Can we use First Order Logic (FOL) instead?Yes, although with some notational “effort”. Example:“Always P(x , y)” = ∀t(Instant(t)→ P(x , y , (t))
Notice how we indexed all predicates wrt t (time has been reified)
. . . In fact, modal logics can be translated to FOL.
Then, which is their utility?Notation and deduction methods are more comfortableWe lose expressivity but we gain in restricting the reasoningmethodsUsually, propositional version is decidable
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 59 / 123
![Page 175: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/175.jpg)
Survey of NMR formalisms Autoepistemic Logic
Relation to First Order Logic
Can we use First Order Logic (FOL) instead?Yes, although with some notational “effort”. Example:“Always P(x , y)” = ∀t(Instant(t)→ P(x , y , (t))
Notice how we indexed all predicates wrt t (time has been reified)
. . . In fact, modal logics can be translated to FOL.
Then, which is their utility?Notation and deduction methods are more comfortableWe lose expressivity but we gain in restricting the reasoningmethodsUsually, propositional version is decidable
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 59 / 123
![Page 176: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/176.jpg)
Survey of NMR formalisms Autoepistemic Logic
Relation to First Order Logic
Can we use First Order Logic (FOL) instead?Yes, although with some notational “effort”. Example:“Always P(x , y)” = ∀t(Instant(t)→ P(x , y , (t))
Notice how we indexed all predicates wrt t (time has been reified)
. . . In fact, modal logics can be translated to FOL.
Then, which is their utility?
Notation and deduction methods are more comfortableWe lose expressivity but we gain in restricting the reasoningmethodsUsually, propositional version is decidable
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 59 / 123
![Page 177: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/177.jpg)
Survey of NMR formalisms Autoepistemic Logic
Relation to First Order Logic
Can we use First Order Logic (FOL) instead?Yes, although with some notational “effort”. Example:“Always P(x , y)” = ∀t(Instant(t)→ P(x , y , (t))
Notice how we indexed all predicates wrt t (time has been reified)
. . . In fact, modal logics can be translated to FOL.
Then, which is their utility?Notation and deduction methods are more comfortableWe lose expressivity but we gain in restricting the reasoningmethodsUsually, propositional version is decidable
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 59 / 123
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Survey of NMR formalisms Autoepistemic Logic
Syntax
We introduce two new unary operators:L α or �α = “α is necessary”M α or ♦α = “α is possible”
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 60 / 123
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Survey of NMR formalisms Autoepistemic Logic
Axiomatization: inference rules
Logical rulesModus Ponens (MP)
` α, ` α→ β
` β
Uniform Substitution (US)
` α` α[β1/p1, . . . , βn/pn]
Necessitation (N)
` α` Lα
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Survey of NMR formalisms Autoepistemic Logic
Some basic axioms
More frequent axioms:
K L(p → q)→ (Lp → Lq)T Lp → pD Lp → Mp4 Lp → LLpB p → LMp
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Survey of NMR formalisms Autoepistemic Logic
Example: system K
Its the most elementary one and included in all the rest.Induced by axiom K.
K L(p → q)→ (Lp → Lq)
These are a pair of theorems in K
K1 L(p ∧ q)→ (Lp ∧ Lq)
K2 Lp ∧ Lq → L(p ∧ q)
Derived rule:
DR1 `α→β`Lα→Lβ
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 63 / 123
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Survey of NMR formalisms Autoepistemic Logic
Kripke Semantics: possible worlds
In propositional logic we have
v : Atoms −→ {T ,F}
that provide the truth value for each atom and for evaluation offormulas. Ex: v(p → ¬q).
Keypoint: handle several worlds, each one with its owninterpretation.
We will have a current world as a reference + a relation statingwhich worlds are visible from which ones.
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Survey of NMR formalisms Autoepistemic Logic
Kripke Semantics: possible worlds
In propositional logic we have
v : Atoms −→ {T ,F}
that provide the truth value for each atom and for evaluation offormulas. Ex: v(p → ¬q).
Keypoint: handle several worlds, each one with its owninterpretation.
We will have a current world as a reference + a relation statingwhich worlds are visible from which ones.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 64 / 123
![Page 184: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/184.jpg)
Survey of NMR formalisms Autoepistemic Logic
Kripke Semantics: possible worlds
In propositional logic we have
v : Atoms −→ {T ,F}
that provide the truth value for each atom and for evaluation offormulas. Ex: v(p → ¬q).
Keypoint: handle several worlds, each one with its owninterpretation.
We will have a current world as a reference + a relation statingwhich worlds are visible from which ones.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 64 / 123
![Page 185: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/185.jpg)
Survey of NMR formalisms Autoepistemic Logic
Kripke frames
Definition (Kripke frame)
Is a pair 〈W ,R〉 where W = {w1,w2, . . . } is a set of worlds andR ⊆W ×W a relation among them.
Definition (Kripke Model)
Is a triple 〈W ,R, v〉 where 〈W ,R〉 is a Kripke frame andv : W × Atoms → {T ,F} an atoms valuation for each world.
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Survey of NMR formalisms Autoepistemic Logic
Kripke frames
Definition (Kripke frame)
Is a pair 〈W ,R〉 where W = {w1,w2, . . . } is a set of worlds andR ⊆W ×W a relation among them.
Definition (Kripke Model)
Is a triple 〈W ,R, v〉 where 〈W ,R〉 is a Kripke frame andv : W × Atoms → {T ,F} an atoms valuation for each world.
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Survey of NMR formalisms Autoepistemic Logic
Satisfactibility
Let M = 〈W ,R, v〉 a Kripke model. We write M,w |= α to point outthat M satisfies a formula α in world w ∈W .
Definition (M,w |= α)
M,w |= > (M,w 6|= ⊥)M,w |= p of v(w ,p) = T (with p ∈ Atoms)M,w |= α ∧ β if M,w |= α and M,w |= β
M,w |= α ∨ β if M,w |= α or M,w |= β
M,w |= ¬α if M,w 6|= α
M,w |= L α if for all w ′ such that wRw ′: M,w ′ |= α.M,w |= M α if for some w ′ such that wRw ′: M,w ′ |= α.
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Survey of NMR formalisms Autoepistemic Logic
Satisfactibility
Let M = 〈W ,R, v〉 a Kripke model. We write M,w |= α to point outthat M satisfies a formula α in world w ∈W .
Definition (M,w |= α)
M,w |= > (M,w 6|= ⊥)M,w |= p of v(w ,p) = T (with p ∈ Atoms)M,w |= α ∧ β if M,w |= α and M,w |= β
M,w |= α ∨ β if M,w |= α or M,w |= β
M,w |= ¬α if M,w 6|= α
M,w |= L α if for all w ′ such that wRw ′: M,w ′ |= α.M,w |= M α if for some w ′ such that wRw ′: M,w ′ |= α.
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Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
Depending on the axioms we add, we obtain different visibilityrelations.
System K =axiom K, induces R as any arbitrary relation.
System T . T = K + T, induces a reflexive relation R, that is wRwfor all w .
System D (Deontic). D = K + D, where
D Lp → Mp
induces (serial frames): for all w , there exists w ′, wRw ′.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 67 / 123
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Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
Depending on the axioms we add, we obtain different visibilityrelations.
System K =axiom K, induces R as any arbitrary relation.
System T . T = K + T, induces a reflexive relation R, that is wRwfor all w .
System D (Deontic). D = K + D, where
D Lp → Mp
induces (serial frames): for all w , there exists w ′, wRw ′.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 67 / 123
![Page 191: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/191.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
Depending on the axioms we add, we obtain different visibilityrelations.
System K =axiom K, induces R as any arbitrary relation.
System T . T = K + T, induces a reflexive relation R, that is wRwfor all w .
System D (Deontic). D = K + D, where
D Lp → Mp
induces (serial frames): for all w , there exists w ′, wRw ′.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 67 / 123
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Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
Depending on the axioms we add, we obtain different visibilityrelations.
System K =axiom K, induces R as any arbitrary relation.
System T . T = K + T, induces a reflexive relation R, that is wRwfor all w .
System D (Deontic). D = K + D, where
D Lp → Mp
induces (serial frames): for all w , there exists w ′, wRw ′.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 67 / 123
![Page 193: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/193.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S4 (epistemic), S4 = T + 4 = K + T + 4, where
4 Lp → LLp
Lp used to model “belief” (agent believes p).
Kripke frames: R reflexive and transitive.
Very important relation to intuitionistic logic established by[Gödel32].
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 68 / 123
![Page 194: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/194.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S4 (epistemic), S4 = T + 4 = K + T + 4, where
4 Lp → LLp
Lp used to model “belief” (agent believes p).
Kripke frames: R reflexive and transitive.
Very important relation to intuitionistic logic established by[Gödel32].
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 68 / 123
![Page 195: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/195.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S4 (epistemic), S4 = T + 4 = K + T + 4, where
4 Lp → LLp
Lp used to model “belief” (agent believes p).
Kripke frames: R reflexive and transitive.
Very important relation to intuitionistic logic established by[Gödel32].
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 68 / 123
![Page 196: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/196.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S4 (epistemic), S4 = T + 4 = K + T + 4, where
4 Lp → LLp
Lp used to model “belief” (agent believes p).
Kripke frames: R reflexive and transitive.
Very important relation to intuitionistic logic established by[Gödel32].
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 68 / 123
![Page 197: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/197.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S5, S5 = T + 5 = K + T + 5, where
5 Lp → LMp
Kripke frames: R is an equivalence: reflexive, symmetric andtransitive.S5 usually captures an agent’s knowledge:Lp → p “if I know p then p is true”.When Lp is a belief, it does not imply p.Logics for dealing with beliefs: stronger than S4 butnon-reflexive.In particular . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 69 / 123
![Page 198: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/198.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S5, S5 = T + 5 = K + T + 5, where
5 Lp → LMp
Kripke frames: R is an equivalence: reflexive, symmetric andtransitive.
S5 usually captures an agent’s knowledge:Lp → p “if I know p then p is true”.When Lp is a belief, it does not imply p.Logics for dealing with beliefs: stronger than S4 butnon-reflexive.In particular . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 69 / 123
![Page 199: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/199.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S5, S5 = T + 5 = K + T + 5, where
5 Lp → LMp
Kripke frames: R is an equivalence: reflexive, symmetric andtransitive.S5 usually captures an agent’s knowledge:Lp → p “if I know p then p is true”.
When Lp is a belief, it does not imply p.Logics for dealing with beliefs: stronger than S4 butnon-reflexive.In particular . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 69 / 123
![Page 200: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/200.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S5, S5 = T + 5 = K + T + 5, where
5 Lp → LMp
Kripke frames: R is an equivalence: reflexive, symmetric andtransitive.S5 usually captures an agent’s knowledge:Lp → p “if I know p then p is true”.When Lp is a belief, it does not imply p.
Logics for dealing with beliefs: stronger than S4 butnon-reflexive.In particular . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 69 / 123
![Page 201: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/201.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System S5, S5 = T + 5 = K + T + 5, where
5 Lp → LMp
Kripke frames: R is an equivalence: reflexive, symmetric andtransitive.S5 usually captures an agent’s knowledge:Lp → p “if I know p then p is true”.When Lp is a belief, it does not imply p.Logics for dealing with beliefs: stronger than S4 butnon-reflexive.In particular . . .
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 69 / 123
![Page 202: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/202.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System KD45. We change T by D + 4KD45 = K + D + 4︸ ︷︷ ︸+5
Kripke frames: R transitive, serial and euclidean (if wRu and wRvthen uRv ).
w
It can be used as underlying framework for Autoepistemic Logic.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 70 / 123
![Page 203: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/203.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System KD45. We change T by D + 4KD45 = K + D + 4︸ ︷︷ ︸+5
Kripke frames: R transitive, serial and euclidean (if wRu and wRvthen uRv ).
w
It can be used as underlying framework for Autoepistemic Logic.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 70 / 123
![Page 204: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/204.jpg)
Survey of NMR formalisms Autoepistemic Logic
Systems and their accessibility relation
System KD45. We change T by D + 4KD45 = K + D + 4︸ ︷︷ ︸+5
Kripke frames: R transitive, serial and euclidean (if wRu and wRvthen uRv ).
w
It can be used as underlying framework for Autoepistemic Logic.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 70 / 123
![Page 205: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/205.jpg)
Survey of NMR formalisms Autoepistemic Logic
Back to Autoepistemic Logic
How do we transform a modal logic into a nonmonotonicframework?
We can use a syntactic fixpoint definition [McDermott 80]. ForAEL we have
Definition (Expansion)A logically closed theory E is an expansion of a theory Γ iff:
E = Cn(Γ ∪ {Lα | α ∈ E} ∪ {¬Lα | α 6∈ E})
Cn means propositional consequences; logically closed meansusing propositional logic and dealing with Lα as atoms.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 71 / 123
![Page 206: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/206.jpg)
Survey of NMR formalisms Autoepistemic Logic
Back to Autoepistemic Logic
How do we transform a modal logic into a nonmonotonicframework?
We can use a syntactic fixpoint definition [McDermott 80]. ForAEL we have
Definition (Expansion)A logically closed theory E is an expansion of a theory Γ iff:
E = Cn(Γ ∪ {Lα | α ∈ E} ∪ {¬Lα | α 6∈ E})
Cn means propositional consequences; logically closed meansusing propositional logic and dealing with Lα as atoms.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 71 / 123
![Page 207: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/207.jpg)
Survey of NMR formalisms Autoepistemic Logic
Back to Autoepistemic Logic
How do we transform a modal logic into a nonmonotonicframework?
We can use a syntactic fixpoint definition [McDermott 80]. ForAEL we have
Definition (Expansion)A logically closed theory E is an expansion of a theory Γ iff:
E = Cn(Γ ∪ {Lα | α ∈ E} ∪ {¬Lα | α 6∈ E})
Cn means propositional consequences; logically closed meansusing propositional logic and dealing with Lα as atoms.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 71 / 123
![Page 208: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/208.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Or we can provide a semantic, models selection criterion.
Simplified semantics: we have interpretations like (S, I) where I isa propositional interpretation, and S a set of them.
Satisfaction:1 (S, I) |= p iff p ∈ I.2 ∧,∨,→,¬ as always.3 (S, I) |= Lα iff for all J ∈ S, (S, J) |= α.
The semantic counterpart of expansion E is a set ofinterpretations S such that:
S = {I | (S, I) |= Γ}
Relating E to S:α ∈ E iff for all J in S, (S, J) |= α
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 72 / 123
![Page 209: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/209.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Or we can provide a semantic, models selection criterion.
Simplified semantics: we have interpretations like (S, I) where I isa propositional interpretation, and S a set of them.
Satisfaction:1 (S, I) |= p iff p ∈ I.2 ∧,∨,→,¬ as always.3 (S, I) |= Lα iff for all J ∈ S, (S, J) |= α.
The semantic counterpart of expansion E is a set ofinterpretations S such that:
S = {I | (S, I) |= Γ}
Relating E to S:α ∈ E iff for all J in S, (S, J) |= α
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 72 / 123
![Page 210: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/210.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Or we can provide a semantic, models selection criterion.
Simplified semantics: we have interpretations like (S, I) where I isa propositional interpretation, and S a set of them.
Satisfaction:1 (S, I) |= p iff p ∈ I.
2 ∧,∨,→,¬ as always.3 (S, I) |= Lα iff for all J ∈ S, (S, J) |= α.
The semantic counterpart of expansion E is a set ofinterpretations S such that:
S = {I | (S, I) |= Γ}
Relating E to S:α ∈ E iff for all J in S, (S, J) |= α
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 72 / 123
![Page 211: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/211.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Or we can provide a semantic, models selection criterion.
Simplified semantics: we have interpretations like (S, I) where I isa propositional interpretation, and S a set of them.
Satisfaction:1 (S, I) |= p iff p ∈ I.2 ∧,∨,→,¬ as always.
3 (S, I) |= Lα iff for all J ∈ S, (S, J) |= α.
The semantic counterpart of expansion E is a set ofinterpretations S such that:
S = {I | (S, I) |= Γ}
Relating E to S:α ∈ E iff for all J in S, (S, J) |= α
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 72 / 123
![Page 212: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/212.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Or we can provide a semantic, models selection criterion.
Simplified semantics: we have interpretations like (S, I) where I isa propositional interpretation, and S a set of them.
Satisfaction:1 (S, I) |= p iff p ∈ I.2 ∧,∨,→,¬ as always.3 (S, I) |= Lα iff for all J ∈ S, (S, J) |= α.
The semantic counterpart of expansion E is a set ofinterpretations S such that:
S = {I | (S, I) |= Γ}
Relating E to S:α ∈ E iff for all J in S, (S, J) |= α
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 72 / 123
![Page 213: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/213.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Or we can provide a semantic, models selection criterion.
Simplified semantics: we have interpretations like (S, I) where I isa propositional interpretation, and S a set of them.
Satisfaction:1 (S, I) |= p iff p ∈ I.2 ∧,∨,→,¬ as always.3 (S, I) |= Lα iff for all J ∈ S, (S, J) |= α.
The semantic counterpart of expansion E is a set ofinterpretations S such that:
S = {I | (S, I) |= Γ}
Relating E to S:α ∈ E iff for all J in S, (S, J) |= α
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 72 / 123
![Page 214: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/214.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
Or we can provide a semantic, models selection criterion.
Simplified semantics: we have interpretations like (S, I) where I isa propositional interpretation, and S a set of them.
Satisfaction:1 (S, I) |= p iff p ∈ I.2 ∧,∨,→,¬ as always.3 (S, I) |= Lα iff for all J ∈ S, (S, J) |= α.
The semantic counterpart of expansion E is a set ofinterpretations S such that:
S = {I | (S, I) |= Γ}
Relating E to S:α ∈ E iff for all J in S, (S, J) |= α
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 72 / 123
![Page 215: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/215.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
An example. Take Γ:
german ∧ ¬L ab → drinks german
Let’s check that S = {{german,drinks}, {german,ab,drinks}} isan expansion.
From S we get the beliefs L german, L drinks and ¬L ab.
This together with Γ allows concluding german,drinks. Atom ab isleft free: we have two possible models.
Check that, for instance, thatS′ = {{german,ab}, {german,ab,drinks}} is not an expansion.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 73 / 123
![Page 216: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/216.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
An example. Take Γ:
german ∧ ¬L ab → drinks german
Let’s check that S = {{german,drinks}, {german,ab,drinks}} isan expansion.
From S we get the beliefs L german, L drinks and ¬L ab.
This together with Γ allows concluding german,drinks. Atom ab isleft free: we have two possible models.
Check that, for instance, thatS′ = {{german,ab}, {german,ab,drinks}} is not an expansion.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 73 / 123
![Page 217: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/217.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
An example. Take Γ:
german ∧ ¬L ab → drinks german
Let’s check that S = {{german,drinks}, {german,ab,drinks}} isan expansion.
From S we get the beliefs L german, L drinks and ¬L ab.
This together with Γ allows concluding german,drinks. Atom ab isleft free: we have two possible models.
Check that, for instance, thatS′ = {{german,ab}, {german,ab,drinks}} is not an expansion.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 73 / 123
![Page 218: An Introduction to Knowledge Representation and Nonmonotonic Reasoningpsantos/slidesIA/Cabalar-KR.pdf · 2010-07-02 · An Introduction to Knowledge Representation and Nonmonotonic](https://reader033.vdocuments.net/reader033/viewer/2022053013/5f0ff0637e708231d446a389/html5/thumbnails/218.jpg)
Survey of NMR formalisms Autoepistemic Logic
Autoepistemic Logic
An example. Take Γ:
german ∧ ¬L ab → drinks german
Let’s check that S = {{german,drinks}, {german,ab,drinks}} isan expansion.
From S we get the beliefs L german, L drinks and ¬L ab.
This together with Γ allows concluding german,drinks. Atom ab isleft free: we have two possible models.
Check that, for instance, thatS′ = {{german,ab}, {german,ab,drinks}} is not an expansion.
P. Cabalar ( Depto. Computación University of Corunna, SPAIN )Intro to KR and NMR June 29, 2010 73 / 123