notes 7: knowledge representation, the propositional calculus

Notes 7: Knowledge Representation, The Propositional Calculus

ICS 270A Winter 2003

ICS-270A: Notes 7: 2

Outline

Representing knowledge using logic Agent that reason logically

A knowledge based agent

Using constraints on feature values

A rich and implicit representation of the world state.

Representing and reasoning with logic Propositional logic

Syntax

Semantic

validity and models

Rules of inference for propositional logic

Resolution

Complexity of propositional inference.

Reading: Nillson Chapters 13,14, Russel and Norvig, Chapter 7


Why knowledge-base

The state of the world may require lots of information..

The agent knowledge of the state of the world: If s is world state K(s) is what the agent knows.

For economy: Not everything explicitly specified. Some facts can be inferred.

Agent may infer whatever he does not know explicitly.

Nillson: Constraints on feature values Block A is not on the floor

Issues: In what language to express what the agent knows about the world. How

explicit to make this knowledge. How to infer.

Description of the worldAgent knowledge of state

Agent explicit specificationof what he knows


Knowledge RepresentationDefined by: syntax, semantix

Assertions Conclusions(knowledge base)

Facts Facts

Inference

Imply

Computer

Real-World

Semantics

Reasoning: in the syntactic levelExample: zxzyyx |,


Constraints on the world

World so far were described by feature values: On(block,floor) On(A,B) Clear(C)

But some information is more complex Law: all human are mortal, all blue box are pushable

Negative information: block a is not on the floor

Either A or B are pushable

Examples: A lifting robot: features: Bat_ok, liftable, moves

Constraints on the worlds can be written in logic: Bat_ok and liftable moves

If moves is false and Bat_ok is true, we infer liftable is false.

Logical languages involve Syntax, the grammar

Semantics: the meaning of words and sentences

Inference rules: deriving new information that is correct.


The party example

If Alex goes, then Beki goes: A B

If Chris goes, then Alex goes: C A

Beki does not go: not B

Chris goes: C

Query: Is it possible to satisfy all these conditions?

Should I go to the party?


Example of Languages for Representation

Programming languages: Formal languages, not ambiguous, but cannot express partial

information. Not expressive enough.

Natural languages: very expressive but ambiguous: ex: small dogs and cats.

Good representation language: Both formal and can express partial information, can accommodate

inference

Main approach used in AI: Logic-based languages.


Propositional Logic

Syntax : Alphabet: {true,false,P,Q}

Connectives:

Well-Formed formulas: (wffs or sentences): w1, w2

If Alex_goes Beki_goes

Semantics: True means true

False means false

Symbols means objects in the world and they are true or false relative to a scenario, or a world, we refer to.

Meaning of a sentence is derived from its parts as defined by truth-tables.

},,,{ Q

)( PQP

SRQP )(


Truth tables for the logical connectives


A grammer for sentences in propositional logic


Truth Tables

Truth tables can be used to compute the truth value of any wff.

Can be used to find the truth of

Given n features there are 2n different worlds, different

interpretations.

Interpretation: any assignment of true and false to atoms

An interpretation satisfies a wff if the wff is assigned true under

the interpretation

A model: An interpretation is a model of a wff if the wff is satisfied in that

interpretation.

Satisfiability of a wff can be determined by the truth-table

Bat_on and turns-key_on Engine-starts

Wff is unsatisfiable or inconsistent it has no models

SQRP ))((

)( PP

)()()()( QPQPQPQP


Validity


Validity, Equivalence

Validity: A wff is Valid if it is true in all interpretations P P

Equivalence: two wffs are equivalent iff they have the same models. DeMorgan laws, law of contrapositive

If w1 is equivalent to w2 then: w1 w2 and w2 w1

Associative

Distributive

DeMorgan’s

)( 1221 wwww

321321321 )()( wwwwwwwww

)()()( 5121521 wwwwwww

))(

))(

2121

2121

wwww

wwww


Logical Entailment:truth in the world

KB ( ) entails a sentence, iff all the models of KB are models of alpha (in other words, any interpretation that satisfies KB satisfies alpha.)

If some sentences are true in the world it implies that some other sentences are true.

statement P is true whenever some other set KB of statements is true, then “KB entails P”.

Whenever means: In any possible world (model) in which every sentence of KB is true.

PQP

PP


Producing an additional wffs from a set of wffs From alpha infer beta

Sound inference rule: The conclusion is true whenever the premises are true.

Examples Modus ponens: { A and A B |-- B} is sound, resolution is sound.

Proof: A sequence of inference rules generating the desired conclusion from

the KB. Example: KB = From From KB

Rules of Inference

221 www

},{},,{ BAACBBAAC ABBA inferandCAAC inferand

C


Rules of inference


Resolution in Propositional Calculus

Using clauses as wffs Literal, clauses, conjunction of clauses (cnfs)

Resolution rule: Resolving (P V Q) and (P V Q) P

Generalize modus ponens, chaining . Resolving a literal with its negation yields empty clause.

Resolution is sound Resolution is NOT complete:

P and R entails P V R but you cannot infer P V R From (P and R) by resolution

Resolution is complete for refutation: adding (P) and (R) to (P and R) we can infer the empty clause.

Decidability of propositional calculus by resolution refutation: if a wff w is not entailed by KB then resolution refutation will terminate without generating the empty clause.

)( RQP


Soundness of resolution


The party example

If Alex goes, then Beki goes: A B

If Chris goes, then Alex goes: C A

Beki does not go: not B

Chris goes: C

Query: Is it possible to satisfy all these conditions?

Should I go to the party?


Example of proof by Refutation

Assume the claim is false and prove inconsistency: Example: can we prove that Chris will not come to the party?

Prove by generating the desired goal.

Prove by refutation: add the negation of the goal and prove no model

Proof:

Refutation:

AC

BBA

CinferAACfrom

AinferBBAfrom

,

,

)( CACBBA

AC


The moving robot examplebat_ok,liftable moves~moves, bat_ok


Converting wffs to Conjunctive clauses

1. Eliminate implications: ~(PQ) or (R P)

2. Reduce the scope of negation sign

3. Convert to cnfs using the associative and distributive laws


Converting wffs to Conjunctive clauses

1. Eliminate implications

2. Reduce the scope of negation sign

3. Convert to cnfs using the associative and distributive laws

QPQP )( )()( PRQP

)()( QPQP )()( PRQP

)()( QPQP )()( PRQP

)()( PRQPRP

)(),( PRQRP


Proof by refutation

Given a database in clausal normal form KB Find a sequence of resolution steps from KB to the

empty clauses

Use the search space paradigm:

• States: current cnf KB + new clauses

• Operators: resolution

• Initial state: KB + negated goal

• Goal State: a database containing the empty clause

• Search using any search method


Proof by refutation (contd.)

Or: Prove that KB has no model - PSAT

A cnf theory is a constraint satisfaction problem:

• variables: the propositions

• domains: true, false

• constraints: clauses (or their truth tables)

• Find a solution to the csp. If no solution no model.

• This is the satisfiability question

• Methods: Backtracking arc-consistency unit resolution, local search


Resolution refutation search strategies

Ordering strategies Breadth-first, depth-first

I-level resolvents are generated from level-(I-1) or less resolvents

Unit-preference: prefer resolutions with a literal

Set of support: Allows reslutions in which one of the resolvents is in the set of support

The set of support: those clauses coming from negation of the theorem or their decendents.

The set of support strategy is refutation complete

Linear input Restricted to resolutions when one member is in the input clauses

Linear input is not refutation complete

Example: (PVQ) (P V not Q) (not P V Q) (not P V not Q) have no model


Complexity of propositional inference

Checking truth tables is exponential

Satisfiability is NP-complete

However, frequently generating proofs is easy.

Propositional logic is monotonic If you can entail alpha from knowledge base KB and if you add sentences

to KB, you can infer alpha from the extended knowledge-base as well.

Inference is local Tractable Classes: Horn, 2-SAT

Horn theories:

Q <-- P1,P2,...Pn

Pi is an atom in the language, Q can be false.

Solved by modus ponens or “unit resolution”.


Summary

Representing knowledge using logic Using logic to represent and reason about knowledge

Logic, syntax, semantics and proof theory

Representing and reasoning with logic Propositional logic

Syntax

Semantic

validity and models

Rules of inference for propositional logic

Complexity of propositional inference.

Reading: Nillson Chaters 13, 14 Russel and Norvig Chapter 7.

notes 7: knowledge representation, the propositional calculus

Documents

outlinerepresenting

knowledgebasethe state

new information

partial information

formal languages

example of languages

lots of information

pushablenegative information