announcements: assignments announcements: …torsten/csc384-f11/lectures/csc384f...1 torsten...

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1Torsten Hahmann, CSC384, University of Toronto, Fall 2011

Announcements: Assignments

1. A1: will not be marked before the midterm; maybe back next Monday (no promises!)

– Direct remarking requests first to Alexandra; only if it cannot be corrected/clarified, talk to me.

– Submit remarking requests in writing (email is sufficient) !

2. A2: Will be out by tomorrow

– You are advised to look at the CSP questions (5-8) before the midterm and at question 1 (Interpretations) before, it will be good preparation for the midterm

– Not much programming; should take significantly less time that A1

3. A2 Help Sessions:

• Wednesday, Nov 2nd : 2:30 – 3:30PM

• Tuesday, Nov 8th 5:30 – 6:30 PM


Announcements: Midterm & Final

• Midterm next week in class: October 24th, 6pm• BE ON TIME! Start writing at 6:10pm, 80 min long

• Afterwards: 1h lecture or a tutorial on common problems with A1

• If you have a conflict, let me know by Wed.

• If you miss the midterm for medical reasons, you need the officalUofT medical certificate

• Midterm: No aids allowed!

• Some sample midterms on the website

• Two office hours this week: • Tuesday, Oct. 18th 5:15 – 6:45 PM (BA4261 – standard room)

• Wednesday, Oct. 19th 2:30 – 3:30PM (BA3289)


Midterm Topics

Essentially everything up to now.• Introduction (What is AI, Agents)• Focus: Search (Uninformed, Heuristic, Game-tree,

Backtracking/CSP)• Need to know how the algorithms work, their

properties, their advantages & disadvantages, problem formulation, etc.

• Some first-order logic: interpretations and models(as covered today and question 1 of Assignment 2)

• Do not need to:• Reproduce complex proofs (but unterstand the ideas)• Write (pseudo)code for algorithms or Prolog code


Other announcements

• Date for the final exam has been announced (see Exam Calendar)

• I believe Dec. 13th (check yourself!)

• Midterm feedback: Let us improve the

learning experience together.

• please fill out during break

• need a volunteer to collect them

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Some unfinished business: CSP

• Conversion of n-ary constraints into binary constraints. – Remember: finite CSPs have finite number of variables

and finite number of domain values for variable (no real values for example)

• Library of problems: csplib.org

• Some constraint solvers:– Choco (Java)

– Minion (C++)

• CSP research here at UofT:– Fahiem Bacchus (also extensions to QBF)

– Chris Beck (MIE): scheduling & optimization


Where are we?

1. Heuristic Search: “Formulate problem as search problem”

� Requires domain-dependent heuristic to efficiently guide the search

2. Games: “Restricted to specific applications of interest to AI”

� Special kind of search over utilities

3. Constraint Satisfaction Problems: “Wider class of AI problems”

� More general: good for a large class of problems; do need to encode algorithms and heuristics for each new problem

4. Logic-based Knowledge representation

� Even more general, we can encode knowledge and reuse general purpose inference engines:

5. Planning (specific kind of logic-based knowledge representation)

6. Reasoning under uncertainty

7. If time permits: Decision making under uncertainty.


Knowledge Representation (KR)

• This material is covered in chapters 7– 9 and 12 of the textbook.

• Chapter 7 provides useful motivation for logic, and an introduction to some basic ideas. It also introduces propositional logic, which is a good background for first-order logic.

• Instead we focus immediately on the first-order case as covered in Chapter 8.

• Chapter 8 contains more examples that we won’t be able to cover in class

• We focus on resolution (9.2 and 9.5) as first-order inference mechanism, other mechanisms covered in Chapter 9 are forward and backward chaining.

• Chapter 12 covers some of the additional notions that have to be dealt with when using knowledge representation in AI. These are geared towards modern uses of logic for logic representation.


What is KNOWLEDGE?Easier question: how do we talk about it?

• We say “John knows that …” and fill in the blank

(can be true/false, right/wrong)

• Contrast: “John fears that …”

(same content, different attitude)

Other forms of knowledge:

- know how, who, what, when, …

- sensorimotor: typing, riding a bicycle

- affective: deep understanding

Belief: not necessarily true and/or held for appropriate reasons

Here we make no distinction between knowledge and belief

MAIN IDEA

Taking the world to be one way and not another

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What is REPRESENTATION?Symbols standing for things in the world

� first aid

� women

“John” � John

“John loves Mary” � the proposition that John loves Mary

Knowledge Representation:

Symbolic encoding of propositions believed

(by some agent)


What is REASONING?

Reasoning:

Manipulation of symbols encoding propositions to

produce new representations of new propositions

Analogy: arithmetic “1011” + “10” → “1101”

↓ ↓ ↓

eleven two thirteen

Modus ponens (forward chaining):

All men are mortals, socrates is a man. Thus, socrates is mortal

All foobars are bliffs, bebong is a foobar. Thus, bebong is a bliff


WHY Knowledge?Taking an intentional stance (Daniel Dennet):

For sufficiently complex systems, it is sometimes compelling to describe

systems in terms of beliefs, goals, fears, intentions

E.g., in a game-playing program

“because it believed its queen was in danger, but wanted to still

control the centre of the board”

More useful than description about actual techniques used for

deciding how to move

“because evaluation procedure P using minimax returned a value

of +7 for this position.”

Is KR just a convenient way of talking about complex systems?

- sometimes such anthropomorphizing is inappropriate (e.g., thermostat)

- can also be very misleading!

fooling users into thinking a system knows more than it does


• We can reuse general knowledge in many different

applications or domains

– Knowledge of time, of general process, of spatial features, of

physics, etc. are useful for many different applications

– E.g. “a door can usually be opened and used to move into a building

or room”

• This is the idea of ontologies = reusable and shareable

formalisations of knowledge

WHY Knowledge?

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WHY Representation?Taking Dennet’s intentional stance says nothing about what is or is not represented symbolically

e.g., in game playing, perhaps the board position is represented but the goal of getting a knight out early is not

KR Hypothesis (Brian Smith)“Any mechanically embodied intelligent process will be comprised of structural

ingredients that a) we as external observers naturally take to represent a propositional account of the knowledge that the overall process exhibits, and b) independent of such external semantic attribution, play a formal but causal and essential role in engendering the behaviour that manifests that knowledge”

Two issues: existence of structures that

• we can interpret propositionally

• determine how the system behaves

KNOWLEDGE-BASED SYSTEM: one designed this way!


Two ExamplesBoth of the following systems can be described intentionally.

However, only the 2nd has a separate collection of symbolic structures (following the KR Hypothesis)

Example 2 has a knowledge base and thus the system is a knowledge-based system.

Example 1:

printColour(snow) :- !, write(`Ìt’s white.’’).

printColour(grass) :- !, write(`Ìt’s green. ’’).

printColour(sky) :- !, write(`Ìt’s yellow. ’’).

printColour(X) :- !, write(``Beats me. ’’).

Example 2:

printColour(X) :- colour(S,Y),!,

``````````````````write(`Ìt’s’’), write(Y), write (``. ’’)

printColour(X) :- !, write(``Beats me. ’’).

colour(snow,white).

colour(sky,yellow).

colour(X,Y) :- madeof(X,Z), colour(Z,Y).

madeof(grass,vegetation).

colour(vegetation,green).


KR and AIMuch of AI involves building systems that are knowledge based

Ability derives in part from reasoning over explicitly represented knowledge

• language understanding,

• planning,

• diagnosis, etc.

Some to a certain extent

game-playing, vision, etc.

Some to a lesser extent

speech, motor control, etc.

Current research question:

how much of intelligent behaviour is knowledge-based?


Why bother?Why not “compile out” knowledge into specialized procedures?

• distribute KB to procedures that need it (as in Example 1)

• almost always achieves better performance

No need to think. Just do it!

• riding a bike

• driving a car

• playing chess?

• doing math?

• staying alive?

Skills (Hubert Dreyfus)

• novices think, experts react

• Compare to “expert systems” which are knowledge based.

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AdvantageKnowledge-based system most suitable for open-ended tasks

can structurally isolate reasons for particular behaviour

Good for

• explanation and justification

• “Because grass is a form of vegetation”

• informality: debugging the KB

• “No the sky is not yellow. It’s blue.”

• extensibility: new relations (add new info)

• “Canaries are yellow”

• extensibility: new applications (use your KB for different tasks)

• Return a list of all the white things

• Painting pictures


Cognitive Penetrability

Cognitive penetrability (Zenon Pylyshyn)

Actions that are conditioned by what is currently believed

An example:

we normally leave the room if we hear a fire alarm

we do not leave the room on hearing a fire alarm if we believe that the alarm is being tested/tampered

- can come to this belief in very many ways

so this action is cognitively penetrable

A non-example:

blinking reflex

Hallmark of knowledge-based system:

The ability to be told facts about the world and adjust our behaviour

correspondingly E.g.,

read a book about canaries learn they’re yellow and add it to the KB


Why reasoning?

Want knowledge to affect action

not do action A if sentence P is in KB

but do action A if world believed in satisfies P

Difference:

P may not be explicitly represented

Need to apply what is known in general to the particulars of the

situation

Example:

“patient x is allergic to medication m.”

“Anybody allergic to medication m is also allergic to n.”

Is it OK to prescribe n for x?

Usually need more than just DB-style retrieval of facts in the KB!


Using Logic

No universal language/semantics

• Why not English?

• Different tasks/worlds

• Different ways to carve up the world

No universal reasoning scheme

• Geared to language

• Sometimes want “extralogical” reasoning

Start with first-order predicate calculus (FOL)

• Invented by philosopher Frege for the formalization of mathematics

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Why Knowledge Representation?

An Example &

An Argument for Logic


Why Knowledge Representation? An Example

• Consider the task of understanding a simple story.

• How do we test understanding?

• Not easy, but understanding at least entails some ability to answer simple questions about the story.


Example.

• Three little pigs: Mother sends them to “seek their fortune”

�1st pig builds a house of straw

�2nd pig builds a house of sticks

�3rd pig builds a house of bricks


Example.

• Three little pigs

�Wolf blows down the straw house, and eats the pig!

�Wolf blows down the sticks house, and eats the pig!

�Wolf cannot huff and poof the brick house!

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Example

• Why couldn’t the wolf blow down the house made of bricks?

• What background knowledge are we applying to come to that conclusion?• Brick structures are stronger than straw and stick structures.

• Objects, like the wolf, have physical limitations. The wolf can only blow so hard.


Why Knowledge Representation?

• Large amounts of knowledge are used to understand the

world around us, and to communicate with others.

• We also have to be able to reason with that knowledge

• Our knowledge won’t be about the blowing ability of wolfs in particular,

it is about physical limits of objects in general.

• We have to employ reasoning to make conclusions about the wolf.

• More generally, reasoning provides an exponential or more

compression in the knowledge we need to store. I.e., without

reasoning we would have to store a infeasible amount of information:

e.g., Elephants can’t fit into teacups.


Logical Representations

• AI typically employs logical representations of knowledge.

• Logical representations useful for a number of

reasons:


Logical Representations

• They are mathematically precise, thus we can analyze their limitations, their properties, the complexity of inference etc.

• They are formal languages, thus computer programs can manipulate sentences in the language.

• They come with both a formal syntax and a formal semantics.

• Typically, have well developed proof theories: formal procedures for reasoning (achieved by manipulating sentences).

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The Knowledge Base

• The Knowledge Base is a set of sentences.

• Syntactically well-formed

• Semantically meaningful

• A user can perform two actions to the KB:

� Tell the KB a new fact

� Ask the KB a question


Syntax of Sentences

English acceptable an one is sentence This.

vs.

This English sentence is an acceptable one.

- P ¬ . Q R

vs.

P - ¬Q . R


Semantics of Sentences

This hungry classroom is a jobless moon.

• Why is this syntactically correct sentence not meaningful?

P - (¬Q . R)

• Represents a world where either P is true, or Q is not true and R is true.


Entailments

α ⊨⊨⊨⊨ β

• read as “α entails β”, or “β follows logically from α”

• meaning that in any world in which α is true, β is

true as well.

For example

(P . Q) ⊨ (P V R)

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Syntactical Derivation

α ⊢ β• read as “α derives β”

• meaning “from sentence α, following the syntactical derivation rules, we can obtain sentence β”

For example

¬(A - B) ⊢ (¬A . ¬B)

Usually with respect to a proof procedure.


The Reasoning Process


Desired Properties of Reasoning

Soundness

• KB ⊢ f → KB ⊨ f• i.e all conclusions arrived at via the proof procedure

are correct: they are logical consequences.

Completeness

•KB ⊨ f → KB ⊢ f• i.e. every logical consequence can be generated by

the proof procedure.


KR with Propositional Logic

• We assume that you have learned Propositional Logic from a previous course, e.g. CSC165 or

CSC236.

• Read Chapter 7 in the textbook for a review!

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Pros and cons of propositional logic

☺ Propositional logic is declarative

☺ Propositional logic allows partial/disjunctive/negated information

• (unlike most data structures and databases)

☺ Propositional logic is compositional:

• meaning of B1,1 ∧ P1,2 is derived from meaning of B1,1 and of P1,2

☺ Meaning in propositional logic is context-independent

• (unlike natural language, where meaning depends on context)

� Propositional logic has very limited expressive power

• (unlike natural language)

• E.g., cannot say "pits cause breezes in adjacent squares“

• except by writing one sentence for each square


First-Order Logic (FOL)

• Whereas propositional logic assumes the world contains facts,

• first-order logic (like natural language) assumes the world contains• Objects: people, houses, numbers, colors, baseball

games, wars, …

• Relations: red, round, prime, brother of, bigger than, part of, comes between, …

• Functions: father of, best friend, one more than, plus, …


First-Order Logic

Syntax: A grammar specifying what are legal syntactic constructs of the representation.

Semantics: A formal mapping from syntactic constructs to set theoretic assertions.


Syntax of FOL: Basic elements

• Constants sheila, 2, blockA,...

• Predicates colour, on, brotherOf, >,...

• Functions sqrt, leftLegOf,...

• Variables x, y, a, b,...

• Connectives ¬, ⇒, ∧, ∨, ⇔

• Equality =

• Quantifiers ∀, ∃

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Syntax of Propositional Logic

• Constants sheila, 2, blockA,...

• Predicates colour, on, brotherOf, >,...

• Functions sqrt, leftLegOf,...

• Variables x, y, a, b,...

• Connectives ¬, ⇒, ∧, ∨, ⇔

• Equality =

• Quantifiers ∀, ∃

Propositions a-ison-b, b-ison-c, …


First Order Syntax

Start with a set of primitive symbols.

• constant symbols.

• function symbols.

• predicate symbols (for predicates and relations).

• variables.

Each function and predicate symbol has a specific arity(determines the number of arguments it takes).


First Order Syntax—Building up.

A term is either:• a variable

• a constant

• an expression of the form f(t1, … tk) where

(a) f is a function symbol;

(b) k is its arity;

(c) each ti is a term



An atom is

An expression of the form p(t1, … tk) where

• p is a predicate symbol;

• k is its arity;

• each ti is a term

Note:

• constants are functions taking zero arguments.

• In slides, UPPER CASE denotes variables, lower case denote function/constant/predicate symbols.

• In Assignment 2: vice versa!

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Semantic Intuition (formalized later)

Terms denote individuals: • constants denote specific individuals;

• functions map tuples of individuals to other individuals

• bill, jane, father(jane), father(father(jane))

• X, father(X), hotel7, rating(hotel7), cost(hotel7)

Atoms denote facts that can be true or false

about the world• father_of(jane, bill), female(jane), system_down()

• satisfied(client15), satisfied(C)

• desires(client15,rome,week29), desires(X,Y,Z)

• rating(hotel7, 4), cost(hotel7, 125)



Atoms are formulas. (Atomic formulas).

• The negation (NOT) of a formula is a new formula

¬ f

Asserts that f is false.

• The conjunction (AND) of a set of formulas is a formula.

f1 Λ f2 Λ … Λ fn where each fi is formula

Asserts that each formula fi is true.



• The disjunction (OR) of a set of formulas is a formula.

f1 V f2 V … V fn where each fi is formula

Asserts that at least one formula fi is true.

• Existential Quantification ∃. ∃X. f where X is a variable and f is a formula.

Asserts there is some individual such that f under than binding will be true.

• Universal Quantification ∀. ∀X.f where X is a variable and f is a formula.

Assets that f is true for every individual.


First Order Syntax—abbreviations.

• Implication (→):

f1→ f2 is equivalent to ¬f1 V f2.

• Other symbols used for implication:

• f1⇒ f2

• f1⊃ f2

• If and only if (↔):

f1 ↔ f2 is equivalent to (f1 → f2) . (f2 → f1).

• Other symbols used for implication:

• f1⇔ f2

• f1≡ f2

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Semantics.

• Formulas (syntax) can be built up recursively, and can become arbitrarily complex.

• Intuitively, there are various distinct formulas (viewed as strings) that really are asserting the same thing∀X,Y. elephant(X) Λ teacup(Y) → largerThan(X,Y)

∀X,Y. teacup(Y) Λ elephant(X) → largerThan(X,Y)

• To capture this equivalence and to make sense of complex formulas we utilize the semantics.


Semantics.

• A formal mapping from formulas to semantic entities (individuals, sets and relations over individuals, functions over individuals).

• The mapping mirrors the recursive structure of the syntax, so we can give any formula, no

matter how complex a mapping to semantic entities.


Semantics—Formal Details

First, we must fix the particular first-order language we are

going to provide semantics for. The primitive symbols included in the syntax defines the particular language.L(F,P,V)

• F = set of function (and constant symbols)

• Each symbol f in F has a particular arity.

• P = set of predicate symbols.

• Each symbol p in P has a particular arity.

• V = an infinite set of variables.


Semantics—Formal Details

An interpretation (model) is a tuple

⟨D, Φ, Ψ,v⟩

•D is a non-empty set (domain of individuals)

• Φ (Phi) is a mapping: Φ(f) : Dk→ D

• maps k-ary function symbol f, to a function from k-ary tuples of

individuals to individuals.

• Ψ (Psi) is a mapping: Ψ(p) : Dk → True/False

• maps k-ary predicate symbol p, to an indicator function over k-ary

tuples of individuals (a subset of Dk)

•v is a variable assignment function. v(X) = d ∈ D (it maps every variable to some individual)

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Intuitions: Domain of Discourse

•Domain D: d∈D is an individual

E.g., { craig, jane, grandhotel, le-fleabag,

rome, portofino, 100, 110, 120 …}

•Underlined symbols denote domain individuals (as opposed

to symbols of the first-order language)

•Domains often infinite, but we’ll use finite models to prime our

intuitions


Intuitions: Φ (individual denoted by fn.)

Φ(f): Dk→ D

Given k-ary function f, k individuals,

what individual does f(d1, …, dk) denote

• 0-ary functions (constants) are mapped to specific

individuals in D.

• Φ(client17) = craig, Φ(hotel5) = le-fleabag,Φ (rome) = rome

• Note that we can have:(rome) = rome and (italian_capital) = rome


Intuitions: Φ (individual denoted by fn.)

Φ(f): Dk→ D

• 1-ary functions are mapped to functions in D → D

• Φ(minquality)=f_minquality:

f_minquality(craig) = 3stars

• Φ(rating)=f_rating:

f_rating(grandhotel) = 5stars

• 2-ary functions are mapped to functions from D2 → D

• Φ(distance)=f_distance:

f_distance(toronto, sienna) = 3256

• n-ary functions are mapped similarly.


Intuitions: Ψ (truth or falsity of formula)

Ψ(p): Dk → True/False

Given k-ary predicate, k individuals,

does the relation denoted by p hold of these?

Ψ(p)(<d1, … dk>) = true?

•0-ary predicates are mapped to true or false.Ψ(rainy) = True Ψ(sunny) = False

•1-ary predicates are mapped indicator functions (True/False) of subsets of D. • Ψ(satisfied) = p_satisfied:

p_satisfied(craig) = True • Ψ(privatebeach) = p_privatebeach:

p_privatebeach(le-fleabag) = False

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Intuitions: Ψ (truth or falsity of formula)

Ψ(p): Dk → True/False

• 2-ary predicates are mapped to indicator functions over D2

• Ψ(location) = p_location: p_location(grandhotel, rome) = Truep_location(grandhotel, sienna) = Falsep_location(grandhotel, portofino) = False

• Ψ(in_country) = p_in_country: p_in_country (rome, italy) = Truep_in_country (sienna, italy) = Truep_in_country (portofino, italy) = Truep_in_country (rome, craig) = False…

• Ψ(available) = p_available: p_available(grandhotel, week29) = True

• n-ary predicates..


Intuitions: v

v exists to take care of quantification.

As we will see the exact mapping it specifies will not matter.

Notation: v[X/d] is a new variable assignment function.

• Exactly like v, except that it maps the variable X to the

individual d.

• Maps every other variable exactly like v:

v’(Y) = v[X/d](Y)


Semantics—Building up

Given language L(F,P,V), and an interpretation I = ⟨D, Φ, Ψ,v⟩

a) Constant c (0-ary function) denotes an individual

I(c) = Φ(c) ∈ D

b) Variable X denotes an individual

I(X) = v(X) ∈ D (variable assignment function).

c) term t = f(t1,…, tk) denotes an individual

I(t) = Φ(f)(I(t1),… I(tk)) ∈ D

We recursively find the denotation of each term, then we apply the function denoted by f to get a new individual.

Hence terms always denote individuals under an interpretation I



Formulas

a) atom a = p(t1,… tk) has truth value

I(a) = Ψ(p)(I(t1), …, I(tk)) ∈ { True, False }

We recursively find the individuals denoted by

the ti, then we check to see if this tuple of

individuals is in the relation denoted by p.

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Formulas

a) Negated formulas ¬f has truth value

I(¬f) = True if I(f) = FalseI(¬f) = False if I(f) = True

b) And formulas f1 Λ f2 Λ … Λ fn have truth value

I(f1 Λ f2 Λ … Λ fn) = True if every I(fi) = True.I(f1 Λ f2 Λ … Λ fn) = False otherwise.

c) Or formulas f1 v f2 v … v fn have truth value

I(f1 v f2 v … v fn) = True if any I(fi) = True.I(f1 v f2 v … v fn) = False otherwise.



Formulas

a) Existential formulas ∃X. f have truth value

I(∃X. f) = True

if there exists a d ∈ D such that I’(f) = True

where I’ = ⟨D, Φ, Ψ,v[X/d]⟩

I(∃X. f) = False otherwise.

I’ is just like I except that its variable assignment function now maps X to d. “d” is the individual of which “f” is true.



Formulas

a) Universal formulas ∀X.f have truth value

I(∀X.f ) = True if for all d ∈ D I’(f) = True

where I’ = ⟨D, Φ, Ψ,v[X/d]⟩

I(∀X.f ) = False otherwise.

Now “f” must be true of every individual “d”.

Hence formulas are always either True or False under an interpretation I


Example

D = {bob, jack, fred}I(∀X.happy(X))

1. Ψ(happy)(v[X/bob](X)) = Ψ(happy)(bob) = True

2. Ψ(happy)(v[X/jack](X)) = Ψ(happy)(jack) = True

3. Ψ(happy)(v[X/fred](X)) = Ψ(happy)(fred) = True

Therefore I(∀X.happy(X)) = True.

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Models – Examples.

A

B

C E

Environment� Constants: a,b,c,e

� Functions:

�No function

� Predicates:

�on: binary

�above: binary

�clear: unary

�ontable: unary

Language (Syntax)


Models—Examples.

A possible Model I1 (semantics)

�D = {A, B, C, E}

�Φ(a) = A, Φ(b) = B, Φ(c) = C, Φ(e) = E.

�Ψ(on) = {(A,B),(B,C)}

�Ψ(above)=

{(A,B),(B,C),(A,C)}

�Ψ(clear)={A,E}

�Ψ(ontable)={C,E}

� Constants: a,b,c,e

� Predicates:

�on (binary)

�above (binary)

�clear (unary)

� ontable(unary)

Language (syntax)

Think of it as possible way the world could be


Models—Examples.

Model I1

�D = {A, B, C, E}

�Φ(a) = A, Φ(b) = B, Φ(c) = C, Φ(e) = E.

�Ψ(on) = {(A,B),(B,C)}

�Ψ(above) = {(A,B),(B,C),(A,C)}

�Ψ(clear)={A,E}


A

B

C E

Environment


Aside on Notation.

Model I1

�D = {A, B, C, E}

�Φ(a) = A, Φ(b) = B, Φ(c) = C, Φ(e) = E.

�Ψ(on) = {(A,B),(B,C)}

�Ψ(above) = {(A,B),(B,C),(A,C)}

�Ψ(clear)={A,E}


Comment on notation:

To this point we have represented an interpretation as a tuple,

⟨D, Φ, Ψ,v⟩. It is also common

practice to abbreviate this and

refer to an interpretation function

as σ and to denote the application of the interpretation function by a superscripting of σ.

E.g.,

aσ = A, bσ =B, etc.

onσ = {(A,B),(B,C)},

aboveσ = {(A,B),(B,C),(A,C)}clearσ = {A,E}

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Models—Formulas true or false?

Model I1

�D = {A, B, C, E}

�Φ(a) = A, Φ(b) = B, Φ(c) = C, Φ(e) = E.

�Ψ(on) = {(A,B),(B,C)}

�Ψ(above) = {(A,B),(B,C),(A,C)}

�Ψ(clear)={A,E}


∀X,Y. on(X,Y)→above(X,Y)

�X=A, Y=B

�X=C, Y=A � …

∀X,Y. above(X,Y)→on(X,Y)

�X=A, Y=B

× X=A, Y=C


Models—Examples.

Model I1

�D = {A, B, C, E}

�Φ(a) = A, Φ(b) = B, Φ(c) = C, Φ(e) = E.

�Ψ(on) = {(A,B),(B,C)}

�Ψ(above) = {(A,B),(B,C),(A,C)}

�Ψ(clear)={A,E}


∀X∃Y. (clear(X) ∨ on(Y,X))� X=A� X=C, Y=B� …

∃Y∀X.(clear(X) ∨ on(Y,X)) × Y=A ? No! (X=C)× Y=C? No! (X=B)

× Y=E? No! (X=B)

× Y=B ? No! (X=B)


KB—many models

KB

1.on(b,c)

2.clear(e)

A

B

C E

B

C E

B

C E

A


Models

• Let our knowledge base KB, consist of a set of formulas.

• We say that I is a model of KB or that I satisfies KB

if every formula f ∈ KB is true under I

• We write I ⊨ KB if I satisfies KB, and I⊨f if f is

true under I.

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What’s Special About Models?

• When we write KB, we have an intended interpretation for it – a way that we think the real

world will be.

• This means that every statement in KB is true in the real world.

• Note however, that not every thing true in the real world need be contained in KB. We might

have only incomplete knowledge.


Models support reasoning.

• Suppose formula f is not mentioned in KB, but is true in every model of KB; i.e.,

For all I, I ⊨ KB → I ⊨ f

then we say that f is a logical consequence of KB or that KB entails f and we write KB ⊨ f.

• Since the real world is a model of KB, f must be true in the real world.

• This means that entailment is a way of finding new true facts that were not explicitly mentioned in KB.

? If KB doesn’t entail f, is f false in the real world?


Logical Consequence Example

• elephant(clyde)

• the individual denoted by the symbol clyde in the set denoted

by elephant (has the property that it is an elephant).

• teacup(cup)

• cup is a teacup.

• Note that in both cases a unary predicate specifies a set

of individuals. Asserting a unary predicate to be true of a term means that the individual denoted by that term is in the specified set.

• Formally, we map individuals to TRUE/FALSE (this is an

indicator function for the set).



∀X,Y.elephant(X) Λ teacup(Y) →largerThan(X,Y)

• For all pairs of individuals if the first is an elephant and the second is a teacup, then the pair of objects are

related to each other by the largerThan relation.

• For pairs of individuals that are not elephants and

teacups, the formula is immediately true.

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∀X,Y.largerThan(X,Y) → ¬fitsIn(X,Y)

• For all pairs of individuals if X is larger than Y (the pair is in the largerThan relation) then we cannot have that

X fits in Y (the pair cannot be in the fitsIn relation).

• (The relation largerThan has a empty intersection with

the fitsIn relation).


• ¬fitsIn(clyde,cup)

• We know largerThan(clyde,teacup) from the first implication. Thus we know this from the second implication.



Elephants × teacups

largerThan

(clyde , cup)

¬fitsIn

fitsIn




• If an interpretation satisfies KB, then the set of pairs

elephant X teacup must be a subset of largerThan, which is disjoint from fitsIn.

• Therefore, the pair (clyde,cup) must be in the complement of the set fitsIn.

• Hence, ¬fitsIn(clyde,cup) must be true in every interpretation that satisfies KB.

• ¬fitsIn(clyde,cup) is a logical consequence of KB.

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Logical Consequences

∃X.elephant(X) → ¬extinct(s_elephant)

• “If there exists something that is an elephant, then the elephant_species is not extinct.”

• When is this sentence true?

(1) ¬extinct(s_elephant):

Φ(s_elephant) = elephant_species, and

Ψ(extinct) = p_extinct: p_extinct(elephant_species) = False

(2) No elephant exists and

Φ(s_elephants) = f_elephant, and

Ψ(extinct) = p_extinct: p_extinct(f_elephant) = True

Not just true when there exists an elephant!


Propositional KB: a, c → b, b → c, d → b, ¬b → ¬c

a b c d

a b c ¬d

a ¬b ¬c ¬d

Set of All Interpretations

Models of KB

Models: A Propositional Example


Models and Interpretations

• the more sentences in KB, the fewer models (satisfying interpretations) there are.

•The more you write down (as long as it’s all true!), the “closer” you get to the “real world”! Because each sentence in KB rules out certain unintended

interpretations.

•This is called axiomatizing the domain


Computing logical consequences

• We want procedures for computing logical consequences that can be implemented in our

programs.

• This would allow us to reason with our knowledge

• Represent the knowledge as logical formulas

• Apply procedures for generating logical consequences

• These procedures are called proof procedures.

• Covered in next lecture

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