Cse@buffalo Symbolic Representation and Reasoning an Overview Stuart C. Shapiro Department of Computer Science and Engineering, Center for Multisource

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SymbolicRepresentation and Reasoning

an Overview

Stuart C. Shapiro Department of Computer Science and Engineering,

Center for Multisource Information Fusion,

and Center for Cognitive Science

University at Buffalo, The State University of New York

201 Bell Hall, Buffalo, NY 14260-2000

[email protected]

http://www.cse.buffalo.edu/~shapiro/

September, 2004 S. C. Shapiro 2

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Introduction

Knowledge Representation

Reasoning

Symbols

Logics


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Knowledge RepresentationA subarea of Artificial Intelligence

Concerned with understanding, designing, and implementing ways of representing information in computers

So that programs can use this information toderive information that is implied by it,

to converse with people in natural languages,

to plan future activities,

to solve problems in areas that normally require human expertise.


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Reasoning

Deriving information that is implied by the information already present is a form of reasoning.

Knowledge representation schemes are useless without the ability to reason with them.

So, Knowledge Representation and Reasoning


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Knowledge vs. Belief

Knowledge: Justified True Belief

KR systems operate the same whether or not the information stored is justified or true.

So, Belief Representation and Reasoning would be better.

But we’ll stick with KR.


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What Is a Symbol?

“A symbol token is a pattern that can be compared to some other symbol token and judged equal with it or different from it…

Symbols may be formed into symbol structures by means of a set of relations…

The `objects’ that symbols designate may include … objects in an external environment of sensible (readable) stimuli.”

[Newell & Simon, Concise Encyclopedia of CS, 2004]


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What Is Logic?

The study of correct reasoning.

Not a particular KR language.

There are many systems of logic.

With slight abuse, we call a system of logic a logic.

KR research may be seen as the search for the correct logic(s) to use in intelligent systems.


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Parts of Specifying a Logic

Syntax

Semantics

Proof Theory


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Syntax

The specification of a set of atomic symbols, and the grammatical rules for combining them into well-formed expressions (symbol-structures).


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Syntactic ExpressionsAtomic symbols

Individual constants: Tom, Betty, whiteVariables: x, y, zFunction symbols: motherOfPredicate symbols: Person, Elephant, ColorPropositions: P, Q, BdT

TermsIndividual constants: Tom, Betty, whiteVariables : x, y, zFunctional terms: motherOf(Fred)

Well-formed formulas (wffs)Propositions (Proposition symbols) : P, Q, BdTAtomic formulas: Color(x, white), Duck(motherOf(Fred))Non-atomic formulas: TdB Td Bp


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Semantics

The specification of the meaning (designation) of the atomic symbols, and the rules for determining the meanings of the well-formed expressions from the meanings of their parts.


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Semantic ValuesTerms could denote

Objects

Categories of objects

Properties…

Wffs could denotePropositions

Truth values


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Truth ValuesCould be 2, 3, 4, …, ∞ different truth values.Some truth values are “distinguished”Needn’t have anything to do with truth in the real world.By default, we’ll assume 2 truth values.

Call distinguished one True (T)Call other False (F)


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Proof Theory

The specification of a set of rules, which, given an initial collection of well-formed expressions, specify what other well-formed expressions can be added to the collection.


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Proof / Knowledge BaseThe collection could be

A proofA knowledge base

The initial collection could beAxiomsHypothesesAssumptionsDomain facts & rules

The added expressions could beTheoremsDerived facts & rules


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Example

Logic: Standard Propositional LogicDomain: CarPool WorldAtomic Proposition Symbols:

BdT, TdB, Bd, Td, Bp, TpUnary wff-forming connective: Binary wff-forming connectives: , , ,


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Intended Interpretation(Intensional Semantics)

BdT: Betty drives TomTdB: Tom drives BettyBd: Betty is the driverTd: Tom is the driverBp: Betty is the passengerTp: Tom is the passenger


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Extensional (Denotational) Semantics

BdT T T T F F

TdB T T F T F

Bd T T T F F

Td T F F T F

Bp T F F T F

Tp T F T F F

Bd Tp T F T F F

Td Td T T T T T

Td Td F F F F F

5 of 26 = 64 possible situations


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T in some situation

BdT T T T F F

TdB T T F T F

Bd T T T F F

Td T F F T F

Bp T F F T F

Tp T F T F F

Bd Tp T F T F F

Td Td T T T T T

Td Td F F F F F


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alo Properties of WffsContingent

T in some, F in some

BdT T T T F F

TdB T T F T F

Bd T T T F F

Td T F F T F

Bp T F F T F

Tp T F T F F

Bd Tp T F T F F

Td Td T T T T T

Td Td F F F F F


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alo Properties of WffsValid

T in all situations

BdT T T T F F

TdB T T F T F

Bd T T T F F

Td T F F T F

Bp T F F T F

Tp T F T F F

Bd Tp T F T F F

Td Td T T T T T

Td Td F F F F F


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alo Properties of WffsContradictory

T in no situation

BdT T T T F F

TdB T T F T F

Bd T T T F F

Td T F F T F

Bp T F F T F

Tp T F T F F

Bd Tp T F T F F

Td Td T T T T T

Td Td F F F F F


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Logical Implication

P1, …, Pn logically imply Q

P1, …, Pn |= Q

In every situation that P1, …, Pn are True,

so is Q.


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Example: CarPool World KB

Let KBCPW =

Bd Bp

Td Tp

BdT Bd Tp

TdB Td Bp

TdB BdT


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Extensional (Denotational) Semantics

BdT T F

TdB F T

Bd T F

Td F T

Bp F T

Tp T F

Only 2 of the 64 situations where KBCPW are TSo, e.g.,

KBCPW, BdT |= Bd Bp

This is how a KB constrains a model to the domain we want.


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Proof TheorySome Rules of Inference

P

Q

P Q P

Q

P Q

P

QPP Q

P Q

Modus Ponens or Elimination Elimination

Elimination Introduction


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Derivation from Assumptions

Q is derivable from P1, …, Pn

P1, …, Pn |- Q

Starting from the collection P1, …, Pn,

one can repeatedly apply rules of inference,

and eventually get Q.


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Example: CarPool World ProofBdT Bd TpBdT

Bd Tp

Bd

Bd Bp

Bp

So, KBCPW, BdT |- Bd Bp

Bd Bp


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Theoremhood

If Q is derivable from no assumptions,

|- Q

We say that Q is provable,

and that Q is a theorem.


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Deduction Theorem

P1, …, Pn |= Q iff |= (P1 · · · Pn ) Q

P1, …, Pn |- Q iff |- (P1 · · · Pn ) Q

So theorem-proving is relevant to reasoning.


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Properties of Logics

SoundnessIf |- P then |= P

(If P is a provable, then P is valid.)

CompletenessIf |= P then |- P

(If P is valid, then P is a provable.)


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Soundness vs. Completeness

Soundness is the essence of correct reasoning

Completeness is less important because it doesn’t indicate how long it might take.


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Sound and Complete Logics

P1, …, Pn |= Q |= (P1 · · · Pn ) Q

P1, …, Pn |- Q |- (P1 · · · Pn ) Q

completeness

completenessso

undn

ess

soun

dnes

sSo, whenever you want one, you can do another.


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Use of Commutativity DiagramRefutation proof techniques,

such as resolution refutationor semantic tableaux,

prove that there can be no situation

in which P1, …, and Pn are Trueand Q is False.

These are semantic proof techniques.


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Decision ProcedureA procedure that is guaranteed

to terminate

and tell whether or not P is provable.


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Semidecision ProcedureA procedure that, if P is a theorem

is guaranteedto terminate

and say so.

Otherwise, it may not terminate.


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A Tour ofSome Classes of Logics

Propositional Logics

Elementary Predicate Logics

Full First-Order Logics


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Propositional LogicsSmallest Unit: Proposition/Sentence

propositional logics that areSound

Complete

Have decision procedures


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What You Can Dowith Propositional Logic

• BettyDrivesTom TomDrivesBetty

• BettyDrivesTom NearTomBetty

• TomDrivesBetty NearTomBetty NearTomBetty

Can derive conclusions

even though the “facts” aren’t entirely known.


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Elementary Predicate LogicsPropositions plusPredicate (Relation) symbols,Individual terms, variables, quantifiers elementary predicate logics that are

SoundCompleteHave decision procedures


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What You Can Say withElementary Predicate Logic

x[Elephant(x) HasA(x, trunk)]

Can state generalities

before all individuals are known.

x[Elephant(x) Color(x, white)]

Can describe individuals

Even when they are not specifically known.


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Full First-Order LogicsElementary predicate logic plus

Function symbols/ functional terms

full first-order logics that areSound

None areComplete

Have decision procedures


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What You Can Say withFull First-Order Logic

p[HasProp(0, p)

x[HasProp(x, p) HasProp(x+1, p)]

x HasProp(x, p)]

Principle of induction.


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Example of Undecidability

• Large KB about ducks, etc.x[y (Duck(y) WalksLike(x,y))

y (Duck(y) TalksLike(x,y))

Duck(x)]x Duck(motherOf(x)) Duck(x)

• Duck(Fred)?• If Fred is not a duck, possible ∞ loop.


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Unsound ReasoningInduction

FromRaven(a) Black(a)

Raven(b) Black(b)

Raven(c) Black(c)

Raven(d) Black(d)

…

Raven(n) Black(n)

Tox[Raven(x) Black(x)]


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Unsound ReasoningAbduction

Fromx[Person(x) Injured(x) Bandaged(x)]

Person(Tom)

Bandaged(Tom)

ToInjured(Tom)


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What’s “First-Order” aboutFirst-Order Logics

Can’t quantify overFunction symbols

Predicate symbols

Propositions


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Examples ofSNePS

Reasoning

Using a Logic

Designed for KRR


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SNePS, A “Higher-Order” Logic: all(R)(Transitive(R) => (all(x,y,z)(R(x,y) and R(y,z) => R(x,z)))).

: Bigger(elephants, lions).

: Bigger(lions, mice).

: Transitive(Bigger).

: Bigger(elephants, mice)? Bigger(elephants,mice)

Really a higher-order language for a first-order logic


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“Higher-Order” Example 2

: all(source)(Trusted(source) => all(p)(Says(source, p) => p)).

: Trusted(Agent007).

: Says(Agent007, Dangerous(Dr_No)).

: Dangerous(Dr_No)? Dangerous(Dr_No)


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Designing New Connectives

: andor(1,1){OnFloor(G2), OnFloor(G1), OnFloor(1), OnFloor(2)}.

: OnFloor(G1).

: OnFloor(?where)? ~OnFloor(G2) ~OnFloor(1) ~OnFloor(2) OnFloor(G1)


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Belief Change

: andor(1,1){OnFloor(G2), OnFloor(G1), OnFloor(1), OnFloor(2)}.

: {OnFloor(G2), OnFloor(G1)} => {Location(belowGround)}.

: {OnFloor(1), OnFloor(2)} => {Location(aboveGround)}.

: perform believe(OnFloor(G2))

: Location(?where)? Location(belowGround)

: perform believe(OnFloor(2))

: Location(?where)? Location(aboveGround)


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Summary 1

Symbolic KRR uses logic.

There are many logics.The question is which to use.


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Summary 2A logic has a

SyntaxSemanticsProof Theory

Logics mayBe soundBe completeHave a decision procedure


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Summary 3

Logics provide non-atomic wffsThat can describe situations

Without knowing all specifics


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Summary 4

One can design and buildUseful new logics

And reasoning systems using them.

Documents

Cse@buffalo Symbolic Representation and Reasoning an Overview Stuart C. Shapiro Department of Computer Science and Engineering, Center for Multisource