Cse@buffalo A Logic of Arbitrary and Indefinite Objects Stuart C. Shapiro Department of Computer Science and Engineering, and Center for Cognitive Science

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A Logic of Arbitraryand Indefinite Objects

Stuart C. Shapiro Department of Computer Science and Engineering,

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/

January, 2005 S. C. Shapiro 2

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Based On

Stuart C. Shapiro, A Logic of Arbitrary and Indefinite Objects. In D. Dubois, C. Welty, & M. Williams, Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR2004), AAAI Press, Menlo Park, CA, 2004, 565-575.


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Collaborators

Jean-Pierre Koenig

David R. Pierce

William J. Rapaport

The SNePS Research Group


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What Is It?A logic

For KRR systems

Supporting NL understanding & generation

And commonsense reasoning

LA

Sound & complete (via translation to Standard FOL)

Based on Arbitrary Objects, Fine (’83, ’85a, ’85b)

And ANALOG, Ali (’93, ’94), Ali & Shapiro (’93)


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Outline of Talk

Introduction and Motivations

Informal Introduction to LA

with Examples

Examples of Proof Theory

Implementation as Logic of SNePS 3


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Basic Idea

Arbitrary Terms(any x R(x))

Indefinite Terms(some x (y1 … yn) R(x))


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Motivation 1Uniform Syntax

Standard FOL (Ls ):

Dolly is white.

White(Dolly)

Every sheep is white.

x(Sheep(x) White(x))

Some sheep is white.

x(Sheep(x) White(x))


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FOL with Restricted Quantifiers (LR ):

Dolly is white.

White(Dolly)


xSheep White(x)


xSheep White(x)


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LA :

Dolly is white.

White(Dolly)


White(any x Sheep(x))


White(some x ( ) Sheep(x))


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Motivation 2Locality of Phrases

Every elephant has a trunk.

Standard FOLx(Elephant(x) y(Trunk(y) Has(x,y))

LR:

xElephant yTrunk Has(x,y))


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Motivation 2Locality of Phrases

Every elephant has a trunk.

• Logical Form,

or FOL with “complex terms” (LC):

Has(<x Elephant(x)>, <yTrunk(y)>)

LA:

Has(any x Elephant(x), some y (x) Trunk(y))


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Motivation 3Prospects for Generalized Quantifiers

Most elephants have two tusks.

Standard FOL??

LA:

Has(most x Elephant(x), two y Tusk(y))

(Currently, just notation.)


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Motivation 4Structure Sharing

any x Elephant(x)

some y ( ) Trunk(y)

Has( , ) Flexible( )

Every elephant has a trunk. It’s flexible.

Quantified terms are “conceptually complete”.Fixed semantics (forthcoming).


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Motivation 5Term Subsumption

Hairy(any x Mammal(x))

Mammal(any y Elephant(y)) Hairy(any y Elephant(y))

Pet(some w () Mammal(w))

Hairy(some z () Pet(z))

Hairy

Mammal

Elephant

Pet


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Outline of Talk



with Examples




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Quantified Terms

Arbitrary terms:

(any x [R(x)])

Indefinite terms:

(some x ([y1 … yn]) [R(x)])


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(Q v ([a1 … an]) [R(v)]) (Q u ([a1 … an]) [R(u)])

(Q v ([a1 … an]) [R(v)]) (Q v ([a1 … an]) [R(v)])

Compatible Quantified Terms

differentor

same

All quantified terms in an expression must be compatible.


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Quantified Terms in an Expression Must be Compatible

• Illegal:

White(any x Sheep(x)) Black(any x Raven(x))

• Legal

White(any x Sheep(x)) Black(any y Raven(y))

White(any x Sheep(x)) Black(any x Sheep(x))


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Capture

White(any x Sheep(x)) Black(x)

White(any x Sheep(x)) Black(x)

bound free

same

Quantifiers take wide scope!


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Examples of DependencyHas(any x Elephant(x), some(y (x) Trunk(y))

Every elephant has (its own) trunk.

(any x Number(x)) < (some y (x) Number(y))

Every number has some number bigger than it.

(any x Number(x)) < (some y ( ) Number(y))

There’s a number bigger than every number.


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Closure

x … contains the scope of x

Compatibility and capture rules

only apply within closures.


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Closure and NegationWhite(any x Sheep(x))Every sheep is not white.

x White(any x Sheep(x)) It is not the case that every sheep is white.

White(some x () Sheep(x))Some sheep is not white.

x White(some x () Sheep(x)) No sheep is white.


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Closure and Capture

Odd(any x Number(x)) Even(x)

Every number is odd or even.

x Odd(any x Number(x))

x Even(any x Number(x))

Every number is odd or every number is even.


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Tricky Sentences:Donkey Sentences

Every farmer who owns a donkey beats it.

Beats(any x Farmer(x)

Owns(x, some y (x) Donkey(y)),

y)


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Tricky Sentences:Branching Quantifiers

Some relative of each villager and some relative of each townsman hate each other.

Hates(some x (any v Villager(v)) Relative(x,v),

some y (any u Townsman(u)) Relative(y,u))


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Closure & Nested Beliefs(Assumes Reified Propositions)

There is someone whom Mike believes to be a spy.

Believes(Mike, Spy(some x ( ) Person(x))

Mike believes that someone is a spy.

Believes(Mike, xSpy(some x ( ) Person(x))

There is someone whom Mike believes isn’t a spy.

Believes(Mike, Spy(some x ( ) Person(x))

Mike believes that no one is a spy.

Believes(Mike, xSpy(some x ( ) Person(x))


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Outline of Talk



with Examples




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Proof Theory:anyE (abbreviated)

From B(any x A(x))

and A(a)

conclude B(a)


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Proof Theory:anyI (abbreviated)

From A(a) as Hyp

and derive B(a)

Conclude B(any x A(x))


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Example ProofFrom

Every woman is a person.

Every doctor is a professional.

Some child of every person all of whose sons are professionals is busy.

ConcludeSome child of every woman all of whose sons are

doctors is busy.

[Based on an example of W. A. Woods]


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Example Proof1. Person(any x Woman(x))2. Professional(any y Doctor(y))3. Busy(some u (v)

childOf(u, any v Person(v) Professional(any w

sonOf(w,v))))4. Woman(a) Hyp5. Doctor(any z sonOf(z,a)) Hyp6. Person(a) anyE,1,47. Professional(any z sonOf(z,a)) anyE,2,68. Busy(some u ( ) childOf(u,a)) anyE3,679. Busy(some u (v)

childOf(u, any v Woman(v) Doctor(any w

sonOf(w,v))))anyI,45—8 QED


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Syllogistic Reasoningas Subsumption

(Derived Rules of Inference)

Barbara:

From A(any x B(x))

and B(any y C(y))

conclude A(any y C(y))


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Syllogistic Reasoningas Subsumption

(Derived Rules of Inference)

Darii:

From A(any x B(x))

and C(some y φ B (y))

conclude A(some y φ C(y))


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Outline of Talk



with Examples




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Current Implementation Status

Partially implemented as the logic of SNePS 3


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SNePS 3 Examplesnepsul(25): #L#!(build object (any x (build member x class Mammal))

property hairy)Is((any Arb1 Isa(Arb1, Mammal)), hairy)

snepsul(26): #L#!(build member (any y (build member y class Elephant)) class Mammal)

Isa((any Arb2 Isa(Arb2, Elephant)), Mammal)

snepsul(27): #L#?(build object (any y (build member y class Elephant)) property hairy)

Is((any Arb2 Isa(Arb2, Elephant)), hairy)

snepsul(28): #L#!(build member Clyde class Elephant)Isa(Clyde, Elephant)

snepsul(29): #L#?(build object Clyde property hairy)Is(Clyde, hairy)


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Summary

LA is

A logic

For KRR systems

Supporting NL understanding & generation

And commonsense reasoning

Uses arbitrary and indefinite terms

Instead of universally and existentially quantified variables.


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Arbitrary & Indefinite Terms

Provide for uniform syntax

Promote locality of phrases

Provide prospects for generalized quantifiers

Are conceptually complete

Allow structure sharing

Support subsumption reasoning.


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Closure

Contains wide-scoping of quantified terms


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Implementation Status

Partially implemented as the logic of SNePS 3


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For More Information

The SNePS Research Group web site:

http://www.cse.buffalo.edu/sneps/

The SNePS 3 Project page:

http://www.cse.buffalo.edu/sneps/Projects/sneps3.html

Documents

Cse@buffalo A Logic of Arbitrary and Indefinite Objects Stuart C. Shapiro Department of Computer Science and Engineering, and Center for Cognitive Science