A Logic of Arbitrary and Indefinite Objects

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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

shapiro@cse.buffalo.edu

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

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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 logicFor KRR systemsSupporting NL understanding & generationAnd commonsense reasoning

LA

Sound & complete via translation to Standard FOLBased on Arbitrary Objects, Fine (’83, ’85a, ’85b)And ANALOG, Ali (’93, ’94), Ali & Shapiro (’93)

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Outline of PaperIntroduction and MotivationsIntroduction to Arbitrary ObjectsInformal Introduction to LA

Formal Syntax of LA

Translations Between and LA Standard FOLSemantics of LA

Proof Theory of A

Soundness & Completeness ProofsSubsumption Reasoning in LA

MRS and LA

Implementation Status

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Outline of Talk

Introduction and Motivations

Informal Introduction to LA

with examples

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Basic Idea

Arbitrary Terms(any x R(x))

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

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Motivations

See paper for other logics

that each satisfy some of these motivations

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Motivation 1Uniform Syntax

Standard FOL:White(Dolly)

x(Sheep(x) White(x))

x(Sheep(x) White(x))

LA:

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))

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

Introduction and Motivations

Informal Introduction to LA

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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Current Implementation Status

Partially implemented as the logic of SNePS 3

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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