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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

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

Collaborators

- Jean-Pierre Koenig
- David R. Pierce
- William J. Rapaport
- The SNePS Research Group

S. C. Shapiro

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)

S. C. Shapiro

Outline of Paper

- Introduction and Motivations
- Introduction to Arbitrary Objects
- Informal Introduction to LA
- Formal Syntax of LA
- Translations Between and LA Standard FOL
- Semantics of LA
- Proof Theory of A
- Soundness & Completeness Proofs
- Subsumption Reasoning in LA
- MRS and LA
- Implementation Status

S. C. Shapiro

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))

S. C. Shapiro

Motivation 2Locality of Phrases

- Every elephant has a trunk.
- Standard FOL

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

- LA:

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

S. C. Shapiro

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.)

S. C. Shapiro

Motivation 4Structure Sharing

- Every elephant has a trunk. It’s flexible.
- Quantified terms are “conceptually complete”.
- Fixed semantics (forthcoming).

Has( , )

Flexible( )

some y ( ) Trunk(y)

any x Elephant(x)

S. C. Shapiro

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

Pet

Elephant

S. C. Shapiro

Quantified Terms

- Arbitrary terms:

(any x [R(x)])

- Indefinite terms:

(some x ([y1 … yn]) [R(x)])

S. C. Shapiro

Compatible Quantified Terms

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

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

different

or

same

All quantified terms in an expression must be compatible.

S. C. Shapiro

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))

S. C. Shapiro

Capture

free

bound

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

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

same

Quantifiers take wide scope!

S. C. Shapiro

Examples of Dependency

Has(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.

S. C. Shapiro

Closure

x … contains the scope of x

Compatibility and capture rules

only apply within closures.

S. C. Shapiro

Closure and Negation

White(any x Sheep(x))

Every sheep is not white.

xWhite(any x Sheep(x))

It is not the case that every sheep is white.

- White(some x () Sheep(x))

Some sheep is not white.

- xWhite(some x () Sheep(x))

No sheep is white.

S. C. Shapiro

Closure and Capture

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

Every number is odd or even.

xOdd(any x Number(x))

xEven(any x Number(x))

Every number is odd or every number is even.

S. C. Shapiro

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)

S. C. Shapiro

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))

S. C. Shapiro

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))

S. C. Shapiro

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.

S. C. Shapiro

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.

S. C. Shapiro

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