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A Logic of Arbitraryand Indefinite Objects: 

A Logic of Arbitrary and 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/

Collaborators: 

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

What Is It?: 

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)

Outline of Paper: 

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

Outline of Talk: 

Outline of Talk Introduction and Motivations Informal Introduction to LA with examples

Basic Idea: 

Basic Idea Arbitrary Terms (any x R(x)) Indefinite Terms (some x (y1 … yn) R(x))

Motivations : 

Motivations See paper for other logics that each satisfy some of these motivations

Motivation 1Uniform Syntax: 

Motivation 1 Uniform 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))

Motivation 2Locality of Phrases: 

Motivation 2 Locality 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))

Motivation 3Prospects for Generalized Quantifiers: 

Motivation 3 Prospects for Generalized Quantifiers Most elephants have two tusks. Standard FOL ?? LA: Has(most x Elephant(x), two y Tusk(y)) (Currently, just notation.)

Motivation 4Structure Sharing: 

Motivation 4 Structure 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).

Motivation 5Term Subsumption: 

Motivation 5 Term 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

Outline of Talk: 

Outline of Talk Introduction and Motivations Informal Introduction to LA with examples

Quantified Terms: 

Quantified Terms Arbitrary terms: (any x [R(x)]) Indefinite terms: (some x ([y1 … yn]) [R(x)])

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)]) Compatible Quantified Terms different or same All quantified terms in an expression must be compatible.

Quantified Terms in an Expression Must be Compatible: 

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))

Capture: 

Capture White(any x Sheep(x)) Black(x) White(any x Sheep(x))  Black(x) bound free same Quantifiers take wide scope!

Examples of Dependency: 

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.

Closure: 

Closure x …  contains the scope of x Compatibility and capture rules only apply within closures.

Closure and Negation: 

Closure and Negation White(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.

Closure and Capture: 

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.

Tricky Sentences:Donkey Sentences: 

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)

Tricky Sentences:Branching Quantifiers: 

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))

Closure & Nested Beliefs(Assumes Reified Propositions): 

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))

Current Implementation Status: 

Current Implementation Status Partially implemented as the logic of SNePS 3

Summary: 

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.

Arbitrary & Indefinite Terms: 

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.

Closure: 

Closure Contains wide-scoping of quantified terms