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

June, 2004S. C. Shapiro2 Collaborators Jean-Pierre Koenig David R. Pierce William J. Rapaport The SNePS Research Group

June, 2004S. C. Shapiro3 What Is It? A logic For KRR systems Supporting NL understanding & generation And commonsense reasoning LALA Sound & complete via translation to Standard FOL Based on Arbitrary Objects, Fine (’83, ’85a, ’85b) And ANALOG, Ali (’93, ’94), Ali & Shapiro (’93)

June, 2004S. C. Shapiro4 Outline of Paper Introduction and Motivations Introduction to Arbitrary Objects Informal Introduction to L A Formal Syntax of L A Translations Between and L A Standard FOL Semantics of L A Proof Theory of A Soundness & Completeness Proofs Subsumption Reasoning in L A MRS and L A Implementation Status

June, 2004S. C. Shapiro5 Outline of Talk Introduction and Motivations Informal Introduction to L A with examples

June, 2004S. C. Shapiro6 Basic Idea Arbitrary Terms (any x R (x)) Indefinite Terms (some x (y 1 … y n ) R (x))

June, 2004S. C. Shapiro7 Motivations See paper for other logics that each satisfy some of these motivations

June, 2004S. C. Shapiro8 Motivation 1 Uniform Syntax Standard FOL: White(Dolly)  x(Sheep(x)  White(x))  x(Sheep(x)  White(x)) L A : White(Dolly) White(any x Sheep(x)) White(some x ( ) Sheep(x))

June, 2004S. C. Shapiro9 Motivation 2 Locality of Phrases Every elephant has a trunk. Standard FOL  x(Elephant(x)   y(Trunk(y)  Has(x,y)) L A : Has(any x Elephant(x), some y (x) Trunk(y))

June, 2004S. C. Shapiro10 Motivation 3 Prospects for Generalized Quantifiers Most elephants have two tusks. Standard FOL ?? L A : Has(most x Elephant(x), two y Tusk(y)) (Currently, just notation.)

June, 2004S. C. Shapiro11 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).

June, 2004S. C. Shapiro12 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

June, 2004S. C. Shapiro13 Outline of Talk Introduction and Motivations Informal Introduction to L A with examples

June, 2004S. C. Shapiro14 Quantified Terms Arbitrary terms: (any x [ R (x)]) Indefinite terms: (some x ([y 1 … y n ]) [ R (x)])

June, 2004S. C. Shapiro15 (Q v ([a 1 … a n ]) [ R (v)]) (Q u ([a 1 … a n ]) [ R (u)]) (Q v ([a 1 … a n ]) [ R (v)]) Compatible Quantified Terms different or same All quantified terms in an expression must be compatible.

June, 2004S. C. Shapiro16 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))

June, 2004S. C. Shapiro17 Capture White(any x Sheep(x)) Black(x) White(any x Sheep(x))  Black(x) bound free same Quantifiers take wide scope!

June, 2004S. C. Shapiro18 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.

June, 2004S. C. Shapiro19 Closure  x …  contains the scope of x Compatibility and capture rules only apply within closures.

June, 2004S. C. Shapiro20 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.

June, 2004S. C. Shapiro21 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.

June, 2004S. C. Shapiro22 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)

June, 2004S. C. Shapiro23 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))

June, 2004S. C. Shapiro24 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,  x Spy(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,   x Spy(some x ( ) Person(x)  )

June, 2004S. C. Shapiro25 Current Implementation Status Partially implemented as the logic of SNePS 3

June, 2004S. C. Shapiro26 Summary L A 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.

June, 2004S. C. Shapiro27 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.

June, 2004S. C. Shapiro28 Closure Contains wide-scoping of quantified terms