# alo cse f buf A Logic of Arbitrary

- Slides: 28

alo @ cse f buf 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 [email protected] buffalo. edu http: //www. cse. buffalo. edu/~shapiro/ June, 2004 S. C. Shapiro

alo @ cse f buf Collaborators Jean-Pierre Koenig David R. Pierce William J. Rapaport The SNe. PS Research Group 2 June, 2004 S. C. Shapiro

alo f buf @ cse 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, ’ 85 a, ’ 85 b) And ANALOG, Ali (’ 93, ’ 94), Ali & Shapiro (’ 93) 3 June, 2004 S. C. Shapiro

alo @ cse f buf 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 4 June, 2004 S. C. Shapiro

alo @ cse f buf Outline of Talk Introduction and Motivations Informal Introduction to LA with examples 5 June, 2004 S. C. Shapiro

alo @ cse f buf Basic Idea Arbitrary Terms (any x R(x)) Indefinite Terms (some x (y 1 … yn) R(x)) 6 June, 2004 S. C. Shapiro

alo @ cse f buf Motivations See paper for other logics that each satisfy some of these motivations 7 June, 2004 S. C. Shapiro

alo @ cse f buf Motivation 1 Uniform Syntax Standard FOL: White(Dolly) x(Sheep(x) White(x)) L A: White(Dolly) White(any x Sheep(x)) White(some x ( ) Sheep(x)) 8 June, 2004 S. C. Shapiro

alo @ cse f buf 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)) 9 June, 2004 S. C. Shapiro

alo @ cse f buf 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. ) 10 June, 2004 S. C. Shapiro

alo @ cse f buf Motivation 4 Structure Sharing Every elephant has a trunk. It’s flexible. Has( , ) Flexible( ) some y ( ) Trunk(y) any x Elephant(x) Quantified terms are “conceptually complete”. Fixed semantics (forthcoming). 11 June, 2004 S. C. Shapiro

alo @ cse f buf 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 Pet Elephant 12 June, 2004 S. C. Shapiro

alo @ cse f buf Outline of Talk Introduction and Motivations Informal Introduction to LA with examples 13 June, 2004 S. C. Shapiro

alo @ cse f buf Quantified Terms Arbitrary terms: (any x [R(x)]) Indefinite terms: (some x ([y 1 … yn]) [R(x)]) 14 June, 2004 S. C. Shapiro

alo f buf @ cse Compatible Quantified Terms (Q v ([a 1 … an]) [R(v)]) (Q u ([a 1 … an]) [R(u)]) different or same (Q v ([a 1 … an]) [R(v)]) All quantified terms in an expression must be compatible. 15 June, 2004 S. C. Shapiro

alo f buf @ cse 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)) 16 June, 2004 S. C. Shapiro

alo f buf @ cse Capture free bound White(any x Sheep(x)) Black(x) White(any x Sheep(x)) Black(x) same Quantifiers take wide scope! 17 June, 2004 S. C. Shapiro

alo f buf @ cse 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. 18 June, 2004 S. C. Shapiro

alo @ cse f buf Closure x … contains the scope of x Compatibility and capture rules only apply within closures. 19 June, 2004 S. C. Shapiro

alo @ cse f buf 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. 20 June, 2004 S. C. Shapiro

alo @ cse f buf 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. 21 June, 2004 S. C. Shapiro

alo f buf @ cse 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) 22 June, 2004 S. C. Shapiro

alo f buf @ cse 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)) 23 June, 2004 S. C. Shapiro

alo f buf @ cse 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) ) 24 June, 2004 S. C. Shapiro

alo @ cse f buf Current Implementation Status Partially implemented as the logic of SNe. PS 3 25 June, 2004 S. C. Shapiro

alo f buf @ cse 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. 26 June, 2004 S. C. Shapiro

alo @ cse f buf 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. 27 June, 2004 S. C. Shapiro

alo @ cse f buf Closure Contains wide-scoping of quantified terms 28 June, 2004 S. C. Shapiro