a logic of arbitrary and indefinite objects
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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] - PowerPoint PPT PresentationTRANSCRIPT
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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/
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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