Post on 21-Dec-2015
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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
shapiro@cse.buffalo.edu
http://www.cse.buffalo.edu/~shapiro/
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Based On
Stuart C. Shapiro, A Logic of Arbitrary and Indefinite Objects. In D. Dubois, C. Welty, & M. Williams, Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR2004), AAAI Press, Menlo Park, CA, 2004, 565-575.
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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 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)
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Outline of Talk
Introduction and Motivations
Informal Introduction to LA
with Examples
Examples of Proof Theory
Implementation as Logic of SNePS 3
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Basic Idea
Arbitrary Terms(any x R(x))
Indefinite Terms(some x (y1 … yn) R(x))
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Motivation 1Uniform Syntax
Standard FOL (Ls ):
Dolly is white.
White(Dolly)
Every sheep is white.
x(Sheep(x) White(x))
Some sheep is white.
x(Sheep(x) White(x))
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Motivation 1Uniform Syntax
FOL with Restricted Quantifiers (LR ):
Dolly is white.
White(Dolly)
Every sheep is white.
xSheep White(x)
Some sheep is white.
xSheep White(x)
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Motivation 1Uniform Syntax
LA :
Dolly is white.
White(Dolly)
Every sheep is white.
White(any x Sheep(x))
Some sheep is white.
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))
LR:
xElephant yTrunk Has(x,y))
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Motivation 2Locality of Phrases
Every elephant has a trunk.
• Logical Form,
or FOL with “complex terms” (LC):
Has(<x Elephant(x)>, <yTrunk(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
Examples of Proof Theory
Implementation as Logic of SNePS 3
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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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Outline of Talk
Introduction and Motivations
Informal Introduction to LA
with Examples
Examples of Proof Theory
Implementation as Logic of SNePS 3
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Proof Theory:anyE (abbreviated)
From B(any x A(x))
and A(a)
conclude B(a)
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Proof Theory:anyI (abbreviated)
From A(a) as Hyp
and derive B(a)
Conclude B(any x A(x))
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Example ProofFrom
Every woman is a person.
Every doctor is a professional.
Some child of every person all of whose sons are professionals is busy.
ConcludeSome child of every woman all of whose sons are
doctors is busy.
[Based on an example of W. A. Woods]
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Example Proof1. Person(any x Woman(x))2. Professional(any y Doctor(y))3. Busy(some u (v)
childOf(u, any v Person(v) Professional(any w
sonOf(w,v))))4. Woman(a) Hyp5. Doctor(any z sonOf(z,a)) Hyp6. Person(a) anyE,1,47. Professional(any z sonOf(z,a)) anyE,2,68. Busy(some u ( ) childOf(u,a)) anyE3,679. Busy(some u (v)
childOf(u, any v Woman(v) Doctor(any w
sonOf(w,v))))anyI,45—8 QED
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Syllogistic Reasoningas Subsumption
(Derived Rules of Inference)
Barbara:
From A(any x B(x))
and B(any y C(y))
conclude A(any y C(y))
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Syllogistic Reasoningas Subsumption
(Derived Rules of Inference)
Darii:
From A(any x B(x))
and C(some y φ B (y))
conclude A(some y φ C(y))
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Outline of Talk
Introduction and Motivations
Informal Introduction to LA
with Examples
Examples of Proof Theory
Implementation as Logic of SNePS 3
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Current Implementation Status
Partially implemented as the logic of SNePS 3
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SNePS 3 Examplesnepsul(25): #L#!(build object (any x (build member x class Mammal))
property hairy)Is((any Arb1 Isa(Arb1, Mammal)), hairy)
snepsul(26): #L#!(build member (any y (build member y class Elephant)) class Mammal)
Isa((any Arb2 Isa(Arb2, Elephant)), Mammal)
snepsul(27): #L#?(build object (any y (build member y class Elephant)) property hairy)
Is((any Arb2 Isa(Arb2, Elephant)), hairy)
snepsul(28): #L#!(build member Clyde class Elephant)Isa(Clyde, Elephant)
snepsul(29): #L#?(build object Clyde property hairy)Is(Clyde, hairy)
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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
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Implementation Status
Partially implemented as the logic of SNePS 3
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For More Information
The SNePS Research Group web site:
http://www.cse.buffalo.edu/sneps/
The SNePS 3 Project page:
http://www.cse.buffalo.edu/sneps/Projects/sneps3.html