SNePS: A Logic for Natural Language Understanding and Commonsense Reasoning Stuart C. Shapiro
SNePS: A Logic for Natural
Language Understanding and
Commonsense Reasoning
Stuart C. Shapiro
Department of Computer Science and Engineering and Center for Cognitive Science
State University of New York at Buffalo shapiro@cse.buffalo.edu
S. C. Shapiro
Based on
Stuart C. Shapiro, “SNePS: A Logic for Natural
Language Understanding and Commonsense
Reasoning,” in Lucja Iwanska and Stuart C. Shapiro, Eds.,
Natural Language Processing and Knowledge
Representation: Language for Knowledge and
Knowledge for Language ,
AAAI Press/The MIT Press, 2000.
S. C. Shapiro
S. C. Shapiro
Abstract
Design a better logic
Than FOPL
For NLU and CSR
SNePS
Presentation Approach
•
Problem
•
Difficulties of FOPL
•
Solution in SNePS
•
Use SNePSLOG .
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Twenty Questions
•
Everything is an animal, a vegetable, or a mineral.
–
Squash is a vegetable
• Is squash an animal?
• a mineral?
–
Marble is neither an animal nor a vegetable.
• Is marble a mineral?
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Twenty Questions in FOPL?
x[Animal(x)
Vegetable(x)
Mineral(x)] but don’t want inclusive or
x[Animal(x) Vegetable(x) Mineral(x)]
T T
T
F
T
So don’t want exclusive or either
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andor andor(i, j){P i
, ..., P n
}
True iff at least i, and at most j of the P i are True
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Twenty Questions in SNePSLOG
: all(x)(andor(1,1){animal(x), vegetable(x), mineral(x)}).
: vegetable(squash)!
VEGETABLE(SQUASH)
~ANIMAL(SQUASH)
~MINERAL(SQUASH)
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Twenty Questions in SNePSLOG
II
: andor(0,0){animal(marble), vegetable(marble)}!
~ANIMAL(MARBLE)
~VEGETABLE(MARBLE)
MINERAL(MARBLE)
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Equivalent Statements
•
For every object, the following statements are equivalent:
– It is human. It is a featherless biped.
It is a rational animal.
•
Socrates is human.
– Is Socrates a featherless biped? A rational animal?
•
Snoopy is not a featherless biped.
–
Is Snoopy a rational animal? A human?
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Equivalent Statements in FOPL?
x[Human(x)
Featherless-Biped(x)
Rational-Animal(x)] wrong:
F
F
T
T
T
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thresh thresh(i, j){P i
, ..., P n
}
True iff either fewer than i, or more than j of the P i are True
Note: thresh(i, j) ~andor(i, j)
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thresh Abbreviation thresh(i){P i
, ..., P n
} for thresh(i, n-1){P i
, ..., P n
}
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Equivalent Statements in SNePSLOG 1
: all(x)(thresh(1){human(x), featherless-biped(x), rational-animal(x)}).
: human(Socrates)!
HUMAN(SOCRATES)
FEATHERLESS-BIPED(SOCRATES)
RATIONAL-ANIMAL(SOCRATES)
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Equivalent Statements in SNePSLOG 2
: ~featherless-biped(Snoopy)!
~RATIONAL-ANIMAL(SNOOPY)
~FEATHERLESS-BIPED(SNOOPY)
~HUMAN(SNOOPY)
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Putative Inclusive or
•
If Hilda is in Boston or Kathy is in Las Vegas, then Eve is in Providence.
•
What if Hilda is in Boston and Kathy is in Las
Vegas?
•
Rips 1983: ___
___
___
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or-entailment
{P i
, ..., P n
} v=> {Q i
, ..., Q n
}
True iff for all i, j P i
Q j
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Hilda and Kathy in SNePSLOG
: {in(Hilda, Boston), in(Kathy, Las_Vegas)} v=> {in(Eve, Providence)}.
: in(Hilda, Boston)!
Since
{IN(HILDA,BOSTON),IN(KATHY,LAS_VE
GAS)} v=> {IN(EVE,PROVIDENCE)} and IN(HILDA,BOSTON)
I infer IN(EVE,PROVIDENCE)
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Disappointed Voter Problem
“If someone votes for X and someone votes for Y, one of them will be disappointed”
: all(u,v,x,y)(
{votesfor(u,x), votesfor(v,y)}
&=>
{andor(1,1){disappointed(u), disappointed(v)}}).
: all(u,x)({votesfor(u,x),wins(x)}
&=> {~disappointed{u}}).
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Hillary and Elizabeth Vote
: votesfor(Hillary, Bill).
: votesfor(Elizabeth, Bob).
: wins(Bill).
: disappointed(?x)?
DISAPPOINTED(ELIZABETH)
~DISAPPOINTED(HILLARY)
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FOPL Disappointment
{votesfor(Hillary, Bill), votesfor(Hillary, Bill)}
&=>
{andor(1,1)
{disappointed(Hillary), disappointed(Hillary)}}
disappointed(Hillary)
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Unique Variable Binding Rule
(UVBR)
Two variables in one wff cannot be replaced by the same term.
or
Two terms in an mgu cannot be equal.
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Sisters
“Mary, Sue, and Sally are sisters.”
Sisters in FOPL
Sisters(Mary, Sue)
Sisters(Sue, Sally)
(x,y)[Sisters(x,y)
Sisters(y,x)]
(x,z)[x
z
(
(y)[sisters(x,y)
sisters(y,z)]
Sisters(x,z))]
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Reduction Inference
P(s
1
,...,s i
, ,s i+1
,...,s m
), ’
P(s
1
,...,s i
, ’,s i+1
,...,s m
)
P(s
1
,...,s i
,{t
1
,…,t n
},s i+1
,...,s m
)
P(s
1
,...,s i
,t i
,s i+1
,...,s m
)
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Sisters in SNePSLOG
: sisters({Mary, Sue, Sally}).
: all(x,y)(sisters({x,y}) =>
{likes(x,y), likes(y,x)}).
: likes(?x,?y)?
LIKES(SUE,MARY)
LIKES(MARY,SUE)
LIKES(SALLY,SUE)
LIKES(SUE,SALLY)
LIKES(MARY,SALLY)
LIKES(SALLY,MARY)
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Inadequacy of F OPL 1
•
If R is a transitive relation and R(x, y) and R(y, z) then R(x, z).
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Term Logic
Every expression in SNePS is a term.
SNePSLOG Transitivity Rule
: all(R)(Transitive(R) => all(x,y,z)({R(x,y), R(y,z)}
&=> {R(x,z)})).
: Transitive(bigger).
: bigger(elephant,lion).
: bigger(lion,mouse).
: bigger(elephant,mouse)?
BIGGER(ELEPHANT,MOUSE)
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Inadequacy of F OPL 2
•
Everything Bob believes is true.
•
Bob believes everything Bill believes.
• Bill believes Kevin’s favorite proposition.
• Kevin’s favorite proposition is that John is taller than Mary.
•
Is John taller than Mary?
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Quantifying Over Propositions
: all(p)(Believes(Bob, p) => p).
: all(p)(Believes(Bill, p)
=> Believes(Bob, p)).
: all(p)
(Favorite-proposition(Kevin, p)
=> Believes(Bill, p)).
: Favorite-proposition(Kevin,
Taller(John, Mary)).
: Taller(John, Mary)?
TALLER(JOHN,MARY)
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Opaque Contexts
“George IV wished to know whether Scott was the author of Waverly
”
[Russell 1906]
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Intensional Representation
Every SNePS term represents (denotes) an intensional (mental) entity.
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Uniqueness Principle
No two SNePS terms denote the same entity.
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McCarthy’s
Telephone Number Problem
: all(R)(Transparent(R)
=> all(a,x,y)({R(a,x), =({x,y})}
&=> {R(a,y)})).
: Transparent(Dial).
: =({Telephone(Mike),
Telephone(Mary)}).
: Know(Pat, Telephone(Mike)).
: Dial(Pat, Telephone(Mike)).
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Correct Answer to Telephone
Number Problem
: ?what(Pat, ?which)?
DIAL(PAT,TELEPHONE(MARY))
KNOW(PAT,TELEPHONE(MIKE))
DIAL(PAT,TELEPHONE(MIKE))
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Reasoning by Elimination
•
No one has more than one mother.
–
Jane is John's mother.
– Is Mary John's mother?
•
The committee members are Chris, Leslie,
Pat, and Stevie. At least two of them are women.
–
Leslie and Stevie are men.
–
Is Pat a man or a woman?
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Numerical Quantifiers nexists(i,j,k)(x)
({P
1
(x),..., P n
(x)}: {Q(x)})}
There are k individuals that satisfy
P
1
(x)
...
P n
(x) and, of them, at least i and at most j also satisfy
Q(x)
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Numerical Quantifier
Rule of Inference 1
• If j individuals are known that satisfy
P
1
(x)
…
P n
(x)
Q(x) then every other individual that satisfies
P
1
(x)
...
P n
(x) also satisfies
~Q(x)
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Numerical Quantifier
Rule of Inference 2
• If k-i individuals are known that satisfy
P
1
(x)
…
P n
(x)
~ Q(x) then every other individual that satisfies
P
1
(x)
…
P n
(x) also satisfies
Q(x)
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Numerical Quantifier Forms
• nexists(i,j,k)
• nexists(_,j,_)
• nexists(i,_,k)
Reasoning by Elimination in SNePSLOG 1
: all(x)(Person(x) => nexists(_,1,_)(y)({Person(y)}:
{Mother(y,x)})).
: Person({John, Jane, Mary}).
: Mother(Jane, John).
: Mother(Mary, John)?
~MOTHER(MARY,JOHN)
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Reasoning by Elimination in SNePSLOG 2
: Member({Chris, Leslie, Pat,
Stevie}).
: nexists(2,_,4)(x)
({Member(x)}: {Woman(x)}).
: all(x)(Member(x) => andor(1,1){Man(x), Woman(x)}).
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Committee Problem Answer
: Man({Leslie, Stevie}).
: ?What(Pat)?
WOMAN(PAT)
MEMBER(PAT)
~MAN(PAT)
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Contexts
•
Pegasus is a winged horse in mythology
•
But a normal horse in the real world.
•
So Bellerophon travels by air in mythology
• but by ground in the real world.
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Pegasus in the Real and
Mythological Worlds
: set-context real-world ()
: set-default-context real-world
: all(x)({Bird(x), Beast(x)} v=> {Animal(x)}).
: all(x)(Animal(x) => andor(1,1){Bird(x), Beast(x)}).
: all(x)(Horse(x) => Beast(x)).
: Horse(Pegasus).
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Bellerophon in the Real and
Mythological Worlds
: all(x,y)(Rides(x,y) => andor(1,1){Travelsby(x, air),
Travelsby(x, ground)}).
: all(x,y)(Rides(x,y) => thresh(1,1){Winged(y),
Travelsby(x, air)}).
: Rides(Bellerophon, Pegasus).
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Initialize Mythology
: describe-context
((ASSERTIONS
(WFF1 WFF2 WFF3 WFF4 WFF5 WFF6
WFF7))
(RESTRICTION NIL)
(NAMED (REAL-WORLD)))
: set-context mythology
(WFF1 WFF2 WFF3 WFF4 WFF5 WFF6
WFF7)
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Riding in the Real World
: all(x)(Winged(x) <=> Bird(x)).
: Travelsby(Bellerophon,?what)?
TRAVELSBY(BELLEROPHON,GROUND)
~TRAVELSBY(BELLEROPHON,AIR)
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Riding in Mythology
: set-default-context mythology
: Winged(Pegasus).
: Travelsby(Bellerophon,?what)?
~TRAVELSBY(BELLEROPHON,GROUND)
TRAVELSBY(BELLEROPHON,AIR)
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Relevance Logic:
A Paraconsistent Logic
•
A contradiction should not imply anything whatsoever.
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Opus Flies and Doesn’t
: all(x)(Flies(x)=>Feathered(x)).
: all(x)(~Flies(x) => Swims(x)).
: Flies(Opus).
: ~Flies(Opus).
A contradiction was detected within context DEFAULT-DEFAULTCT.
...
~FLIES(OPUS)
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Opus is Feathered and Swims, but the Earth isn’t Flat
: Feathered(Opus)?
FEATHERED(OPUS)
: Swims(Opus)?
SWIMS(OPUS)
: Flat(Earth)?
:
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The Inconsistent Belief Space
: list-asserted-wffs all(X)(FLIES(X) => FEATHERED(X)) all(X)((~FLIES(X)) => SWIMS(X))
FLIES(OPUS)
~FLIES(OPUS)
FEATHERED(OPUS)
SWIMS(OPUS) but the earth isn’t flat.
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Information Provided by Real
People
• is circular
• and left- and right-recursive
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Circular Definitions
: all(x,y)(thresh(1,1)
{North-of(x,y), Southof(y,x)}).
: North-of(Seattle, Portland).
: South-of(San_Francisco,
Portland).
: North-of(San_Francisco,
Los_Angeles).
: South-of(San_Diego,
Los_Angeles).
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Using a Circular Definition
: North-of(?x, ?y)?
NORTH-OF(SEATTLE,PORTLAND)
NORTH-OF
(SAN_FRANCISCO,LOS_ANGELES)
NORTH-OF(LOS_ANGELES,SAN_DIEGO)
NORTH-OF(PORTLAND,SAN_FRANCISCO)
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Recursive Rules
: all(x,y)(parent(x,y) => ancestor(x,y)).
: all(x,y,z)({ancestor(x,y), ancestor(y,z)}
&=> {ancestor(x,z)}).
: parent(John, Mary).
: ancestor(Mary, George).
: ancestor(George, Sally).
: parent(Sally, Jimmy).
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Using a Recursive Rule
: ancestor(John, ?y)?
ANCESTOR(JOHN,JIMMY)
ANCESTOR(JOHN,MARY)
ANCESTOR(JOHN,GEORGE)
ANCESTOR(JOHN,SALLY)
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In the Other Direction
: ancestor(?x, Jimmy)?
ANCESTOR(SALLY,JIMMY)
ANCESTOR(JOHN,JIMMY)
ANCESTOR(GEORGE,JIMMY)
ANCESTOR(MARY,JIMMY)
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Outline
•
Set-Oriented Logical Connectives
•
The Unique Variable Binding Rule
•
Set Arguments
• “Higher-Order” Logic
•
Intensional Representation
•
The Numerical Quantifiers
•
Contexts
•
Relevance Logic
•
Circular and Recursive Rules
•
Practical Donkey Sentences
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Practical Donkey Sentences
• Donkey Sentence:
– Every farmer who owns a donkey beats it.
• Practical Donkey Sentence:
– If you have a quarter in your pocket, put it in the meter.
[Lenhart Schubert]
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FOPL Practical Sentence
•
(x)[Quarter(x)
In(x, pocket)
PutIn(x, meter)]
– puts all quarters in pocket in the meter
•
(x) )[Quarter(x)
In(x, pocket)]
PutIn(x, meter)
– last x unbound
•
(x) )[Quarter(x)
In(x, pocket)]
(x) )[Quarter(x)
In(x, pocket)
PutIn(x, meter)]
– Finds one, may put in another
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Withsome withsome(var, suchthat, do,[else])
Term denoting an act: if there is some a satisfying suchthat{a/x} perform do{a/x} else perform else.
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Practical Sentence in SNePSLOG
(define-primaction Putin
(object place)
(format t
"~&I am putting ~{~A~}~ in the ~{~A~}.~%” object place))
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Practical Sentence in SNePSLOG 2
: Nickel(N1).
: Nickel(N2).
: Quarter(Q1).
: Quarter(Q2).
: Quarter(Q3).
: In(N1, pocket).
: In(Q1, pocket).
: In(Q2, pocket).
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Practical Sentence in SNePSLOG 3
: perform withsome(?x,
Quarter(?x) and In(?x, pocket),
Putin(?x, meter))
I am putting Q1 in the METER.
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Summary
SNePS Logic has been designed specifically to support Natural Language Understanding and CommonSense Reasoning.
The aspects of that design fall under several categories:
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Basic SNePS Principles
Term Logic:
Every expression in SNePS is a term.
Intensional Representation:
Every SNePS term represents (denotes) an intensional (mental) entity.
Uniqueness Principle:
No two SNePS terms denote the same entity.
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Specialized Syntax
•
Term Logic
•
Set Arguments
•
Set-Oriented Logical Connectives
•
Numerical Quantifiers
• “Higher-Order” Logic
•
Acts
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Specialized Inference Rules
•
The Unique Variable Binding Rule
•
Relevance Logic
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Specialized Semantics
•
Intensional Representation
•
Uniqueness Principle
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Specialized Inference Mechanism
•
Contexts
•
Belief Revision
•
Use of circular and recursive rules
•
Integrated reasoning and acting
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For More Information
•
Personnel
•
Manual
•
Tutorial
•
Bibliography
• ftp’able SNePS source code
• etc.
http://www.cse.buffalo.edu/sneps/
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