Predicate%20Calculus

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Predicate Calculus

• Representing meaning

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Revision

• First-order predicate calculus
• Typical semantic representation
• Quite distant from syntax
• But still clearly a linguistic level of
  representation (it uses words, sort of)

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Types of representation
5. Predicate calculus
The man shot an elephant with his gun
An elephant was shot by the man with his gun
The man used his gun to shoot an elephant
The man owned the gun which he used to shoot an
elephant
The man used the gun which he owned to shoot an
elephant
event(e) time(e,past) pred(e,shoot) man(a)
the(a) ?(b) dog(b) shoot(a,b) ?(c)
gun(c) own(a,c) use(a,c,e)
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First-order predicate calculus

• Computationally tractable
• Well understood, mathematically sound
• Therefore useful for inferencing, expressing
  equivalence
• Can be made quite shallow (almost like a deep
  structure), or quite abstract
• Good for expressing facts and relations
• Therefore good for question-answering,
  information retrieval

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First-order predicate calculus

• Predicates express relationships between
  objects, e.g. father(x,y), or properties of
  objects, e.g. man(x)
• Functions can be evaluated to objects, e.g.
  fatherof(x)
• Constants specific objects in the world being
  described
• Operators (and, or, implies, not) and quantifiers
  (?, ?)

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Logic operators and quantifiers

• Universal quantifier ? (all)
• All dogs are mammals
• ?x dog(x) ? mammal(x)
• Dogs are mammals, The dog is a mammal
• A dog is a mammal
• Existential quantifier ? (there exists)
• John has a car ?x car(x) own(john,x)

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Quantifier scope

• Every man loves a woman
• Ambiguous in natural language
• ?x man(x) ?x woman(y) love(x,y)
• ?x woman(y) ?x man(x) love(x,y)
• Every farmer who owns a donkey beats it
• What does it refer to?
• ?x (farmer(x) ?y donkey(y) own(x,y)) ?
  beat(x,y)

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Quantifiers

• Natural language has many and various
  quantifiers, some of which are difficult to
  express in FOPC
• many, most, some, few, one, three, at least one,
  ...
• often, usually, might, ...

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Ambiguity with negatives

• Every student did not pass an exam
• ?x student(x) ?x exam(y) ?pass(x,y)
• ?y exam(y) ?x student(x) ?pass(x,y)
• ??x student(x) ?x exam(y) pass(x,y)
• All women dont love fur coats
• No smoking seats are available
• I dont think he will come (neg raising)
• I dont know he will come I know he wont come

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Combinatorial explosion

• Quantifier ambiguities can be compounded
• Many people feel that most sentences exhibit too
  few quantifier scope ambiguities for much effort
  to be devoted to this problem, but a casual
  inspection of several sentences from any text
  should convince almost everyone otherwise.
  (Jerry Hobbs)
• On top of other ambiguities (e.g. attachment)

Quantifiers Readings
4 14
5 42
6 132
7 429
8 1430

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First-order predicate calculus

• In a quite shallow FOPC representation we can
  closely map verbs, nouns and adjectives onto
  predicates
• man(x), fat(x), standup(x), see(x,y), give(x,y,z)
• Proper names map onto objects, e.g. man(john),
  see(john,mary)

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• Grammatical meanings can be expressed as
  predicates
• e.g. A man eats icecream with a spoon
• ?X man(x) ?y icecream(y) ?z spoon(z)
  eats(x,y) uses(x,z)
• A man shot an elephant in his pyjamas
• ?x man(x) ?y elephant(y) shot(x,y) ?z
  pyjamas(z) owns(x,z) ...
• wearing(x,z)
• loc(y,z)

wearing(y,z)
loc(x,z)
(wearing(x,z) wearing(y,z) loc(y,z))
loc(x,z))
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First-order predicate calculus

• We can use operators of predicate calculus to
  express aspects of meaning that are implicit, and
  thereby extract new meaning from new utterances
• e.g. eats(x,_) uses(x,y) ? holds(x,y)
• Or make inferences
• e.g. gives(x,y,z) ? has(x,z) ? has(x,y)

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Tense and time

• Representing text, we need to represent tense
• John eats a cake
• ?X cake(X) eats(john,X)
• John ate a cake
• ?X cake(X) ate(john,X)

?X cake(X) eats(john,X,past) ?X cake(X)
eats(john,X,pres)

• event(E) eating(E) agent(E,john)
• X cake(X) object(E,X)

time(E,past)
past(E)
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Tense and time

• Relationship between tense and time by no means
  straightforward
• I fly to Delhi on Monday
• I fly to Delhi on Mondays
• I fly to Delhi and find they have lost my luggage
• I fly to Delhi if I win the competition
• He will be in Delhi now
• You might want a deeper representation rather
  than just a mirror of the surface tense

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Tense and time

• Reichenbachs approach
• Tense is determined by three perspectives
• Event time
• Reference time
• Utterance time
• These can be ordered relative to time
• Also, they can be points or durations

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Tense and time
I had eaten E lt R lt U
I ate ER lt U
I have eaten E lt RU
I eat ERU
I will eat UR lt E
I will have eaten U lt E lt R
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Linguistic issues

• There are many other similarly tricky linguistic
  phenomena
• Modality (could, should, would, must, may)
• Aspect (completed, ongoing, resulting)
• Determination (the, a, some, all, none)
• Fuzzy sets (often, some, many, usually)

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Semantic analysis

• Syntax-driven semantic analysis
• Compositionality
• Semantic grammars
• Procedural view of semantics

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Syntax-driven semantic analysis

• Based on syntactic grammars
• CFG rules augmented by semantic annotations
• Compositionality
• Meaning of the whole is the sum of the meaning of
  its parts
• But not just the parts, but also the way they fit
  together

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Pipeline architecture
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Semantic augmentations to PSG rules - example

• NP ? det, adj, n
• sem(NP,X) qtf(det,X) sem(adj,X)
  sem(n,X)
• a det qtf(X,exists(X))
• fat adj sem(X,fat(X))
• man n sem(X,male(X) hum(X)
• a fat man
• exists(X) fat(X) male(X) hum(X)

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Semantic augmentations to PSG rules - example

• S ? NP, VP
• sem(S,X,Y) sem(NP,X) sem(VP,X,Y)
• NP ? det, adj, n
• sem(NP,X) qtf(det,X) sem(adj,X)
  sem(n,X)
• VP ? v, NP
• sem(VP,X,Y) sem(v,X,Y) sem(NP,Y)
• eats v sem(X,Y,eats(X,Y) tense(pres)
• cake n sem(X,cake(X)
• a fat man eats a cake
• exists(X) fat(X) male(X) hum(X) exists(Y)
  cake(Y) eats(X,Y) tense(pres)

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How to do this

• Quite complex
• Fortunately, there is a mechanism
• Lambda calculus (Church 1940)
• See JM -)
• Such representations often called quasi logical
  forms because of their (too) close relation to
  syntax