Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
1.2 Propositional prolog
The programming language prolog is based on a theorem prover for a subset of first order logic. A pure prolog
“program” is a theory, that is, a finite set of sentences. An execution of the program is an attempt to prove some
theorem from the theory. (Sometimes we introduce “impurities” to do things like produce outputs.) I prefer to
introduce prolog from this pure perspective, and introduce the respects in which it acts like a programming
language later.
(Notation) Let {a − z} ={a, b, c, . . . , z}
Let {a − zA − Z0 − 9_} ={a,b,c,...,z,A,B,...,Z,0, 1,...,9, _}
For any set S, letS∗be the set of all strings of elements of S.
For any set S, letS + be the set of all non-empty strings of elements of S.
For any sets S,T,letST be the set of all strings st for s ∈ S,t ∈ T .
language:
atomic formulas p ={a − z}{a − zA − Z0 − 9_} ∗ | ′ {a − zA − Z0 − 9_ @#$% ∗ ()} ∗′
conjunctions C ::= ɛ. | p, C
goals G ::= ?-C
definite clauses D ::= p:-C
(Notation) Definite clauses p:-q1,...,qn,ɛ.are written p:-q1,...,qn.
And definite clauses p:-ɛ. are written p.
The consequent p of a definite clause is the head, the antecedent is the body.
(Notation) The goal ?-ɛ. is written . This is the contradiction.
(Notation) Parentheses can be added: ((p:-q)). isjustthesameasp:-q.
inference: G, Γ ⊢ G [axiom] for any set of definite clauses Γ and any goal G
G, Γ ⊢ (?-p, C)
G, Γ ⊢ (?-q1,...,qn,C)
if (p:-q1,...,qn) ∈ Γ
semantics: amodelM=〈2,[[·]]〉 where 2 ={0, 1} and [[·]] is a valuation of atomic formulas that extends
compositionally to the whole language:
[[p]] ∈ 2, for atomic formulas p
[[A, B]] = min{[[A]], [[B]]}
1 if [[A]] ≤ [[B]]
[[B:-A]] =
0 otherwise
[[?-A]] = 1 − [[A]]
[[ɛ ]] = 1
metatheory: For any goals G, A and any definite clause theory Γ ,
Soundness: G, Γ ⊢ A only if G, Γ ⊨ A,
Completeness: G, Γ ⊢ A if G, Γ ⊨ A
So we can establish whether C follows from Γ with a “reductio” argument by deciding: (?-C,Γ ⊢ ?-ɛ.)
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