D. W. Loveland's Automated Theorem Proving. A Logical Basis PDF

By D. W. Loveland

ISBN-10: 0720404991

ISBN-13: 9780720404999

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Example text

Y(yn))=S(x). δ(χ) is an element of D^ which "makes y(B) true". Then i^(3xBz)Bf9 y) = T in {sé\ y). 5] PREPARATION OF FORMULAS 45 assignment y similar mod y to φ where y is defined over y. This defines an w-ary function over D^ which we take as (p'(f). Moreover, 3xB^>Bf is valid in (se', φ') as just shown. Thus, (se', φ') is a model for A, by definition of φ', and also of 3XBZDB/. 1 we have (se', φ') as a model of A'. Now we want to show that A has a model if A' has a model. Let (se, φ) be a model of ^4'.

The matrix clauses of A are Pzy and ~ Qf(x)Pax. Typical ground clauses are Pf(f(a))a and ~Qf(a)Paf(a). 6 If S is a set of clauses, then SH denotes the set of all ground clauses generated from S by replacing variables by Herbrand terms of S uniformly within each clause. ) Consider, for example, S={Px, Qxy). Here H(S) = {a} and SH={Pa, Qaa}. If a function symbol occurs in S9 clearly SH is an infinite set unless S contains only ground clauses. If S is a set of ground clauses, then S= SH. If S contains only propositional letters, then SH=S.

In the conversion rules above, C is said to be subsumed by B. Clause C is a subsumed clause and the rules are of a type called subsumption rules. Examples. 14) remains the same. Step 8. ] For convenience we adopt the following shorthand notation. The universal quantifiers are deleted, as are the v and & symbols. All parentheses are dropped except for the parentheses around the function arguments. g. the constant 12. ) Commas are sometimes used between function arguments and are used to separate clauses on the same line.

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Automated Theorem Proving. A Logical Basis by D. W. Loveland

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