Completeness of Resolution for Definite Answers

We investigate the problem of nding a computable witness for the exis-tential quantiier in a formula of the classical rst-order predicate logic. The A-resolution calculus based on the program derivation algorithm A of C-L. Chang, R. C-T. Lee and R.Waldinger is used for nding a deenite substitution t for an existentially bound variable y in some formula F , such that F ft=yg is provable. The term t is built of the function and predicate symbols in F , plus Boolean functions and a case splitting function i f , deened in the standard way: i f (True; x; y) = x and i f (False; x; y) = y. We prove that the A-resolution calculus is complete in the following sense: if such a deenite substitution exists, then the A-calculus derives a clause giving such a substitution. The result is strengthened by allowing the usage of liftable criterias R of a certain type, prohibiting the derivation of the substitution terms t for which R(t) fails. This enables us to specify, for example, that the substitution t must be in some special signature or must be type-correct, without losing completeness. We will also consider ordering restrictions for the A-calculus.