Abstract
We investigate the problem of finding a computable witness for the existential quantifier in a formula of the classical first-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 (a subsystem of the Manna-Waldinger calculus) is used for finding a definite substitution t for an existentially bound variable y in some formula F, such that F{t/y} is provable. The term t is built of the function and predicate symbols in F, plus Boolean functions and a case splitting function if, defined in the standard way: if(True,x,y) = x and if(False,x,y) = y. We prove that the A-resolution calculus is complete, i.e. if such a definite 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.
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Tammet, T. (1995). Completeness of resolution for definite answers with case analysis. In: Pacholski, L., Tiuryn, J. (eds) Computer Science Logic. CSL 1994. Lecture Notes in Computer Science, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022265
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DOI: https://doi.org/10.1007/BFb0022265
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