A Linear Axiomatization of Negation as Failure

This paper is concerned with the axiomatization of success and failure in propositional logic programming. It deals with the actual implementation of SLDNF in PROLOG, as opposed to the general nondeterministic SLDNF evaluation method. Given any propositional program P, a linear theory LTP is defined (the linear translation of P) and the following results are proved for any literal A: soundness of PROLOG evaluation (if the goal A PROLOG-succeeds on P, then LTP⊢lin A, and if A PROLOG-fails on P, then LTP⊢lin A⊥), and completeness of PROLOG evaluation (if LTP⊢lin A, then the goal A PROLOG-succeeds on P, and if LTP⊢lin A⊥, then A PROLOG-fails on P). Here ⊢lin means provability in linear logic, and A⊥ is the linear negation of A.

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