Index sets for Finite Normal Predicate Logic Programs

Abstract: <Q>e is the effective list of all finite predicate logic programs. <T_e> is the list of recursive trees. We modify constructions of Marek, Nerode, and Remmel [25] to construct recursive functions f and g such that for all indices e, (i) there is a one-to-one degree preserving correspondence between the set of stable models of Q_e and the set of infinite paths through T{f(e)} and (ii) there is a one-to-one degree preserving correspondence between the set of infinite paths through T_e and the set of stable models of Q_{g(e)}. We use these two recursive functions to reduce the problem of finding the complexity of the index set I_P for various properties P of normal finite predicate logic programs to the problem of computing index sets for primitive recursive trees for which there is a large variety of results [6], [8], [16], [17], [18], [19]. We use our correspondences to determine the complexity of the index sets of all programs and of certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having at least one stable model, (iii) having exactly c stable models for any given positive integer c, (iv) having only finitely many stable models, or (vi) having infinitely many stable models.