Index sets for Finite Normal Predicate Logic Programs

Let L be a computable first order predicate language with infinitely many constant symbols and infinitely many n-ary predicate symbols and n-ary functions symbols for all n ≥ 1 and let Q0, Q1, . . . be an effective list all the finite normal predicate logic programs over L. Given some property P of finite normal predicate logic programs over L, we define the index set IP to be the set of indices e such that Qe has property P . Let T0, T1, . . . be an effective list of all primitive recursive trees contained in ω. Then [T0], [T1], . . . is an effective list of all Π 0 1 classes where for any tree T ⊆ ω, [T ] denotes the set of infinite paths through T . We modify constructions of Marek, Nerode, and Remmel [25] to construct recursive functions f and g such that for all e, (i) there is a one-to-one degree preserving correspondence between the set of stable models of Qe and the set of infinite paths through Tf(e) and (ii) there is a one-to-one degree preserving correspondence between the set of infinite paths through Te and the set of stable models Qg(e). We shall use these two recursive functions to reduce the problem of finding the complexity of the index Corresponding author. Email:[email protected] Email: [email protected] Email: [email protected]