Decision Problems in Predicate Logic

Abstract

ABSTRACT In this paper we survey fundamental methods and results about the decision problem for classes of first order logical formulae. We begin with a historical account which tells the main steps in the development of the field from Hilbert's formulation of the Entscheidungs-problem to today. We then discuss in more detail a muthod due to Aanderaa and myself which builds upon and extends ideas of Turing and Büchi and is particularly well suited for logical descriptions of computational problems ; we explain how by this method (and variants there of) structural properties of computation formalisms and of their describing formulae are intimately correlated in such a way that many recursion and complexity theoretical properties by this reduction are easily carried over from the combinatorial decision problems to the corresponding logical decision problems. As example we produce by slight and natural variations of that method uniform (and easy) proofs for: NP-resp. resp. n-resp. - completeness of the decision problem for propositional (Cook) resp. frist order logic (Church, Turing) resp. of the emptiness (Trachtenbrot, Büchi) resp. the infinity problem for first order iptctna , the characterization of the latter (Scholz's problem) as the NEXPTIME-acceptable sets (Bennett, Rödding, Schwichtenberg, Jones, Selman) resp. of the generalized spectra as the NP-sets (Fagin), simple axioms for essentially undecidable and Incomplete. theories , resp. satisfiable. formulae. without recursive. models describing enumeration programs for -unseparable r.e. sets, lower complexity bounds , and indeed completeness results for many natural solvable cases of first order logical decision problems as subrecursive analogues to the undecidable reduction classes, and other complexity results for first order or propositional logic problems like a natural logical characterization of network or Turing machine complexity o& boolean functions , which is strongly related to the P = NP-problem. Our main concern is to reveal the deep structural and combinatorial simularities between computations and logical deductions, which bring out explicitely the fundamental and uniform reason for many undecidability and complexity results for combinatorial and for logical decision problems (see the above cited examples).