Random Models and the Godel Case of the Decision Problem

In a paper of 1933 Godel proved that every satisfiable first-order V2?3* sentence has a finite model. Actually he constructed a finite model in an ingenious and sophisticated way. In this paper we use a simple and straightforward probabilistic argument to establish existence of a finite model of an arbitrary satisfiable V123* sentence. ?0. Introduction. We consider the usual first-order logic of textbooks. A firstorder formula is called a sentence if it has no free individual variables. In this paper we restrict our attention to formulas without function symbols or individual constants. Unless we explicitly note otherwise, our formulas are without occurrences of equality. THEOREM 1. Let q=VVNVV2HV3... 3 v1q(vj, ..., v1) where 0i is quantifier free. If 5 has a model it has afinite model. Theorem 1 was proved in Gddel [3]. It was proved independently in Schiitte [9], [10]. Gddel's proof is much cleaner and easier than that of Schiitte. Still it is very sophisticated. Its overall scheme is simple, however. Gbdel formulates a syntactical criterion and proves that the criterion is necessary for satisfiability and sufficient for finite satisfiability. It is the proof of sufficiency that is difficult. In ?1 we give the simple part of Gddel's proof. In ?2 we define random finite structures and prove the sufficiency result in a straightforward way. Let us mention that the idea of random structures is not a perfect novelty: see Fagin [2]. COROLLARY. There is an algorithm that decides satisfiability of V23* sentences. The Corollary was independently proved in Kalmar [6]. When our proof gives an easier proof of Theorem 1, G6del's proof gives a better upper bound on the size of a minimal model of q5. Additional information about models of V23* sentences can be found in Dreben and Goldfarb [1]. Lewis [8] gives lower and upper bounds on the computational complexity of algorithms that decide satisfiability of V23* sentences. Both Theorem 1 and the Corollary are easily generalized to 3*V23* sentences, which form one of the maximal decidable for satisfiability classes of prenex sentences that is defined by type of prefix; see Lewis [7]. Even the V33 class with unary predicates and at most one binary predicate is undecidable for satisfiability. In Received January 18, 1982. 'This paper was written essentially during the Logic Year 1980-81 in Jerusalem when both authors were fellows in the Institute for Advanced Studies of the Hebrew University. ? 1984, Association for Symbolic Logic 0022-4812/83/4804-001 8/$01 .50