Efficient decision procedures for locally finite theories II

Abstract

Let T be a finitely axiomatized, universal theory in a finite, first-order language L, and suppose T has a model companion T′ with only finitely many countable models. Then by [We1], T is uniformly locally finite, say with generating function g: ℕ → ℕ. We show the existence of a further function am: ℕ → ℕ measuring the extent to which Mod(T) fails to satisfy the amalgamation property. The main result is as follows: There exist explicitly described uniform decision and quantifier elimination procedures for T′, whose asymptotic complexity can be bounded from above by an elementary recursive function in g and am, without any further reference to T or T′. A corresponding result (with g replaced by d) holds, if T is not finitely axiomatized, provided there is a function d: ℕ → ℕ bounding the size of suitable descriptions of n-generated T-models. Our results generalize those in [We2], which deal with the special case that Mod (T) has the amalgamation property, i.e. that am is the zero function. Applications include the following theories T: The theory of trees, the theory of commutative monoids of bounded exponent, universal Horn theories generated by finitely many finite structures, in particular the theory of distributive p-algebras in the finite Lee-classes and the theory of N-colourable graphs. In each of these cases (see [Pa1], [Bu], [Sch], [Wh]), the best previously known upper complexity bounds were primitive recursive.