On the Complexity of the Theories of Weak Direct Products

Abstract

Let N be the set of nonnegative integers and let < N ,+> be the weak direct product of < N,+> with itself. Mostowski[ 9 ] shows that the theory of < N ,*> is decidable, but his decision procedure isn't elementary recursive. We present here a more efficient procedure which operates within space 2 2 . As corollaries we obtain the same upper bound for the theory of finite abelian groups, the theory of finitely generated abelian groups, and the theory of the structure < N ,' > of positive ...