On the complexity of the theories of weak direct powers

Journal of Symbolic Logic 41 (3):561-573 (1976)
Abstract
Mostowski [11] shows that if a structure has a decidable theory, then its weak direct power has one as well; his proof however never produces decision procedures which are elementary recursive. Some very general results are obtained here about the nature of the weak direct power of a structure, which in most cases lead to elementary recursive decision procedures for weak direct powers of structures which themselves have elementary recursive procedures. In particular, it is shown that $\langle N^\ast, +\rangle$ , the weak direct power of $\langle N, +\rangle$ , can be decided in space 2 2 2 cn for some constant c. As corollaries, the same upper bound is obtained for the theory of the structure $\langle N^+, \cdot\rangle$ of positive integers under multiplication, and for the theory of finite abelian groups. Fischer and Rabin [7] have shown that the theory of $\langle N^\ast, +\rangle$ requires time 2 2 2 dn even on nondeterministic Turing machines
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