Journal of Symbolic Logic

On the Complexity of the Theories of Weak Direct Powers

Charles Rackoff

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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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J. Symbolic Logic, Volume 41, Issue 3 (1976), 561-573.

First available in Project Euclid: 6 July 2007

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Rackoff, Charles. On the Complexity of the Theories of Weak Direct Powers. J. Symbolic Logic 41 (1976), no. 3, 561--573.

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