Journal of Symbolic Logic

On the Complexity of the Theories of Weak Direct Powers

Charles Rackoff

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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.

Article information

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183739822

Mathematical Reviews number (MathSciNet)
MR485308

Zentralblatt MATH identifier
0383.03011

JSTOR
links.jstor.org

Citation

Rackoff, Charles. On the Complexity of the Theories of Weak Direct Powers. J. Symbolic Logic 41 (1976), no. 3, 561--573. https://projecteuclid.org/euclid.jsl/1183739822


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