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On the complexity of the theories of weak direct powers

Published online by Cambridge University Press:  12 March 2014

Charles Rackoff
Charles Rackoff*
Affiliation:
University of Toronto, Toronto, Canada M55 1A7
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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 〈N*, +〉, the weak direct power of 〈N, +〉, can be decided in space

for some constant c. As corollaries, the same upper bound is obtained for the theory of the structure 〈N +, ·〉 of positive integers under multiplication, and for the theory of finite abelian groups. Fischer and Rabin [7] have shown that the theory of 〈N,* +〉 requires time

even on nondeterministic Turing machines.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 3 , September 1976 , pp. 561 - 573
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

*

This research was supported by the National Science Foundation under research grant GJ–34671, while the author was at the Massachusetts Institute of Technology, Project MAC and the Department of Computer Science, University of Toronto.

References

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