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On the solvability of a subclass of the surányi reduction class1

Published online by Cambridge University Press:  12 March 2014

Richard Goldberg
Richard Goldberg*
Affiliation:
International Business Machines Corporation, Thomas J. Watson Research Center, Yorktown Heights, New York
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Extract

In [1] Dreben showed that the subclass K′ (described below) of the Suranyi reduction class is recursively solvable by showing that the subclass is finitely controllable; that is, by showing that any member S of K′ is satisfiable only if it is finitely satisfiable. Dreben's argument is very complex, but much of the complexity is due to his proving not merely solvability, but the deeper property of finite controllability. In the present note, by exploiting certain features of Dreben's technique, a simpler, direct proof of the solvability of K′ is obtained — that is, a proof in which the question of satisfiability in a finite domain plays no role.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 3 , September 1963 , pp. 237 - 244
Copyright
Copyright © Association for Symbolic Logic 1964

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Footnotes

1

I am greatly indebted to Burton Dreben for the many clarifying emendments he made to the first draft of this paper. I also wish to thank the referee for suggesting many improvements in the presentation.

References

[1]Dreben, B., Solvable Surányi subclasses: an introduction to the Herbrand theory. Annals of the Computation Laboratory of Harvard University, Vol. 31, 1962, pp. 32–47.Google Scholar
[2]Surányi, J., Reduktionstheorie des Entscheidungsproblems im Prädikatenkalkül der Ersten Stufe, Budapest, 1959, pp. 80–81.Google Scholar
[3]Dreben, B., Kahr, A. S. and Wang, H., Classification of AEA formulas by letter atoms. Bulletin of the American Mathematical Society, Vol. 68, No. 5, 1962.CrossRefGoogle Scholar