Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T06:59:51.764Z Has data issue: false hasContentIssue false
  • English
  • Français

An undecidable problem in the algebra of truth-tables

Published online by Cambridge University Press:  12 March 2014

Jan Kalicki
Jan Kalicki*
Affiliation:
University of California, Berkeley
Get access Rights & Permissions [Opens in a new window]

Extract

In a previous paper I have described a decision method for testing whether or not two arbitrary finite truth-tables are “equal”, i.e., determine the same set of tautologies. The same problem for the case of infinite truth-tables remained open.

In the present note we shall show that the answer to the problem in the case of infinite truth-tables is negative; in fact we shall prove that there exists neither a decision method for testing the equality of arbitrary truth-tables, nor even one for testing the equality of what we call “recursive” truth-tables.

The terminology of Kalicki [3] will be used; for brevity's sake the discussion involving the notions of recursiveness and recursive enumerability uses the informal terminology and mode of argumentation employed by Post in [9].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 3 , September 1954 , pp. 172 - 176
Copyright
Copyright © Association for Symbolic Logic 1954

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

[1]Church, A., An unsolvable problem of elementary number theory, American Journal of Mathematics, vol. 17 (1936), pp. 345–363.CrossRefGoogle Scholar
[2]Church, A., Introduction to mathematical logic, Part I, Annals of Mathematics Studies, Princeton University Press, 1944, vi + 118 pp.Google Scholar
[3]Kalicki, J., A test for the equality of truth-tables, this Journal, vol. 17 (1952), pp. 161–163.Google Scholar
[4]Kalicki, J., On the comparison of finite algebras, Proceedings of the American Mathematical Society, vol. 3 (1952), pp. 36–40.CrossRefGoogle Scholar
[5]Kleene, S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1953), pp. 41–73.CrossRefGoogle Scholar
[6]Linial, S. and Post, E. L., Recursive unsolvability of the deducibility, Tarski's completeness, and independence of axioms problems of propositional calculus, Bulletin of the American Mathematical Society, vol. 55 (1949), p. 50.Google Scholar
[7]McKinsey, J. C. C., A solution of the decision problem for the Lewis systems S2 and 54, with an application to topology, this Journal, vol. 6 (1941), pp. 117–134.Google Scholar
[8]Mostowski, A., Sentences undecidable in formalized arithmetic, an exposition of the theory of Kurt Gödel, Studies in Logic, North Holland Publishing Company, Amsterdam 1952, viii + 117 pp.Google Scholar
[9]Post, E. L., Recursively enumerable sets of positive integers, and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284–316.CrossRefGoogle Scholar