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The complexity of decision procedures in relevance logic II

Published online by Cambridge University Press:  12 March 2014

Alasdair Urquhart
Alasdair Urquhart*
Affiliation:
Department of Philosophy, University of Toronto, 215 Huron Street, Toronto, Ontario, Canada, M5S 1A1, E-mail: urquhart@theory.toronto.edu
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Extract

In this paper, we show that there is no primitive recursive decision procedure for the implication-conjunction fragments of the relevant logics R, E and T, as well as for a family of related logics. The lower bound on the complexity is proved by combining the techniques of an earlier paper on the same subject [20] with a method used by Lincoln, Mitchell, Scedrov and Shankar in proving that propositional linear logic is undecidable.

The decision problem for the pure implicational fragments of E and R were solved by Saul Kripke in a tour de force of combinatorial reasoning, published only as an abstract [9]. Belnap and Wallace extended Kripke's decision procedure to the implication-negation fragment of E in [3]; an account of their decision method is to be found in [1, pp. 124–139]. The decision method extends immediately to the implication/negation fragment of R. In fact, in the case of R we can go farther: Meyer in his thesis [13] showed how to translate the logic LR, which results from R by omitting the distribution axiom, into R→⋀, so that the decision procedure can be extended to all of LR. This decision procedure has been implemented as a program Kripke by Thistlewaite, McRobbie and Meyer [17]. The program is not simply a straightforward implementation of the decision procedure; finite matrices are used extensively to prune invalid nodes from the search tree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1774 - 1802
Copyright
Copyright © Association for Symbolic Logic 1999

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References

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