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Proof Search in Hajek's Basic Logic

S. Bova·F. Montagna
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Abstract

We introduce a proof system for Hajek's logic BL based on a relational hypersequents framework. We prove that the rules of our logical calculus, called RHBL, are sound and invertible with respect to any valuation of BL into a suitable algebra, called omega[0,1]. Refining the notion of reduction tree that arises naturally from RHBL, we obtain a decision algorithm for BL provability whose running time upper bound is 2^O(n), where n is the number of connectives of the input formula. Moreover, if a formula is unprovable, we exploit the constructiveness of a polynomial time algorithm for leaves validity for providing a procedure to build countermodels in omega[0,1]. Finally, since the size of the reduction tree branches is O(n^3), we can describe a polynomial time verification algorithm for BL unprovability.

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S. Bova, F. Montagna. 2006-05-22. Proof Search in Hajek's Basic Logic. https://arxiv.org/abs/cs/0605094

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