Proof Search in Hajek's Basic Logic

Abstract

We introduce a proof system for H&#225;jek's logic <b>BL</b> based on a relational hypersequents framework. We prove that the rules of our logical calculus, called <b>RHBL</b>, are sound and invertible with respect to any valuation of <b>BL</b> into a suitable algebra, called (&#969;)&lsqb;0,1&rsqb;. Refining the notion of reduction tree that arises naturally from <b>RHBL</b>, we obtain a decision algorithm for <b>BL</b> provability whose running time upper bound is 2<sup><i>O</i>(<i>n</i>)</sup>, where <i>n</i> 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 (&#969;)&lsqb;0, 1&rsqb;. Finally, since the size of the reduction tree branches is <i>O</i>(<i>n</i><sup>3</sup>), we can describe a polynomial time verification algorithm for <b>BL</b> unprovability.

DOI: 10.1145/1352582.1352589

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Cite this paper

@article{Bova2008ProofSI, title={Proof Search in Hajek's Basic Logic}, author={Simone Bova and Franco Montagna}, journal={ACM Trans. Comput. Log.}, year={2008}, volume={9}, pages={21:1-21:26} }