Proof search in Hájek's basic logic

Abstract

We introduce a proof system for Hájek'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 (ω)[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 (ω)[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.