We introduce a proof system for Hájek'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 (ω)[0,1]. 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 (ω)[0, 1]. 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.