Proof-Search in Hilbert Calculi

It is well-known [7] that the standard formalizations of classical and intuitionistic logic based on Hilbert calculi, sequent calculi and natural deduction are equivalent. In spite of this, proof-search has been mainly developed around the notion of sequent calculi almost neglecting the cases of natural deduction and Hilbert calculi. This is primarily motivated by the fact that the latters lack the “deep symmetries” of sequent calculi which can be immediately exploited to control and reduce the search space (see, e.g., [2,3,6] for an accurate discussion). However, as we have shown in [3], in the case of natural deduction it is possible to design proof-search procedures with structural properties and complexity comparable with those based on sequent calculi. In this paper we begin an analogous investigation concerning Hilbert calculi. In particular, we consider the {→,¬}-fragment of the Hilbert calculus for Classical Propositional Logic defined in [5] and we describe a terminating proof-search procedure for it.