A Contraction-Free Focused Sequent Calculus for Classical Propositional Logic

Existing focused proof systems for classical and intuitionistic logic allow contraction for exactly those formulas chosen for focus. For proof-search applications, contraction is undesirable, as we risk traversing the same path multiple times. We present here a contraction-free focused sequent calculus for classical propositional logic, called LKFCF, which is a modification of the recently developed proof system LKF. We prove that our system is sound and complete with respect to LKF, and therefore it is also sound and complete with respect to propositional classical logic. LKF can be justified with a compilation into focused proofs for linear logic; in this work we show how to do a similar compilation for LKFCF, but into focused proofs for linear logic with subexponentials instead. We use two subexponentials, neither allowing contraction but one allowing weakening. We show how the focused proofs for linear logic can then simulate proofs in LKFCF. Returning to proof-search, we end this work with a small experimental study showing that a proof-search implementation based on LKFCF performs well compared to implementations based on leanTAP and several variants and optimizations on LK and LKF. 1998 ACM Subject Classification F.4.1 Mathematical Logic