A cirquent calculus system with clustering and ranking

Cirquent calculus is a new proof-theoretic and semantic approach introduced by G. Japaridze for the needs of his theory of computability logic (CoL). The earlier article “From formulas to cirquents in computability logic” by Japaridze generalized formulas in CoL to circuit-style structures termed cirquents. It showed that, through cirquents with what are termed clustering and ranking, one can capture, refine and generalize independence-friendly (IF) logic. Specifically, the approach allows us to account for independence from propositional connectives in the same spirit as IF logic accounts for independence from quantifiers. Japaridze's treatment of IF logic, however, was purely semantical, and no deductive system was proposed. The present paper syntactically constructs a cirquent calculus system with clustering and ranking, sound and complete w.r.t. the propositional fragment of cirquent-based semantics. Such a system captures the propositional version of what is called extended IF logic, thus being an axiomatization of a nontrivial fragment of that logic.