On the concurrent computational content of intermediate logics

A typed parallel lambda-calculus via 1-depth intermediate proofs

“…Since these communications are higher-order, they can transmit arbitrary simply typed λ-terms as messages, provided their free variables are not bound in the surrounding context. Unlike in [4], messages are thus not restricted to values and communication channels have directions. The reduction rules of λ comprise two groups of rules: those for the simply typed λ-calculus (intuitionistic reductions), and those that deal with process communication (cross reductions).…”

“…We introduced λ , a simple and yet expressive typed parallel λ-calculus based on (suitable fragments of) propositional intermediate logics. λ is based on and greatly simplifies the calculi in [4,3,2]. The basic communication reductions in the calculi λ CL [3] and λ G [2] implement particular λ communications: the message passing mechanisms based on classical logic and on Gödel logic, respectively.…”

“…λ CL and λ G also contain additional reductions (permutations and full cross), needed for proving weak normalization and the subformula property. These calculi were generalized in [4], that introduces a Curry-Howard correspondence for all propositional intermediate logics characterized by classical disjunctive tautologies, the axioms for classical and Gödel logic being particular cases. The reductions in [4] are the same as for λ CL and λ G but their activation procedure is based on transmitting values and thus it is logic independent.…”

“…A recent addition to them is the discovery in [4] of the connection between propositional logics intermediate between classical logic and IL, and concurrent extensions of the simply typed λ-calculus. More precisely, the considered logics extend IL with the classical disjunctive tautologies interpreted as synchronization schemata in [14], i.e.…”

“…The aim of this work is to present a version of the calculi in [4,2,3] suitable for programming. We show that a simplified version of these calculi is expressive enough to model arbitrary process network topologies and to encode interesting parallel algorithms taken from [22].…”