A typed parallel {\lambda}-calculus via 1-depth intermediate proofs

We introduce a Curry-Howard correspondence for a large class of intermediate logics characterized by intuitionistic proofs with non-nested applications of rules for classical disjunctive tautologies (1-depth intermediate proofs). The resulting calculus, we call it $\lambda_{\parallel}$, is a strongly normalizing parallel extension of the simply typed $\lambda$-calculus. Although simple, the $\lambda_{\parallel}$ reduction rules can model arbitrary process network topologies, and encode interesting parallel programs ranging from numeric computation to algorithms on graphs.