The parallel versus branching recurrences in computability logic

TL;DR

This paper compares the logical strength of parallel and branching recurrences in Computability Logic, confirming that parallel recurrence forms a proper superset of branching recurrence and is strictly weaker.

Contribution

It verifies Japaridze's conjecture that the logic induced by parallel recurrence is a proper superset of that induced by branching recurrence, and shows their logical relationship.

Findings

Parallel recurrence logic is a proper superset of branching recurrence logic.

Branching recurrence logically implies parallel recurrence, but not vice versa.

Parallel recurrence is strictly weaker than branching recurrence.

Abstract

This paper shows that the basic logic induced by the parallel recurrence of Computability Logic is a proper superset of the basic logic induced by the branching recurrence. The latter is known to be precisely captured by the cirquent calculus system CL15, conjectured by Japaridze to remain sound---but not complete---with parallel recurrence instead of branching recurrence. The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that parallel recurrence is strictly weaker than branching recurrence in the sense that, while the latter logically implies the former, vice versa does not hold.