A System of Interaction and Structure III: The Complexity of BV and Pomset Logic

TL;DR

This paper compares pomset logic and BV, two non-commutative logics extending linear logic, demonstrating they are not equivalent and revealing significant differences in their computational complexities.

Contribution

It proves that pomset logic and BV are not the same logic and analyzes their complexity, showing BV is NP-complete while pomset logic is a2_2^p-complete.

Findings

Pomset logic and BV are distinct logics.

Provability in BV is NP-complete.

Provability in pomset logic is a2_2^p-complete.

Abstract

Pomset logic and BV are both logics that extend multiplicative linear logic (with Mix) with a third connective that is self-dual and non-commutative. Whereas pomset logic originates from the study of coherence spaces and proof nets, BV originates from the study of series-parallel orders, cographs, and proof systems. Both logics enjoy a cut-admissibility result, but for neither logic can this be done in the sequent calculus. Provability in pomset logic can be checked via a proof net correctness criterion and in BV via a deep inference proof system. It has long been conjectured that these two logics are the same. In this paper we show that this conjecture is false. We also investigate the complexity of the two logics, exhibiting a huge gap between the two. Whereas provability in BV is NP-complete, provability in pomset logic is $\Sigma_2^p$-complete. We also make some observations with respect to possible sequent systems for the two logics.