Complexity Bounds for Sum-Product Logic via Additive Proof Nets and Petri Nets

Abstract

Abstract-We investigate efficient algorithms for the additive fragment of linear logic. This logic is an internal language for categories with finite sums and products, and describes concurrent two-player games of finite choice. In the context of session types, typing disciplines for communication along channels, the logic describes the communication of finite choice along a single channel.We give a simple linear time correctness criterion for unit-free propositional additive proof nets via a natural construction on Petri nets. This is an essential ingredient to linear time complexity of the second author's combinatorial proofs for classical logic.For full propositional additive linear logic, including the units, we give a proof search algorithm that is linear-time in the product of the source and target formula, and an algorithm for proof net correctness that is of the same time complexity. We prove that proof search in first-order additive linear logic is NP-complete.

Conflict nets

“…Coalescence for MALL departs from coalescence in purely additive linear logic [18]. Conflict nets restricted to ALL coincide with ALL proof nets, with a single node covering the axiom links.…”

“…Geometric correctness is coalescence, a form of rewriting that can be thought of as abstract, top-down sequentialization, generalizing additive coalescence [18] and multiplicative contractibility [8]. A conflict net is correct if it coalesces to a single link on all formula roots.…”

“…Proof nets for MLL [13] and ALL in binary-relation formulation [18,21,25,26] are consummate categorically, representing the free unitfree star-autonomous category [19,20] and the free binary productcoproduct category [21]. Conflict nets remain faithful to both, by including them as sub-systems: an MLL /ALL net is a conflict net…”

“…Recent work has emphasised the interplay between canonicity and complexity. Correctness of MLL proof nets is linear-time [15], as is correctness for ALLU proof nets [18], while the problem is NLcomplete for strongly canonical MALL proof nets [9]. The interplay is particularly strong for proof equivalence problems, which may be decided by via translation to canonical proof nets: for MLL with units the problem is PSPACE-complete [17], effectively ruling out canonical proof nets, while for MALL it is LOGSPACE-complete [3].…”

“…For example, the net above is invariant upon commuting adjacent ⊗ rules. Similarly, proof nets for ALL (unit-free additive linear logic) in binary-relation formulation [18,21,25,26] are canonical geometric abstractions of ALL proofs. For example, the two ALL proofs • Problem: are there efficient canonical proof nets for MALL (unitfree multiplicative-additive linear logic)?…”