Proof Diagrams for Multiplicative Linear Logic

The original idea of proof nets can be formulated by means of interaction nets syntax. Additional machinery as switching, jumps and graph connectivity is needed in order to ensure correspondence between a proof structure and a correct proof in sequent calculus. In this paper we give an interpretation of proof nets in the syntax of string diagrams. Even though we lose standard proof equivalence, our construction allows to define a framework where soundness and well-typeness of a diagram can be verified in linear time.Comment: In Proceedings LINEARITY 2016, arXiv:1701.0452

The Complexity of Local Proof Search in Linear Logic (Extended Abstract)

AbstractProof search in linear logic is known to be difficult: the provability of propositional linear logic formulas is undecidable. Even without the modalities, multiplicative-additive fragment of propositional linear logic, mall, is known to be PSPACE-complete, and the pure multiplicative fragment, mll, is known to be np-complete. However, this still leaves open the possibility that there might be proof search heuristics (perhaps involving randomization) that often lead to a proof if there is one, or always lead to something close to a proof. One approach to these problems is to study strategies for proof games. A class of linear logic proof games is developed, each with a numeric score that depends on the number of certain preferred axioms used in a complete or partial proof tree. Using recent techniques for proving lower bounds on optimization problems, the complexity of these games is analyzed for the fragment mll extended with additive constants and for the fragment MALL. It is shown that no efficient heuristics exist unless there is an unexpected collapse in the complexity hierarchy

Two paradigms of logical computation in affine logic?

We propose a notion of symmetric reduction for a system of proof nets for multiplicative Affine Logic with Mix. We prove that such a reduction has the strong normalization and Church-Rosser properties. A notion of irrelevance in a proof net is defined and the possibility of cancelling the irrelevant parts without erasing the entire net is taken as one of the correctness conditions. Therefore purely local cut-reductions are given, minimizing cancellation and suggesting a paradigm of "computation without garbage collection". Reconsidering Ketonen and Weyhrauch's decision procedure for affine logic, the use od the mix rule is related to the non-determinism of classical proof theory. The question arises whether these features of classical cut-elimination are really irreducible to the familiar paradigm of cut-elimination in intuitionistic and linea logic

Proof equivalence in MLL is PSPACE-complete

MLL proof equivalence is the problem of deciding whether two proofs in multiplicative linear logic are related by a series of inference permutations. It is also known as the word problem for star-autonomous categories. Previous work has shown the problem to be equivalent to a rewiring problem on proof nets, which are not canonical for full MLL due to the presence of the two units. Drawing from recent work on reconfiguration problems, in this paper it is shown that MLL proof equivalence is PSPACE-complete, using a reduction from Nondeterministic Constraint Logic. An important consequence of the result is that the existence of a satisfactory notion of proof nets for MLL with units is ruled out (under current complexity assumptions). The PSPACE-hardness result extends to equivalence of normal forms in MELL without units, where the weakening rule for the exponentials induces a similar rewiring problem.Comment: Journal version of: Willem Heijltjes and Robin Houston. No proof nets for MLL with units: Proof equivalence in MLL is PSPACE-complete. In Proc. Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science, 201

Canonical Proof nets for Classical Logic

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK\ast in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of [23]), we give a full proof of (c) for expansion nets with respect to LK\ast, and in addition give a cut-elimination procedure internal to expansion nets - this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and Computation