Multiplicative-Additive Proof Equivalence is Logspace-complete, via Binary Decision Trees

Given a logic presented in a sequent calculus, a natural question is that of equivalence of proofs: to determine whether two given proofs are equated by any denotational semantics, ie any categorical interpretation of the logic compatible with its cut-elimination procedure. This notion can usually be captured syntactically by a set of rule permutations. Very generally, proofnets can be defined as combinatorial objects which provide canonical representatives of equivalence classes of proofs. In particular, the existence of proof nets for a logic provides a solution to the equivalence problem of this logic. In certain fragments of linear logic, it is possible to give a notion of proofnet with good computational properties, making it a suitable representation of proofs for studying the cut-elimination procedure, among other things. It has recently been proved that there cannot be such a notion of proofnets for the multiplicative (with units) fragment of linear logic, due to the equivalence problem for this logic being Pspace-complete. We investigate the multiplicative-additive (without unit) fragment of linear logic and show it is closely related to binary decision trees: we build a representation of proofs based on binary decision trees, reducing proof equivalence to decision tree equivalence, and give a converse encoding of binary decision trees as proofs. We get as our main result that the complexity of the proof equivalence problem of the studied fragment is Logspace-complete.Comment: arXiv admin note: text overlap with arXiv:1502.0199

Session Types in Abelian Logic

There was a PhD student who says "I found a pair of wooden shoes. I put a coin in the left and a key in the right. Next morning, I found those objects in the opposite shoes." We do not claim existence of such shoes, but propose a similar programming abstraction in the context of typed lambda calculi. The result, which we call the Amida calculus, extends Abramsky's linear lambda calculus LF and characterizes Abelian logic.Comment: In Proceedings PLACES 2013, arXiv:1312.221

Multiplicative-Additive Focusing for Parsing as Deduction

Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on identifying the essential mathematical structure of derivations. This is trivial in the case of context free grammar, where the parse structures are ordered trees; in the case of categorial grammar, the parse structures are proof nets. However, with respect to multiplicatives intrinsic proof nets have not yet been given for displacement calculus, and proof nets for additives, which have applications to polymorphism, are involved. Here we approach multiplicative-additive spurious ambiguity by means of the proof-theoretic technique of focalisation.Comment: In Proceedings WoF'15, arXiv:1511.0252

A language for multiplicative-additive linear logic

A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence is decidable by showing that the cut elimination rewrites supply a confluent rewriting system modulo equations.Comment: 16 pages without appendices, 30 with appendice

Multiparty Sessions based on Proof Nets

We interpret Linear Logic Proof Nets in a term language based on Solos calculus. The system includes a synchronisation mechanism, obtained by a conservative extension of the logic, that enables to define non-deterministic behaviours and multiparty sessions.Comment: In Proceedings PLACES 2014, arXiv:1406.331

Constructing Fully Complete Models of Multiplicative Linear Logic

The multiplicative fragment of Linear Logic is the formal system in this family with the best understood proof theory, and the categorical models which best capture this theory are the fully complete ones. We demonstrate how the Hyland-Tan double glueing construction produces such categories, either with or without units, when applied to any of a large family of degenerate models. This process explains as special cases a number of such models from the literature. In order to achieve this result, we develop a tensor calculus for compact closed categories with finite biproducts. We show how the combinatorial properties required for a fully complete model are obtained by this glueing construction adding to the structure already available from the original category.Comment: 72 pages. An extended abstract of this work appeared in the proceedings of LICS 201