Proof nets for additive linear logic with units

Abstract—Additive linear logic, the fragment of linear logic concerning linear implication between strictly additive formu-lae, coincides with sum-product logic, the internal language of categories with free finite products and coproducts. Deciding equality of its proof terms, as imposed by the categorical laws, is complicated by the presence of the units (the initial and terminal objects of the category) and the fact that in a free setting products and coproducts do not distribute. The best known desicion algorithm, due to Cockett and Santocanale (CSL 2009), is highly involved, requiring an intricate case analysis on the syntax of terms. This paper provides canonical, graphical representations of the categorical morphisms, yielding a novel solution to this decision problem. Starting with (a modification of) existing proof nets, due to Hughes and Van Glabbeek, for additive linear logic without units, canonical forms are obtained by graph rewriting. The rewriting algorithm is remarkably simple. As a decision procedure for term equality it matches the known complexity of the problem. A main technical contribution of the paper is the substantial correctness proof of the algorithm. I

Graphical representation of canonical proof: two case studies

An interesting problem in proof theory is to find representations of proof that do not distinguish between proofs that are ‘morally’ the same. For many logics, the presentation of proofs in a traditional formalism, such as Gentzen’s sequent calculus, introduces artificial syntactic structure called ‘bureaucracy’; e.g., an arbitrary ordering of freely permutable inferences. A proof system that is free of bureaucracy is called canonical for a logic. In this dissertation two canonical proof systems are presented, for two logics: a notion of proof nets for additive linear logic with units, and ‘classical proof forests’, a graphical formalism for first-order classical logic. Additive linear logic (or sum–product logic) is the fragment of linear logic consisting of linear implication between formulae constructed only from atomic formulae and the additive connectives and units. Up to an equational theory over proofs, the logic describes categories in which finite products and coproducts occur freely. A notion of proof nets for additive linear logic is presented, providing canonical graphical representations of the categorical morphisms and constituting a tractable decision procedure for this equational theory. From existing proof nets for additive linear logic without units by Hughes and Van Glabbeek (modified to include the units naively), canonical proof nets are obtained by a simple graph rewriting algorithm called saturation. Main technical contributions are the substantial correctness proof of the saturation algorithm, and a correctness criterion for saturated nets. Classical proof forests are a canonical, graphical proof formalism for first-order classical logic. Related to Herbrand’s Theorem and backtracking games in the style of Coquand, the forests assign witnessing information to quantifiers in a structurally minimal way, reducing a first-order sentence to a decidable propositional one. A similar formalism ‘expansion tree proofs’ was presented by Miller, but not given a method of composition. The present treatment adds a notion of cut, and investigates the possibility of composing forests via cut-elimination. Cut-reduction steps take the form of a rewrite relation that arises from the structure of the forests in a natural way. Yet reductions are intricate, and initially not well-behaved: from perfectly ordinary cuts, reduction may reach unnaturally configured cuts that may not be reduced. Cutelimination is shown using a modified version of the rewrite relation, inspired by the game-theoretic interpretation of the forests, for which weak normalisation is shown, and strong normalisation is conjectured. In addition, by a more intricate argument, weak normalisation is also shown for the original reduction relation

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

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

MALL proof equivalence is Logspace-complete, via binary decision diagrams

Proof equivalence in a logic is the problem of deciding whether two proofs are equivalent modulo a set of permutation of rules that reflects the commutative conversions of its cut-elimination procedure. As such, it is related to the question of proofnets: finding canonical representatives of equivalence classes of proofs that have good computational properties. It can also be seen as the word problem for the notion of free category corresponding to the logic. It has been recently shown that proof equivalence in MLL (the multiplicative with units fragment of linear logic) is PSPACE-complete, which rules out any low-complexity notion of proofnet for this particular logic. Since it is another fragment of linear logic for which attempts to define a fully satisfactory low-complexity notion of proofnet have not been successful so far, we study proof equivalence in MALL- (multiplicative-additive without units fragment of linear logic) and discover a situation that is totally different from the MLL case. Indeed, we show that proof equivalence in MALL- corresponds (under AC0 reductions) to equivalence of binary decision diagrams, a data structure widely used to represent and analyze Boolean functions efficiently. We show these two equivalent problems to be LOGSPACE-complete. If this technically leaves open the possibility for a complete solution to the question of proofnets for MALL-, the established relation with binary decision diagrams actually suggests a negative solution to this problem.Comment: in TLCA 201

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