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

The Functional Machine Calculus

This paper presents the Functional Machine Calculus (FMC) as a simple model of higher-order computation with "reader/writer" effects: higher-order mutable store, input/output, and probabilistic and non-deterministic computation. The FMC derives from the lambda-calculus by taking the standard operational perspective of a call-by-name stack machine as primary, and introducing two natural generalizations. One, "locations", introduces multiple stacks, which each may represent an effect and so enable effect operators to be encoded into the abstraction and application constructs of the calculus. The second, "sequencing", is known from kappa-calculus and concatenative programming languages, and introduces the imperative notions of "skip" and "sequence". This enables the encoding of reduction strategies, including call-by-value lambda-calculus and monadic constructs. The encoding of effects into generalized abstraction and application means that standard results from the lambda-calculus may carry over to effects. The main result is confluence, which is possible because encoded effects reduce algebraically rather than operationally. Reduction generates the familiar algebraic laws for state, and unlike in the monadic setting, reader/writer effects combine seamlessly. A system of simple types confers termination of the machine

Un Lambda-Calcul Atomique

International audienceNous introduisons un lambda-calcul avec partage explicite, le lambda-calcul atomique, dans lequel la duplication des sous-termes est faite pas à pas en fonction des constructeurs. Nous donnons une fonction de dénotation du lambda-calcul atomique dans le lambda-calcul et montrons que le lambda-calcul atomique simule la -réduction et préserve la normalisation forte. Nous donnons aussi un système de type pour le lambda-calcul atomique et montrons que la réduction préserve le type

A deep quantitative type system

We investigate intersection types and resource lambda-calculus in deep-inference proof theory. We give a unified type system that is parametric in various aspects: it encompasses resource calculi, intersection-typed lambda-calculus, and simply-typed lambda-calculus; it accommodates both idempotence and non-idempotence; it characterizes strong and weak normalization; and it does so while allowing a range of algebraic laws to determine reduction behaviour, for various quantitative effects. We give a parametric resource calculus with explicit sharing, the "collection calculus", as a Curry-Howard interpretation of the type system, that embodies these computational properties

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

A deep quantitative type system

We investigate intersection types and resource lambda-calculus in deep-inference proof theory. We give a unified type system that is parametric in various aspects: it encompasses resource calculi, intersection-typed lambda-calculus, and simply-typed lambda-calculus; it accommodates both idempotence and non-idempotence; it characterizes strong and weak normalization; and it does so while allowing a range of algebraic laws to determine reduction behaviour, for various quantitative effects. We give a parametric resource calculus with explicit sharing, the “collection calculus”, as a Curry–Howard interpretation of the type system, that embodies these computational properties

Proof Nets for First-Order Additive Linear Logic

We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. Both capture sequent calculus cut-elimination as a one-step global composition operation