Proof nets, garbage, and computations

A cornerstone of the theory of proof nets for unit-free multiplicative linear logic (MLL) is the abstract representation of cut-free proofs modulo inessential rule commutation. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cut-free monomial proof nets can correspond to the same cut-free proof. Thus, the problem of finding a satisfactory notion of proof net for unit-free multiplicative-additive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory.

This paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.

The deep inference presentation of multiplicative exponential linear logic (MELL) benefits from a rich combinatoric analysis with many more proofs in comparison to its sequent calculus presentation. In the deep inference setting, all the sequent calculus proofs are preserved. Moreover, many other proofs become available, and some of these proofs are much shorter. However, proof search in deep inference is subject to a greater nondeterminism, and this nondeterminism constitutes a bottleneck for applications. To this end, we address the problem of reducing nondeterminism in MELL by refining and extending our technique that has been previously applied to multiplicative linear logic and classical logic. We show that, besides the nondeterminism in commutative contexts, the nondeterminism in exponential contexts can be reduced in a proof theoretically clean manner. The method conserves the exponential speed-up in proof construction due to deep inference, exemplified by Statman tautologies. We validate the improvement in accessing the shorter proofs by experiments with our implementations.

We consider the following decision problems:ProofNet: Is a given multiplicative linear logic (MLL) proof structure a proof net?EssNet: Is a given essential net (of an intuitionistic MLL sequent) correct?In this article we show how to obtain linear-time algorithms for EssNet. As a corollary, by showing that ProofNet is linear-time reducible to EssNet (by the Trip Translation), we obtain a linear-time algorithm for ProofNet.We show further that it is possible to optimize the verification so that each node of the input structure is visited at most once. Finally, we present linear-time algorithms for sequentializing proof nets and essential nets, that is, for finding derivations of the underlying sequents.

We present a syntax for MALL (multiplicative additive linear logic without units) proof nets which refines Girard's one. It is also based on the use of monomial weights for identifying additive components (slices). Our generalization gives the possibility of representing a kind of sharing of nodes which does not exist in Girard's nets. This sharing leads to the definition of a strong cut elimination procedure for MALL. We give a correctness criterion which is proved to be stable by reduction and to give a sequentialization theorem with respect to the sequent calculus. Sequentialization is proved by showing that an expansion procedure allows us to unfold any of our proof nets into a Girard proof net.

Proof nets provide permutation-independent representations of proofs and are used to investigate coherence problems for monoidal categories. We investigate a coherence problem concerning Second Order Multiplicative Linear Logic (MLL2), that is, the one of characterizing the equivalence over proofs generated by the interpretation of quantifiers by means of ends and coends. We provide a compact representation of proof nets for a fragment of MLL2 related to the Yoneda isomorphism. By adapting the "rewiring approach" used in coherence results for star-autonomous categories, we define an equivalence relation over proof nets called "re-witnessing". We prove that this relation characterizes, in this fragment, the equivalence generated by coends.

Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded into a differential net, which is its Taylor expansion. We prove that two different MELL proof-nets have two different Taylor expansions. As a corollary, we prove a completeness result for MELL: We show that the relational model is injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the relational model is exactly axiomatized by cut-elimination.

It is well known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarised briefly herein, we have provided a class of models calledclassical categoriesthat is sound and complete and avoids this collapse by interpreting cut reduction by a poset enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatisation of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models calledDummett categories. Examples of these include, besides the classical categories mentioned above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices, which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic by applying it to obtain a classical category from a Dummett category.Along the way, we gain detailed insights into the changes that proofs undergo during cut elimination in the presence of weakening and contraction.

We give an event structures based true-concurrency characterization of deep inference proofs. The method is general to all deep inference systems that can be expressed as term rewriting systems. This delivers three consequences in a spectrum from theoretical to practical: the event structure characterization i provides a qualification of proof identity akini¾?to proof nets for multiplicative linear logic and to atomic flows for classical logic; ii provides a concurrency theoretic interpretation for applications in logic programming; iii reduces the length of the proofs, and thereby extends the margin of proof search applications.

Constant-only multiplicative linear logic is NP-complete

We define two non-idempotent intersection type systems for the linear substitution calculus, a calculus with partial substitutions acting at a distance that is a computational interpretation of linear logic proof-nets. The calculus naturally express linear-head reduction, a notion of evaluation of proof nets that is strongly related to abstract machines. We show that our first (resp. second) quantitave type system characterizes linear-head, head and weak (resp. strong) normalizing sets of terms. All such characterizations are given by means of combinatorial arguments, i.e. there is a measure based on type derivations which decreases with respect to each reduction relation considered in the paper.

Sharing graphs are a way of representing linear logic proof-nets in such a way that their reduction never duplicates a redex. In their usual presentations, they present a problem of coherence: if the proof-net N reduces by standard cut-elimination to N′, then, by reducing the sharing graph of N we do not obtain the sharing graph of N′. We solve this problem by changing the way the information is coded into sharing graphs and introducing a new reduction rule (absorption). The rewriting system is confluent and terminating.

We propose a notion of symmetric reduction for a system of proof-nets for Multiplicative Affine Logic with Mix (MAL + Mix) (namely, multiplicative linear logic with the mix-rule the unrestricted weakening-rule). 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 of a proof-net 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 Weyhranch's decision procedure for affine logic [15, 4], the use of 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 for intuitionistic and linear logic.

We introduce graph reduction technology that implements functional languages with control, such as Scheme with call/cc, where continuations can be manipulated explicitly as values, and can be optimally reduced in the sense of LEvy. The technology is founded on proofnets for multiplicative-exponential linear logic, extending the techniques originally proposed by Lamping, where we adapt the continuation-passing style transformation to yield a new understanding of sharable values. Confluence is maintained by returning multiple answers to a (shared) continuation. Proofnets provide a concurrent version of linear logic proofs, eliminating structurally irrelevant sequentialization, and ignoring asymmetric distinctions between inputs and outputs--dually, expressions and continuations. While Lamping's graphs and their variants encode an embedding of intuitionistic logic into linear logic, our construction implicitly contains an embedding of classical logic into linear logic. We propose a family of translations, produced uniformly by beginning with a continuation-passing style semantics for the languages, employing standard codings into proofnets using call-by-value, call-by-name--or hybrids of the two--to locate proofnet boxes, and converting the proofnets to direct style. The resulting graphs can be reduced simply (cut elimination for linear logic), have a consistent semantics that is preserved by reduction (geometry of interaction, via the so-called context semantics), and allow shared, incremental evaluation of continuations (optimal reduction).

We present a demonstration of Andreoli’s focusing theorem for proofs of linear logic (MALL) that avoids directly reasoning on sequent calculus proofs. Following Andreoli-Maieli’s strategy, exploited in the MLL case, we prove the focusing theorem as a particular sequentialization strategy for MALL proof nets that are in canonical form. Canonical proof nets satisfy the property that asynchronous links are always ready to sequentialization while synchronous focusing links represent clusters of links that are hereditarily ready to sequentialization.Keywordslinear logicsequent calculusfocusing proofsproof nets