Proof Nets, Coends and the Yoneda Isomorphism

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.

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.

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, garbage, and computations

A graph-theoretic characterization theorem for multiplicative fragment of non-commutative linear logic

Given an intuitionistic proof net of linear logic, we abstract an order between its atomic formulas. From this order, we represent intuitionistic multiplicative proof nets in the more compact form of models of directed acyclic graph descriptions. If we restrict the logical framework to the implicative fragment of intuitionistic linear logic, we show that proof nets reduce to models of tree descriptions.KeywordsDirected Acyclic GraphConstraint Satisfaction ProblemAtomic FormulaLinear LogicSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We propose to represent the syntax and semantics of natural languages with labelled proof nets in the implicative fragment of intuitionistic linear logic. Resource-sensitivity of linear logic is used to express all dependencies between the syntactic constituents of a sentence in the form of a proof net. Phonological and semantic labelling of the proof net from its inputs to the unique output are used to produce the well-formed phonological form and the semantic representation of a sentence from entries of a lexicon. In this way we obtain a linguistic model of great flexibility because labelling is not completely determined logically: it is used to introduce linguistic constraints which go beyond the underlying logic. 1

We refine the simulation technique introduced in Di Cosmo and Kesner (1997) to show strong normalisation of $\l$-calculi with explicit substitutions via termination of cut elimination in proof nets (Girard 1987). We first propose a notion of equivalence relation for proof nets that extends the one in Di Cosmo and Guerrini (1999), and show that cut elimination modulo this equivalence relation is terminating. We then show strong normalisation of the typed version of the $\ll$-calculus with de Bruijn indices (a calculus with full composition defined in David and Guillaume (1999)) using a translation from typed $\ll$ to proof nets. Finally, we propose a version of typed $\ll$ with named variables, which helps to give a better understanding of the complex mechanism of the explicit weakening notation introduced in the $\ll$-calculus with de Bruijn indices (David and Guillaume 1999).

Abstract.This paper presents a new correctness criterion for marked Danos-Reginer graphs (D-R graphs, for short) of Multiplicative Cyclic Linear Logic MCLL and Abrusci's non-commutative Linear Logic MNLL.As a corollary we obtain an affirmative answer to the open question whether a known quadratic-time algorithm for the correctness checking of proof nets for MCLL and MNLL can be improved to linear-time.

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 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

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.

The topics of structural proof theory and logic programming have influenced each other for more than three decades. Proof theory has contributed the notion of sequent calculus, linear logic, and higher-order quantification. Logic programming has introduced new normal forms of proofs and forced the examination of logic-based approaches to the treatment of bindings. As a result, proof theory has responded by developing an approach to proof search based on focused proof systems in which introduction rules are organized into two alternating phases of rule application. Since the logic programming community can generate many examples and many design goals (e.g., modularity of specifications and higher-order programming), the close connections with proof theory have helped to keep proof theory relevant to the general topic of computational logic.

Constant-only multiplicative linear logic is NP-complete

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use ofstructural proof theoryas an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.