Proof Nets Research Articles

Proof nets Construction and Automated Deduction in Non-Commutative Linear Logic: extended abstract

Proof nets can be seen as a multiple conclusion natural deduction system for Linear Logic (LL) and form a good formalism to analyze some computation mechanisms, for instance in type-theoretic interpretations. This paper presents an algorithm for automated proof nets construction in the non-commutative multiplicative linear logic that is useful for applications including planning, concurrency or sequentiality. The properties of this algorithm can be proved from a recently defined graph-theoretic characterization of non-commutative proof nets. Involving simple construction principles improved in the commutative case, it leads also to a new proof search method for the non-commutative fragment. Moreover because of the relationships between the non-commutative linear logic and the Lambek calculus we can derive from it an alternate method for automatic construction of proof nets in this calculus.

A New Correctness Criterion for Cyclic Proof Nets

We define proof nets for cyclic multiplicative linear logic as edge bi-coloured graphs. Our characterization is purely graph theoretical and works without further complication for proof nets with cuts, which are usually harder to handle in the non-commutative case. This also provides a new characterization of the proof nets for the Lambek calculus (with the empty sequence) which simply are a restriction on the formulas to be considered (which are asked to be intuitionistic).

On the Jordan-Hölder decomposition of proof nets

Having defined a notion of homology for paired graphs, Metayer ([Ma]) proves a homological correctness criterion for proof nets, and states that for any proof net $G$ there exists a Jordan-Holder decomposition of ${\mathsf H}_0(G)$ . This decomposition is determined by a certain enumeration of the pairs in $G$ . We correct his proof of this fact and show that there exists a 1-1 correspondence between these Jordan-Holder decompositions of ${\mathsf H}_0(G)$ and the possible ‘construction-orders’ of the par-net underlying $G$ .

Interaction Combinators

It is shown that a very simple system of interaction combinators , with only three symbols and six rules, is a universal model of distributed computation, in a sense that will be made precise. This paper is the continuation of the author's work on interaction nets , inspired by Girard's proof nets for linear logic , but no preliminary knowledge of these topics is required for its reading.

Natural deduction and coherence for weakly distributive categories

This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the two-tensor structure ( times/par) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansion-reduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion-reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories. The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending similar results for monoidal categories to include the treatment of the tensor units. Second, we extend these coherence results to the full theory of ∗-autonomous categories — providing a decision procedure for the maps of free symmetric ∗-autonomous categories. Third, we derive a conservative extension result for the passage from weakly distributive categories to ∗-autonomous categories. We show strong categorical conservativity, in the sense that the unit of the adjunction between weakly distributive and ∗-autonomous categories is fully faithful.

Perfect matchings and series-parallel graphs: multiplicatives proof nets as R&B-graphs: [Extended Abstract

A graph-theoretical look at multiplicative proof nets lead us to two new descriptions of a proof net, both as a graph endowed with a perfect matching.The first one is a rather conventional encoding of the connectives which nevertheless allows us to unify various sequentialisation techniques as the corollaries of a single graph theoretical result.The second one is more exciting: a proof net simply consists in the set of its axioms — the perfect matching — plus one single series-parallel graph which encodes the whole syntactical forest of the sequent. We thus identify proof nets which only differ because of the commutativity or associativity of the connectives, or because final par have been performed or not. We thus push further the program of proof net theory which is to get closer to the proof itself, ignoring as much as possible the syntactical “bureaucracy”.

A Graph-Theoretic Characterization Theorem for Multiplicative Fragment of Non-Commutative Linear Logic (Extended Abstract)

It is well-known that every proof net of a non-commutative version of MLL (Multiplicative fragment of Commutative Linear Logic) can be drawn as a plane Danos-Regnier graph (drawing) satisfying the switching condition of Danos-Regnier [3]. In this paper, we study the reverse direction; we introduce a system MNCLL logically equivalent to the multiplicative fragment of Cyclic Linear Logic introduced by Yetter [9], and show that any plane Danos-Regnier graph drawing satisfying the switching condition represents a unique non-commutative proof net (i.e., a proof net of MNCLL) modulo cyclic shifts. In the course of proving this, we also give the characterization of the non-commutative proof nets by means of the notion of strong planity, as well as the notion of a certain long-trip condition, called the stack-condition, of a Danos-Regnier graph, the latter of which is related to Abrusci balanced long-trip condition [2].

From Proof Nets to Games: extended abstract

We give a class of proof nets for Intuitionistic Linear Logic with the connectives -o, !, prove a correctness criterion for them and show that a games semantics can be directly derived from these nets, along with a full completeness theorem.

Causal dependencies in multiplicative linear logic with MIX

A new correctness criterion for discriminating Proof Nets among Proof Structures of Multiplicative Linear Logic with the MIX rule is provided. This criterion is inspired by an original interpretation of Proof Structures as distributed systems, and logical formulae as processes. The computation inside a system corresponds to the logical flow of information inside a proof, that is, roughly speaking, a distributed version of Girard's token trip. Proof Nets are then characterised as deadlock free Proof Structures (deadlock free distributed systems). This result follows by explicitly considering the causal dependencies among logical formulae inside proofs, and it provides a new understanding of notions such as acyclicity, chains and empires in terms of concurrent computations.

Proof nets of PN as graphs

Proof nets of PN as graphs

Proof nets of PN as graphs

Proof nets of PN as graphs

Head linear reduction and pure proof net extraction

Proof net calculus introduced in Girard (1987) has been extended in the paradigm of Scott's domains equation D= D→ D which generates a logical point of view for pure λ-calculus in Donos (1989). Methodologically speaking, in this paper the proof theoretic counterpart of Böhm's theorem given in Böhm (1968) for pure λ-calculus, is proposed as extension of the Curry-Howard paradigm. Technically speaking, as the extraction of a subterm using the β-reduction is possible also subnet extraction can be internalized by cut-elimination, using proof nets realizes managing and better understanding the procedure of extraction.

Interpolation in fragments of classical linear logic

AbstractWe study interpolation for elementary fragments of classical linear logic. Unlike in intuitionistic logic (see [Renardel de Lavalette, 1989]) there are fragments in linear logic for which interpolation does not hold. We prove interpolation for a lot of fragments and refute it for the multiplicative fragment (→, +), using proof nets and quantum graphs. We give a separate proof for the fragment with implication and product, but without the structural rule of permutation. This is nearly the Lambek calculus. There is an appendix explaining what quantum graphs are and how they relate to proof nets.

Linear logic, coherence and dinaturality

A general coherence theorem for monoidal closed structures is obtained by modifying the logical approach to coherence questions, due to Lambek [1969, 1990] by making use of linear logic. Linear logic, introduced by Girard, has many advantages which are of use in studying coherence. Most notably, its resource-sensitive nature makes it ideal for studying monoidal closed structures. The logical approach is also modified by using natural deduction rather than sequent calculus. The natural deduction system in question is proof nets, also introduced by Girard. Proof nets have several important properties which are exploited to prove the coherence theorem. In particular, the cut elimination procedure is confluent and strongly normalizing. The approach to coherence is to define a general structure, the autonomous deductive system, for defining many theories of monoidal closed categories. An autonomous deductive system is a deductive system with several added features, which are suggested by the properties of proof nets. It is then possible to give a straightforward criterion for whether a given theory of monoidal closed categories, specified by an autonomous deductive system, is coherent. Finally, a relationship is established between coherence and the composition problem for dinatural transformations. Thus, the dinatural approach to modelling polymorphic types, due to Bainbridge et al. [1990], can be extended to linear polymorphism.

Generating plans in linear logic: II. A geometry of conjunctive actions

After the proof-theoretic study given in Masseron (1991), we propose a new geometric characterization of actions, in the spirit of Girard's proof nets and Bibel's connection graphs, in the conjunctive case.

Constructive logics Part I: A tutorial on proof systems and typed λ-calculi

The purpose of this paper is to give an exposition of material dealing with constructive logics, typed λ-calculi, and linear logic. The emergence in the past ten years of a coherent field of research often named “logic and computation” has had two major (and related) effects: firstly, it has rocked vigorously the world of mathematical logic; secondly, it has created a new computer science discipline, which spans a range of subjects from what is traditionally called the theory of computation, to programming language design. Remarkably, this new body of work relies heavily on some “old” concepts found in mathematical logic, like natural deduction, sequent calculus, and λ-calculus (but often viewed in a different light), and also on some newer concepts. Thus, it may be quite a challenge to become initiated to this new body of work (but the situation is improving, and there are now some excellent texts on this subject matter). This paper attempts to provide a coherent and hopefully “gentle” initiation to this new body of work. We have attempted to cover the basic material on natural deduction, sequent calculus, and typed λ-calculus, but also to provide an introduction to Girard's linear logic, one of the most exciting developments in logic these past six years. The first part of these notes gives an exposition of the background material (with some exceptions, such as “contraction-free” systems for intuitionistic propositional logic and the Girard translation of classical logic into intuitionistic logic, which is new). The second part is devoted to more current topics such as linear logic, proof nets, the geometry of interaction, and unified systems of logic ( LU).

Proof Nets for Lambek Calculus

The proof nets of linear logic are adapted to the non-commutative Lambek calculus. A different criterion for soundness of proof nets is given, which gives rise to new algorithms for proof search. The order sensitiveness of the Lambek calculus is reflected by the planarity condition on proof nets; this gives rise to a new non-provability check: balance.