The Structure of Interaction.

Interaction nets provide a graphical paradigm of computation based on net rewriting. By encoding the cut-elimination process of linear logic they have proved successful in understanding the dynamics of reduction in the /spl lambda/-calculus. G. Gonthier et al. (1992) gave an optimal infinite system of interaction nets for linear logic by removing the global boxes. However efficient implementations of optimal reduction have had to break away from the interaction net paradigm. Here we give an efficient new finite interaction net encoding of linear logic which is not optimal, but overcomes many of the inefficiencies caused by the bookkeeping operations in the implementations of optimal reduction. We code the global operations on boxes (contraction, weakening, dereliction, commutative cut) in a local way, keeping the box structure, which results in a system allowing a great deal of sharing with an extremely low overhead. We believe that this implementation is the most faithful of all the extant interaction net encodings of linear logic.

Interaction nets for linear logic

This article aims to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. It was presented at the 36th International Conference on Logic Programming (ICLP).

An implementation model of the typed λ-calculus based on Linear Chemical Abstract Machine

A linear process is a system of events and states related by an inner product, on which are defined the behaviorally motivated operations of tensor product or orthocurrence, sum or concurrence, sequence, and choice. Linear process algebra or LPA is the theory of this framework. LPA resembles Girard's linear logic with the differences attributable to its focus on behavior instead of proof. As with MLL the multiplicative part can be construed via the Curry-Howard isomorphism as an enrichment of Boolean algebra. The additives cater for independent concurrency or parallel play. The traditional sequential operations of sequence and choice exploit process-specific state information catering for notions of transition and cancellation.

Focusing is a proof-search strategy, originating in linear logic, that elegantly eliminates inessential nondeterminism, with one byproduct being a correspondence between focusing proofs and programs with explicit evaluation order. Higher-order abstract syntax (HOAS) is a technique for representing higher-order programming language constructs (e.g., λ's) by higher-order terms at the"meta-level", thereby avoiding some of the bureaucratic headaches of first-order representations (e.g., capture-avoiding substitution).

In this paper we deal the recursion and corecursion in functional programming. We discuss about the morphisms which express the recursion or corecursion, resp. We apply the linear logic which provides a logical perspective on computational issues such as control of resources and order of evaluation. The most important feature of linear logic is that formulae are considered as actions and its truth value depends on an internal state of a dynamic system. In this paper we present an alternative way of computation based on algebras and coalgebras. The correctness of our approaches we show by Curry-Howard correspondence.

When designing languages for functional reactive programming (FRP) the main challenge is to provide the user with a simple, flexible interface for writing programs on a high level of abstraction while ensuring that all programs can be implemented efficiently in a low-level language. To meet this challenge, a new family of modal FRP languages has been proposed, in which variants of Nakano's guarded fixed point operator are used for writing recursive programs guaranteeing properties such as causality and productivity. As an apparent extension to this it has also been suggested to use Linear Temporal Logic (LTL) as a language for reactive programming through the Curry-Howard isomorphism, allowing properties such as termination, liveness and fairness to be encoded in types. However, these two ideas are in conflict with each other, since the fixed point operator introduces non-termination into the inductive types that are supposed to provide termination guarantees. In this paper we show that by regarding the modal time step operator of LTL a submodality of the one used for guarded recursion (rather than equating them), one can obtain a modal type system capable of expressing liveness properties while retaining the power of the guarded fixed point operator. We introduce the language Lively RaTT, a modal FRP language with a guarded fixed point operator and an `until' type constructor as in LTL, and show how to program with events and fair streams. Using a step-indexed Kripke logical relation we prove operational properties of Lively RaTT including productivity and causality as well as the termination and liveness properties expected of types from LTL. Finally, we prove that the type system of Lively RaTT guarantees the absence of implicit space leaks.

Linear logic has recently been introduced by Girard as a logic of actions that seems well suited for concurrent computation. In this paper, we establish a systematic correspondence between Petri nets, linear logic theories, and linear categories. Such a correspondence sheds new light on the relationships between linear logic and concurrency, and on how both areas are related to category theory. Categories are here viewed as concurrent systems the objects of which are states, and the morphisms of which are transitions. This is an instance of the Lambek-Lawvere correspondence between logic and category theory that cannot be expressed within the more restricted framework of the Curry-Howard correspondence.

A decision procedure revisited: notes on direct logic, linear logic and its implementation

Pomset logic introduced by Retor\'e is an extension of linear logic with a self-dual noncommutative connective. The logic is defined by means of proof-nets, rather than a sequent calculus. Later a deep inference system BV was developed with an eye to capturing Pomset logic, but equivalence of system has not been proven up to now. As for a sequent calculus formulation, it has not been known for either of these logics, and there are convincing arguments that such a sequent calculus in the usual sense simply does not exist for them. In an on-going work on semantics we discovered a system similar to Pomset logic, where a noncommutative connective is no longer self-dual. Pomset logic appears as a degeneration, when the class of models is restricted. Motivated by these semantic considerations, we define in the current work a semicommutative multiplicative linear logic}, which is multiplicative linear logic extended with two nonisomorphic noncommutative connectives (not to be confused with very different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets and show how this logic degenerates to Pomset logic. However, a more interesting problem than just finding yet another noncommutative logic is to find a sequent calculus for this logic. We introduce decorated sequents, which are sequents equipped with an extra structure of a binary relation of reachability on formulas. We define a decorated sequent calculus for semicommutative logic and prove that it is cut-free, sound and complete. This is adapted to "degenerate" variations, including Pomset logic. Thus, in particular, we give a variant of sequent calculus formulation for Pomset logic, which is one of the key results of the paper.

In this paper a strongly normalizing cut-elimination procedure is presented for classical logic. The procedure adapts the standard cut transformations, see for example []. In particular our cutelimination procedure requires no special annotations on formulae. We design a term calculus for a variant of Kleene’s sequent calculus G3 via the Curry-Howard correspondence and the cut-elimination steps are given as rewrite rules. In the strong normalization proof we adapt the symmetric reducibility candidates developed by Barbanera and Berardi.

Constructive linear-time temporal logic: Proof systems and Kripke semantics

The graph rewriting system of interaction nets has been very successful for the implementation of the lambda calculus. In this paper we show how the ideas can be extended and simplified to encode Godel’s System \(\mathcal {T}\)—the simply typed \(\lambda \)-calculus extended with numbers. Surprisingly, using some results about System \(\mathcal {T}\), we obtain a very simple system of interaction nets that is significantly more efficient than a direct encoding for the evaluation of programs.

The main result of this paper is a normalizing system of natural deduction for the full language of intuitionistic linear logic. No explicit weakening or contraction rules for \(\)-formulas are needed. By the systematic use of general elimination rules a correspondence between normal derivations and cut-free derivations in sequent calculus is obtained. Normalization and the subformula property for normal derivations follow through translation to sequent calculus and cut-elimination.