. This tutorial paper provides an introduction to intuitionistic logic and linear logic, and shows how they correspond to type systems for functional languages via the notion of `Propositions as Types'. The presentation of linear logic is simplified by basing it on the Logic of Unity. An application to the array update problem is briefly discussed. 1 Introduction Some of the best things in life are free; and some are not. Truth is free. Having proved a theorem, you may use this proof as many times as you wish, at no extra cost. Food, on the other hand, has a cost. Having baked a cake, you may eat it only once. If traditional logic is about truth, then linear logic is about food. In traditional logic, if a fact is used to conclude another fact, the first fact is still available. For instance, given that A implies B and given A, one may deduce both A and B. In symbols, this is written as the judgement A ! B; A ` A \Theta B (i) where A ! B is read `A implies B', and A \Theta B is read `...

An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.

Girard's original definition of proof nets for linear logic involves boxes. The box is the unit for erasing and duplicating fragments of proof nets. It imposes synchronization, limits sharing, and impedes a completely local view of computation. Here we describe an implementation of proof nets without boxes. Proof nets are translated into graphs of the sort used in optimal -calculus implementations; computation is performed by simple graph rewriting. This graph implementation helps in understanding optimal reductions in the -calculus and in the various programming languages inspired by linear logic. 1 Beyond the -calculus The -calculus is not entirely explicit about the operations of erasing and duplicating arguments. These operations are important both in the theory of the - calculus and in its implementations, yet they are typically treated somewhat informally, implicitly. The proof nets of linear logic [1] provide a refinement of the -calculus where these operations become explici...

this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \Delta ` \Gamma as asserting that the multiplicative conjunction of the formulas in \Delta together imply the multiplicative disjunction of the formulas in \Gamma. A sequent calculus proof rule consists of a set of hypothesis sequents, displayed above a horizontal line, and a single conclusion sequent, displayed below the line, as below: Hypothesis1 Hypothesis2 Conclusion 4 Connections to Other Logics

. This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth (emphasized in classical logic) and proof construction (emphasized in intuitionistic logic) remain important in linear logic, it might be said that the emphasis in linear logic is on state. Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2]. As an intuitive motivation, let us consider reading logical deductions so tha...