We show how linear typing can be used to obtain functional programs which modify heap-allocated data structures in place. We present this both as a "design pattern" for writing C-code in a functional style and as a compilation process from linearly typed first-order functional programs into malloc()-free C code. The main technical result is the correctness of this compilation. The crucial innovation over previous linear typing schemes consists of the introduction of a resource type # which controls the number of constructor symbols such as cons in recursive definitions and ensures linear space while restricting expressive power surprisingly little. While the space e#ciency brought about by the new typing scheme and the compilation into C can also be realised by with state-of-the-art optimising compilers for functional languages such as Ocaml [16], the present method provides guaranteed bounds on heap space which will be of use for applications such as languages for embedd...

We formulate a typed formalism for concurrency where types denote freely composable structure of dyadic interaction in the symmetric scheme. The resulting calculus is a typed reconstruction of name passing process calculi. Systems with both the explicit and implicit typing disciplines, where types form a simple hierarchy of types, are presented, which are proved to be in accordance with each other. A typed variant of bisimilarity is formulated and it is shown that typed fi-equality has a clean embedding in the bisimilarity. Name reference structure induced by the simple hierarchy of types is studied, which fully characterises the typable terms in the set of untyped terms. It turns out that the name reference structure results in the deadlock-free property for a subset of terms with a certain regular structure, showing behavioural significance of the simple type discipline. 1 Introduction This is a preliminary study of types for concurrency. Types here denote freely composable structur...

. In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky [1]) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these and consider their computational content. 1 Intuitionistic Linear Logic Girard's Intuitionistic Linear Logic [3] is a refinement of Intuitionistic Logic where formulae must be used exactly once. Given this restriction the familiar logical connectives become divided into multiplicative and additive versions. Within this paper, we shall only consider the multiplicatives. Int...

The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives# of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the asynchronous connectives -#, & and #.

. While types for name passing calculi have been studied extensively in the context of sorting of polyadic ß-calculus [5, 34, 9, 28, 32, 19, 33, 10, 17], the same type abstraction is not possible in the monadic setting, which was left as an open issue by Milner [21]. We solve this problem with an extension of sorting which captures dynamic aspects of process behaviour in a simple way. Equationally this results in the full abstraction of the standard encoding of polyadic ß-calculus into the monadic one: the sorted polyadic ß-terms are equated by a basic behavioural equality in the polyadic calculus if and only if their encodings are equated in a basic behavioural equality in the typed monadic calculus. This is the first result of this kind we know of in the context of the encoding of polyadic name passing, which is a typical example of translation of high-level communication structures into ß- calculus. The construction is general enough to be extendable to encodings of calculi with mo...

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

In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We define a simple (but more general than previous proposals) categorical model for Intuitionistic Linear Logic and show how this can be used to derive the term assignment system. We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these, as well as with the equations which follow from our categorical model.

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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...

Introduction If we consider the interpretation of proofs as programs, say in intuitionistic logic, the question of equality between proofs becomes crucial: The syntax introduces meaningless distinctions whereas the (denotational) semantics makes excessive identifications. This question does not have a simple answer in general, but it leads to the notion of proof-net, which is one of the main novelties of linear logic. This has been already explained in [Gir87] and [GLT89]. The notion of interaction net introduced in [Laf90] comes from an attempt to implement the reduction of these proof-nets. It happens to be a simple model of parallel computation, and so it can be presented independently of linear logic, as in [Laf94]. However, we think that it is also useful to relate the exact origin of interaction nets, especially for readers with some knowledge in linear logic. We take this opportunity to give a survey of the theory of proof-nets, including a new proof of the sequentializ