The deep symmetry of Linear Logic [18] makes it suitable for providing abstract models of computation, free from implementation details which are, by nature, oriented and non symmetrical. I propose here one such model, in the area of Logic Programming, where the basic computational principle is Computation = Proof search.

When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus during the bottom-up search for a cut-free proof contexts, represented as the left-hand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goal-directed interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model. 1

Gamma is a minimal language based on local multiset rewriting with an elegant chemical reaction metaphor. The virtues of this paradigm in terms of systematic program construction and design of parallel programs have been argued in previous papers. Gamma can also be seen as a notation for coordinating independent programs in a larger application. In this paper, we study a notion of refinement for programs involving parallel and sequential composition operators, and derive a number of programming laws. The calculus thus obtained is applied in the development of a generic "pipelining" transformation, which enables certain sequential compositions to be refined into parallel compositions. Keywords: Gamma, Multiset Rewriting, Program Transformation. 1 Introduction We first describe the general motivation of the work presented here before summarising the main results developed in the body of the paper. 1.1 Motivation The notion of sequential computation has played a central role in the des...

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Exploiting the full potential of massively parallel systems requires programming models that explicitly deal with the concurrency of cooperation among very large numbers of active entities that comprise a single application. In practice, the concurrent applications of today essentially use a set of ad hoc templates to coordinate the cooperation of their active components. This shows the lack of proper coordination languages that can be used to explicitly describe complex cooperation protocols in terms of simple primitives and structuring constructs. In this paper we present a generic model of communication and describe a specific control-oriented coordination language based on this model. The important characteristics of this model include compositionality, which it inherits from the data-flow model, anonymous communication, and separation of computation concerns from communication concerns. These characteristics lead to clear advantages in large concurrent applications....

The agent expressions of the π-calculus can be translated into a theory of linear logic in such a way that the reflective and transitive closure of π-calculus (unlabeled) reduction is identified with “entailed-by”. Under this translation, parallel composition is mapped to the multiplicative disjunct (“par”) and restriction is mapped to universal quantification. Prefixing, non-deterministic choice (+), replication (!), and the match guard are all represented using non-logical constants, which are specified using a simple form of axiom, called here a process clause. These process clauses resemble Horn clauses except that they may have multiple conclusions; that is, their heads may be the par of atomic formulas. Such multiple conclusion clauses are used to axiomatize communications among agents. Given this translation, it is nature to ask to what extent proof theory can be used to understand the meta-theory of the π-calculus. We present some preliminary results along this line for π0, the “propositional ” fragment of the π-calculus, which lacks restriction and value passing (π0 is a subset of CCS). Using ideas from proof-theory, we introduce co-agents and show that they can specify some testing equivalences for π0. If negation-as-failure-to-prove is permitted as a co-agent combinator, then testing equivalence based on co-agents yields observational equivalence for π0. This latter result follows from observing that co-agents directly represent formulas in the Hennessy-Milner modal logic. 1

Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspace-complete. We also establish membership in np for the multiplicative fragment, np-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic. 1 Introduction Linear logic, introduced by Girard [14, 18, 17], is a refinement of classical logic which may be derived from a Gentzen-style sequent calculus axiomatization of classical logic in three steps. The resulting sequent system Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Labo...

The theory of cut-free sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, λProlog and its linear logic refinement, Lolli [12], provide data types, higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) [2] provides for concurrency but lacks abstraction mechanisms. In this paper we present Forum, a logic programming presentation of all of linear logic that modularly extends the languages λProlog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. As a meta-language, Forum greatly extends the expressiveness of these other logic programming languages. To illustrate its expressive strength, we specify in Forum a sequent calculus proof system and the operational semantics of a functional programming language that incorporates such nonfunctional features as counters and references. 1

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