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

Abstract: A universally quantified goal can be interpreted intensionally, that is, the goal ∀x.G(x) succeeds if for some new constant c, the goal G(c) succeeds. The constant c is, in a sense, given a scope: it is introduced to solve this goal and is “discharged ” after the goal succeeds or fails. This interpretation is similar to the interpretation of implicational goals: the goal D ⊃ G should succeed if when D is assumed, the goal G succeeds. The assumption D is discharged after G succeeds or fails. An interpreter for a logic programming language containing both universal quantifiers and implications in goals and the body of clauses is described. In its non-deterministic form, this interpreter is sound and complete for intuitionistic logic. Universal quantification can provide lexical scoping of individual, function, and predicate constants. Several examples are presented to show how such scoping can be used to provide a Prolog-like language with facilities data types, and encapsulation of state.

this paper, we do this by adding the #-quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important

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A powerful and declarative means of specifying computations containing abstractions involves meta-level, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, ∇, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with ∇ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the π-calculus and the encoding of objectlevel provability.

. We exhibit a methodology for formulating and verifying metatheorems about deductive systems in the Elf language, an implementation of the LF Logical Framework with an operational semantics in the spirit of logic programming. It is based on the mechanical verification of properties of transformations between deductions, which relies on type reconstruction and schema-checking. The latter is justified by induction principles for closed LF objects, which can be constructed over a given signature. We illustrate our technique through several examples, the most extensive of which is an interpretation of classical logic in minimal logic through a continuation-passing-style transformation on proofs. 1 Introduction Formal deductive systems have become an important tool in computer science. They are used to specify logics, type systems, operational semantics and other aspects of languages. The role of such specifications is three-fold. Firstly, inference rules serve as a high-level notation w...

Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. In both cases, it is possible to relate transition paths to proofs in sequent calculus. In neither encoding, however, does it seem possible to capture properties, such as simulation and bisimulation, that need to consider all possible transitions or all possible computation paths. We consider augmenting both intuitionistic and linear logics with a proof theoretical treatment of definitions. In both cases, this addition allows proving various judgments concerning simulation and bisimulation (especially for noetherian transition systems). We also explore the use of infinite proofs to reason about infinite sequences of transitions. Finally, combining definitions and induction into sequent calculus proofs makes it possible to reason more richly about properties of transition systems completely within the formal setting of sequent calculus.

The theory of partial inductive definitions is a mathematical formalism which has proved to be useful in a number of different applications. The fundamentals of the theory is shortly described. Partial inductive definitions and their associated calculi are essentially infinitary. To implement them on a computer, they must be given a formal finitary representation. We present such a finitary representation, and prove its soundness. The finitary representation is given in a form with and without variables. Without variables, derivations are unchanging entities. With variables, derivations can contain logical variables that can become bound by a binding environment that is extended as the derivation is constructed. The variant with variables is essentially a generalization of the pure GCLA programming language.

We present a meta-logic that contains a new quantifier (for encoding "generic judgment") and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite π-calculus within this meta-logic. Since we

In this report we introduce a new proof-theoretic approach to the semantics of Constraint Logic Programming, based on an intuitionistic first-order modal logic, called QLL. The distinguishing feature of this new approach is that the logic calculus of QLL is used not only to capture the usual extensional aspects of Logic Programming, i.e. "which queries are successful, " but also some of the intensional aspects, i.e. "what is the answer constraint and how is it constructed." It provides for a direct link between the model-theoretic and the operational semantics following a formulas-as-programs and proofs-as-constraints principle. This approach makes use of logic in a different way than other approaches based on logic calculi. On the one side it is to be distinguished from the well-known provability semantics which is concerned merely with what is derivable as opposed to how it is derivable, paying attention to the fact that it is the how that determines the answer constraint. ...