We propose a novel formalism, called Frame Logic (abbr., F-logic), that accounts in a clean and declarative fashion for most of the structural aspects of object-oriented and frame-based languages. These features include object identity, complex objects, inheritance, polymorphic types, query methods, encapsulation, and others. In a sense, F-logic stands in the same relationship to the objectoriented paradigm as classical predicate calculus stands to relational programming. F-logic has a model-theoretic semantics and a sound and complete resolution-based proof theory. A small number of fundamental concepts that come from object-oriented programming have direct representation in F-logic; other, secondary aspects of this paradigm are easily modeled as well. The paper also discusses semantic issues pertaining to programming with a deductive object-oriented language based on a subset of F-logic.

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.

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers # new and # new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predic...

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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

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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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 [15], provide for various forms of abstraction (modules, abstract data types, and higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) [2] provides some primitives for concurrency but lacks abstraction mechanisms. In this paper we present Forum, a logic programming presentation of all of linear logic that modularly extends λProlog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. To illustrate the new expressive strengths of Forum, we specify in it a sequent calculus proof system and the operational semantics of a programming language that incorporates references and concurrency. We also show that the meta theory of linear logic can be used to prove properties of the objectlanguages specified in Forum.

We describe Elf, a metalanguage for proof manipulation environments that are independent of any particular logical system. Elf is intended for meta-programs such as theorem provers, proof transformers, or type inference programs for programming languages with complex type systems. Elf unifies logic definition (in the style of LF, the Edinburgh Logical Framework) with logic programming (in the style of Prolog). It achieves this unification by giving types an operational interpretation, much the same way that Prolog gives certain formulas (Horn-clauses) an operational interpretation. Novel features of Elf include: (1) the Elf search process automatically constructs terms that can represent object-logic proofs, and thus a program need not construct them explicitly, (2) the partial correctness of meta-programs with respect to a given logic can be expressed and proved in Elf itself, and (3) Elf exploits Elliott's unification algorithm for a -calculus with dependent types. This research was...

In this paper we consider the problem of identifying logic programming languages for linear logic. Our analysis builds on a notion of goal-directed provability, characterized by the so-called uniform proofs, previously introduced for minimal and intuitionistic logic. A class of uniform proofs in linear logic is identified by an analysis of the permutability of inferences in the linear sequent calculus. We show that this class of proofs is complete (for logical consequence) for a certain (quite large) fragment of linear logic, which thus forms a logic programming language. We obtain a notion of resolution proof, in which only one left rule, of clause-directed resolution, is required. We also consider a translation, resembling those of Girard, of the hereditary Harrop fragment of intuitionistic logic into our framework. We show that goal-directed provability is preserved under this translation. 1 Introduction An interesting recent development in logic of some significance for theoretica...