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

There has been considerable work aimed at enhancing the expressiveness of logic programming languages. To this end logics other than classical first order logic have been considered, including intuitionistic, relevant, temporal, modal and linear logic. Girard's linear logic has formed the basis of a number of logic programming languages. These languages are successful in enhancing the expressiveness of (pure) Prolog and have been shown to provide natural solutions to problems involving concurrency, natural language processing, database processing and various resource oriented problems. One of the richer linear logic programming languages is Lygon. In this paper we investigate the implementation of Lygon. Two significant problems that arise are the division of resources between sub-branches of the proof and the selection of the formula to be decomposed. We present solutions to both of these problems.

Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negation-as-failure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgements, we adapt the idea of elimination of negation introduced in [21] for Horn logic to a fragment of higher-order HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.

. Two years experience with programming in Linear Logic has shown that while some problems require the full power of linear context management, for many this much control is too much. In such cases a restriction on either weakening or contraction, but not both, is most appropriate. In this article we introduce a refinement of the system proposed by Hodas and Miller in which each of these constraints is independently available. This enables programs to be more succinct, understandable, and efficient. 1 Introduction Sequential logic programming based on linear logic was first proposed by Hodas and Miller in 1991 [8]. The motivating idea was that the context (database) management provided by traditional languages based on intuitionistic logic--- such as Prolog, Prolog [12], N-Prolog [2], and others---was insufficient for many applications. Therefore, a new language was introduced which extended Prolog by using two separate contexts. Clauses in the ordinary, intuitionistic, context conti...

This paper is a study of circumscription, not in classical logic, as usual, but in intuitionistic logic. We first review the intuitionistic circumscription of Horn clause logic programs, which was discussed in previous work, and we then consider the larger class of embedded implications . The ordinary circumscription axiom turns out to be inappropriate for this class of rules, and we analyze two alternatives: (1) prioritized circumscription, which works for stratified embedded implications; and (2) partial circumscription, which is independent of the stratification. We then show that these two approaches coincide by identifying a single structure that serves as the final Kripke model for both circumscription axioms. This means that prioritized circumscription and partial circumscription entail exactly the same set of implicational queries. Several applications of these ideas are described, including: (1) an interpretation of negation-asfailure; (2) a formalization of indefinite reasoni...

Abstract. This work investigates and develops a backtracking algorithm with a novel approach to enumerating and traversing search space. A basic logical framework inspired by relevant logics is presented, highlighting relationships between search and refutation proof construction. Mechanisation of a relevance aware Davis Putnam Logemann Loveland procedure is investigated, and this yields an intelligent backtracking algorithm with abilities similar to other mechanisms including extended freedom in manipulating search space or rearranging refutation proof construction. Simplicity is achieved by a separation of concerns of the underlying logic and the construction of a sound and complete algorithm. The key advantage in the method is that it captures the notion of proof construction and relevant causality, and empirical analysis shows that this is an effective approach. 1

. As a knowledge base grows large, it can be inconsistent in many ways. However, until now, there is no logic programming systems that have enough ability to deal with inconsistent knowledge. Blair and Subrahmanian proposed an approach, named "Annotated Logic Programming" (ALP for short), to describe inconsistent knowledge. However, ALP does not provide sufficient facilities to help users to deal with inconsistent knowledge. We have developed a new annotated logic programming system with hypothetical implications, named ALPS-HI, which can help users to deal with inconsistent knowledge. This paper presents the design and implementation of ALPS-HI and shows an example to demonstrate its applications. 1. Background and Motivation If a knowledge base directly or deductively includes something that is regarded to be both true and false simultaneously, then we say that the knowledge base includes a contradiction and therefore it is inconsistent. As a knowledge base grows large, it may be ...