Logic Programming in a Fragment of Intuitionistic Linear Logic

Abstract

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 G 1 ⊗ G 2 , from the context ∆, the latter must be split into disjoint contexts ∆ 1 and ∆ 2 for which G 1 follows from ∆ 1 and G 2 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.

Efficient resource management for linear logic proof search

“…The programming language Lolli [Hod94,HM94] is based on the fragment of linear logic freely generated by the operators , &, −•, ⊃ and ∀. The connective ⊃ is called intuitionistic implication and is defined as A ⊃ B ≡ !A −• B.…”

“…In particular, many problems centered around the notion of a state that evolves as a computation proceeds fall into this category. Consequently, several logic programming languages based on linear logic have been designed in recent years [AP91,HP91,HM94,Mil96]. Others are the subject of extensive research.…”

“…Each proposal is accompanied by interesting theoretical results that show its computational relevance, and by numerous examples that prove its practical significance. However, to our knowledge, usable implementations have thus far been released only for Lolli [HM94] and Lygon [HP91].…”

“…In this paper, we begin by reiterating the deterministic solution to this problem presented in the original work on Lolli [Hod94,HM94]. We will then concentrate on less apparent issues in context management involving the additive connectives and constants.…”

“…Similarly, we do not consider orthogonal issues in linear proof search, such as the permutability of inference rules, which have been treated exhaustively elsewhere [GP94,Min93,Tam94]. We will focus our attention on the language Lolli [Hod94,HM94], that we used to test the techniques described below [CHMP]. However, our results have already been applied to prototype implementations [HP96,LP97] of a programming language based on Miller's specification logic Forum [Mil96].…”