This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and object-oriented programming and some other applications of LL, like semantics of negation in LP, non-monotonic issues in AI planning, etc. Although the overview covers pretty much the state-of-the-art in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...

Recently there has been significant interest in the logic programming community in linear logic, a logic designed with bounded resources in mind. As linear logic is a generalisation of classical logic, a logic programming language based on linear logic subsumes and extends (pure) Prolog. One such language is Lygon, a language based on a certain kind of proof in the linear sequent calculus. However these proofs, whilst providing a logical characterization of the language, still retain some of the non-determinism of the sequent calculus, and hence require further analysis before an implementation can be attempted. In this report we define and discuss a more detailed proof system, which is more deterministic than the original. In particular, this system handles the allocation of resources to different branches of the proof in a lazy manner. The resulting system differs significantly from the original sequent calculus, and so we discuss its properties in some detail. We prove the soundness...

Proof Graph . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.3 Refinement of Abstract Proof Graph . . . . . . . . . . . . . . . . . 37 1 7 Performance Measurements 42 7.1 The Sorting Program . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.2 A Petri Net for Producer-Consumer System . . . . . . . . . . . . . 45 7.3 A Producer-Consumer Program . . . . . . . . . . . . . . . . . . . . 46 8 Conclusion 48 A A Lemma of the Leveled I/O model 54 B A Proof of the Equivalence of the the Leveled I/O model and L ## 56 2 Abstract In this thesis, we propose a new static analysis method which is applicable for a classical linear logic programming language. Andreoli et al. proposed a static analysis method for the classical linear logic programming language LO, but their method did not cover the multiplicative conjunction which is the most important connective for a resource-sensitive feature of linear logic. Our method, in contrast, covers the multiplicative conjunction in addition to the multiplicative disjunction and linear implication. In classical linear logic programming language, such as Forum, the execution of a program is sometimes highly non-deterministic. For example, the execution time of a sorting program written in Forum rises hyper-exponentially with list length. The reason is that most of the execution time is spent for trying to prove unprovable sequents. Therefore, it is very important to find unprovable sequents before the execution of programs, and this is realizable by a static analysis performing an abstract proof search on an abstract proof. In order to develop abstract proof search in linear logic, we need to map the sequents into finite sets. we introduce a finite monoid #M,+, 0# homomorphic to N (a set of natural numbers including 0) t...

There have been several proposals for logic programming language based on linear logic: Lolli [8], Lygon [7], LO [3], LinLog [2], Forum [11], HACL [10]. In these languages, it is possible to create and consume resources dynamically as logical formulas. The e#cient handling of resource formulas is, therefore, an important issue in the implementation of these languages. Lolli, Lygon, and Forum are implemented as interpreter systems; Lolli is on SML and #Prolog, Lygon is on Prolog, Forum is on SML, #Prolog and Prolog. However, none of them have been implemented in Java. In this