Grammatical Framework (GF) is a special-purpose functional language for defining grammars. It uses a Logical Framework (LF) for a description of abstract syntax, and adds to this a notation for defining concrete syntax. GF grammars themselves are purely declarative, but can be used both for linearizing syntax trees and parsing strings. GF can describe both formal and natural languages. The key notion of this description is a grammatical object, which is not just a string, but a record that contains all information on inflection and inherent grammatical features such as number and gender in natural languages, or precedence in formal languages. Grammatical objects have a type system, which helps to eliminate run-time errors in language processing. In the same way as an LF, GF uses...

An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.

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...

We present a system of natural deduction and associated term calculus for intuitionistic non-commutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment.

We present a sequent calculus for intuitionistic non-commutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cut-free derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of non-commutative natural deduction and its associated -calculus. INCLL conservatively extends linear logic with means to express sequencing, which has applications in functional programming, logical frameworks, logic programming, and natural language parsing. 1 Introduction Linear logic [11] has been described as a logic of state because it views linear hypotheses as resources which may be consumed in the course of a deduction. It thereby significantly extends the expressive power of both classical and intuitionistic logics, yet it does not offer means to express sequencing. Th...

The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambda-calculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.-Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...

Consider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structurefree logic. The aim of this work is to give such a condition by using algebraic semantics. We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set R of structural rules, the cut elimination theorem holds for FL enriched with R if and only if R satisfies the propagation property. As an application, we show that any set R of structural rules can be ”completed” into another set R ⋆ , so that the cut elimination theorem holds for FL enriched with R ⋆ , while the provability remains the same. 1

Abstract. We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically.

xth Annual Symposium on Logic in Computer Science, Amsterdam, July 15-18, 1991. [LS94a] P. Lincoln and A. Scedrov. First order linear logic without modalities is NEXPTIMEhard. Theoretical Computer Science, 1994. [LS94b] P. Lincoln and N. Shankar. Proof search in first-order linear logic and other cut-free sequent calculi. In Logic in Computer Science (LICS '94), 1994. [LW92] P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NPComplete, September 1992. Manuscript. [Mac93] I. Mackie. Lilac - a functional programming language based on linear logic. Journal of Functional Programming, 1993. [Mil92] D. Miller. The ß-calculus as a theory in linear logic: Preliminary results. In E. Lamma and P. Mello, editors, Proceedings of the 1992 Workshop on Extensions to Logic Programming, pages 242--265. Springer-Verlag, 1992. LNCS 660. [Mil94] D. Miller. A multiple-conclusion meta-logic. In S. Abramsky