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Grammatical Framework: A TypeTheoretical Grammar Formalism
, 2003
"... Grammatical Framework (GF) is a specialpurpose 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 lineariz ..."
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Cited by 77 (19 self)
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Grammatical Framework (GF) is a specialpurpose 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 runtime errors in language processing. In the same way as an LF, GF uses...
A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 53 (8 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
Applications of Linear Logic to Computation: An Overview
, 1993
"... 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 objectoriented programming and some other applications of LL, li ..."
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Cited by 42 (3 self)
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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 objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart 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...
Natural Deduction for Intuitionistic NonCommutative Linear Logic
 Proceedings of the 4th International Conference on Typed Lambda Calculi and Applications (TLCA'99
, 1999
"... We present a system of natural deduction and associated term calculus for intuitionistic noncommutative 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. ..."
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Cited by 34 (16 self)
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We present a system of natural deduction and associated term calculus for intuitionistic noncommutative 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.
Relating Natural Deduction and Sequent Calculus for Intuitionistic NonCommutative Linear Logic
, 1999
"... We present a sequent calculus for intuitionistic noncommutative 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 cutfree derivations, and arbitrary natura ..."
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Cited by 28 (15 self)
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We present a sequent calculus for intuitionistic noncommutative 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 cutfree derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of noncommutative 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...
Towards a Semantic Characterization of cutelimination
, 2006
"... We introduce necessary and sufficient conditions for a (singleconclusion) sequent calculus to admit (reductive) cutelimination. Our conditions are formulated both syntactically and semantically. ..."
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Cited by 18 (5 self)
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We introduce necessary and sufficient conditions for a (singleconclusion) sequent calculus to admit (reductive) cutelimination. Our conditions are formulated both syntactically and semantically.
Chu Spaces as a Semantic Bridge Between Linear Logic and Mathematics
 Theoretical Computer Science
, 1998
"... 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 selfdual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soun ..."
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Cited by 12 (2 self)
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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 selfdual 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 lambdacalculus, 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...
Which Structural Rules Admit Cut Elimination? — An Algebraic Criterion
 Journal of Symbolic Logic
"... 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 ..."
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Cited by 9 (4 self)
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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
Anaphoric Linking at Run Time: A TypeLogical Account of Discourse Representation ∗
"... A number of attempts have been made to combine Discourse Representation Theory with typelogical grammar in order to obtain a compositional (bottomup) framework for discourse representation (e.g., Muskens[Mus94], van Eijck & Kamp[vEK97]). In those attempts, however, it is usually assumed that a ..."
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A number of attempts have been made to combine Discourse Representation Theory with typelogical grammar in order to obtain a compositional (bottomup) framework for discourse representation (e.g., Muskens[Mus94], van Eijck & Kamp[vEK97]). In those attempts, however, it is usually assumed that anaphoric links are given in advance of the construction of discourse representation structures (DRSs). This assumption causes a difficulty, typically dealing with plural anaphora, because generally suitable referents of plural pronouns are not present before the construction, but should be created through the construction of DRSs. To settle this difficulty, we shall propose a new typelogical framework of DRT in which anaphoric linking takes place during the construction of DRSs (Run Time Anaphoric Linking). The construction is bottomup in the sense that it is based on the sequent calculus proof search of Lambek calculus. We exploit quantifier types to represent the run time anaphoric linking mechanism. We shall briefly illustrate how plural anaphora are treated in our framework. In particular, we shall show that Summation and Abstraction, whichareusedtocreate suitable antecedent discourse referents for plural pronouns in [KR93], are available in our typelogical setting using the linear logic modality. 1
Semantic Characterizations for Reachability and Trace Equivalence in a Linear LogicBased Process Calculus (Preliminary Report)
"... We give semantic characterizations for the notions of reachability and trace equivalence in a linearlogic based framework of asyncronous concurrent process calculus. Usually the reachability relation in linear logicbased concurrent process calculi is characterized by the logical notion of provabil ..."
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We give semantic characterizations for the notions of reachability and trace equivalence in a linearlogic based framework of asyncronous concurrent process calculus. Usually the reachability relation in linear logicbased concurrent process calculi is characterized by the logical notion of provability, which is in turn characterized by modeltheoretic semantics such as the phase semantics. The standard phase semantics is, however, too abstract to give concrete meanings to processes due to the presence of the closure operation. To remedy this, we introduce a simplification of the phase semantics, which we call the naive phase semantics, and show that the reachability relation is also characterized by the completeness with respect to the naive phase semantics. On the other hand, the logical provability does not provide any satisfactory notion of equivalence over processes. We consider the trace equivalence (Hoare[Hoa80]) for our process calculus and introduce certain algebraic models, which we call the trace semantics. Then the trace equivalence is characterized by the completeness with respect to the trace semantics.