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An Explicit Formula for the Free Exponential Modality of Linear Logic [chapter]

Paul-André Melliès, Nicolas Tabareau, Christine Tasson
2009 Lecture Notes in Computer Science  
The exponential modality of linear logic associates a commutative comonoid !A to every formula A in order to duplicate it. Here, we explain how to compute the free commutative comonoid !  ...  This algebraic approach enables to unify for the rst time apparently dierent constructions of the exponential modality in spaces and games.  ...  The authors would like to thank Thomas Ehrhard and Martin Hyland for enlightening discussions and feedbacks.  ... 
doi:10.1007/978-3-642-02930-1_21 fatcat:ld3cgburhbfftdhkp2fce6hiqu

An explicit formula for the free exponential modality of linear logic

2017 Mathematical Structures in Computer Science  
The exponential modality of linear logic associates to every formula A a commutative comonoid !A which can be duplicated in the course of reasoning.  ...  This algebraic description unifies for the first time a number of apparently different constructions of the exponential modality in spaces and games.  ...  The authors would like to thank Thomas Ehrhard and Martin Hyland for enlightening discussions and feedbacks.  ... 
doi:10.1017/s0960129516000426 fatcat:hztrwhdopfdljobpz26jwhjgjq

An analysis of (linear) exponentials based on extended sequents

S Guerrini
1998 Logic Journal of the IGPL  
We apply the 2-sequents approach to the analysis of several logics derived from linear logic.  ...  Comments Abstract: We apply the 2-sequents approach to the analysis of several logics derived from linear logic.  ...  A 2-sequence is an expression 0 1 . . . n in which each i is an ordinary (possibly empty) sequence of linear formulas. The formulas of i are at level i.  ... 
doi:10.1093/jigpal/6.5.735 fatcat:vkd5mqlgtfbqvaw7whpk72zzba

Decision Problems for Second-Order Linear Logic [chapter]

P. D. Lincoln, A. Scedrov, N. Shankar
1997 Logic and Scientific Methods  
The decision problem is studied for fragments of second-order linear logic without modalities.  ...  Among the consequences are the undecidability of the intuitionistic second-order fragment of propositional multiplicative linear logic and the undecidability of multiplicative linear logic with rst-order  ...  The decidability of mell, the fragment of quanti er-free propositional logic that contains multiplicatives and exponentials (modalities) but not additives, remains an interesting open problem.  ... 
doi:10.1007/978-94-017-0487-8_7 fatcat:sp5afdrfa5hythn66hgfja5nkq

From linear proofs to direct logic with exponentials [chapter]

Enno Sandner
1997 Lecture Notes in Computer Science  
Following the idea of Linear Proofs presented in 4] we introduce the Direct Logic 14] with exponentials (DLE). The logic combines Direct Logic with the exponentials of Linear Logic.  ...  For a well-chosen subclass of formulas of this logic we provide a matrix-characterization which can be used as a foundation for proof-search methods based on the connection calculus.  ...  Herbrand theorem for Direct Logic. Let 2 F nnf be an exponential and universal quanti er free sentence.  ... 
doi:10.1007/3540634932_10 fatcat:fi62ng247fgghmcygtnfa6tgly

Resource modalities in tensor logic

Paul-André Melliès, Nicolas Tabareau
2010 Annals of Pure and Applied Logic  
The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics  ...  linear, affine, relevant or exponential.  ...  Acknowledgement This work is supported by the ANR Curry-Howard Correspondence and Concurrency Theory (CHOCO).  ... 
doi:10.1016/j.apal.2009.07.018 fatcat:4jqktwfiava3lhfi7ay47kj7lq

Page 8788 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews  
As a corollary, it is shown that a linear Kripke models which satisfies the principle of excluded middle for formulas in prenex normal form also satisfies the principle of excluded middle for all formulas  ...  The other result is a characterization of those linear Kripke structures which decide all Ao-formulas, and in which forcing and satisfaction for Il,, formulas coincide.  ... 

Logic of predicates with explicit substitutions [chapter]

Marek A. Bednarczyk
1996 Lecture Notes in Computer Science  
We present a non-commutative linear logic | the logic of predicates with equality and explicit substitutions.  ...  Linear logic is not an alternative logic ; it should rather be seen as an extension of usual logic. This paper aims at supporting the same idea.  ...  The research presented here was partially sponsored by The State Committee for Scienti c Research, Grant No 2P301 007 04.  ... 
doi:10.1007/3-540-61550-4_148 fatcat:7srloqu57vdvzhn3qcllmwck2y

Substructural logics on display

R Gore
1998 Logic Journal of the IGPL  
from tense logic, the exponentials of linear logic, the converse operator familiar from relation algebra, four negations, and two unusual modalities corresponding to the non-classical analogues of Sheffer's  ...  The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way.  ...  logical modalities, quite at odds with the usual exponentials of linear logic.  ... 
doi:10.1093/jigpal/6.3.451 fatcat:erqlhxit7fe25a4gmybgimeuzy

Correctness and Worst-Case Optimality of Pratt-Style Decision Procedures for Modal and Hybrid Logics [chapter]

Mark Kaminski, Thomas Schneider, Gert Smolka
2011 Lecture Notes in Computer Science  
Our development is modular and we extend it stepwise from modal logic with eventualities to the full logic. ⋆ This is an author version of the contribution to Proc. TABLEAUX 2011.  ...  The main theorem states that a formula is satisfiable if and only if it is contained in a demo. From this theorem the correctness of the decision procedure is easily obtained.  ...  We thank the anonymous reviewers for helpful comments. References  ... 
doi:10.1007/978-3-642-22119-4_16 fatcat:a3et5obo7zazbm3v2u5435lgue

Finite Model Reasoning in Expressive Fragments of First-Order Logic

Lidia Tendera
2017 Electronic Proceedings in Theoretical Computer Science  
the image of the standard translation of modal logic to first-order logic.  ...  When presenting the material special attention is given to decision procedures for the finite satisfiability problems, as many of the fragments discussed contain infinity axioms.  ...  In fact in [50] an exponential reduction from UNFP to the modal µ-calculus is presented that additionally preserves finiteness of the models.  ... 
doi:10.4204/eptcs.243.4 fatcat:47pmommxorc7hbjp7y2v3fxiuu

Beth Definability for the Guarded Fragment [chapter]

Eva Hoogland, Maarten Marx, Martin Otto
1999 Lecture Notes in Computer Science  
The Beth property for GF is here established on the basis of a limited form of interpolation, which more closely resembles the interpolation property that is usually studied in modal logics.  ...  The guarded fragment (GF) was introduced in [1] as a fragment of first order logic which combines a great expressive power with nice modal behavior.  ...  The same for explicit definitions. Description Logics Description logics were designed for the purpose of knowledge representation.  ... 
doi:10.1007/3-540-48242-3_17 fatcat:mpkg77qddzgn5kwjkm2fz4lisa

Symbolic Model Checking of Non-regular Properties [chapter]

Martin Lange
2004 Lecture Notes in Computer Science  
This paper presents a symbolic model checking algorithm for Fixpoint Logic with Chop, an extension of the modal µ-calculus capable of defining non-regular properties.  ...  Some empirical data about running times of a naive implementation of this algorithm are given as well.  ...  Acknowledgments We would like to thank the anonymous referee who suggested the variable ordering used in this version. It hugely improved the performance compared to the previous version.  ... 
doi:10.1007/978-3-540-27813-9_7 fatcat:gizk33jb3jghdndfjrebkx25iu


2016 Mathematical Structures in Computer Science  
An obvious remark is that LL yielded a first logical quantitative account of the use of resources: the logical distinction between linear and non-linear formulas through the introduction of the exponential  ...  Linear Logic (DiLL) and the differential λ-calculus.  ...  The last paper of the volume, 'An explicit formula for the free exponential modality of LL by Paul-André Melliès, Nicolas Tabareau and Christine Tasson, attacks the general question of the categorical  ... 
doi:10.1017/s096012951600044x fatcat:rdlnum3ddvetxh3p5arwzlp2gy

Reasoning with Global Assumptions in Arithmetic Modal Logics [chapter]

Clemens Kupke, Dirk Pattinson, Lutz Schröder
2015 Lecture Notes in Computer Science  
Unlike earlier results of this kind, we do not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics.  ...  Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities.  ...  Catesbeiana for remarks on unsatisfiability. Work of the first author forms part of the DFG project GenMod2 (SCHR 1118/5-2).  ... 
doi:10.1007/978-3-319-22177-9_28 fatcat:ggkvnu5drjahtp4cdtwq24x7hu
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