DOCSLIB.ORG

Linear Logic and Sub-Polynomial Classes of Complexity Clément Aubert

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Logic and Sub-Polynomial Classes of Complexity Clément Aubert Linear Logic and Sub-polynomial Classes of Complexity Clément Aubert To cite this version: Clément Aubert. Linear Logic and Sub-polynomial Classes of Complexity. Computational Complexity [cs.CC]. Université Paris-Nord - Paris XIII, 2013. English. tel-00957653v2 HAL Id: tel-00957653 https://tel.archives-ouvertes.fr/tel-00957653v2 Submitted on 9 Nov 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - ShareAlike| 4.0 International License Université Paris :3 Sorbonne Paris Cité Laboratoire d’Informatique de Paris Nord Logique linéaire et classes de complexité sous-polynomiales k Linear Logic and Sub-polynomial Classes of Complexity Thèse pour l’obtention du diplôme de Docteur de l’Université de Paris :3, Sorbonne Paris-Cité, en informatique présentée par Clément Aubert <[email protected]> F = f Mémoire soutenu le mardi 26 novembre 20:3, devant la commission d’examen composée de : M. Patrick Baillot C.N.R.S., E.N.S. Lyon Rapporteur M. Arnaud Durand Université Denis Diderot - Paris 7 Président M. Ugo Dal Lago I.N.R.I.A., Università degli Studi di Bologna Rapporteur Mme. Claudia Faggian C.N.R.S., Université Paris Diderot - Paris 7 M. Stefano Guerrini Institut Galilée - Université Paris :3 Directeur M. Jean-Yves Marion Lo.R.I.A., Université de Lorraine M. Paul-André Melliès C.N.R.S., Université Paris Diderot - Paris 7 M. Virgile Mogbil Institut Galilée - Université Paris :3 Co-encadrant = f F 1 Créteil, le 7 novembre 20:4 1 Errata ome minor editions have been made since the defence of this thesis (improvements in the bibliog- S raphy, fixing a few margins and some typos): they won’t be listed here, as those errors were not altering the meaning of this work.: However, while writing an abstract for TERMGRAPH 20:4, I realized that the simulation of Proof Circuits by an Alternating Turing Machine, in the Section 2.4 of Chapter 2 was not correct. More precisely, bPCCi is not an object to be considered (as it is trivially equal to PCCi) and Theorem 2.4.3, p. 5:, that states that “For all i > 0, bPCCi STA(log, ,logi)”, is false. The ATM ⊆ ∗ constructed do normalize the proof circuit given in input, but not within the given bounds. This flaw comes from a mismatch between the logical depth of a proof net (Definition 2.:.:0) and the height of a piece (Definition 2.2.6), which is close to the notion of depth of a Boolean circuit (Definition 2.:.2). First, remark that the class bPCCi (Definition 2.2.8) is ill-defined: recall that (Lemma 2.3.:) in the case of k and k pieces, one entry is at the same logical depth than the output, and all the other Disj Conj entries are at depth 3 plus the depth of the output. In the process of decomposing a single n-ary operation in n 1 binary operations, the “depth-efficient − way” is not the “logical depth-efficient way”. Let us consider a tree with n leafs (f) and two ways to organize it: f f f f f f f f f f f f Let us take the distance to be the number of edges between a leaf and the root of the tree. Then, on the left tree, every leaf is at the same distance log(n), and on the right tree, the greatest distance is n. Everything differs if one consider that every binary branching is a 2 or 2 piece and consider the Disj Conj logical depth rather than the distance. In that case, on the left-hand tree, the ith leaf (starting from the left) is at logical depth 3 times the number of 1 in the binary encoding of i.2 On the right-hand tree, every leaf is at logical depth 3, except for the red leaf (the left-most one), which is at logical depth 0. So the logical depth is in fact independent of the fan-in of the pieces, and it has the same “logical cost” in terms of depth to compute i or i for any i ¾ 2. A proof circuit of PCCi is a proof circuit of Disj Conj bPCCi: every piece of can be obtained from pieces of without increasing the logical depth. So Pu Pb we have that for all i N, ∈ PCCi = bPCCi. Secondly, the proof of Theorem 2.4.3 was an attempt to adapt the function value(n, g, p) [:35, p. 65] that prove that an ATM can normalize a Boolean circuit with suitable bounds. The algorithm is correct, :Except maybe for a silly mistake in Proposition 4.3.:, spotted by Marc Bagnol. 2Sic. This sequence is known as A000:20 and starts with 0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,.... iii iv but not the bounds: it is stated that “Each call to f uses a constant number of alternations, and O(d(Pn)) i i calls are made. As Pn is of depth log (n), O(log (n)) calls are made.”, and this argument is wrong. Consider that P PCCi (this is harmful taking the first remark into account), when the ATM calls f n ∈ to evaluate the output of a k or k piece, it makes k 1 calls to f at a lesser depth, and one call at Disj Conj − the same depth. The only bound on the fan-in of the pieces, k, is the size of the proof circuit, that is to say a polynomial in the size of the input: this is a disaster. tl;dr: bPCCi should not be considered as a pertinent object of study and the correspondence between the proof circuits and the Alternating Turing Machines is partially wrong. This does not affect anyhow the rest of this work and this part has never been published nor submitted. Créteil, 7 November 20:4. c This thesis is dedicated to the free sharing of knowledge. Cette thèse est dédiée au libre partage des savoirs. b vi Disclaimer: It is not possible for me to write these acknowledgements in a language other than French. Remerciements. Il est d’usage de commencer par remercier son directeur de thèse, et je me plie à cet usage avec une certaine émotion : Stephano Guerrini m’a accordé une grande confiance en acceptant de m’encadrer dans cette thèse en Informatique alors que mon cursus ne me pré-disposait pas tellement à cette aventure. Par ses conseils, ses intuitions et nos discussions, il a su m’ouvrir des pistes, en fermer d’autres, conseiller et pointer de fructueuses lectures. Virgile Mogbil a co-encadré cette thèse avec une attention et un soin qui m’ont touchés. Il a su m’accompagner dans tous les aspects de celles-ci, m’initier à la beauté de la recherche sans me cacher ses défauts, me guider dans le parcours académique et universitaire bien au-delà de ma thèse. Il su faire tout cela dans des conditions délicates, ce qui rend son dévouement encore plus précieux. Patrick Baillot fût le rapporteur de cette thèse et comme son troisième directeur : il a su durant ces trois années me conseiller, me questionner, chercher à comprendre ce que je ne comprenais pas moi-même. C’est un honneur pour moi qu’il ait accepté de rapporter ce travail avant autant de justesse et de clairvoyance. Je suis également ravi qu’Ugo Dal Lago ait accepté de rapporter ce travail, et m’excuse pour l’avalanche de coquilles qui ont dû lui abîmer l’œil. Sa place centrale dans notre communauté donne une saveur toute particulière à ses remarques, conseils et critiques. Je suis honoré qu’Arnaud Durand — qui fût mon enseignant lorsque je me suis reconverti aux Mathématiques — ait accepté d’être le président de ce jury. La présence de Claudia Faggian, Jean-Yves Marion et Paul-André Melliès place cette thèse à l’interscetion de communautés auxquelles je suis ravi d’appartenir. Je les remercie chaleureusement pour leur présence, leurs conseils, leurs encouragements. Une large partie de ce travail n’aurait pas pu voir le jour sans la patience et la générosité de Thomas Seiller, avec qui j’ai toujours eu plaisir à rechercher un accès dans la compréhension des travaux de Jean-Yves Girard. § Mon cursus m’a permis de rencontrer de nombreux enseignants qui ont tous, d’une façon ou d’une autre, rendu ce cheminement possible. J’ai une pensée particulière pour Louis Allix, qui m’a fait découvrir la logique et pour Jean-Baptiste Joinet, qui a accompagné, encouragé et rendu possible ce cursus. De nombreux autres enseignants, tous niveaux et matières confondus, m’ont fait aimer le savoir, la réflexion et le métier d’enseignant, qu’ils m’excusent de ne pouvoir tous les nommer ici, ils n’en restent pas moins présents dans mon esprit. § J’ai été accueilli au Laboratoire d’Informatique de Paris Nord par Paulin Jacobé de Naurois (qui encadrait mon stage), Pierre Boudes et Damiano Mazza (qui ont partagé leur bureau avec moi), et je ne pouvais rêver de meilleur accueil.
Load more

Recommended publications
  • Logic Programming in a Fragment of Intuitionistic Linear Logic ∗
    Logic Programming in a Fragment of Intuitionistic Linear Logic ¤ Joshua S. Hodas Dale Miller Computer Science Department Computer Science Department Harvey Mudd College University of Pennsylvania Claremont, CA 91711-5990 USA Philadelphia, PA 19104-6389 USA [email protected] [email protected] 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 [ fDg. 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 G1 ­ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2.
    [Show full text]
  • Relevant and Substructural Logics
    Relevant and Substructural Logics GREG RESTALL∗ PHILOSOPHY DEPARTMENT, MACQUARIE UNIVERSITY [email protected] June 23, 2001 http://www.phil.mq.edu.au/staff/grestall/ Abstract: This is a history of relevant and substructural logics, written for the Hand- book of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods.1 1 Introduction Logics tend to be viewed of in one of two ways — with an eye to proofs, or with an eye to models.2 Relevant and substructural logics are no different: you can focus on notions of proof, inference rules and structural features of deduction in these logics, or you can focus on interpretations of the language in other structures. This essay is structured around the bifurcation between proofs and mod- els: The first section discusses Proof Theory of relevant and substructural log- ics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, Dunn's algebraic models [76, 77] Urquhart's operational semantics [267, 268] and Routley and Meyer's rela- tional semantics [239, 240, 241] arrived decades after the initial burst of ac- tivity from Alan Anderson and Nuel Belnap. The same goes for work on the Lambek calculus: although inspired by a very particular application in lin- guistic typing, it was developed first proof-theoretically, and only later did model theory come to the fore.
    [Show full text]
  • Linear Logic : Its Syntax and Semantics
    LINEAR LOGIC : ITS SYNTAX AND SEMANTICS Jean-Yves Girard Laboratoire de Math´ematiques Discr`etes UPR 9016 { CNRS 163, Avenue de Luminy, Case 930 F-13288 Marseille Cedex 09 [email protected] 1 THE SYNTAX OF LINEAR LOGIC 1.1 The connectives of linear logic Linear logic is not an alternative logic ; it should rather be seen as an exten- sion of usual logic. Since there is no hope to modify the extant classical or intuitionistic connectives 1, linear logic introduces new connectives. 1.1.1 Exponentials : actions vs situations Classical and intuitionistic logics deal with stable truths : if A and A B, then B, but A still holds. ) This is perfect in mathematics, but wrong in real life, since real implication is causal. A causal implication cannot be iterated since the conditions are modified after its use ; this process of modification of the premises (conditions) is known in physics as reaction. For instance, if A is to spend $1 on a pack of cigarettes and B is to get them, you lose $1 in this process, and you cannot do it a second time. The reaction here was that $1 went out of your pocket. The first objection to that view is that there are in mathematics, in real life, cases where reaction does not exist or can be neglected : think of a lemma which is forever true, or of a Mr. Soros, who has almost an infinite amount of dollars. 1: Witness the fate of non-monotonic \logics" who tried to tamper with logical rules with- out changing the basic operations : : : 1 2 Jean-Yves Girard Such cases are situations in the sense of stable truths.
    [Show full text]
  • Categorical Models for Intuitionistic and Linear Type Theory
    Categorical Models for Intuitionistic and Linear Type Theory Maria Emilia Maietti, Valeria de Paiva, and Eike Ritter? School of Computer Science, University of Birmingham Edgbaston, Birmingham B15 2TT, United Kingdom mem,vdp,exr @cs.bham.ac.uk { } Abstract. This paper describes the categorical semantics of a system of mixed intuitionistic and linear type theory (ILT). ILT was proposed by G. Plotkin and also independently by P. Wadler. The logic associ- ated with ILT is obtained as a combination of intuitionistic logic with intuitionistic linear logic, and can be embedded in Barber and Plotkin’s Dual Intuitionistic Linear Logic (DILL). However, unlike DILL, the logic for ILT lacks an explicit modality ! that translates intuitionistic proofs into linear ones. So while the semantics of DILL can be given in terms of monoidal adjunctions between symmetric monoidal closed categories and cartesian closed categories, the semantics of ILT is better presented via fibrations. These interpret double contexts, which cannot be reduced to linear ones. In order to interpret the intuitionistic and linear iden- tity axioms acting on the same type we need fibrations satisfying the comprehension axiom. 1 Introduction This paper arises from the need to fill a gap in the conceptual development of the xSLAM project. The xSLAM project is concerned with the design and implementation of abstract machines based on linear logic. For xSLAM we ini- tially developed a linear λ-calculus by adding explicit substitutions to Barber and Plotkin’s DILL [GdPR00]. We then considered the categorical models one obtains for both intuitionistic and linear logic with explicit substitutions on the style of Abadi et al.
    [Show full text]
  • A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics
    FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 1 Contents: Preface by C. Le: 3 0. Introduction: 9 1. Neutrosophy - a new branch of philosophy: 15 2. Neutrosophic Logic - a unifying field in logics: 90 3. Neutrosophic Set - a unifying field in sets: 125 4. Neutrosophic Probability - a generalization of classical and imprecise probabilities - and Neutrosophic Statistics: 129 5. Addenda: Definitions derived from Neutrosophics: 133 2 Preface to Neutrosophy and Neutrosophic Logic by C.
    [Show full text]
  • Linear Logic Programming Dale Miller INRIA/Futurs & Laboratoire D’Informatique (LIX) Ecole´ Polytechnique, Rue De Saclay 91128 PALAISEAU Cedex FRANCE
    1 An Overview of Linear Logic Programming Dale Miller INRIA/Futurs & Laboratoire d’Informatique (LIX) Ecole´ polytechnique, Rue de Saclay 91128 PALAISEAU Cedex FRANCE Abstract Logic programming can be given a foundation in sequent calculus by viewing computation as the process of building a cut-free sequent proof bottom-up. The first accounts of logic programming as proof search were given in classical and intuitionistic logic. Given that linear logic allows richer sequents and richer dynamics in the rewriting of sequents during proof search, it was inevitable that linear logic would be used to design new and more expressive logic programming languages. We overview how linear logic has been used to design such new languages and describe briefly some applications and implementation issues for them. 1.1 Introduction It is now commonplace to recognize the important role of logic in the foundations of computer science. When a major new advance is made in our understanding of logic, we can thus expect to see that advance ripple into many areas of computer science. Such rippling has been observed during the years since the introduction of linear logic by Girard in 1987 [Gir87]. Since linear logic embraces computational themes directly in its design, it often allows direct and declarative approaches to compu- tational and resource sensitive specifications. Linear logic also provides new insights into the many computational systems based on classical and intuitionistic logics since it refines and extends these logics. There are two broad approaches by which logic, via the theory of proofs, is used to describe computation [Mil93]. One approach is the proof reduction paradigm, which can be seen as a foundation for func- 1 2 Dale Miller tional programming.
    [Show full text]
  • Linear Logic
    App eared in SIGACT Linear Logic Patrick Lincoln lincolncslsricom SRI and Stanford University Linear Logic Linear logic was intro duced by Girard in Since then many results have supp orted Girards claims such as Linear logic is a resource conscious logic Increasingly computer scientists have recognized linear logic as an expressive and p owerful logic with deep connec tions to concepts from computer science The expressive p ower of linear logic is evidenced by some very natural enco dings of computational mo dels such as Petri nets counter machines Turing machines and others This note presents an intuitive overview of linear logic some recent theoretical results and some interesting applications of linear logic to computer science Other intro ductions to linear logic may b e found in Linear Logic vs Classical and Intuitionistic Logics Linear logic diers from classical and intuitionistic logic in several fundamental ways Clas sical logic may b e viewed as if it deals with static prop ositions ab out the world where each prop osition is either true or false Because of the static nature of prop ositions in classical logic one may duplicate prop ositions P implies P and P Implicitly we learn that one P is as go o d as two Also one may discard prop ositions P and Q implies P Here the prop osition Q has b een thrown away Both of these sentences are valid in classical logic for any P and Q In linear logic these sentences are not valid A linear logician might ask Where did the second P come from and Where did the Q go Of course these
    [Show full text]
  • 13 QUESTIONS ABOUT UNIVERSAL LOGIC 13 Questions to Jean-Yves B´Eziau, by Linda Eastwood
    Bulletin of the Section of Logic Volume 35:2/3 (2006), pp. 133–150 Jean-Yves B´eziau 13 QUESTIONS ABOUT UNIVERSAL LOGIC 13 questions to Jean-Yves B´eziau, by Linda Eastwood The expression “universal logic” prompts a number of misunderstandings pressing up against to the confusion prevailing nowadays around the very notion of logic. In order to clear up such equivocations, I prepared a series of questions to Jean-Yves B´eziau,who has been working for many years on his project of universal logic, recently in the University of Neuchˆatel, Switzerland. 1. Although your proposal to develop a universal logic is very appealing, isn’t it a utopian one? Isn’t it an absurd, or even dangerous thing to believe that it would be possible to develop a unique logic accounting for everything? Let us immediately reject some misunderstanding; universal logic, as I understand it, is not one universal logic. In fact, from the viewpoint of universal logic the existence of one universal logic is not even possible, and this is a result that can easily be shown. One might thus say somehow ironically the following: according to universal logic there is no universal logic. Some people in some countries have always tried to elaborate a uni- versal system that would account for any sort of reasoning, or reasoning as a whole. Aristotelian logic was depicted itself as a universal one. More recently, first-order classical logic appeared to some as a universal system accounting for mathematical reasoning as well as current one, that is, the one used to buy your bread at the bakery.
    [Show full text]
  • Linear Logic Lecture Notes for CIS 670
    Linear Logic Lecture Notes for CIS 670 Steve Zdancewic University of Pennsylvania February 25, 2014 Linear Logic Lecture Notes Draft of: February 25, 2014 Contents 1 Overview 5 1.1 Introduction . .5 2 Intuitionistic Linear Logic 7 2.1 The Multiplicative Fragment . .8 2.1.1 Multiplicative Conjunction . 10 2.1.2 Multiplicative Conjunction Unit . 11 2.1.3 Linear Implication . 11 2.1.4 Summary . 12 2.2 The Additive Fragment . 13 2.2.1 Additive Products . 13 2.2.2 Additive Product Unit . 15 2.2.3 Additive Sums . 15 2.2.4 Additive Sum Unit . 16 2.3 Persistence and the Exponential Modality . 16 3 Terms for ILL 21 3.1 Evaluations . 21 3.2 Call-by-Name Translation of Intuitionistic Logic . 27 3.3 Call-by-Value Translation of Intuitionistic Logic . 27 4 Proof Search and ILL 29 4.1 Normal Proofs . 29 4.2 Sequent Calculus . 30 Linear Logic Lecture Notes Draft of: February 25, 2014 Linear Logic Lecture Notes Draft of: February 25, 2014 Chapter 1 Overview 1.1 Introduction • Linear logic was introduced in 1987 by Jean-Yves Girard [62]. – Spawned hundreds of subsequent papers, with many(!) variants (substructural logics) – Very influential paper: cited > 4250 times according to Google Scholar. Why? – Linear logic is a generalization of “ordinary” logic in such a way that it becomes “resource conscious”. * Key idea: a resource is a “hypothesis” that can be used only once (hence it is used up) * Weakening and Contraction rules of ordinary propositional logic * Also possible to ephasize the “communication” behavior: parts of the proof that can “interact”
    [Show full text]
  • Linear Logic and Noncommutativity in the Calculus of Structures
    Linear Logic and Noncommutativity in the Calculus of Structures Dissertation zur Erlangung des akademischen Grades Doktor rerum naturalium (Dr. rer. nat.) vorgelegt an der Technischen Universit¨at Dresden Fakult¨at Informatik eingereicht am 26. Mai 2003 von Diplom-Informatiker Lutz Straßburger geboren am 21. April 1975 in Dresden Betreuer: Dr. Alessio Guglielmi Betreuender Hochschullehrer: Prof. Dr. rer. nat. habil. Steffen H¨olldobler Gutachter: Prof. Dr. rer. nat. habil. Steffen H¨olldobler Dr. Fran¸cois Lamarche Prof. Dr. rer. nat. habil. Horst Reichel Disputation am 24. Juli 2003 Acknowledgements Iamparticularlyindebted to Alessio Guglielmi. Whithout his advice and guidance this thesis would not have been written. He awoke my interest for thefascinating field of proof theory and introduced me to the calculus of structures. I benefited from many fruitful and inspiring discussions with him, and in desperate situations he encouraged me to keep going. He also provided me with his TEXmacrosfor typesetting derivations. IamgratefultoSteffenH¨olldobler for accepting the supervision of this thesis and for providing ideal conditions for doing research. He also made many helpful suggestions forimproving the readability. I am obliged to Fran¸cois Lamarche and Horst Reichel for accepting to be referees. Iwouldlike to thank Kai Br¨unnler, Paola Bruscoli, Charles Stewart, and Alwen Tiu for many fruitful discussion during the last three years. In particular, I am thankful to Kai Br¨unnler for struggling himself through a preliminary version and making helpful comments. Jim Lipton and Charles Stewart made valuable suggestions for improving the introduction. Additionally, I would like to thank Claus J¨urgensen for the fun we had in discussing TEXmacros.Inparticular, the (self-explanatory) macro \clap,whichisused on almost every page, came out of such a discussion.
    [Show full text]
  • Logic Programming in a Fragment of Intuitionistic Linear Logic: Extended Abstract
    University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science April 1991 Logic Programming in a Fragment of Intuitionistic Linear Logic: Extended Abstract Joshua S. Hodas University of Pennsylvania Dale Miller University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Joshua S. Hodas and Dale Miller, "Logic Programming in a Fragment of Intuitionistic Linear Logic: Extended Abstract", . April 1991. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-91-32. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/371 For more information, please contact [email protected]. Logic Programming in a Fragment of Intuitionistic Linear Logic: Extended Abstract Abstract Logic programming languages based on fragments of intuitionistic logic have recently been developed and studied by several researchers. In such languages, implications are permitted in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G in a context τ leads to an attempt to prove the goal G in the extended context τ ∪{D}. While an intuitionistic notion of context has many uses, it has turned out to be either too powerful or too limiting in several settings. We refine the intuitionistic notion of context by using a fragment of Girard's linear logic that includes additive and multiplicative conjunction, linear implication, universal quantification, the of" course" exponential, and the constants 1 (the empty context) and T (for "erasing" contexts). After presenting our fragment of linear logic, which contains the hereditary Harrop formulas, we show that the logic has a goal-directed interpretation.
    [Show full text]
  • Linear Logic Properly Displayed
    Linear Logic Properly Displayed GiuseppeGreco AlessandraPalmigiano∗ Abstract We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut-elimination and subformula property. Based on the same design, we introduce a variant of Lambek cal- culus with exponentials, aimed at capturing the controlled application of exchange and asso- ciativity. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main interest and added value of the present proposal, and allows for the smoothest proof of cut-elimination. Our proposal builds on an algebraic and order-theoretic analysis of linear logic, and applies the guidelines of the multi-type methodology in the design of display calculi. Keywords: Linear logic, substructural logics, algebraic proof theory, sequent calculi, cut elim- ination, display calculi, multi-type calculi, proof calculi for categorial grammar. 2010 Math. Subj. Class. 03F52, 03F05, 03G10. Contents 1 Introduction 2 2 Precursors of the multi-type approach to linear logic 3 3 Multi-type semantic environment for linear logic 6 3.1 Linear algebras and their kernels . ....... 6 3.2 The interaction of exponentials in paired linear algebras.............. 11 3.3 Reverse-engineering exponentials . ........ 13 3.4 Canonical extensions of linear algebras and their kernels ............. 16 4 Multi-type Hilbert-style presentation for linear logic 19 5 Proper display calculi for linear logics 21 arXiv:1611.04181v1 [math.LO] 13 Nov 2016 5.1 Language ...................................... 21 5.2 Rules ........................................ 22 5.3 Linear negations as primitive connectives . .......... 25 6 Properties 29 6.1 Soundness...................................... 29 6.2 Completeness ...................................
    [Show full text]