DOCSLIB.ORG

Linear Type Theories, Semantics and Action Calculi

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Type Theories, Semantics and Action Calculi Linear Type Theories, Semantics and Action Calculi Andrew Graham Barber LFCS, Department of Computing Science, Edinburgh University Doctor of Philosophy University of Edinburgh 1997 To my wife, for her trust Abstract In this thesis, we study linear type-theories and their semantics. We present a general framework for such type-theories, and prove certain decidability properties of its equality. We also present intuitionistic linear logic and Milner’s action calculi as instances of the framework, and use our results to show decidability of their respective equality judgements. Firstly, we motivate our development by giving a split-context logic and type- theory, called dual intuitionistic linear logic (DILL), which is equivalent at the level of term equality to the familiar type-theory derived from intuitionistic linear logic (ILL). We give a semantics for the type-theory DILL, and prove soundness and completeness for it; we can then deduce these results for the type-theory ILL by virtue of our translation. Secondly, we generalise DILL to obtain a general logic, type-theory and se- mantics based on an arbitrary set of operators, or general natural deduction rules. We again prove soundness and completeness results, augmented in this case by an initiality result. We introduce Milner’s action calculi, and present example instances of our framework which are isomorphic to them. We extend this iso- morphism to three significant higher-order variants of the action calculi, having functional properties, and compare the induced semantics for these action calculi with those given previously. Thirdly, motivated by these functional extensions, we define higher-order in- stances of our general framework, which are equipped with functional structure, proceeding as before to give logic, type-theory and semantics. We show that the logic and type-theory DILL arise as a higher-order instance of our general frame- work. We then define the higher-order extension of any instance of our framework, and use a Yoneda lemma argument to show that the obvious embedding from an instance into its higher-order extension is conservative. This has the corollary that the embeddings from the action calculi into the higher-order action calculi are all conservative, extending a result of Milner. Finally, we introduce relations, a syntax derived from proof-nets, for our gen- eral framework, and use them to show that certain instances of our framework, including some higher-order instances, have decidable equality judgements. This immediately shows that the equalities of DILL, ILL, the action calculi and the higher-order action calculi are decidable. Acknowledgements I am hugely grateful to my supervisor, Gordon Plotkin, for his insight and ability to communicate it, his encouragement, and for his detailed comments at each stage of my thesis. I also thank Philippa Gardner, my second supervisor, for her attention, her enthusiasm and for pointing me in the direction of many interesting problems. Other than my supervisors, I have benefited greatly during my time at Edin- burgh from the insight of many people here, particularly Masahito Hasegawa, John Power, Alex Simpson, Paul-Andre Mellies and Marcello Fiore, whom I thank, and from the good-humour of my office partner Thomas Schreiber. Finally, I thank my wife for all her support, encouragement and proof-reading– her efforts have immeasurably improved this thesis and my time in Edinburgh. Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Andrew Graham Barber LFCS, Department of Computing Science, Edinburgh University) Table of Contents Chapter 1 Introduction 4 1.1MotivatingSketch........................... 5 1.2Background.............................. 11 1.3Development............................. 24 1.4ChapterSummary.......................... 29 1.5HistoricalNotes............................ 31 Chapter 2 An Alternative Formulation of ILL 32 2.1 The Logic DILL ............................ 32 2.2 The Type Theory DILL(C)...................... 35 2.3 The Type Theory ILL(C)....................... 42 2.4 Relating DILL and ILL ........................ 45 Chapter 3 The Semantics of DILL 60 3.1TheInterpretation.......................... 60 3.2 Soundness . 65 3.3TheTermModel........................... 68 3.4Completeness............................. 78 3.5 ILL andOtherModels........................ 79 Chapter 4 Linear Type Theories 81 4.1Introduction.............................. 81 4.2TheGeneralisedLogic........................ 84 4.3 The Typing System Lin(O)...................... 87 4.4 The Type Theory Lin(O, A) ..................... 91 Chapter 5 The Semantics of Lin(O, A) 95 5.1TheInterpretation.......................... 95 5.2 Soundness . 99 5.3TheTermModel........................... 101 1 5.4Completeness............................. 107 5.5Initiality................................ 109 5.6OutputNaturality.......................... 114 Chapter 6 Action Calculi and Extensions 120 6.1ActionCalculi............................. 121 6.2GeneralisedLinearTypeTheory................... 124 6.3TheTranslations........................... 126 6.4Extensions............................... 134 6.5Semantics............................... 143 Chapter 7 Higher-Order Type Theories 148 7.1Higher-OrderTypeTheories..................... 148 7.2Higher-OrderSemantics....................... 153 7.3Conservativity............................. 159 7.4Corollaries............................... 165 7.5ActionCalculi............................. 168 7.6LinearLogic.............................. 170 Chapter 8 Normal Forms for Lin(O, A) 178 8.1ProofNets............................... 178 8.2Relations................................ 185 8.3RelationalEquality.......................... 196 8.4TermstoRelations.......................... 199 8.5RelationstoTerms.......................... 204 Chapter 9 Normal Forms and Decidability 215 9.1Proof-Nets............................... 215 9.2 The F -Equalities........................... 216 9.3DecidingtheEquality......................... 218 9.4TheHigher-OrderCase........................ 227 9.5Corollaries............................... 236 Chapter 10 Conclusions 239 10.1Results................................. 239 10.2 Extending Lin(O, A) ......................... 241 10.3AimsandObjectives......................... 246 10.4FinalRemarks............................. 249 2 Appendix A Basic Definitions 250 A.1SyntacticPreliminaries........................ 250 A.2CategoricalDefinitions........................ 251 Bibliography 255 Index to Notation 265 Index to Definitions 267 3 Chapter 1 Introduction The central concept in this thesis is that of linearity. We can say that linearity is an attribute of systems in which unrestricted duplication of computations or information is not allowed. There are many practical situations in which this is the case—computations with side effects, where duplicating the computation also duplicates the side-effect, manipulating large data-structures, where a duplication may bear an unacceptable computational cost, or simply efficient implementations where duplicating computation is to be avoided except where necessary. Areas as diverse as optimal λ-calculus implementations, the study of process calculi and denotational semantics can all utilise linearity in this broad sense. Although such systems have been studied for many years, a new and striking development occurred when Girard introduced linear logic [Gir87]. Unlike con- ventional intuitionistic and classical logics, linear logic is built on the underlying idea that assumptions should be treated linearly, which is to say that an assump- tion should not be used twice in a deduction. Starting from this basis, Girard built a complete classical linear logic, which, just as in the conventional case, has both minimal and intuitionistic fragments. Since the introduction of linear logic, many researchers have used it as an un- derlying logic for systems which exhibit linearity in our general sense, for example in the study of optimal λ-calculus reductions [GAL92a], in the analysis of imple- mentation issues such as garbage collection, references and others [Wad90, GH90] and to control interference in imperative languages [OTPT95]. It has become the logic of choice for such applications in much the same way as conventional logic underpins a huge range of theory in computer science. In the case of conventional logics, one large section of applications are those which are based on conventional intuitionistic logic as a typing system for a syntax of some sort. A variety of sys- tems can be typed in this way, and the general principles underlying such typings are well understood. 4 In this thesis, we present a first step towards a similar unifying programme for systems with linearity. We give a general operator theory, logic, and type-theory over a linear underlying structure, and show how it can be used to present a number of typing systems exhibiting linearity. We also give a semantics for our theory, and prove some general results about it. As an introduction, we give a motivating summary in a simple setting, provide brief tutorials and summaries of related work on the concepts we work with, and then outline the work contained in the thesis. A chapter summary can be found in section 1.4. We assume a basic understanding of elementary intuitionistic logic, its type theory, simple category theory, and the various relationships of a familiar triangle,
Load more

Recommended publications
  • Linear Type Theory for Asynchronous Session Types
    JFP 20 (1): 19–50, 2010. c Cambridge University Press 2009 19 ! doi:10.1017/S0956796809990268 First published online 8 December 2009 Linear type theory for asynchronous session types SIMON J. GAY Department of Computing Science, University of Glasgow, Glasgow G12 8QQ, UK (e-mail: [email protected]) VASCO T. VASCONCELOS Departamento de Informatica,´ Faculdade de Ciencias,ˆ Universidade de Lisboa, 1749-016 Lisboa, Portugal (e-mail: [email protected]) Abstract Session types support a type-theoretic formulation of structured patterns of communication, so that the communication behaviour of agents in a distributed system can be verified by static typechecking. Applications include network protocols, business processes and operating system services. In this paper we define a multithreaded functional language with session types, which unifies, simplifies and extends previous work. There are four main contributions. First is an operational semantics with buffered channels, instead of the synchronous communication of previous work. Second, we prove that the session type of a channel gives an upper bound on the necessary size of the buffer. Third, session types are manipulated by means of the standard structures of a linear type theory, rather than by means of new forms of typing judgement. Fourth, a notion of subtyping, including the standard subtyping relation for session types (imported into the functional setting), and a novel form of subtyping between standard and linear function types, which allows the typechecker to handle linear types conveniently. Our new approach significantly simplifies session types in the functional setting, clarifies their essential features and provides a secure foundation for language developments such as polymorphism and object-orientation.
    [Show full text]
  • 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]
  • A General Framework for the Semantics of Type Theory
    A General Framework for the Semantics of Type Theory Taichi Uemura November 14, 2019 Abstract We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L¨oftype theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory: every type theory has a bi-initial model; every model of a type theory has its internal language; the category of theories over a type theory is bi-equivalent to a full sub-2-category of the 2-category of models of the type theory. 1 Introduction One of the key steps in the semantics of type theory and logic is to estab- lish a correspondence between theories and models. Every theory generates a model called its syntactic model, and every model has a theory called its internal language. Classical examples are: simply typed λ-calculi and cartesian closed categories (Lambek and Scott 1986); extensional Martin-L¨oftheories and locally cartesian closed categories (Seely 1984); first-order theories and hyperdoctrines (Seely 1983); higher-order theories and elementary toposes (Lambek and Scott 1986). Recently, homotopy type theory (The Univalent Foundations Program 2013) is expected to provide an internal language for what should be called \el- ementary (1; 1)-toposes". As a first step, Kapulkin and Szumio (2019) showed that there is an equivalence between dependent type theories with intensional identity types and finitely complete (1; 1)-categories. As there exist correspondences between theories and models for almost all arXiv:1904.04097v2 [math.CT] 13 Nov 2019 type theories and logics, it is natural to ask if one can define a general notion of a type theory or logic and establish correspondences between theories and models uniformly.
    [Show full text]
  • Implicit Versus Explicit Knowledge in Dialogical Logic Manuel Rebuschi
    Implicit versus Explicit Knowledge in Dialogical Logic Manuel Rebuschi To cite this version: Manuel Rebuschi. Implicit versus Explicit Knowledge in Dialogical Logic. Ondrej Majer, Ahti-Veikko Pietarinen and Tero Tulenheimo. Games: Unifying Logic, Language, and Philosophy, Springer, pp.229-246, 2009, 10.1007/978-1-4020-9374-6_10. halshs-00556250 HAL Id: halshs-00556250 https://halshs.archives-ouvertes.fr/halshs-00556250 Submitted on 16 Jan 2011 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. Implicit versus Explicit Knowledge in Dialogical Logic Manuel Rebuschi L.P.H.S. – Archives H. Poincar´e Universit´ede Nancy 2 [email protected] [The final version of this paper is published in: O. Majer et al. (eds.), Games: Unifying Logic, Language, and Philosophy, Dordrecht, Springer, 2009, 229-246.] Abstract A dialogical version of (modal) epistemic logic is outlined, with an intuitionistic variant. Another version of dialogical epistemic logic is then provided by means of the S4 mapping of intuitionistic logic. Both systems cast new light on the relationship between intuitionism, modal logic and dialogical games. Introduction Two main approaches to knowledge in logic can be distinguished [1]. The first one is an implicit way of encoding knowledge and consists in an epistemic interpretation of usual logic.
    [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]
  • From Axioms to Rules — a Coalition of Fuzzy, Linear and Substructural Logics
    From Axioms to Rules — A Coalition of Fuzzy, Linear and Substructural Logics Kazushige Terui National Institute of Informatics, Tokyo Laboratoire d’Informatique de Paris Nord (Joint work with Agata Ciabattoni and Nikolaos Galatos) Genova, 21/02/08 – p.1/?? Parties in Nonclassical Logics Modal Logics Default Logic Intermediate Logics (Padova) Basic Logic Paraconsistent Logic Linear Logic Fuzzy Logics Substructural Logics Genova, 21/02/08 – p.2/?? Parties in Nonclassical Logics Modal Logics Default Logic Intermediate Logics (Padova) Basic Logic Paraconsistent Logic Linear Logic Fuzzy Logics Substructural Logics Our aim: Fruitful coalition of the 3 parties Genova, 21/02/08 – p.2/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Genova, 21/02/08 – p.3/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Genova, 21/02/08 – p.3/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Linear Logic: Cut Elimination Genova, 21/02/08 – p.3/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Linear Logic: Cut Elimination A logic without cut elimination is like a car without engine (J.-Y. Girard) Genova, 21/02/08 – p.3/?? Outcome We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is defined based on Polarity (Linear Logic). Genova, 21/02/08 – p.4/?? Outcome We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is defined based on Polarity (Linear Logic). Give an automatic procedure to transform axioms up to level ¼ È È ¿ ( , in the absense of Weakening) into Hyperstructural ¿ Rules in Hypersequent Calculus (Fuzzy Logics).
    [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]
  • Bunched Hypersequent Calculi for Distributive Substructural Logics
    EPiC Series in Computing Volume 46, 2017, Pages 417{434 LPAR-21. 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning Bunched Hypersequent Calculi for Distributive Substructural Logics Agata Ciabattoni and Revantha Ramanayake Technische Universit¨atWien, Austria fagata,[email protected]∗ Abstract We introduce a new proof-theoretic framework which enhances the expressive power of bunched sequents by extending them with a hypersequent structure. A general cut- elimination theorem that applies to bunched hypersequent calculi satisfying general rule conditions is then proved. We adapt the methods of transforming axioms into rules to provide cutfree bunched hypersequent calculi for a large class of logics extending the dis- tributive commutative Full Lambek calculus DFLe and Bunched Implication logic BI. The methodology is then used to formulate new logics equipped with a cutfree calculus in the vicinity of Boolean BI. 1 Introduction The wide applicability of logical methods and their use in new subject areas has resulted in an explosion of new logics. The usefulness of these logics often depends on the availability of an analytic proof calculus (formal proof system), as this provides a natural starting point for investi- gating metalogical properties such as decidability, complexity, interpolation and conservativity, for developing automated deduction procedures, and for establishing semantic properties like standard completeness [26]. A calculus is analytic when every derivation (formal proof) in the calculus has the property that every formula occurring in the derivation is a subformula of the formula that is ultimately proved (i.e. the subformula property). The use of an analytic proof calculus tremendously restricts the set of possible derivations of a given statement to deriva- tions with a discernible structure (in certain cases this set may even be finite).
    [Show full text]
  • Applications of Non-Classical Logic Andrew Tedder University of Connecticut - Storrs, [email protected]
    University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 8-8-2018 Applications of Non-Classical Logic Andrew Tedder University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Tedder, Andrew, "Applications of Non-Classical Logic" (2018). Doctoral Dissertations. 1930. https://opencommons.uconn.edu/dissertations/1930 Applications of Non-Classical Logic Andrew Tedder University of Connecticut, 2018 ABSTRACT This dissertation is composed of three projects applying non-classical logic to problems in history of philosophy and philosophy of logic. The main component concerns Descartes’ Creation Doctrine (CD) – the doctrine that while truths concerning the essences of objects (eternal truths) are necessary, God had vol- untary control over their creation, and thus could have made them false. First, I show a flaw in a standard argument for two interpretations of CD. This argument, stated in terms of non-normal modal logics, involves a set of premises which lead to a conclusion which Descartes explicitly rejects. Following this, I develop a multimodal account of CD, ac- cording to which Descartes is committed to two kinds of modality, and that the apparent contradiction resulting from CD is the result of an ambiguity. Finally, I begin to develop two modal logics capturing the key ideas in the multi-modal interpretation, and provide some metatheoretic results concerning these logics which shore up some of my interpretive claims. The second component is a project concerning the Channel Theoretic interpretation of the ternary relation semantics of relevant and substructural logics. Following Barwise, I de- velop a representation of Channel Composition, and prove that extending the implication- conjunction fragment of B by composite channels is conservative.
    [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]
  • Dependable Atomicity in Type Theory
    Dependable Atomicity in Type Theory Andreas Nuyts1 and Dominique Devriese2 1 imec-DistriNet, KU Leuven, Belgium 2 Vrije Universiteit Brussel, Belgium Presheaf semantics [Hof97, HS97] are an excellent tool for modelling relational preservation prop- erties of (dependent) type theory. They have been applied to parametricity (which is about preservation of relations) [AGJ14], univalent type theory (which is about preservation of equiv- alences) [BCH14, Hub15], directed type theory (which is about preservation of morphisms) and even combinations thereof [RS17, CH19]. Of course, after going through the endeavour of con- structing a presheaf model of type theory, we want type-theoretic profit, i.e. we want internal operations that allow us to write cheap proofs of the `free' theorems [Wad89] that follow from the preservation property concerned. While the models for univalence, parametricity and directed type theory are all just cases of presheaf categories, approaches to internalize their results do not have an obvious common ances- tor (neither historically nor mathematically). Cohen et al. [CCHM16] have used the final type extension operator Glue to prove univalence. In previous work with Vezzosi [NVD17], we used Glue and its dual, the initial type extension operator Weld, to internalize parametricity to some extent. Before, Bernardy, Coquand and Moulin [BCM15, Mou16] have internalized parametricity using completely different `boundary filling’ operators Ψ (for extending types) and Φ (for extending func- tions). Unfortunately, Ψ and Φ have so far only been proven sound with respect to substructural (affine-like) variables of representable types (such as the relational or homotopy interval I). More 1 recently, Licata et al. [LOPS18] have exploited the fact that the homotopyp interval I is atomic | meaning that the exponential functor (I ! xy) has a right adjoint | in order to construct a universe of Kan-fibrant types from a vanilla Hofmann-Streicher universe [HS97] internally.
    [Show full text]