DOCSLIB.ORG

An Internal Semantics for Modal Logic: Preliminary Report

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Internal Semantics for Modal Logic: Preliminary Report An Internal Semantics for Modal Logic: Preliminary Report Ronald Fagin IBM Research Laboratory, 5600 Cottle Road, San Jose, CA 95193 Moshe Y.Vardi CSLI, Ventura Hall, Stanford University, Stanford, CA 94305 Abstract: In Kripke semantics for modal logic, “pos- and culminating in the book Symbolic Logic with sible world^" and the possibility relation are both Langford in 1932. Since then modal logic has been primitive notions. This has both technical and con- extensively studied by logicians and philosophers ceptual shortcomings. From a technical point of ([Chl is a good textbook). More recently, modal view, the mathematics associated with Kripke. se- logic has been applied in several areas of computer mantics is often quite complicated. From a concep- science, such as artificial intelligence [MH]. program tual point of view, it is not clear how to use Kripke verification and synthesis [MW.Pn,Pral]. hardware structures to model know!edge and belief, where one specification [Bo.RS], protocol specification and ver- wants a clearer understanding of the notions that ification [CES.SM], database theory [ CCF,Li], and are primitive in Kripke semantics. We introduce distributed computing [HMl]. modal structures as models for modal logic. We use Lewis’ semantics for modal logic was of an al- the idea of possible worlds, but by directly describing gebraic cast. Algebraic semantics, however, though the “internal semantics” of each possible world. It technically adequate (cf. [Gu,Mc,McT,Ts]), is nev- is much easier to study the standard logical questions, ertheless not very intuitive. In 194.6 Carnap such as completeness, decidability, and compactness, [Cal,Ca2] suggested using the more intuitive ap- ushg modal structures. Furthermore, modal struc- proach of possible wort% to assign semantics to mo- tures offer a much more intuitive approach to mod- dalities. According to this approach, one starts with elling knowledge and belief. a set of possible worlds. Then statements of the form up(i.e., p 3 wessuri/y true) are interpreted in 1. Introduction the following way: upis true if p is true in every Modal bgic can be described briefly as the logic “possible world”. (The idea that necessity is truth of necessity and possibility, of “must be” and “may in all possible worlds is often credited to Leibniz be”. (One should not take “necessity” and “possi- [Bat], though this is historically debatable.) Possible- bility” literally. “Necessarily” can mean “according worlds semantics was further developed indepen- to the laws of physics” or “according to my beliefs”, dently by several researchers [Bay, Hil, Hi2, Ka, or even “after the program terminates”.) Modal Krl, Me, Mol, Pril, reaching its current form with logic was discussed by several authors in ancient Kripke [Kr21. The basic idea of the development is times, notably Aristotle in De Inrerpretatioone and Prior to consider, instead of a set of worlds that are Amlystia, and by medieval logicians, but like most possible outright. a set of worlds that are, or are work before the modern period. it was non-symbolic, not. possible with respect to each other. Kripke structures and not particularly systematic in approach. The capture this intuitive idea. A Kripke structure can first symbolic and systematic approach to the subject be viewed as a labeled directed graph: the nodes are appears to be the work of Lewis beginning in 1912 the possible worlds labeled by truth assignments, and a world Y is possible with respect to a world u if there is an edge from I( to v. Permission to copy without fee all or part of this material is granted Kripke structures were immensely successful provided that the copies are not made or distributed for direct mathematical. tools and served as the basis for ex- commercial advantage, the ACM copyright notice and the title of the tremely fertile research. Nevertheless, they do suffer publication and its date appear, and notice is given that copying is by jxrrnission of the Association for Computing Machinery. To copy from both technical and conceptual shortcomings. otherwise, or to republish, requires a fee and/or specific permission. From a technical point of view, the mathematics associated with them is often quite complicated. For @ 1985 ACM 0-89791-151-2/85/005/0305 $00.75 example, completeness proofs are either non-elegant 305 [Kr2] (Kripke himself described his completeness of depth 2 is essentially a set of worlds of depth 1; proof as “rather messy”), or non-constructive etc. Modal structures are worlds of depth O, and [Ma,Lem2]. (Only in 1975 did Fine come with an their recursive structure enables us to assign meaning elegant and constructive proof [Fi], but his proof is to iterated modalities. far from straightforward.) Also. the standard tech- Having introduced modal structures, we investi- nique for proving decidability is to show that the gate their relationship to Kripke structures. It turns logic has thefinitemdelpropzny, which is not straight- out that modal structures model individual nodes in forward at all. Furthermore, in order to model dif- Kripke structures, while Kripke structures model ferent modal logics, certain graph-theoretic contraints collections of modal structures. Thus, modal struc- on the possibility relation between possible worlds tures can be seen as duals to Kripke structures. have to be imposed; these constraints are very often Nevertheless, it is much easier to study the standard far from intuitive, and sometimes they are not even logical questions, such as completeness, decidability, first-order definable [Go]. and compactness, using modal structures. The crucial From a conceptual point of view, it is not clear point is that satisfaction of a formula in a modal that Kripke structures are as intuitive as they are structure depends only on a certain finite part of supposed to be. The basic problem is that in Kripke that structure. Furthermore, the “size” of that part semantics the notion of a possible world is a pimitin? depends on the “size” of the formula. Thus, the notion. (Indeed possible worlds are called reference proofs of decidability and compactness are almost pints in [Mo2] and indices in [Sc].) This works well straightforward, and the completeness proof is both in applications where it is intuitively clear what a elegant and constructive. We urge the reader to possible world is. For example, m dmicbgic a compare our proofs to previous proofs (e.g., [Fr. possible world is just a program state [Prall, i.e.. an Kr2, Lem2, Ma, Mc]) in order to appreciate their assignment of values to the variables and to the elegance. location counter, and in temporal bgic a possible Beyond the technical usefulness of modal struc- world is just a point in time [Bu]. But in applications tures, we claim that they are more intuitive and where it is not clear what a possible world is, e.g., more appropriate to conceptual modelling. For ex- in eNtemic logic, the logic of knowledge and belief, ample, the simple scenarios in distributed environ- how can we construct a Kripke structure without ments mentioned above can be modelled by modal understandir.g its basic constituents? Indeed, in dy- structures in a straightforward way [FHV]. We also r:amic and temporal logic one constructs first the demonstrate the intuitiveness of modal structures by structures, and then proceeds to find the axioms modelling belief and by modelling joint knowledge, [Bu,KP], while in epistemic logic one first selects then using our techniques to prove decidability and axioms and then tailors the structures to the axioms completeness in these cases (and compactness in the [HM2.Re]. case of belief; compactness fails for joint knowledge). Furthermore, if we want to upp& modal logic it The double perspective that we have now on is often necessary to construct models for particular modal logic, namely. Kripke structures and modal situations. But if we have no means of explicitly structures, turns out to be very useful in proving describing the possible worlds, how can we construct optimal upper bounds for the complexity of the Kripke structures to model particular situations? decision problem. By their graph-theoretic nature, Indeed, as pointed out in [FHV], there are simple Kripke structures are amenable to automata-theoretic scenarios in distributed environments that cannot techniques [ES,Str,VW]. By combining our results be easily modelled by Kripke structures. for modal structures with a new automata-theoretic We believe that our approach, which is to describe technique for Kripke structures, we prove that sev- explicitly the “internal” semantics of a possible eral of the logics that we study are complete in world, is a much more intuitive approach to mod- SPACE. elling. We introduce ~aklstructuresas models for 2. Modal structures modal logic. We use the idea of possible worlds, but in Carnap’s style rather than Kripke’s style. hic definifions. We now define structures that Thus, we define a modal structure to be essentially capture the essence of the “possible worlds” ap a set of modal structures. This is of course a circular proach. (For the sake of simplicity, we restrict ourselves here to mml modal logics [Ch]. Never- definition, and to make it meaningful, we define work& inductively, by constructing worlds of greater theless, this is not an inherent limitation of our approach.) In anticipation of subsequent develop- and greater depth. A world of depth 0 is a description of reality. i.e., a truth assignment; a world of depth ments where modalities correspond to “attitudes” of agents or players, we allow multiple modalities 1 is essentially a set of worlds of depth 0; a world 306 01, ..., 0,. A good way to interpret the statement world and <gQ..,.&-2 > E fk-l(i)] for each j 2 k and Dip is “after program i terminates, p must be true”.
Load more

Recommended publications
  • Semantical Investigations
    Bulletin of the Section of Logic Volume 49/3 (2020), pp. 231{253 http://dx.doi.org/10.18778/0138-0680.2020.12 Satoru Niki EMPIRICAL NEGATION, CO-NEGATION AND THE CONTRAPOSITION RULE I: SEMANTICAL INVESTIGATIONS Abstract We investigate the relationship between M. De's empirical negation in Kripke and Beth Semantics. It turns out empirical negation, as well as co-negation, corresponds to different logics under different semantics. We then establish the relationship between logics related to these negations under unified syntax and semantics based on R. Sylvan's CC!. Keywords: Empirical negation, co-negation, Beth semantics, Kripke semantics, intuitionism. 1. Introduction The philosophy of Intuitionism has long acknowledged that there is more to negation than the customary, reduction to absurdity. Brouwer [1] has al- ready introduced the notion of apartness as a positive version of inequality, such that from two apart objects (e.g. points, sequences) one can learn not only they are unequal, but also how much or where they are different. (cf. [19, pp.319{320]). He also introduced the notion of weak counterexample, in which a statement is reduced to a constructively unacceptable principle, to conclude we cannot expect to prove the statement [17]. Presented by: Andrzej Indrzejczak Received: April 18, 2020 Published online: August 15, 2020 c Copyright for this edition by UniwersytetL´odzki, L´od´z2020 232 Satoru Niki Another type of negation was discussed in the dialogue of Heyting [8, pp. 17{19]. In it mathematical negation characterised by reduction to absurdity is distinguished from factual negation, which concerns the present state of our knowledge.
    [Show full text]
  • Introduction Nml:Syn:Int: Modal Logic Deals with Modal Propositions and the Entailment Relations Among Sec Them
    syn.1 Introduction nml:syn:int: Modal logic deals with modal propositions and the entailment relations among sec them. Examples of modal propositions are the following: 1. It is necessary that 2 + 2 = 4. 2. It is necessarily possible that it will rain tomorrow. 3. If it is necessarily possible that ' then it is possible that '. Possibility and necessity are not the only modalities: other unary connectives are also classified as modalities, for instance, \it ought to be the case that '," \It will be the case that '," \Dana knows that '," or \Dana believes that '." Modal logic makes its first appearance in Aristotle's De Interpretatione: he was the first to notice that necessity implies possibility, but not vice versa; that possibility and necessity are inter-definable; that If ' ^ is possibly true then ' is possibly true and is possibly true, but not conversely; and that if ' ! is necessary, then if ' is necessary, so is . The first modern approach to modal logic was the work of C. I. Lewis,cul- minating with Lewis and Langford, Symbolic Logic (1932). Lewis & Langford were unhappy with the representation of implication by means of the mate- rial conditional: ' ! is a poor substitute for \' implies ." Instead, they proposed to characterize implication as \Necessarily, if ' then ," symbolized as ' J . In trying to sort out the different properties, Lewis identified five different modal systems, S1,..., S4, S5, the last two of which are still in use. The approach of Lewis and Langford was purely syntactical: they identified reasonable axioms and rules and investigated what was provable with those means.
    [Show full text]
  • On the Logic of Two-Dimensional Semantics
    Matrices and Modalities: On the Logic of Two-Dimensional Semantics MSc Thesis (Afstudeerscriptie) written by Peter Fritz (born March 4, 1984 in Ludwigsburg, Germany) under the supervision of Dr Paul Dekker and Prof Dr Yde Venema, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: June 29, 2011 Dr Paul Dekker Dr Emar Maier Dr Alessandra Palmigiano Prof Dr Frank Veltman Prof Dr Yde Venema Abstract Two-dimensional semantics is a theory in the philosophy of language that pro- vides an account of meaning which is sensitive to the distinction between ne- cessity and apriority. Usually, this theory is presented in an informal manner. In this thesis, I take first steps in formalizing it, and use the formalization to present some considerations in favor of two-dimensional semantics. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of two-dimensional semantics. I use this to show that some criticisms of two- dimensional semantics that claim that the theory is incoherent are not justified. I also axiomatize the logic, and compare it to the most important proposals in the literature that define similar logics. To indicate that two-dimensional semantics is a plausible semantic theory, I give an argument that shows that all theorems of the logic can be philosophically justified independently of two-dimensional semantics. Acknowledgements I thank my supervisors Paul Dekker and Yde Venema for their help and encour- agement in preparing this thesis.
    [Show full text]
  • Probabilistic Semantics for Modal Logic
    Probabilistic Semantics for Modal Logic By Tamar Ariela Lando A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Philosophy in the Graduate Division of the University of California, Berkeley Committee in Charge: Paolo Mancosu (Co-Chair) Barry Stroud (Co-Chair) Christos Papadimitriou Spring, 2012 Abstract Probabilistic Semantics for Modal Logic by Tamar Ariela Lando Doctor of Philosophy in Philosophy University of California, Berkeley Professor Paolo Mancosu & Professor Barry Stroud, Co-Chairs We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940’s. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero. In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfed in a model over the algebra if it is assigned to the top element in the algebra—or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propo- sitional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4C, is sound and com- plete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators).
    [Show full text]
  • Kripke Semantics for Basic Sequent Systems
    Kripke Semantics for Basic Sequent Systems Arnon Avron and Ori Lahav? School of Computer Science, Tel Aviv University, Israel {aa,orilahav}@post.tau.ac.il Abstract. We present a general method for providing Kripke seman- tics for the family of fully-structural multiple-conclusion propositional sequent systems. In particular, many well-known Kripke semantics for a variety of logics are easily obtained as special cases. This semantics is then used to obtain semantic characterizations of analytic sequent sys- tems of this type, as well as of those admitting cut-admissibility. These characterizations serve as a uniform basis for semantic proofs of analyt- icity and cut-admissibility in such systems. 1 Introduction This paper is a continuation of an on-going project aiming to get a unified semantic theory and understanding of analytic Gentzen-type systems and the phenomenon of strong cut-admissibility in them. In particular: we seek for gen- eral effective criteria that can tell in advance whether a given system is analytic, and whether the cut rule is (strongly) admissible in it (instead of proving these properties from scratch for every new system). The key idea of this project is to use semantic tools which are constructed in a modular way. For this it is es- sential to use non-deterministic semantics. This was first done in [6], where the family of propositional multiple-conclusion canonical systems was defined, and it was shown that the semantics of such systems is provided by two-valued non- deterministic matrices { a natural generalization of the classical truth-tables. The sequent systems of this family are fully-structural (i.e.
    [Show full text]
  • Kripke Semantics a Semantic for Intuitionistic Logic
    Kripke Semantics A Semantic for Intuitionistic Logic By Jonathan Prieto-Cubides Updated: 2016/11/18 Supervisor: Andr´esSicard-Ram´ırez Logic and Computation Research Group EAFIT University Medell´ın,Colombia History Saul Kripke He was born on November # 13, 1940 (age 75) Philosopher and Logician # Emeritus Professor at # Princeton University In logic, his major # contributions are in the field of Modal Logic Saul Kripke In Modal Logic, we # attributed to him the notion of Possible Worlds Its notable ideas # ◦ Kripke structures ◦ Rigid designators ◦ Kripke semantics Kripke Semantics The study of semantic is the study of the truth Kripke semantics is one of many (see for instance # (Moschovakis, 2015)) semantics for intuitionistic logic It tries to capture different possible evolutions of the world # over time The abstraction of a world we call a Kripke structure # Proof rules of intuitionistic logic are sound with respect to # krikpe structures Intuitionistic Logic Derivation (proof) rules of the ^ connective Γ ` ' ^ ^-elim Γ ` ' 1 Γ ` ' Γ ` ^-intro Γ ` ' ^ Γ ` ' ^ ^-elim Γ ` 2 Intuitionistic Logic Derivation (proof) rules of the _ connective Γ ` ' _-intro1 Γ ` ' _ Γ ` ' _ Γ;' ` σ Γ; ` σ _-elim Γ ` σ Γ ` _-intro Γ ` ' _ 2 Intuitionistic Logic Derivation (proof) rules of the ! connective Γ;' ` Γ ` ' Γ ` ' ! !-intro !-elim Γ ` ' ! Γ ` Intuitionistic Logic Derivation (proof) rules of the : connective where :' ≡ ' !? Γ;' ` ? Γ ` ? :-intro explosion Γ ` :' Γ ` ' Intuitionistic Logic Other derivation (proof) rules Γ ` ' unit assume weaken Γ ` > Γ;' ` ' Γ; ` ' Classical Logic The list of derivation rules are the same above plus the following rule Γ; :' ` ? RAA Γ ` ' Kripke Model Def.
    [Show full text]
  • 1. Kripke's Semantics for Modal Logic
    Ted Sider Kripke Modality seminar 1. Kripke’s semantics for modal logic 2A “Necessarily, A” 3A “Possibly, A” Lewis’s system K: Rules of inference: modus ponens, plus the rule of necessitation, which lets · you infer 2A from A. Axioms: those of propositional logic, plus: · 2(A B) (2A 2B) (K) ! ! ! Lewis’s system S4: Rules of inference: same as K · Axioms: those of K, plus: · 2A A (T) ! 2A 22A (S4) ! Proof-theoretic approach A logically implies B iff you can reason, using speci ed axioms and rules of inference, from A to B Model-theoretic approach A logically implies B iff B is true in every model in which A is true propositional logic model: an assignment of truth-values to sentence letters. predicate logic model: a domain and an interpretation function that assigns denotations to names and extensions to predicates. modal logic model: a set of “possible worlds”, each one containing an entire model of the old sort. 1 2A is true at world w iff A is true at every world accessible from w 3A is true at world w iff A is true at some world accessible from w Promising features of Kripke semantics: • Explaining logical features of modal logic (duality of 2 and 3; logical truth of axiom K) • Correspondence of Lewis’s systems to formal features of accessibility A cautionary note: what good is all this if 2 doesn’t mean truth in all worlds? 2. Kripke’s Naming and Necessity Early defenders of modal logic, for example C. I. Lewis, thought of the 2 as meaning analyticity.
    [Show full text]
  • Kripke Completeness Revisited
    Kripke completeness revisited Sara Negri Department of Philosophy, P.O. Box 9, 00014 University of Helsinki, Finland. e-mail: sara.negri@helsinki.fi Abstract The evolution of completeness proofs for modal logic with respect to the possible world semantics is studied starting from an analysis of Kripke’s original proofs from 1959 and 1963. The critical reviews by Bayart and Kaplan and the emergence of Henkin-style completeness proofs are detailed. It is shown how the use of a labelled sequent system permits a direct and uniform completeness proof for a wide variety of modal logics that is close to Kripke’s original arguments but without the drawbacks of Kripke’s or Henkin-style completeness proofs. Introduction The question about the ultimate attribution for what is commonly called Kripke semantics has been exhaustively discussed in the literature, recently in two surveys (Copeland 2002 and Goldblatt 2005) where the rˆoleof the precursors of Kripke semantics is documented in detail. All the anticipations of Kripke’s semantics have been given ample credit, to the extent that very often the neutral terminology of “relational semantics” is preferred. The following quote nicely summarizes one representative standpoint in the debate: As mathematics progresses, notions that were obscure and perplexing become clear and straightforward, sometimes even achieving the status of “obvious.” Then hindsight can make us all wise after the event. But we are separated from the past by our knowledge of the present, which may draw us into “seeing” more than was really there at the time. (Goldblatt 2005, section 4.2) We are not going to treat this issue here, nor discuss the parallel development of the related algebraic semantics for modal logic (Jonsson and Tarski 1951), but instead concentrate on one particular and crucial aspect in the history of possible worlds semantics, namely the evolution of completeness proofs for modal logic with respect to Kripke semantics.
    [Show full text]
  • First-Order Modal Logic: Frame Definability and Lindström Theorems
    First-Order Modal Logic: Frame Definability and Lindstr¨om Theorems R. Zoghifard∗ M. Pourmahdian† Abstract This paper involves generalizing the Goldblatt-Thomason and the Lindstr¨om characterization theorems to first-order modal logic. Keywords: First-order modal logic, Kripke semantics, Bisimulation, Goldblatt- Thomason theorem, Lindstr¨om theorem. 1 Introduction and Preliminaries The purpose of this study is to extend two well-known theorems of propositional modal logic, namely the Goldblatt-Thomason and the Lindstr¨om characterization theorems, to first-order modal logic. First-order modal logic (FML) provides a framework for incorporating both propositional modal and classical first-order logics (FOL). This, from a model theoretic point of view, means that a Kripke frame can be expanded by a non-empty set as a domain over which the quantified variables range. In particular, arXiv:1602.00201v2 [math.LO] 29 May 2016 first-order Kripke models subsume both propositional Kripke models and classical first- order structures. The celebrated Goldblatt-Thomason theorem provides a model theoretic character- ization of elementary classes of frames which are definable by a set of propositional modal sentences. This theorem states that an elementary class of Kripke frames is definable by a set of propositional modal formulas if and only if it is closed under ∗Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran. E-mail: [email protected]. †Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran. E-mail: [email protected]. 1 bounded morphic images, generated subframes and disjoint unions and reflects ultra- filter extensions.
    [Show full text]
  • Generalized Topological Semantics for First-Order Modal Logic 205 VI.1
    Generalized Topological Semantics for First-Order Modal Logic1 Kohei Kishida 1Draft of November 14, 2010. Draft of November 14, 2010 Abstract. This dissertation provides a new semantics for first-order modal logic. It is philosophi- cally motivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities. (i) The semantic modelling of epistemic modalities, in particular verifiability and falsifiability, cannot be properly achieved by Kripke’s relational notion of accessibility. It requires instead a more general, topological notion of accessibility. (ii) Also, the epistemic reading of modal operators seems to require that we combine modal logic with fully classical first-order logic. For this purpose, however, Kripke’s semantics for quanti- fied modal logic is inadequate; its logic is free logic as opposed to classical logic. (iii) More importantly, Kripke’s semantics comes with a restriction that is too strong to let us se- mantically express, for instance, that the identity of Hesperus and Phosphorus, even if meta- physically necessary, can still be a matter of epistemic discovery. To provide a semantics that accommodates the three desiderata, I show, on the one hand, how the desideratum (i) can be achieved with topological semantics, and more generally neighborhood se- mantics, for propositional modal logic. On the other hand, to achieve (ii) and (iii), it turns out that David Lewis’s counterpart theory is helpful at least technically. Even though Lewis’s own formulation is too liberal—in contrast to Kripke’s being too restrictive—to achieve our goals, this dissertation provides a unification of the two frameworks, Kripke’s and Lewis’s.
    [Show full text]
  • The Epistemic View of Concurrency Theory Sophia Knight
    The Epistemic View of Concurrency Theory Sophia Knight To cite this version: Sophia Knight. The Epistemic View of Concurrency Theory. Logic in Computer Science [cs.LO]. Ecole Polytechnique X, 2013. English. tel-00940413 HAL Id: tel-00940413 https://tel.archives-ouvertes.fr/tel-00940413 Submitted on 31 Jan 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. The Epistemic View of Concurrency Theory Sophia Knight September 2013 1 i Abstract This dissertation describes three distinct but complementary ways in which epistemic reasoning plays a role in concurrency theory. The first and perhaps the one least explored so far is the idea of using epistemic modalities as programming constructs. Logic program- ming emerged under the slogan \Logic as a programming language" and the connection was manifest in a very clear way in the concurrent constraint programming paradigm. In the first part of the present thesis, we explore the role of epistemic, and closely related spatial modalities, as part of the programming language and not just as part of the meta-language for reasoning about protocols. The next part explores a variant of dynamic epistemic logic adapted to labelled transition systems.
    [Show full text]
  • Topos Semantics for a Higher-Order Temporal Logic of Actions
    Topos Semantics for a Higher-order Temporal Logic of Actions Philip Johnson-Freyd Jon Aytac Geoffrey Hulette Sandia National Laboratories Sandia National Laboratories Sandia National Laboratories [email protected] [email protected] [email protected] TLA is a popular temporal logic for writing stuttering-invariant specifications of digital systems. However, TLA lacks higher-order features useful for specifying modern software written in higher- order programming languages. We use categorical techniques to recast a real-time semantics for TLA in terms of the actions of a group of time dilations, or “stutters,” and an extension by a monoid incor- porating delays, or “falters.” Via the geometric morphism of the associated presheaf topoi induced by the inclusion of stutters into falters, we construct the first model of a higher-order TLA. 1 1 Introduction The Temporal Logic of Actions (TLA) is a temporal logic commonly used for specifying digital computer systems [11, 13]. TLA formulae are linear temporal properties invariant under “stuttering.” Stuttering invariant specifications written as TLA formulae are easily composed, using nothing more than conjunc- tion, with no implicit assumptions about synchronization. Stuttering invariance also leads to a simple but powerful notion of “refinement,” that is, showing that a detailed specification implements an abstract one. In [11] Lamport presents TLA as a first-order logic, but, in specifications, higher-order features are often desirable. For example, one would often like to prove a rule of inference that works over all proposi- tions or all predicates. Lamport must introduce special syntax (e.g., for fairness) where in a higher-order context these language features could be replaced with simple functions on propositions.
    [Show full text]