DOCSLIB.ORG

The Joys of Bisimulation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Joys of Bisimulation The Joys of Bisimulation Colin Stirling Department of Computer Science University of Edinburgh Edinburgh EH JZ UK email cpsdcsedacuk Intro duction Bisimulation is a rich concept which app ears in various areas of theoretical com puter science Its origins lie in concurrency theory for instance see Milner and in mo dal logic see for example van Benthem In this pap er we review results ab out bisimulation from b oth the p oint of view of automata and from a logical p oint of view We also consider how bisimulation has a role in nite mo del theory and we oer a new undenability result Basics Lab elled transition systems are commonly encountered in op erational semantics of programs and systems They are just lab elled graphs A transition system a is a pair T S f a Ag where S is a nonempty set of states A is a a nonempty set of lab els and for each a L is a binary relation on S a a We write s s instead of s s Sometimes there is extra structure in a transition system a set of atomic colours Q such that each colour q S the subset of states with colour q Bisimulations were intro duced by Park as a small renement of the b e havioural equivalence dened by Hennessy and Milner in b etween basic CCS pro cesses whose b ehaviour is a transition system Denition A binary relation R b etween states of a transition system is a bisimulation just in case whenever s t R and a A a a if s s then t t for some t such that s t R and a a if t t then s s for some s such that s t R In the case of an enriched transition system with colours there is an extra clause in the denition of a bisimulation that it preserves colours if s t R then for all colours q s q i t q Simple examples of bisimulations are the identity relation and the empty rela tion Two states of a transition system s and t are bisimulation equivalent or bisimilar written s t if there is a bisimulation relation R with s t R One can also present bisimulation equivalence as a game G s t see for 0 0 example which is played by two participants players I and I I A play of G s t is a nite or innite length sequence of the form s t s t 0 0 0 0 i i Player I attempts to show that the initial states are dierent whereas player II wishes to establish that they are equivalent Supp ose an initial part of a play is s t s t The next pair s t is determined by one of the 0 0 j j j +1 j +1 following two moves a s and then player II cho oses a trans Player I cho oses a transition s j +1 j a t ition with the same lab el t j +1 j a t and then player II cho oses a trans Player I cho oses a transition t j +1 j a s ition with the same lab el s j +1 j The play continues with further moves Player I always cho oses rst and then player I I with full knowledge of player Is selection must cho ose a corresp onding transition of the other state A play of a game continues until one of the players wins In a p osition s t if one of these states has an a transition and the other do esnt then s and t are clearly distinguishable and in the case of an enriched transition systems if one of these states has a colour which the other do esnt have then again they are distinguishable Consequently any p osition s t where s and t are n n n n distinguishable counts as a win for player I and are called Iwins A play is won by player I if the play reaches a Iwin p osition Any play that fails to reach such a p osition counts as a win for player I I Consequently player II wins if the play is innite or if the play reaches the p osition s t and neither state has an n n available transition Dierent plays of a game can have dierent winners Nevertheless for each game one of the players is able to win any play irresp ective of what moves her opp onent makes To make this precise the notion of strategy is essential A strategy for a player is a family of rules which tell the player how to move However it turns out that we only need to consider historyfree strategies whose rules do not dep end on what happ ened previously in the play For player I a rule is therefore of the form at p osition s t cho ose transition x where x is a a s s or t t for some a A rule for player II is at p osition s t when a a player I has chosen x cho ose y where x is either s s or t t and y is a corresp onding transition of the other state A player uses the strategy in a play if all her moves ob ey the rules in The strategy is a winning strategy if the player wins every play in which she uses Prop osition For any game G s t either player I or player II has a history free winning strategy Prop osition Player II has a winning strategy for G s t i s t Transition systems are mo dels for basic pro cess calculi such as CCS and CSP Mo dels for richer calculi capturing value passing mobility causality time probability and lo cations have b een develop ed The basic notion of bisimulation has b een generalised often in a variety of dierent ways to cover these extra features Bisimulation also has a nice categorical representation via coalgebras due to Aczel see for example which allows a very general denition It is an interesting question whether all the dierent brands of bisimulation are instances of this categorical account In this pap er we shall continue to examine only the very concrete notion of bisimulation on transition systems Bisimulation closure and invariance It is common to identify a ro ot of a transition system as some sp ecial start state Ab ove we dened a bisimulation on states of the same transition graph Equally we could have dened it b etween states of dierent transition systems When transition systems are ro oted we can then say that two systems are bisimilar if their ro ots are A family of ro oted transition graphs is said to b e closed under bisimulation equivalence when the following holds if T and T T then T Given a ro oted transition system there is a smallest transition system which is bisimilar to it this is its canonical transition graph which is the result of rst removing any states which are not reachable from the ro ot and then identifying bisimilar states using quotienting An alternative p ersp ective on bisimulation closure is from the viewp oint of prop erties of transition systems Prop erties whose transition systems are bisim ulation closed are said to b e bisimulation invariant Over ro oted transition graphs prop erty is bisimulation invariant provided that if T j and T T then T j By T j we mean that is true of the transition graph T On the whole counting prop erties are not bisimulation invariant for example has states or has an even numb er of states In contrast temp oral prop erties are bisimula tion invariant for instance will eventually do an atransition or is never able to do a btransition Other prop erties such as has an Hamiltonian circuit or is colourable are also not bisimulation invariant Later we shall b e interested in parameterised prop erties that is prop erties of arbitrary arity We say that an nary prop erty x x on transition systems is bisimulation invariant 1 n provided that if T j s s and t t are states of T and 1 n 1 n t s for all i i n then T j t t i i 1 n By T j s s we mean that is true of the states s s of T 1 n 1 n An example of a prop erty which is not bisimulation invariant is x x is a 1 n cycle and an example of a bisimulation invariant prop erty is x is language 1 equivalent to x 2 The notions of bismulation closure and invariance have app eared indep end ently in a variety of contexts see for instance Caucals hierarchy Bisimulation equivalence is a very ne equivalence b etween states An interest ing line of enquiry is to reconsider classical results in automata theory replacing language equivalence with bismulation equivalence These results concern den ability closure prop erties and decidabilityundecidability Grammars can b e viewed as generators of transition systems Let b e a nite family of nonterminals and assume that A is a nite set of terminals A a basic transition has the form where and a A A state is then any memb er of and the transition relations on states are dened as the least relations closed under basic transitions and the following prex rule a a PRE if then Given a state we can dene its ro oted transition system whose states are just the ones reachable from In the table b elow is a Caucal hierarchy of transition graph descriptions according to how the family of basic transitions is sp ecied In each case we assume a nite family of rules Typ e captures nitestate graphs Typ e cap 1 in fact tures contextfree grammars in Greibach normal form and Typ e 2 captures pushdown automata For Typ e and b elow this means that in each case there are nitely many basic transitions In the other cases R and R are 1 2 a regular expressions over the alphab et The idea is that each rule R 1 a stands for the p ossibly innite family of basic transitions f R g 1 a a R stands for the family f R and R g For and R 2 1 2 1 a instance a Typ e rule of the form X Y Y includes for each n the a n basic transition X Y Y Basic Transitions a R Typ e R 2 1 a Typ e R 1 a Typ e a 1 Typ e where jj and j j 2 a Typ e X a a Typ e X Y or X This hierarchy is implicit in Caucals work on understanding contextfree graphs and understanding when the monadic secondorder theory of graphs is decidable With resp ect to language equivalence the hierarchy collapses to just two levels the regular and the context
Load more

Recommended publications
  • Bisimulations and the Standard Translation
    1 Philosophy 244: Bisimulations and the Standard Translation Guest Lecture:Cosmo Grant,October 26, 2016 The Plan Sources: Modal Logic for Open Minds, by Johan van Benthem, and Modal Logic, by Blackburn, de Rijke and Venema. We’ll look at some more metatheory of propositional modal logic. Our They’re superb—check them out! previous work on propositional modal logic focused on frames. Today we’ll focus on models. The central new concept is that of a bisimulation between two models. We’ll use bisimulations to prove some results Bisimulations, under the name p- about the expressive power of modal logic. And bisimulations turn out relations, were introduced by Johan van Benthem in his PhD thesis. to be exactly what we need to connect modal logic with first-order logic. Here’s the plan: 1. Bisimulations: definition and examples 2. Bisimulations and modal equivalence 3. Bisimulations and expressive power 4. Bisimulations and the Standard Translation Bisimulations: definition and examples Definition 1 (Modal equivalence). Let’s say that (M, w), (M0, w0) are 0 0 0 0 modally equivalent just if for every formula f, (M, w) j= f iff (M , w ) j= notation: (M, w) ! (M , w ) f. 0 0 When are (M, w), (M , w ) modally equivalent? Well, if there is an iso- A bijection f : W ! W0 is an isomorphism 0 morphism f : W ! W0 then (M, w), (M0, w0) will certainly be modally from M to M just if for all w, v in M: ( ) equivalent. But an isomorphism is over-kill: it guarantees modal 1. w and f w satisfy the same proposi- tion letters equivalence but in a heavy-handed way.
    [Show full text]
  • Profunctors, Open Maps and Bisimulation
    BRICS RS-04-22 Cattani & Winskel: Profunctors, Open Maps and Bisimulation BRICS Basic Research in Computer Science Profunctors, Open Maps and Bisimulation Gian Luca Cattani Glynn Winskel BRICS Report Series RS-04-22 ISSN 0909-0878 October 2004 Copyright c 2004, Gian Luca Cattani & Glynn Winskel. BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory RS/04/22/ Profunctors, Open Maps and Bisimulation∗ Gian Luca Cattani DS Data Systems S.p.A., Via Ugozzolo 121/A, I-43100 Parma, Italy. Email: [email protected]. Glynn Winskel University of Cambridge Computer Laboratory, Cambridge CB3 0FD, England. Email: [email protected]. October 2004 Abstract This paper studies fundamental connections between profunctors (i.e., dis- tributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells.
    [Show full text]
  • On the Origins of Bisimulation and Coinduction
    On the Origins of Bisimulation and Coinduction DAVIDE SANGIORGI University of Bologna, Italy The origins of bisimulation and bisimilarity are examined, in the three fields where they have been independently discovered: Computer Science, Philosophical Logic (precisely, Modal Logic), Set Theory. Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed points, in particular greatest fixed points. Therefore also the appearance of coinduction and fixed points is discussed, though in this case only within Computer Science. The paper ends with some historical remarks on the main fixed-point theorems (such as Knaster-Tarski) that underpin the fixed-point theory presented. Categories and Subject Descriptors: F.4.1 [Mathematical logic and formal languages]: Math- ematical Logic—Computational logic; Modal logic; Set theory; F.4.0 [Mathematical logic and formal languages]: General General Terms: Theory, Verification Additional Key Words and Phrases: Bisimulation, coinduction, fixed points, greatest fixed points, history Contents 1 Introduction 112 2 Background 114 2.1 Bisimulation................................ 114 2.2 Approximants of bisimilarity . 116 2.3 Coinduction................................116 3 Bisimulation in Modal Logic 119 3.1 Modallogics................................119 3.2 Fromhomomorphismtop-morphism . 120 3.3 JohanvanBenthem ........................... 122 3.4 Discussion.................................125 4 Bisimulation in Computer Science 125 4.1 Algebraictheoryofautomata . 125 4.2 RobinMilner ...............................128 Author’s address: Dipartimento di Scienze dell’Informazione Universita’ di Bologna Mura Anteo Zamboni, 7 40126 Bologna, ITALY. Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc.
    [Show full text]
  • A Uniform (Bi-)Simulation-Based Framework for Reducing Tree Automata
    Electronic Notes in Theoretical Computer Science 251 (2009) 27–48 www.elsevier.com/locate/entcs A Uniform (Bi-)Simulation-Based Framework for Reducing Tree Automata Parosh A. Abdulla1 Department of Information Technology University of Uppsala Uppsala, Sweden Luk´aˇsHol´ık5 ,2 Faculty of Information Technology Brno University of Technology Brno, Czech Republic Lisa Kaati3 Department of Information Technology University of Uppsala Uppsala, Sweden Tom´aˇsVojnar5 ,4 Faculty of Information Technology Brno University of Technology Brno, Czech Republic Abstract In this paper, we address the problem of reducing the size of nondeterministic (bottom-up) tree automata. We propose a uniform framework that allows for combining various upward and downward bisimulation and simulation relations in order to obtain a language-preserving combined relation suitable for reducing tree automata without a need to determinise them. The framework generalises and extends several previous works and provides a broad spectrum of different relations yielding a possibility of a fine choice between the amount of reduction and the computational demands. We, moreover, provide a significantly improved way of computing the various combined (bi-)simulation relations. We analyse properties of the considered relations both theoretically as well as through a series of experiments. Keywords: tree automata, bisimulation, simulation, combined relations, size reduction 1571-0661/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.entcs.2009.08.026 28 P.A. Abdulla et al. / Electronic Notes in Theoretical Computer Science 251 (2009) 27–48 1 Introduction Finite tree automata are a natural generalisation of word automata. Since trees (or terms) appear in many areas of computer science and engineering, tree au- tomata are quite broadly applicable—including, for instance, applications in XML manipulation, natural language processing, or formal verification.
    [Show full text]
  • Brzozowski's and Up-To Algorithms for Must Testing
    Erschienen in: Programming Languages and Systems : 11th Asian Symposium, APLAS 2013, Melbourne, VIC, Australia, December 9-11, 2013, Proceedings / Shan, Chung-chieh (Hrsg.). - Cham : Springer, 2013. - (Lecture Notes in Computer Science ; 8301). - S. 1-16 https://dx.doi.org/10.1007/978-3-319-03542-0_1 Brzozowski’s and Up-To Algorithms for Must Testing Filippo Bonchi1, Georgiana Caltais2, Damien Pous1, and Alexandra Silva3, 1 ENS Lyon, U. de Lyon, CNRS, INRIA, UCBL 2 Reykjavik University 3 Radboud University Nijmegen Abstract. Checking language equivalence (or inclusion) of finite au- tomata is a classical problem in Computer Science, which has recently received a renewed interest and found novel and more effective solu- tions, such as approaches based on antichains or bisimulations up-to. Several notions of equivalence (or preorder) have been proposed for the analysis of concurrent systems. Usually, the problem of checking these equivalences is reduced to checking bisimilarity. In this paper, we take a different approach and propose to adapt algorithms for language equiva- lence to check one prime equivalence in concurrency theory, must testing semantics. To achieve this transfer of technology from language to must semantics, we take a coalgebraic outlook at the problem. 1 Introduction Determining whether two systems exhibit the same behavior under a given no- tion of equivalence is a recurring problem in different areas from Computer Science, from compiler analysis, to program verification, to concurrency theory. A widely accepted notion of equivalence is that two systems are equivalent if they behave the same when placed in the same context. We will focus on the equivalence problem in the context of concurrency theory and process calculi.
    [Show full text]
  • Logical Characterization of Bisimulation Metrics
    Logical Characterization of Bisimulation Metrics Valentina Castiglioni Daniel Gebler Simone Tini University of Insubria (IT) VU University Amsterdam (NL) University of Insubria (IT) [email protected] [email protected] [email protected] Bisimulation metrics provide a robust and accurate approach to study the behavior of nondetermin- istic probabilistic processes. In this paper, we propose a logical characterization of bisimulation metrics based on a simple probabilistic variant of the Hennessy-Milner logic. Our approach is based on the novel notions of mimicking formulae and distance between formulae. The former are a weak version of the well known characteristic formulae and allow us to characterize also (ready) proba- bilistic simulation and probabilistic bisimilarity. The latter is a 1-bounded pseudometric on formulae that mirrors the Hausdorff and Kantorovich lifting the defining bisimilarity pseudometric. We show that the distance between two processes equals the distance between their own mimicking formulae. 1 Introduction To verify whether the implementation proposed for a reactive concurrent system (or process) conforms the properties and behavior entailed by its specification, two main approaches have been proposed: equiv- alence checking [1] and model checking [16]. The former is used when both the implementation and specification of the system are modeled by a labeled transition system (LTS) [45] and it consists in veri- fying whether each of the observations enabled by the implementation is allowed by the specification. To this aim, several notions of behavioral equivalence, identifying those processes which are indistinguish- able to external observers, and of behavioral preorder have been proposed. Among those equivalences, the most successfully employed is the bisimulation equivalence [53].
    [Show full text]
  • Logical Characterizations of Simulation and Bisimulation for Fuzzy Transition Systems
    Logical Characterizations of Simulation and Bisimulation for Fuzzy Transition Systems Hengyang Wu1∗ Yuxin Deng2 1Information Engineer College, Hangzhou Dianzi University Hangzhou 310018, P.R. China 2Shanghai Key Laboratory of Trustworthy Computing, East China Normal University Shanghai 200062, P.R. China Abstract Simulations and bisimulations are known to be useful for abstracting and comparing formal systems, and they have recently been introduced into fuzzy systems. In this study, we provide sound and complete logical characterizations for simulation and bisimulation, which are defined over fuzzy labeled transition systems via two variants of the Hennessy-Milner Logic. The logic for character- izing fuzzy simulation has neither negation nor disjunction, which is very differ- ent from the well-known logical characterizations of probabilistic simulations, although the completeness proofs of our characterization results are inspired by relevant research in probabilistic concurrency theory. The logic for characteriz- ing fuzzy bisimulation also deviates from that for probabilistic bisimulations. Keywords: Bisimulation, Completeness, Fuzzy labeled transition system, Logical characterization, Simulation, Soundness. 1 Introduction The analysis of fuzzy systems has been the subject of active research during the last 60 years and many formalisms have been proposed for modeling them, ∗Corresponding author. Email address: wuhengy−[email protected], [email protected]. 1 including fuzzy automata (e.g., see [2, 3, 7, 28, 29, 31, 36, 38, 44]), fuzzy Petri nets [40], fuzzy Markov processes [4], and fuzzy discrete event systems [30, 37, 39]. Recently, a new formal model for fuzzy systems called fuzzy labeled tran- sition systems (FLTSs) was proposed [6, 18, 24]. FLTSs are a natural gener- alization of the classical labeled transition systems in computer science, where after performing some action, a system evolves from one state into a fuzzy set of successor states instead of a unique state.
    [Show full text]
  • Bisimulation Can't Be Traced
    Bisimulation Can’t Be Traced BARD BLOOM Cornell Urzuersity, Ithaca, New York SORIN ISTRAIL Sandta National Laboratories, Albuquerque, New Meuco AND ALBERT R. MEYER Massachusetts Institute of Technology, Cambtidge, Massachusetts Abstract. In the concurrent language CCS, hvo programs are considered the same if they are bzsimilar. Several years and many researchers have demonstrated that the theory of bisimulation is mathematically appealing and useful in practice. However, bisimulation makes too many distinc- tions between programs. We consider the problem of adding operations to CCS to make bisimulation fully abstract. We define the class of GSOS operations, generalizing the style and technical advantages of CCS operations. We characterize GSOS congruence in as a bisimulation- like relation called ready simulation. Bisimulation is strictly finer than ready simulation, and hence not a congruence for any GSOS language. Categories and Subject Descriptors: D.3. 1 [Programming Languages]: Formal Definitions and Theo~—semarztics; D.3.3 [Programming Languages]: Language Constructs and Features—corz- current programming structures; F.3.2 [Logics and Meanings of Programs]: Semantics of Program- ming Languages—algebraic approaches to semarztics; operational semantics; 1.6.2 [Simulation and Modeling]: Simulation Languages General Terms: Languages, Theory, Verification Additional Key Words and Phrases: Bisimulation, structural operational semantics, process algebra, CCS The work of B. Bloom was supported by a National Science Foundation (NSF) Fellowship, also NSF Grant No. 85-1 1190-DCR and Office of Naval Research (ONR) grant no. NOO014-83-K-0125. The work of S. Istrail was supported by NSF Grant CCR 88-1174. This work performed at Sandia National Laboratories supported by the U.S.
    [Show full text]
  • A Parallel Relation-Based Algorithm for Symbolic Bisimulation Minimization
    Leiden University Computer Science A Parallel Relation-Based Algorithm for Symbolic Bisimulation Minimization Name: Richard Huybers Date: 26/10/2018 1st supervisor: Dr. A.W. Laarman 2nd supervisor: Prof.dr. H.A.G. Wijshoff MASTER'S THESIS Leiden Institute of Advanced Computer Science (LIACS) Leiden University Niels Bohrweg 1 2333 CA Leiden The Netherlands A Parallel Relation-Based Algorithm for Symbolic Bisimulation Minimization Richard Huybers Abstract Symbolic computation using BDDs and bisimulation minimization are alternative ways to cope with the state space explosion in model checking. The combination of both techniques opens up many parameters that can be tweaked for further optimization. Most importantly, the bisimulation can either be represented as equivalence classes or as a relation. While recent work argues that storing partitions is more efficient, we show that the relation-based approach is preferable. We do so by presenting a relation-based minimization algorithm based on new coarse-grained BDD operations. These coarse-grained BDD operations are derived directly from their definition using Shannon decomposition, guaranteeing correctness. The implementation demonstrates that the relational approach uses two orders of magnitude fewer memory, performs better and has better parallel scalability. i ii Contents Abstract i 1 Introduction 1 2 Background 4 2.1 Model checking . 4 2.2 Labelled transition systems . 6 2.3 Bisimulation . 7 2.4 Binary decision diagrams . 8 2.4.1 Definition . 9 2.4.2 Operations . 12 2.4.3 Variable Ordering . 16 3 Symbolic Bisimulation 18 3.1 A relation-based symbolic bisimulation algorithm . 18 3.2 Computing pairs of non-bisimilar states . 20 3.3 Computing the converse of a relation .
    [Show full text]
  • Pure Bigraphs
    Technical Report UCAM-CL-TR-614 ISSN 1476-2986 Number 614 Computer Laboratory Pure bigraphs Robin Milner January 2005 15 JJ Thomson Avenue Cambridge CB3 0FD United Kingdom phone +44 1223 763500 http://www.cl.cam.ac.uk/ c 2005 Robin Milner Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet: http://www.cl.cam.ac.uk/TechReports/ ISSN 1476-2986 Pure bigraphs Robin Milner University of Cambridge, Computer Laboratory, JJ Thomson Avenue, Cambridge CB3 0FD, UK Abstract Bigraphs are graphs whose nodes may be nested, represent- ing locality, independently of the edges connecting them. They may be equipped with reaction rules, forming a bigraphical reactive system (Brs) in which bigraphs can reconfigure themselves. Brss aim to unify process calculi, and to model applications —such as pervasive computing— where locality and mobility are prominent. The paper is devoted to the theory of pure bigraphs, which underlie various more refined forms. It begins by de- veloping a more abstract structure, a wide reactive system (Wrs), of which a Brs is an instance; in this context, labelled transitions are defined in such a way that the induced bisimilarity is a congruence. This work is then specialised to Brss, whose graphical structure allows many refinements of the dynamic theory. Elsewhere it is shown that be- havioural analysis for Petri nets, π-calculus and mobile ambients can all be recovered in the uniform framework of bigraphs. The latter part of the paper emphasizes the parts of bigraphical theory that are common to these applications, especially the treatment of dynamics via labelled transitions.
    [Show full text]
  • A Gentle Introduction to Process Algebras?
    A gentle introduction to Process Algebras? Rocco De Nicola IMT - Institute for Advanced Studies Lucca Piazza S. Ponziano, 6 - I-55100 Lucca - Italy [email protected] ABSTRACT: Process Algebras are mathematically rigorous languages with well defined semantics that permit describing and verifying properties of concurrent communicating systems. They can be seen as models of processes, regarded as agents that act and in- teract continuously with other similar agents and with their common environment. The agents may be real-world objects (even people), or they may be artifacts, embodied per- haps in computer hardware or software systems. Many different approaches (operational, denotational, algebraic) are taken for describing the meaning of processes. However, the operational approach is the reference one. By relying on the so called Structural Opera- tional Semantics (SOS), labelled transition systems are built and composed by using the different operators of the many different process algebras. Behavioral equivalences are used to abstract from unwanted details and identify those labelled transition systems that react “similarly” to external experiments. Due to the large number of properties which may be relevant in the analysis of concurrent systems, many different theories of equivalences have been proposed in the literature. The main con- tenders consider those systems equivalent that (i) perform the same sequences of actions, or (ii) perform the same sequences of actions and after each sequence are ready to accept the same sets of actions, or (iii) perform the same sequences of actions and after each sequence exhibit, recursively, the same behavior. This approach leads to many different equivalences that preserve significantly different properties of systems.
    [Show full text]
  • Coalgebra: Basic Concepts
    Coalgebra: Basic Concepts Paul Blain Levy University of Birmingham 1 Transition Systems Integer Ltd. makes integer I/O machines, which have a button and a display. You press the button and it prints an integer. You press the button again and it prints another integer. A machine has a variety of internal states. When you press the button, it's the current state that determines what integer gets printed, and what the new state will be (it could be a different state, or it could be the same state). A machine is described by { a set X (the set of states) { a function ζ : X −! Z × X (what happens when you press the button) { the current state x0 2 X Exercise 1. Machine number 392 has Z × Z as set of states. The behaviour function is ζ : hn; n0i 7! hn + n0; hn0 + 1; n − 2ii. The cur- rent state is h4; 6i. What is printed when you press the button three times? A rival company Integer And Boolean Inc. makes machines with three buttons and a display. If you press the red button or the green button it prints an integer, but if you press the bright pink button it prints a boolean. Such a machine is described by a pair (X; ζ), where { a set X (the set of states) { a function ζred : X −! Z × X (what happens when you press the red button) { a function ζgreen : X −! Z × X (what happens when you press the green button) 2 Paul Blain Levy { a function ζbright pink : X −! B × X (what happens when you press the bright pink button).
    [Show full text]