Table Of Content

John Derrick · Eerke Boiten Refinement Semantics, Languages and Applications fi Re nement John Derrick Eerke Boiten (cid:129) fi Re nement Semantics, Languages and Applications 123 JohnDerrick Eerke Boiten Department ofComputer Science Schoolof Computer Science University of Sheffield andInformatics Sheffield,UK De MontfortUniversity Leicester, Leicestershire, UK ISBN978-3-319-92709-1 ISBN978-3-319-92711-4 (eBook) https://doi.org/10.1007/978-3-319-92711-4 LibraryofCongressControlNumber:2018949887 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors, and the editorsare safeto assume that the adviceand informationin this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To our families: Michelle and Bea; Gwen, Sara, Ivor, Rowan and Sam. Contents Part I Semantics 1 Labeled Transition Systems and Their Refinement. . . . . . . . . . . . . 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Labelled Transition Systems – A Simple Model of Computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Trace Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Completed Trace Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Failures Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Readiness Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7 Failure Trace Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.8 Ready Trace Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.9 Conformance and Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.10 Data and Infinite Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.10.1 Adding Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.10.2 Infinite Trace Refinement . . . . . . . . . . . . . . . . . . . . . . 22 1.10.3 Infinite Completed Trace Refinement. . . . . . . . . . . . . . 23 1.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.12 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Automata - Introducing Simulations. . . . . . . . . . . . . . . . . . . . . . . . 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Refinement and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 Forward and Backward Simulations. . . . . . . . . . . . . . . 30 2.2.2 Completeness Results. . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Bisimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 vii viii Contents 3 Simple State-Based Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 A Basic Model with Refinement . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Termination and Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 More than One Observation. . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5 Towards Total Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Observations Including Initial States . . . . . . . . . . . . . . . . . . . . 44 3.7 State Abstraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.8 State Variables and Observation . . . . . . . . . . . . . . . . . . . . . . . 47 3.9 Programs and Labeled Transitions . . . . . . . . . . . . . . . . . . . . . . 48 3.10 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 A Relational View of Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Relational Data Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Relational Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4 From a Theory of Total Relational Refinement to Partial Relational Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.1 Partial Relations and Their Totalisation . . . . . . . . . . . . 59 4.4.2 Simulation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.3 Completeness of Partial Relational Simulations . . . . . . 63 4.4.4 A Fully Partial Relational Theory . . . . . . . . . . . . . . . . 65 4.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Perspicuity, Divergence, and Internal Operations. . . . . . . . . . . . . . 69 5.1 Perspicuous Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Error Behaviour: Catastrophic or Not? . . . . . . . . . . . . . . . . . . . 72 5.3 Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.4 Internal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.4.1 Internal Operations and Choice in LTS . . . . . . . . . . . . 76 5.4.2 Internal Actions in a Relational Context . . . . . . . . . . . 78 5.4.3 Divergence from Internal Operations . . . . . . . . . . . . . . 79 5.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Part II Refinement in Specification Languages 6 Process Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.1 CSP - The Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.1.1 Sequential Operators. . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.1.2 Building Concurrent Specifications . . . . . . . . . . . . . . . 89 Contents ix 6.2 CSP - The Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2.1 The CSP Failures Model. . . . . . . . . . . . . . . . . . . . . . . 96 6.2.2 The CSP Failures, Divergences and Infinite Traces Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.3 CSP - Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3.1 Trace Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3.2 Stable Failures Refinement . . . . . . . . . . . . . . . . . . . . . 106 6.3.3 FDI Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4 LOTOS - Language, Semantics and Refinement . . . . . . . . . . . . 108 6.4.1 The LOTOS Language . . . . . . . . . . . . . . . . . . . . . . . . 108 6.4.2 LOTOS Semantics and Refinement . . . . . . . . . . . . . . . 110 6.5 CCS - Language, Semantics and Refinement . . . . . . . . . . . . . . 114 6.5.1 The CCS Language . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.5.2 CCS Semantics and Refinement . . . . . . . . . . . . . . . . . 116 6.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7 State-Based Languages: Z and B . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1 Z - The Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1.1 Using Schemas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7.1.2 The Z Schema Calculus . . . . . . . . . . . . . . . . . . . . . . . 124 7.2 Z – Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.2.1 The Relational Semantics of a Z Specification . . . . . . . 126 7.2.2 Data Refinement and Simulations for Z. . . . . . . . . . . . 127 7.2.3 Z Refinement in Practice . . . . . . . . . . . . . . . . . . . . . . 130 7.3 The B-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.3.1 Machine Consistency . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.4 Refinement in the B-Method . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.4.1 Proof Obligations for Refinement . . . . . . . . . . . . . . . . 141 7.4.2 Implementation Machines . . . . . . . . . . . . . . . . . . . . . . 143 7.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8 State-Based Languages: Event-B and ASM. . . . . . . . . . . . . . . . . . . 149 8.1 Event-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.1.1 Specifying Machines in Event-B . . . . . . . . . . . . . . . . . 150 8.1.2 Proof Obligations for Event-B Machines . . . . . . . . . . . 151 8.2 Refinement in Event-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.2.1 A Relational Semantics for Event-B . . . . . . . . . . . . . . 154 8.2.2 Data Refinement and Simulations for Event-B . . . . . . . 158 8.3 ASM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.3.1 ASM Semantics and ASM Refinement . . . . . . . . . . . . 169 8.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 x Contents Part III Relating Notions of Refinement 9 Relational Concurrent Refinement . . . . . . . . . . . . . . . . . . . . . . . . . 179 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 9.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.3 Relating Process Algebraic and Relational Refinement . . . . . . . 181 9.3.1 Trace Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.3.2 Completed Trace Refinement . . . . . . . . . . . . . . . . . . . 185 9.3.3 Failure Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . 188 9.3.4 Failure Trace Refinement . . . . . . . . . . . . . . . . . . . . . . 189 9.3.5 Extension and Conformance . . . . . . . . . . . . . . . . . . . . 190 9.4 Relating Data Refinement to Process Algebraic Refinement. . . . 192 9.4.1 Non-blocking Data Refinement and the Traces-Divergences Semantics. . . . . . . . . . . . . . . . . . . 194 9.4.2 Blocking Data Refinement and the Singleton Failures Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.5 Relating Automata and Relational Refinement . . . . . . . . . . . . . 196 9.5.1 IO Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.6 Internal Events and Divergence . . . . . . . . . . . . . . . . . . . . . . . . 201 9.7 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10 Relating Data Refinement and Failures-Divergences Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.2 The Basic Relational Embedding. . . . . . . . . . . . . . . . . . . . . . . 207 10.3 Dealing with Input and Output . . . . . . . . . . . . . . . . . . . . . . . . 211 10.3.1 Defining the Extended Finalisation . . . . . . . . . . . . . . . 212 10.3.2 Deriving the New Simulation Rules. . . . . . . . . . . . . . . 214 10.3.3 An Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 10.4 Summary of Simulation Conditions . . . . . . . . . . . . . . . . . . . . . 226 10.4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 10.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11 Process Data Types - A Fully General Model of Concurrent Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 11.2 A Relational ADT with Divergence and Blocking . . . . . . . . . . 237 11.2.1 Forward Simulation for Process Data Types. . . . . . . . . 239 11.2.2 Backward Simulation for Process Data Types . . . . . . . 241 11.2.3 Simulations on Process Data Types and Basic Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Contents xi 11.3 Using Process Data Types – Failures-Divergences Refinement with Internal Operations and Divergence . . . . . . . . 243 11.3.1 Embedding a Basic Data Type with Internal Operations into a Process Data Type. . . . . . . . . . . . . . 243 11.3.2 Correctness of the Embedding: Blocking Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 11.3.3 Simulations in the Blocking Approach. . . . . . . . . . . . . 248 11.3.4 The Non-blocking Approach. . . . . . . . . . . . . . . . . . . . 251 11.4 Adding in a Consideration of Outputs . . . . . . . . . . . . . . . . . . . 254 11.4.1 Refinement Conditions for Output Embeddings . . . . . . 254 11.4.2 Outputs in the Blocking Approach . . . . . . . . . . . . . . . 255 11.4.3 Outputs in the Non-blocking Approach . . . . . . . . . . . . 258 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 11.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 12 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

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