Programming Language Semantics A Rewriting Approach Grigore Ros, u University of Illinois at Urbana-Champaign 2 Contents 1 Introduction 7 2 Background and Preliminaries 15 2.1 Mathematical Background . 16 2.1.1 Functions . 16 2.1.2 Partial Functions . 17 2.1.3 Context-Free Grammars and the Mixfix Algebraic Notation . 18 2.1.4 Basic Term Rewriting . 19 2.1.5 On Logics and Logic Deduction . 20 2.2 Basic Computability Elements . 22 2.2.1 Turing Machines . 22 2.2.2 Universal Machines, the Halting Problem, and Decision Problems . 24 2.2.3 The Arithmetic Hierarchy . 25 2.2.4 Notes . 26 2.2.5 Exercises . 26 2.3 Propositional Logic . 27 2.3.1 Syntax and Semantics . 27 2.3.2 Proof System and Soundness . 28 2.3.3 Consistency and Model Existence . 29 2.3.4 Completeness and Compactness . 30 2.3.5 Notes . 30 2.3.6 Exercises . 31 2.4 Equational Logic . 32 2.4.1 Signatures, Algebras, Equations, Satisfaction . 32 2.4.2 Proof System and Soundness . 32 2.4.3 Computation as Equational Deduction . 33 2.4.4 Initial Algebra . 36 2.4.5 Completeness . 37 2.4.6 Defining Data-Types Equationally . 42 2.4.7 Notes . 49 2.4.8 Exercises . 49 2.5 Rewrite logic . 50 2.5.1 Equations versus Rewrite Rules . 50 2.5.2 Proof System and Soundness . 55 2.5.3 Computation as Rewrite Logic Deduction . 58 3 2.5.4 Completeness . 61 2.5.5 Rewrite Logic as a Universal Logical Formalism . 61 2.5.6 Maude: A High Performance Rewrite Logic System . 62 2.5.7 Exercises . 82 2.5.8 Notes . 84 2.6 Complete Partial Orders and the Fixed-Point Theorem . 85 2.6.1 Posets, Upper Bounds and Least Upper Bounds . 85 2.6.2 Complete Partial Orders . 86 2.6.3 Monotone and Continuous Functions . 87 2.6.4 The Fixed-Point Theorem . 89 2.6.5 Equational Framework for Fixed-Points . 90 2.6.6 Exercises . 92 2.6.7 Notes . 92 2.7 Category Theory . 95 3 Executable Semantics Approaches 101 3.1 IMP: A Simple Imperative Language . 109 3.1.1 IMP Syntax . 109 3.1.2 IMP State . 114 3.1.3 Notes . 115 3.2 Big-Step Structural Operational Semantics (Big-Step SOS) . 117 3.2.1 IMP Configurations for Big-Step SOS . 118 3.2.2 The Big-Step SOS Rules of IMP . 119 3.2.3 Big-Step SOS in Rewrite Logic . 123 3.2.4 Defining a Type System for IMP Using Big-Step SOS . 130 3.2.5 Notes . 135 3.2.6 Exercises . 137 3.3 Small-Step Structural Operational Semantics (Small-Step SOS) . 139 3.3.1 IMP Configurations for Small-Step SOS . 140 3.3.2 The Small-Step SOS Rules of IMP . 141 3.3.3 Small-Step SOS in Rewrite Logic . 147 3.3.4 Notes . 153 3.3.5 Exercises . 154 3.4 Denotational Semantics . 157 3.4.1 The Denotational Semantics of IMP . 162 3.4.2 Denotational Semantics in Equational/Rewrite Logic . 162 3.4.3 Notes . 165 3.4.4 Exercises . 168 3.5 IMP++: IMP Extended with Several Features . 171 3.5.1 Adding Variable Increment . 172 3.5.2 Adding Input/Output . 176 3.5.3 Adding Abrupt Termination . 181 3.5.4 Adding Dynamic Threads . 188 3.5.5 Adding Local Variables . 193 3.5.6 Putting Them All Together: First Attempt . 196 3.5.7 Putting Them All Together: Second Attempt . 204 4 3.5.8 Notes . 204 3.5.9 Exercises . 205 3.6 Modular Structural Operational Semantics (MSOS) . 213 3.6.1 The MSOS of IMP . 218 3.6.2 The MSOS of IMP++ ................................. 218 3.6.3 MSOS in Rewrite Logic . 224 3.6.4 Notes . 230 3.6.5 Exercises . 230 3.7 Reduction Semantics with Evaluation Contexts . 233 3.7.1 The Reduction Semantics with Evaluation Contexts of IMP . 236 3.7.2 The Reduction Semantics with Evaluation Contexts of IMP++ ............ 239 3.7.3 Reduction Semantics with Evaluation Contexts in Rewrite Logic . 242 3.7.4 Notes . 263 3.7.5 Exercises . 264 3.8 The Chemical Abstract Machine (CHAM) . 269 3.8.1 The CHAM of IMP . 272 3.8.2 The CHAM of IMP++ ................................. 278 3.8.3 CHAM in Rewrite Logic . 281 3.8.4 Notes . 291 3.8.5 Exercises . 293 3.9 Advanced Denotational Semantics: Continuations, Monads and Powerdomains . 295 3.10 Discussion: Advantages, Disadvantages and Comparisons . 296 3.11 Existing Approaches and Their Limitations for Programming Language Semantics . 296 3.11.1 Structural Operational Semantics (SOS) Frameworks . 296 3.11.2 Rewriting Frameworks . ..