Verification of Systems and Circuits Using LOTOS, Petri Nets, and CCS

1. Introduction 1

1.1 Event-Based Approach 2

1.2 Event-Based Systems 2

1.3 Types of Verification 2

1.4 Toolsets Used 3

1.5 Level-Based Approach 3

1.6 Overview of the Book 3

1.7 References 5

2. Processes 7

2.1 Introduction 7

2.2 Examples of Processes and Basic Concepts 7

2.3 About Prefixing 10

2.4 Process Graphs 10

2.5 Choice Operator 11

2.6 Another Process Example 13

2.7 Equivalences 13

2.7.1 Strong Equivalence 13

2.7.2 Observation Equivalence 14

2.7.3 Some Additional Laws 15

2.8 Labeled Transition Systems (LTSs) 15

2.9 Parallel Operators 16

2.9.1 Parallel Composition 16

2.9.2 Synchronization Operator k (Blot Version) 16

2.9.3 Examples of Parallel Compositions 17

2.9.4 More Laws 17

2.9.5 Sample Proof 18

2.9.6 Interleaving Operator kj 18

2.10 Sequential Composition 18

2.11 Further Reading 19

2.12 Selected Solutions 20

2.13 References 21

3. From Digital Hardware to Processes 23

3.1 The C-Element 23

3.1.1 The 2-Input CEL-Circuit 23

3.1.2 The 3-Input CEL-Circuit 25

3.1.3 The 4-Input CEL-Circuit 26

3.2 The XOR-Gate 26

3.2.1 The 2-Input XOR-Gate 26

3.2.2 The 3-Input XOR-Gate 27

3.3 TOGGLES 29

3.4 Modulo-N Transition Counters 30

3.4.1 Modulo-N Transition Counter Specification 30

3.4.2 Modulo-N Transition Counter Implementations 30

3.5 Modular Networks 31

3.6 Propositional Logic: A Review of Known Concepts 33

3.6.1 Logical Operators 34

3.6.2 Proving Logical Equivalences 35

3.6.3 Tautologies and the EQUIV Operator 36

3.7 Selected Solutions 36

3.8 References 37

4. Introducing LOTOS 39

4.1 From Blot to Basic LOTOS 39

4.1.1 Recursion 40

4.2 Some Semantics 41

4.3 From LTS to LOTOS 42

4.4 Comparing Parallel Operators 43

4.5 Sequential Composition 44

4.6 Hiding 44

4.7 Equivalences and Preorders 44

4.8 About CADP 45

4.8.1 Getting Started with CADP 45

4.8.2 Verifying Equivalences and Preorders Using CADP 46

4.8.3 Generating LTS of Choice Using CADP 47

4.8.4 Generating LTS of Recursion Using CADP 48

4.9 Full LOTOS—An Introduction 49

4.9.1 The Full-LOTOS NOT-Gate Example 49

4.9.2 The Non-Terminating NOT-Gate 51

4.9.3 The Max Specifications 52

4.10 The Regular Mu-Calculus (RMC) 53

4.10.1 Introducing RMC by Examples 53

4.11 Further Reading 55

4.12 Selected Solutions 56

4.13 References 57

5. Introducing Petri Nets 59

5.1 About Petri Nets 59

5.1.1 Petri Graphs and Petri Nets 59

5.1.2 Enabling and Firing 60

5.1.3 Another Definition of Petri Nets 61

5.2 About Languages 61

5.3 About PETRIFY 62

5.4 Illustrating Petri Nets 64

5.5 Labeled Nets 66

5.6 Bounded Nets 68

5.7 Observation Equivalence of LPNs 70

5.8 From Blot to Petri Nets 70

5.9 Liveness and Persistence 72

5.10 Simple Reduction Rules 72

5.11 Marked Graphs 74

5.12 A Simple Net Algebra 75

5.12.1 The Prefix Operator 75

5.12.2 The Choice Operator 77

5.12.3 The Star Operator 77

5.12.4 Parallel Compositions 79

5.13 Arc-Weighted Nets 80

5.13.1 Enabling and Firing in Arc-Weighted Nets 80

5.13.2 Arc-Weighted Versus Non-Labeled Nets 82

5.14 Readers–Writers System 83

5.14.1 A Readers–Writers System Net Representation 83

5.14.2 Verification of a Readers–Writers System 84

5.15 Inhibitor Nets 85

5.15.1 Introduction to Inhibitor Nets 85

5.15.2 The Expressive Power of Inhibitor Nets 85

5.16 True Concurrency 86

5.16.1 The Pi-Language 87

5.16.2 Pi-Equivalence 87

5.16.3 Concurrency-Preserving Synthesis 88

5.17 Further Reading 89

5.18 Selected Solutions 89

5.19 References 93

6. Introducing CCS 95

6.1 About CCS 95

6.2 Operators ‘Prefix’ and ‘Sum’ 95

6.2.1 Semantics 96

6.3 Recursion 97

6.3.1 Semantics 97

6.4 Concurrency Operator 97

6.5 Equivalences 98

6.6 Restriction 98

6.7 CTL 99

6.7.1 Introducing CTL 99

6.8 The Concurrency Workbench (CWB) 100

6.8.1 The ‘sim’ and ‘compile’ Commands 100

6.8.2 Checking Equivalences 102

6.8.3 Checking Restrictions 103

6.8.4 HML Formulas 103

6.8.5 Equivalences—Counterexamples 104

6.8.6 More Equivalence Checking 105

6.8.7 Using the mu-Calculus 106

6.8.8 Using CTL 107

6.9 CCS and CWB Application Examples 109

6.9.1 The CCS XCEL-Circuit Example 109

6.9.2 The CCS CEL3-Circuit Example 112

6.10 Further Reading 113

6.11 Selected Solutions 114

6.12 References 115

7. Verification of Modular Asynchronous Circuits 117

7.1 About Asynchronous Circuits 117

7.1.1 Modular Asynchronous Circuits 117

7.1.2 Edge-Based (Dynamic) Versus Level-Based Behavior 118

7.2 XOR-Gates 118

7.2.1 LOTOS Representation of XOR-Gate 118

7.2.2 Petri Net Representation of XOR-Gate 119

7.2.3 CCS Representation of XOR-Gate 119

7.3 CEL-Circuit 119

7.3.1 LOTOS Representation of CEL-Circuit 120

7.3.2 Petri Net Representation of CEL-Circuit 120

7.3.3 CCS Representation of CEL-Circuit 120

7.4 Other Modules 121

7.4.1 Inverter 121

7.4.2 ICEL-Element 121

7.4.3 TOGGLE 122

7.4.4 CALL 122

7.5 Module Extensions 123

7.5.1 XORk (k . 2) Modules 123

7.5.2 CELk (k . 2) Modules 123

7.5.3 TOGk (k . 2) 124

7.6 Modular Networks 124

7.7 Realizations 125

7.7.1 Introduction to Realization 125

7.7.2 Type-A Realization 125

7.7.3 Type-B Realization 126

7.7.4 Type-C Realization 128

7.7.5 Type-D Realization 130

7.7.6 DI Realization 131

7.8 Verification of Extended Modules 131

7.8.1 Verification of XORk (k . 2) Modules 132

7.8.2 Verification of CELk (k . 2) Modules 135

7.8.3 Verification of TOGk (k . 2) Modules 137

7.9 Verification of Parallel Control Structures 137

7.10 Further Reading 140

7.11 Selected Solutions 140

7.12 References 146

8. Verification of Communication Protocols 147

8.1 Introduction 147

8.2 Two Simple Communication Protocols 147

8.2.1 Simple Communication Protocol SCP 148

8.2.2 Simple Communication Protocol SCP1 148

8.3 The Alternating Bit (AB) Protocol 149

8.3.1 Introduction 149

8.3.2 The Reliable Channel Case 149

8.3.3 The Unreliable Channel Case 151

8.4 Further Reading 156

8.5 Selected Solutions 156

8.6 References 157

9. Verification of Arbiters 159

9.1 Introduction 159

9.2 A Random Arbiter (RGDA) 159

9.2.1 Verifying RGDA Using LOTOS 160

9.2.2 Verifying RGDA Using Petri Nets 163

9.2.3 Verifying RGDA Using CCS 165

9.3 A Token-Ring Arbiter 167

9.3.1 A Petri Net Representation of a Token-Ring Arbiter 167

9.3.2 Verification of a Token-Ring Arbiter Using Petri Net 168

9.4 Further Reading 168

9.5 Selected Solutions 169

9.6 References 171

10. More Verification Case Studies 173

10.1 Verification of Combinational Logic 173

10.1.1 The AND Gate 173

10.1.2 Composite Gates 175

10.2 Verification of Asynchronous Pipeline Controllers 177

10.2.1 Introduction 177

10.2.2 Latch Control Unit 178

10.2.3 Phase Converters 181

10.3 Verification of Producer–Consumer Systems 183

10.3.1 Introduction 183

10.3.2 Verifying Producer–Consumer Systems Using Petri Nets 183

10.3.3 Occurrence Counts 184

10.3.4 Verifying the Producer–Consumer System Using Occurrence Counts 184

10.3.5 Verifying Producer–Consumer Systems Using LOTOS 185

10.3.6 Verification of Multiple-Producer Multiple-Consumer Systems 186

10.4 Verification Based on Design Approaches 188

10.4.1 Synthesis Approach #1 188

10.4.2 Synthesis Approach #2 189

10.4.3 Extending the Synthesis Method by Adding XOR Modules 190

10.4.4 A Decomposition Approach 191

10.5 Verification of Toggles and Transition Counters 193

10.5.1 Verification of k-Toggles 193

10.5.2 Verification of Counters without Outputs 194

10.5.3 Verification of Up–Down Counters with Output 196

10.5.4 Verification of Modulo-N Transition Counters 196

10.6 Vending Machines Verification—Revisited 199

10.6.1 Verifying Vending Machines VeMa Using CCS 199

10.6.2 Verifying Vending Machines VeMa Using LOTOS 200

10.7 Pi-Realizations 201

10.7.1 More on Modular Networks 202

10.7.2 Introducing Circuits 203

10.7.3 Concurrent Behavior of Circuits 204

10.7.4 Pi-Specifications of Circuits 205

10.7.5 Simple Verification Examples 206

10.7.6 Applying Net Algebra 206

10.7.7 Another Verification Example 207

10.7.8 Some Pi Propositions 209

10.8 A Comparison of Equivalence Relations 210

10.8.1 An Equivalence Theorem 210

10.8.2 An Application of the Equivalence Theorem 211

10.9 Selected Solutions 211

10.10 References 223

11. Guide to Further Studies 225

11.1 Verification of Telecommunication Systems 225

11.1.1 Plain Old Telephone System (POTS) 225

11.1.2 Advanced Telephone Systems 226

11.1.3 ISDN Telephony 226

11.2 Verification Using Colored Petri Nets 226

11.3 Verification of Traffic Signal Control Systems 226

11.4 References 227

Index 229