Modal Specifications for the Control Theory of Discrete Event Systems

2 Conjunctive Nu-calculus and Modal Specifications 4 2.1 The Conjunctive Nu-calculus . . . . . . . . . . . . . . . . . . . 5 2.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Modal Specifications and Their Models . . . . . . . . . . . . . 6 2.2.1 Modal Specifications . . . . . . . . . . . . . . . . . . . 7 2.2.2 Graphical Representations . . . . . . . . . . . . . . . . 7 2.2.3 Coherency and S-closure . . . . . . . . . . . . . . . . . 8 2.2.4 The Theorem of Decomposition . . . . . . . . . . . . . 11 2.3 The Theorem of Equivalence . . . . . . . . . . . . . . . . . . . 13 2.3.1 From a Sentence to a Specification . . . . . . . . . . . 14 2.3.2 From a Specification to a Sentence . . . . . . . . . . . 18 2.3.3 Main Proof . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The Lattice of Models . . . . . . . . . . . . . . . . . . . . . . 19