[...]... set of initial and time restricted events of Ψ I X → e : a timed bundle Z(σ, e) : the set of instants where (enabled event) e could happen, after σ where ε is a BES, C is a configuration, Ψ is a TBES, X is a set of events, I is a time interval, e is an event and σ is a timed proving sequence Automata (Chapters 8, 11, 12 and 13) Sets In Communicating Automata, Timed Automata and Timed Automata with Deadlines... however, we do not consider such generalisations and restrict ourselves to considering timing generalisations 1.4 Mathematical Theories The previous sections of this introduction have clarified what we mean by concurrency theory However, this leaves the question of what we mean by a mathematical theory of concurrency In a broad sense, our mathematical theory of concurrency has the same ingredients as the... theories, such as, for example, the theory of (counting) numbers (which is the heart of the mathematical discipline of number theory) We illustrate this by comparing the ingredients of the theory of numbers with those of concurrency theory 1 Values These are the primitive elements in mathematical theories They are the basic objects that the theory is about The values in the theory of numbers are the integers,... book Relations In the same way that we can define relations between values in the theory of numbers, e.g < or >, and hence define relations between expressions, e.g X + 4 < X + 9, we can do the same in concurrency theory In particular, we define preorders and equivalences which characterise relationships between concurrency theory values For example, we define a relation ≤tr (called trace preorder), which... Finally, Part IV of the book considers how the theory presented in previous parts of the book can be enhanced in order to model timed concurrent systems Chapters 9 and 10 consider timed enhancements of the pbLOTOS theory developed in Part II, and the remaining three chapters (11, 12 and 13) consider timed versions of the communicating automata discussed in Chapter 8 and associated theoretical issues 4 Although,... and ×, are used in the theory of numbers to express how numbers can be transformed into other numbers In concurrency theory we also have operators, e.g ||| and [] The former of these 1.4 Mathematical Theories 4 5 6 7 11 denotes a particular form of parallel composition and the second a choice between two alternatives Expressions Arithmetical expressions can be defined in the theory of numbers, e.g (X... Part I Introduction 1 Background on Concurrency Theory 1.1 Concurrency Is Everywhere There are two main axes to mathematical research, 1 Refining and building upon existing mathematical theories, e.g trying to prove or disprove the remaining conjectures in well explored branches of mathematics (Andrew Wiles’s proof of Fermat’s Last Theorem is such an effort [179]), and 2 Developing mathematical theories... important relation is equality For example, in the theory of numbers, (X + 73) × 3 = ((X + 72) × 3) + 3 The fact that these 12 1 Background on Concurrency Theory two expressions are equal is justified by the fact that whatever value you plug in for X, the two expressions evaluate to the same value In a similar way to equality in the theory of numbers, the theory of concurrency defines notions of equality, but... verification arises naturally from the mathematical theory we develop4 and is a major strength of modern concurrency theory It is for this reason that automatic verification plays an important role in our presentation 1.5 Overview of Book The book comprises four parts The first of these just contains this introduction Then Part II presents a basic body of concurrency theory, focused on untimed models of concurrent... computation (of which concurrency theory is a branch) are behaviours of systems: they code the possible executions of systems This computational aspect of underlying mathematical objects is in contrast to standard mathematical theories, where values are static in nature As a reflection of this, we use the term behavioural values to denote the values found in concurrency theory In one sense, standard mathematics . y0 w0 h0" alt="" Concurrency Theory Howard Bowman and Rodolfo Gomez Concurrency Theory Calculi and Automata for Modelling Untimed and Timed Concurrent. is an event and σ is a timed proving sequence. Automata (Chapters 8, 11, 12 and 13) Sets In Communicating Automata, Timed Automata and Timed Automata with Deadlines Act