By Volker Diekert

Parallelism or concurrency is among the primary techniques in desktop technological know-how. yet regardless of its value, theoretical tips on how to deal with concurrency usually are not but sufficiently built. This quantity provides a finished learn of Mazurkiewicz' hint concept from an algebraic-combinatorial viewpoint. This concept is famous as a major instrument for a rigorous mathematical remedy of concurrent platforms. the quantity covers numerous various examine components, and includes not just identified effects but additionally numerous new effects released nowhere else. **Chapter 1** introduces uncomplicated suggestions. **Chapter 2** supplies a directly route to Ochmanski's characterization of recognizable hint languages and to Zielonka's idea of asynchronous automata. **Chapter 3** applies the speculation of strains to Petri nets. a type of morphism among nets is brought which generalizes the concept that of synchronization. **Chapter 4** offers a brand new bridge among the speculation of string rewriting and formal energy sequence. **Chapter 5** is an creation to a combinatorial conception of rewriting on lines which might be used as an summary calculus for reworking concurrent processes.

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Constructor) substitution is a substitution from X to T (C, F ) (resp. T (C)). Since our intention is to interpret a program by a function, an important property is that each term have a unique normal form, if it exists. Henceforth, we just consider conﬂuent programs, that is programs for which the induced relation → is conﬂuent. In particular, our deﬁnition includes orthogonal programs which are conﬂuent [20]. A program is orthogonal if in addition there are no overlapping rules. We now give the semantics of (conﬂuent) programs, based on the “standard” interpretation.

Example 2. 1. By putting if then else ≺F < ≺F insert ≺F sort, we see that the program written in Example 1 terminates by M P O . Eﬃcient First Order Functional Program Interpreter 31 2. The following program computes the exponential. It is convenient to identify 0 with 0 and suc(n) with n + 1. Henceforth, we write n instead of sucn (0). In the same way, n + m stands for sucn+m (0). The program is ordered by M P O with d ≺F exp. d : Nat → Nat and [[d]](n) = n + n d(0) → 0 d(suc(x)) → suc(suc(d(x))) exp : Nat → Nat and [[exp]](n) = 2n exp(0) → suc(0) exp(suc(x)) → d(exp(x)) Deﬁne Prog(M P O ) as the set of programs which terminate by M P O .

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