A Calculus of Distributed and Parallel Processes by Clemens H. Cap

By Clemens H. Cap

This booklet introduces a technique calculus for parallel, allotted and reactive structures. It describes the conceptual foundations in addition to the mathematical thought in the back of a programming language, and a couple of software examples. the selected technique presents a framework for figuring out the semantics of parallel and disbursed platforms. additionally, it may be at once utilized to functional difficulties.

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A process may possess finite and infinite maximal behaviours. 34 Theorem EXISTENCE OF MAXIMAL BEHAVIOURS Every behaviour of a process may be extended to a maximal behaviour7 • PROOF: Let B be a behaviour. According the lemma of KURATOWSKY [DP90j this behaviour is an element of a maximal chain in the extension order. Let r be such a maximal chain. Then r contains equivalence classes of behaviours. In every equivalence class we can find a behaviour in the canonical form of Theorem 29. Let r* be the set of all behaviours in canonical form which may be found in the equivalence classes of r.

Hence, the compositional transition systems may be considered freely genemted and initial, in the sense of algebraic specification [MG85j, [Wir9Qj. It is possible to study compositional systems which identify more transitions. This provides an interesting approach for abstmction from implementation details: 42 Example NOT FREELY GENERATED COMPOSITIONAL TS A transition system is given by S = {add(x,y), T = {lz(z), is (x) I x,y E No} rz(z), li(x, y), ri(x, y), sym(x, y), de(r, s), null I x,y E No,z,r,s E N} These transitions define methods for calculating the addition by a symmetric, left and right zero (null, lz, rz) rule, by a left and right induction rule (li, ri), by symmetrization (sym), and by direct calculation (de), if both operands are different from zero: add(O,z) lz(z) .

Sophisticated algorithms are required to construct so called global system snapshots [TK87]. Conclusion: Transition systems in their present form must be interpreted as sequential, possibly non-deterministic processes. A theory of parallel or distributed behaviour requires an extension of this concept. In the remaining sections of this chapter, the present form of transition systems is studied more closely. In the next chapter, the suggested extensions are provided. 3 Typed and Labeled Systems Transition systems (8, T, ---t) are used to model state changes which are induced by transitions.

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