Abstract
We introduce a formal algebraic semantics for CSP. First we show how algebraic laws can systematically reduce all finite CSP terms to a head normal form, and therefore create an Algebraic Operational Semantics. Then we show how to reduce this to a normal form for the finest CSP model as developed in Chap. 12. We can then show how a small selection of optional laws such as \(P=P\mathbin{\Box}P\) allow this normal form to be transformed into normal forms for any of the main models of CSP. This provides an elegant explanation for why the hierarchy of models developed in the previous two chapters is as it is.
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Notes
- 1.
An amazing proportion of the ‘laws’ stated in [56] are false, and it is possible to find many distinct combinations which prove all processes equal. Some of the laws given in that book make even less sense than simply being false: for example ‘\(P\mathbin{|\!|\!|}Q\wedge Q\mathbin{|\!|\!|}R\Rightarrow P\mathbin{|\!|\!|}R\)’ which is trivially ill-typed since processes are not truth values.
- 2.
The fact that this transformation preserves equivalence over all our CSP models gives an interesting insight into the way in which the models without strict divergence treat div.
- 3.
See Chap. 9.
- 4.
Because all standard CSP operators other than recursion are fully distributive, we have not enumerated all such laws in this book.
- 5.
In the case of the symmetric operators \( \Box \) and \(\mathop{\mathord{\parallel}}\limits_{X}\) this reduces to 15 essentially different pairings.
- 6.
An alternative is only to have div in those choices with no other members.
- 7.
Note that in some presentations of CSP based on failures models only, this has sometimes been given as the definition or \(\rhd\).
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Roscoe, A.W. (2010). The Algebra of CSP. In: Understanding Concurrent Systems. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-84882-258-0_13
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