In C. A. R. Hoare, S. D. Brookes, and A. D. Roscoe (1984, J. Assoc. Comput. Mach. 31(3), 560) an abstract version of Hoare's CSP is defined and a denotational semantics based on the possible failures of processes is given for it. This semantics induces a natural preorder on processes. We define formally this preorder and prove that it can be characterized as the smallest relation satisfying a particular set of axioms. The characterization sheds lights on problems arising from the way divergence and underspecification are handled. After small changes to the semantic domains we propose a new semantics which is closer to the operational intuitions and suggests a possible solution to the above problems. Finally we give an axiomatic characterization for the equivalence induced by the new semantics which leads to fully abstract models in the sense of Scott.
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