A Unary Semigroup Trace Algebra

The Unifying Theories of Programming (UTP) of Hoare and He promote the unification of semantics catering for different concerns, such as, termination, data modelling, concurrency and time. Process calculi like Circus and CSP can be given semantics in the UTP using reactive designs whose traces can be abstractly specified using a monoid trace algebra. The prefix order over traces is defined in terms of the monoid operator. This order, however, is inadequate to characterise a broader family of timed process algebras whose traces are preordered instead. To accommodate these, we propose a unary semigroup trace algebra that is weaker than the monoid algebra. This structure satisfies some of the axioms of restriction semigroups and is a right P-Ehresmann semigroup. Reactive designs specified using it satisfy core laws that have been mechanised so far in Isabelle/UTP. More importantly, our results improve the support for unifying trace models in the UTP.