Concurrent Kleene Algebra with Tests

Abstract

Concurrent Kleene algebras were introduced by Hoare, Möller, Struth and Wehrman in [HMSW09, HMSW09a, HMSW11] as idempotent bisemirings that satisfy a concurrency inequation and have a Kleene-star for both sequential and concurrent composition. Kleene algebra with tests (KAT) were defined earlier by Kozen and Smith [KS97]. Concurrent Kleene algebras with tests (CKAT) combine these concepts and give a relatively simple algebraic model for reasoning about operational semantics of concurrent programs. We generalize guarded strings to guarded series-parallel strings, or gsp-strings, to provide a concrete language model for CKAT. Combining nondeterministic guarded automata [Koz03] with branching automata of Lodaya and Weil [LW00] one obtains a model for processing gsp-strings in parallel, and hence an operational interpretation for CKAT. For gsp-strings that are simply guarded strings, the model works like an ordinary nondeterministic guarded automaton. If the test algebra is assumed to be {0,1} the language model reduces to the regular sets of bounded-width sp-strings of Lodaya and Weil.

Since the concurrent composition operator distributes over join, it can also be added to relation algebras with transitive closure to obtain the variety CRAT. We provide semantics for these algebras in the form of coalgebraic arrow frames expanded with concurrency.