Completeness Theorems for Pomset Languages and Concurrent Kleene Algebras

Pomsets constitute one of the most basic models of concurrency. A pomset is a generalisation of a word over an alphabet in that letters may be partially ordered rather than totally ordered. A term t using the bi-Kleene operations 0, 1,+, · , , ‖,(∗) defines a set [[t]] of pomsets in a natural way. We prove that every valid universal equality over pomset languages using these operations is a consequence of the equational theory of regular languages (in which parallel multiplication and iteration are undefined) plus that of the commutative-regular languages (in which sequential multiplication and iteration are undefined). We also show that the class of rational pomset languages (that is, those languages generated from singleton pomsets using the bi-Kleene operations) is closed under all Boolean operations. An ideal of a pomset p is a pomset using the letters of p, but having an ordering at least as strict as p. A bi-Kleene term t thus defines the set Id([[t]]) of ideals of pomsets in [[t]]. We prove that if t does not contain commutative iteration (∗) (in our terminology, t is bw-rational) then Id([[t]])∩Pomsp, where Pomsp is the set of pomsets generated from singleton pomsets using sequential and parallel multiplication (· and ‖) is defined by a bw-rational term, and if two such terms t, t define the same ideal language, then t = t is provable from the Kleene axioms for 0, 1,+, · , plus the commutative idempotent semiring axioms for 0, 1,+, ‖ plus the exchange law (u ‖ v) · (x ‖ y) ≤ v · y ‖ u · x.