(Synchronous Kleene Algebra) Sample Clauses

(Synchronous Kleene Algebra). A synchronous KA (SKA) is a tuple (A, S, +, ·,∗ , ×, 0, 1) such that (A, +, ·,∗ , 0, 1) is a Kleene algebra and × is a binary operator on A, with S ⊆ A closed under × and (S, ×) a semilattice. Furthermore, the following hold for all e, f, g ∈ A and α, β ∈ S: e × (f + g)= e × f + e × g e × (f × g)= (e × f ) × g e × 0=0 (α · e) × (β · f )= (α × β) · (e × f ) e × f = f × e e × 1= e Note that 0 and 1 need not be elements of S. The semilattice terms, denoted TSL, are given by the following grammar. TSL e e, f :: = a ∈ Σ | e × f The synchronous regular terms, denoted TSKA, are given by the grammar: TSKA e e, f :: = 0 | 1 | a ∈ TSL | e + f | e · f | e × f | e∗ T ⊆ T ≡SKA Thus we have SL SKA. We then define as the smallest congruence on TSKA satisfying the axioms of SKA H˙ ere, TSL plays the role of the semilattice; for instance, for a ∈ TSL we have that a × a ≡SKA a. × ×
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