Kleene Algebra with Hypotheses

Abstract

We study the Horn theories of Kleene algebras and star continuous Kleene algebras, from the complexity point of view. While their equational theories coincide and are PSpace-complete, their Horn theories differ and are undecidable. We characterise the Horn theory of star continuous Kleene algebras in terms of downward closed languages and we show that when restricting the shape of allowed hypotheses, the problems lie in various levels of the arithmetical or analytical hierarchy. We also answer a question posed by Cohen about hypotheses of the form 1 = S where S is a sum of letters: we show that it is decidable.

Concurrent Kleene Algebra with Observations: From Hypotheses to Completeness

“…The first contribution of this paper is to extend Kleene algebra with hypotheses [7] with a parallel operator. The resulting framework, concurrent Kleene algebra with hypotheses (CKAH), is interpreted over pomset languages, a standard model of concurrency.…”

“…If does not occur in any hypothesis, then the definition of sequential closure coincides with the closure operator from [7]. Thus, if L ⊆ Σ * , then L↓ H seq ⊆ Σ * .…”

“…It is not very hard to show that reductions can be chained, as follows. Another way of reducing H is to find two sets of hypotheses H 0 and H 1 , and reduce each of those to another set of hypotheses H ′ [7]. The idea is that a proof of e ≡ H f can be split up in a phase where we find e ′ , f ′ ∈ T such that e ≡ H0 e ′ and f ≡ H0 f ′ , after which we find e ′′ , f ′′ ∈ T with e ′ ≡ H1 e ′′ and f ′ ≡ H1 f ′′ .…”

“…A number of reduction procedures already exist at the level of Kleene algebra [19,7]; ideally, one would like to lift those procedures to CKA.…”

“…the semantics, as well as decidability of equivalence in the semantics. Instead of such a tailor-made proof, however, we take a more general approach by first proposing CKA with hypotheses (CKAH) as a formalism for studying extensions of CKA, akin to how Kleene algebra with hypotheses [5,17,19,7] can be used to extend Kleene algebra. We then apply CKAH to study CKAO, but the meta-theory developed can also be applied to extensions other than CKAO.…”