Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union (+) and iteration ( * ) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa's axiomatization of Kleene Algebra. INTRODUCTIONComputer scientists have long explored the connections between families of programming languages and abstract machines. This dual perspective has furnished deep theoretical insights as well as practical tools. As an example, Kleene's classic result establishing the equivalence of regular expressions and finite automata [22] inspired decades of work across a variety of areas including programming language design, mathematical semantics, and formal verification.Kleene Algebra with Tests (KAT) [25], which combines Kleene Algebra (KA) with Boolean Algebra (BA), is a modern example of this approach. Viewed from the program-centric perspective, a KAT models the fundamental constructs that arise in programs: sequencing, branching, iteration, non-determinism, etc. The equational theory of KAT enables algebraic reasoning and can be finitely axiomatized [30]. Viewed from the machine-centric perspective, a KAT describes a kind of automaton that generates a regular language of traces. This shift in perspective admits techniques from the theory of coalgebras for reasoning about program behavior. In particular, there are efficient algorithms for checking bisimulation, which can be optimized using properties of bisimulations [8,19] or symbolic automata representations [40].KAT has been used to model computation across a wide variety of areas including program transformations [2, 26], concurrency control [10], compiler optimizations [29], cache control [5,9], and more [9]. A prominent recent application is NetKAT [1], a language for reasoning about the packetforwarding behavior of software-defined networks. NetKAT has a sound and complete equational theory, and a coalgebraic decision procedure that can be used to automatically verify many important networking properties including reachability, loop-freedom, and isolation [15]. However, while NetKAT's implementation scales well in practice, the complexity of deciding equivalence for NetKAT is PSPACE-complete in the worst case [1].A natural question to ask is whether there is an efficient fragment of KAT that is reasonably expressive, while retaining a solid foundation. We answer this question positively in this paper with a comprehensive study of Guarded Kleene Algebra with Tests (GKAT), the guarded fragment of KAT. The language is a propositional abstraction of imperative while programs, which have been well-studied in the literature. We establish the fundamental properties of GKAT and develop its ...

Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and Wehrman in 2009 as a framework to reason about concurrent programs. We prove that the axioms for CKA with bounded parallelism are complete for the semantics proposed in the original paper; consequently, these semantics are the free model for this fragment. This result settles a conjecture of Hoare and collaborators. Moreover, the technique developed to this end allows us to establish a Kleene Theorem for CKA, extending an earlier Kleene Theorem for a fragment of CKA. ments.The notion of N-freeness for pomsets is useful for proving the lemmas to come.Definition A.1. Let U = [u] be a pomset. We say that U is N-free if there are no u 0 , u 1 , u 2 , u 3 ∈ S u such that u 0 ≤ u u 1 , u 2 ≤ u u 3 and u 0 ≤ u u 3 and no other relation between them, i.e., the graph of these elements has the shape of an N.Note that N-freeness is well-defined for pomsets, for the presence of an Nshape does not depend on the particular representative u. It is not hard to see that all series-parallel pomsets are N-free. Perhaps surprisingly, this N-freeness provides a complete characterisation of series-parallel pomsets [6].It is also useful to restrict a labelled poset to a part of its carrier, as follows.Definition A.2. Let u be a labelled poset, and let S ⊆ S u . We write u ↾ S for the restriction of u to S, i.e., labelled poset given by S u↾S = S, ≤ u↾S = ≤ u ∩ S × S, and λ u↾S (z) = λ u (z).A.1 Subsumption of empty or primitive pomsets Lemma A.2. Let u be a labelled poset such that u ⊑ 1 or 1 ⊑ u. Then u = 1.Proof. We treat the case where u ⊑ 1; the case where 1 ⊑ u is similar. Let h : 1 → u witness that u ⊑ 1. Then h is a bijection from S 1 = ∅ to S u ; accordingly, S u = ∅. But then u = 1, because the labelled poset with empty carrier is unique.Proof. First, suppose that U = 1. We then have that U = [1] and V = [v] such that u ⊑ 1 or 1 ⊑ u. By Lemma A.2, we find that v = 1 and thus V = [1] = 1.Second, suppose that U = a for some a ∈ Σ. Then U = [u] for some pomset with singleton carrier S u , with λ u (u) = a for all u ∈ S u . Since V = [v] and u ⊑ v or v ⊑ u, we find that u ≃ v by Lemma A.3. This establishes that U = V . A.2 The factorisation lemmaLemma 3.3 (Factorisation). Let U , V 0 , and V 1 be pomsets such that U is subsumed by V 0 · V 1 . Then there exist pomsets U 0 and U 1 such that:Also, if U 0 , U 1 and V are pomsets such that U 0 U 1 ⊑ V , then there exist pomsets V 0 and V 1 such that:Proof. We start with the first claim. Let U , V 0 and V 1 be as in the premise, and write U = [u], V 0 = [v 0 ] and V 1 = [v 1 ]. Without loss of generality, we can assume that v 0 and v 1 are disjoint, that S v0 ∪ S v1 = S u , and that the identity function S v0 ∪ S v1 → S u is the subsumption witnessing that u ⊑ v 0 · v 1 .We then choose u i = u ↾ vi for i ∈ 2, and claim that u 0 · u 1 = u.-For the carrier, we already know that

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Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally misses tests, which are needed to model standard programming constructs such as conditionals and while-loops. It turns out that integrating tests in CKA is subtle, due to their interaction with parallelism. In this paper we provide a solution in the form of Concurrent Kleene Algebra with Observations (CKAO). Our main contribution is a completeness theorem for CKAO. Our result resorts on a more general study of CKA "with hypotheses", of which CKAO turns out to be an instance: this analysis is of independent interest, as it can be applied to extensions of CKA other than CKAO. PreliminariesWe recall basic definitions on pomset languages, used in the semantics of CKA, which generalise languages to allow letters in words to be partially ordered. We fix a (possibly infinite) alphabet Σ. When defining sets parametrised by Σ, say S(Σ), if Σ is clear from the context we use S to refer to S(Σ). Pomsets [9,10] are labelled posets, up to isomorphism. Posets and PomsetsDefinition 2.1 (Labellet poset). A labelled poset over Σ is a tuple u = S, ≤, λ , where S is a finite set (the carrier of u), ≤ u is a partial order on S (the order of u), and λ : S → Σ is a function (the labelling of u).

Over the past decades, coordination languages emerged for modeling and implementing concurrency protocols among components in component-based systems. Coordination languages allow programmers to express concurrency protocols at a higher and more appropriate level of abstraction than what traditional programming and scripting languages offer. In this paper, we extend a significant coordination model-constraint automata-with a mechanism to finitely and compactly deal with infinite data domains, including foundational notions as behavior and equivalence (based on languages), weak and strong congruence (based on bisimulation), and composition. We also address the act of composing a number of simple primitive constraint automata into a complex composite one, by discussing two alternative composition approaches and by analyzing their performance in a number of experiments.