ProbNetKAT is a probabilistic extension of NetKAT with a denotational semantics based on Markov kernels. The language is expressive enough to generate continuous distributions, which raises the question of how to compute effectively in the language. This paper gives an new characterization of ProbNetKAT's semantics using domain theory, which provides the foundation needed to build a practical implementation. We show how to use the semantics to approximate the behavior of arbitrary ProbNetKAT programs using distributions with finite support. We develop a prototype implementation and show how to use it to solve a variety of problems including characterizing the expected congestion induced by different routing schemes and reasoning probabilistically about reachability in a network.

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

We present an Angluin-style algorithm to learn nominal automata, which are acceptors of languages over infinite (structured) alphabets. The abstract approach we take allows us to seamlessly extend known variations of the algorithm to this new setting. In particular we can learn a subclass of nominal non-deterministic automata. An implementation using a recently developed Haskell library for nominal computation is provided for preliminary experiments.

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We extend the Kearns-Vazirani learning algorithm to be able to handle systems that change over time. We present a new learning algorithm that can reuse and update previously learned behavior, implement it in the LearnLib library, and evaluate it on large examples, to which we make small adjustments between two runs of the algorithm. In these experiments our algorithm significantly outperforms both the classic Kearns-Vazirani learning algorithm and the current state-of-the-art adaptive algorithm.⋆ Full implementation and experiment results available at https://github.com/UCL-PPLV/learnlib.