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We prove two completeness results for Kleene algebra with tests and a top element, with respect to guarded string languages and binary relations. While the equational theories of those two classes of models coincide over the signature of Kleene algebra, this is no longer the case when we consider an additional constant ``top'' for the full element. Indeed, the full relation satisfies more laws than the full language, and we show that those additional laws can all be derived from a single additional axiom. We recover that the two equational theories coincide if we slightly generalise the notion of relational model, allowing sub-algebras of relations where top is a greatest element but not necessarily the full relation. We use models of closed languages and reductions in order to prove our completeness results, which are relative to any axiomatisation of the algebra of regular events. For one of our constructions, we extend the concept of finite monoid recognisability to guarded-string languages; this device makes it possible to obtain a PSpace algorithm for the equational theory of binary relations.

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We explore asynchronous programming with algebraic effects. We complement their conventional synchronous treatment by showing how to naturally also accommodate asynchrony within them, namely, by decoupling the execution of operation calls into signalling that an operation's implementation needs to be executed, and interrupting a running computation with the operation's result, to which the computation can react by installing interrupt handlers. We formalise these ideas in a small core calculus and demonstrate its flexibility using examples ranging from a multi-party web application, to pre-emptive multi-threading, to (cancellable) remote function calls, to a parallel variant of runners of algebraic effects. In addition, the paper is accompanied by a formalisation of the calculus's type safety proofs in Agda, and a prototype implementation in OCaml.

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Game semantics is a denotational semantics presenting compositionally the computational behaviour of various kinds of effectful programs. One of its celebrated achievement is to have obtained full abstraction results for programming languages with a variety of computational effects, in a single framework. This is known as the semantic cube or Abramsky's cube, which for sequential deterministic programs establishes a correspondence between certain conditions on strategies (''innocence'', ''well-bracketing'', ''visibility'') and the absence of matching computational effects. Outside of the sequential deterministic realm, there are still a wealth of game semantics-based full abstraction results; but they no longer fit in a unified canvas. In particular, Ghica and Murawski's fully abstract model for shared state concurrency (IA) does not have a matching notion of pure parallel program-we say that parallelism and interference (i.e. state plus semaphores) are entangled. In this paper we construct a causal version of Ghica and Murawski's model, also fully abstract for IA. We provide compositional conditions parallel innocence and sequentiality, respectively banning interference and parallelism, and leading to four full abstraction results. To our knowledge, this is the first extension of Abramsky's semantic cube programme beyond the sequential deterministic world.

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Sequential model synchronisation is the task of propagating changes from one model to another correlated one to restore consistency. It is challenging to perform this propagation in a least-changing way that avoids unnecessary deletions (which might cause information loss). From a theoretical point of view, so-called short-cut (SC) rules have been developed that enable provably correct propagation of changes while avoiding information loss. However, to be able to react to every possible change, an infinite set of such rules might be necessary. Practically, only small sets of pre-computed basic SC rules have been used, severely restricting the kind of changes that can be propagated without loss of information. In this work, we close that gap by developing an approach to compute more complex required SC rules on-the-fly during synchronisation. These higher-order SC rules allow us to cope with more complex scenarios when multiple changes must be handled in one step. We implemented our approach in the model transformation tool eMoflon. An evaluation shows that the overhead of computing higher-order SC rules on-the-fly is tolerable and at times even improves the overall performance. Above that, completely new scenarios can be dealt with without the loss of information.

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As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in proof assistants based on impredicative logics to proof assistants based on predicative logics, whenever impredicativity is not used in an essential way. In this paper we present a transformation for sharing proofs with a core predicative system supporting prenex universe polymorphism. It consists in trying to elaborate each term into a predicative universe-polymorphic term as general as possible. The use of universe polymorphism is justified by the fact that mapping each universe to a fixed one in the target theory is not sufficient in most cases. During the elaboration, we need to solve unification problems in the equational theory of universe levels. In order to do this, we give a complete characterization of when a single equation admits a most general unifier. This characterization is then employed in a partial algorithm which uses a constraint-postponement strategy for trying to solve unification problems. The proposed translation is of course partial, but in practice allows one to translate many proofs that do not use impredicativity in an essential way. Indeed, it was implemented in the tool Predicativize and then used to translate semi-automatically many non-trivial developments from Matita's library to Agda, including proofs of […]

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