A coalgebraic take on regular and $\omega$-regular behaviours

We present a general coalgebraic setting in which we define finite and infinite behaviour with B\"uchi acceptance condition for systems whose type is a monad. The first part of the paper is devoted to presenting a construction of a monad suitable for modelling (in)finite behaviour. The second part of the paper focuses on presenting the concepts of a (coalgebraic) automaton and its ($\omega$-) behaviour. We end the paper with coalgebraic Kleene-type theorems for ($\omega$-) regular input. The framework is instantiated on non-deterministic (B\"uchi) automata, tree automata and probabilistic automata.

Distribution Bisimilarity via the Power of Convex Algebras

Probabilistic automata (PA), also known as probabilistic nondeterministic labelled transition systems, combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of distribution bisimilarity, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull. Comment: Full (extended) version of a CONCUR 2017 paper, minor revision of the LMCS submission

Fixpoint Theory – Upside Down

Knaster-Tarski’s theorem, characterising the greatest fix- point of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form $$\mathbb {M}^Y$$ M Y , where Y is a finite set and $$\mathbb {M}$$ M an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.

Recognition of multiple sequences by subgroups of autonomous probabilistic automata

Recognition of a group of sequences is carried out using a set of k subgroups, each of which includes b autonomous probabilistic automata (APA) set on the basis of ergodic stochastic matrices (ESM) of a certain subclass. Each of the b APA within k subgroups has a common set of states of a given power, and the vector of the initial distribution of each of these APA is determined by the distribution law of the first element of the corresponding sequence. The subgroup input from b APA receives element-wise members of b sequences.. The obtained probability vectors are subjected to clustering by the k-means method, which is based on the previously proposed methodology for a sample of reference object introduction. The clustering technique makes it possible to identify the groups of sequences by determining the probability of identification at each step of the algorithm, which can be implemented in parallel when each sequence is identified from a given set by each APA, parallelizing this process.