174 research outputs found

    Acta Cybernetica : Volume 17. Number 4.

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    Sound and complete axiomatizations of coalgebraic language equivalence

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    Coalgebras provide a uniform framework to study dynamical systems, including several types of automata. In this paper, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalised powerset construction that determinises coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor FTFT, where TT is a monad describing the branching of the systems (e.g. non-determinism, weights, probability etc.), has as a quotient the rational fixpoint of the "determinised" type functor Fˉ\bar F, a lifting of FF to the category of TT-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain non-deterministic automata, where we recover Rabinovich's sound and complete calculus for language equivalence.Comment: Corrected version of published journal articl

    Probabilistic Logic, Probabilistic Regular Expressions, and Constraint Temporal Logic

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    The classic theorems of BĂĽchi and Kleene state the expressive equivalence of finite automata to monadic second order logic and regular expressions, respectively. These fundamental results enjoy applications in nearly every field of theoretical computer science. Around the same time as BĂĽchi and Kleene, Rabin investigated probabilistic finite automata. This equally well established model has applications ranging from natural language processing to probabilistic model checking. Here, we give probabilistic extensions BĂĽchi\\\''s theorem and Kleene\\\''s theorem to the probabilistic setting. We obtain a probabilistic MSO logic by adding an expected second order quantifier. In the scope of this quantifier, membership is determined by a Bernoulli process. This approach turns out to be universal and is applicable for finite and infinite words as well as for finite trees. In order to prove the expressive equivalence of this probabilistic MSO logic to probabilistic automata, we show a Nivat-theorem, which decomposes a recognisable function into a regular language, homomorphisms, and a probability measure. For regular expressions, we build upon existing work to obtain probabilistic regular expressions on finite and infinite words. We show the expressive equivalence between these expressions and probabilistic Muller-automata. To handle Muller-acceptance conditions, we give a new construction from probabilistic regular expressions to Muller-automata. Concerning finite trees, we define probabilistic regular tree expressions using a new iteration operator, called infinity-iteration. Again, we show that these expressions are expressively equivalent to probabilistic tree automata. On a second track of our research we investigate Constraint LTL over multidimensional data words with data values from the infinite tree. Such LTL formulas are evaluated over infinite words, where every position possesses several data values from the infinite tree. Within Constraint LTL on can compare these values from different positions. We show that the model checking problem for this logic is PSPACE-complete via investigating the emptiness problem of Constraint BĂĽchi automata
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