4,373 research outputs found

    On the Problem of Computing the Probability of Regular Sets of Trees

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    We consider the problem of computing the probability of regular languages of infinite trees with respect to the natural coin-flipping measure. We propose an algorithm which computes the probability of languages recognizable by \emph{game automata}. In particular this algorithm is applicable to all deterministic automata. We then use the algorithm to prove through examples three properties of measure: (1) there exist regular sets having irrational probability, (2) there exist comeager regular sets having probability 00 and (3) the probability of \emph{game languages} Wi,kW_{i,k}, from automata theory, is 00 if kk is odd and is 11 otherwise

    Acta Cybernetica : Volume 17. Number 4.

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    Near-Optimal Scheduling for LTL with Future Discounting

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    We study the search problem for optimal schedulers for the linear temporal logic (LTL) with future discounting. The logic, introduced by Almagor, Boker and Kupferman, is a quantitative variant of LTL in which an event in the far future has only discounted contribution to a truth value (that is a real number in the unit interval [0, 1]). The precise problem we study---it naturally arises e.g. in search for a scheduler that recovers from an internal error state as soon as possible---is the following: given a Kripke frame, a formula and a number in [0, 1] called a margin, find a path of the Kripke frame that is optimal with respect to the formula up to the prescribed margin (a truly optimal path may not exist). We present an algorithm for the problem; it works even in the extended setting with propositional quality operators, a setting where (threshold) model-checking is known to be undecidable

    Computing Measures of Weak-MSO Definable Sets of Trees

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    This work addresses the problem of computing measures of recognisable sets of infinite trees. An algorithm is provided to compute the probability measure of a tree language recognisable by a weak alternating automaton, or equivalently definable in weak monadic second-order logic. The measure is the uniform coin-flipping measure or more generally it is generated by a branching stochastic process. The class of tree languages in consideration, although smaller than all regular tree languages, comprises in particular the languages definable in the alternation-free ?-calculus or in temporal logic CTL. Thus, the new algorithm may enhance the toolbox of probabilistic model checking

    Automatic modular abstractions for template numerical constraints

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    We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floating-point variables and containing linear assignments and tests. In addition to loop-free code, the same method also applies for obtaining least fixed points as functions of the precondition, which permits the analysis of loops and recursive functions. Our algorithms are based on new quantifier elimination and symbolic manipulation techniques. Given the specification of an abstract domain, and a program block, our method automatically outputs an implementation of the corresponding abstract transformer. It is thus a form of program transformation. The motivation of our work is data-flow synchronous programming languages, used for building control-command embedded systems, but it also applies to imperative and functional programming

    Flexible Coinduction

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    openRecursive definitions of predicates by means of inference rules are ubiquitous in computer science. They are usually interpreted inductively or coinductively, however there are situations where none of these two options provides the expected meaning. In the thesis we propose a flexible form of coinductive interpretation, based on the notion of corules, able to deal with such situations. In the first part, we define such flexible coinductive interpretation as a fixed point of the standard inference operator lying between the least and the greatest one, and we provide several equivalent proof-theoretic semantics, combining well-founded and non-well-founded derivations. This flexible interpretation nicely subsumes standard inductive and coinductive ones and is naturally associated with a proof principle, which smoothly extends the usual coinduction principle. In the second part, we focus on the problem of modelling infinite behaviour by a big-step operational semantics, which is a paradigmatic example where neither induction nor coinduction provide the desired interpretation. In order to be independent from specific examples, we provide a general, but simple, definition of what a big-step semantics is. Then, we extend it to include also observations, describing the interaction with the environment, thus providing a richer description of the behaviour of programs. In both settings, we show how corules can be successfully adopted to model infinite behaviour, by providing a construction extending a big-step semantics, which as usual only describes finite computations, to a richer one including infinite computations as well. Finally, relying on these constructions, we provide a proof technique to show soundness of a predicate with respect to a big-step semantics. In the third part, we ez face eez the problem of providing an algorithmic support to corules. To this end, we consider the restriction of the flexible coinductive interpretation to regular derivations, analysing again both proof-theoretic and fixed point semantics and developing proof techniques. Furthermore, we show that this flexible regular interpretation can be equivalently characterised inductively by a cycle detection mechanism, thus obtaining a sound and complete (abstract) (semi-)algorithm to check whether a judgement is derivable. Finally, we apply such results to extend logic programming by coclauses, the analogous of corules, defining declarative and operational semantics and proving ez that eez the latter is sound and complete with respect to the regular declarative model, thus obtaining a concrete support to flexible coinduction.openXXXIII CICLO - INFORMATICA E INGEGNERIA DEI SISTEMI/ COMPUTER SCIENCE AND SYSTEMS ENGINEERING - Informatica/computer scienceDagnino, Francesc

    Lukasiewicz mu-Calculus

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    We consider state-based systems modelled as coalgebras whose type incorporates branching, and show that by suitably adapting the definition of coalgebraic bisimulation, one obtains a general and uniform account of the linear-time behaviour of a state in such a coalgebra. By moving away from a boolean universe of truth values, our approach can measure the extent to which a state in a system with branching is able to exhibit a particular linear-time behaviour. This instantiates to measuring the probability of a specific behaviour occurring in a probabilistic system, or measuring the minimal cost of exhibiting a specific behaviour in the case of weighted computations
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