4,373 research outputs found
On the Problem of Computing the Probability of Regular Sets of Trees
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 and
(3) the probability of \emph{game languages} , from automata theory,
is if is odd and is otherwise
Near-Optimal Scheduling for LTL with Future Discounting
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
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
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
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
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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