2,376 research outputs found
Real-time and Probabilistic Temporal Logics: An Overview
Over the last two decades, there has been an extensive study on logical
formalisms for specifying and verifying real-time systems. Temporal logics have
been an important research subject within this direction. Although numerous
logics have been introduced for the formal specification of real-time and
complex systems, an up to date comprehensive analysis of these logics does not
exist in the literature. In this paper we analyse real-time and probabilistic
temporal logics which have been widely used in this field. We extrapolate the
notions of decidability, axiomatizability, expressiveness, model checking, etc.
for each logic analysed. We also provide a comparison of features of the
temporal logics discussed
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
Completeness of Flat Coalgebraic Fixpoint Logics
Modal fixpoint logics traditionally play a central role in computer science,
in particular in artificial intelligence and concurrency. The mu-calculus and
its relatives are among the most expressive logics of this type. However,
popular fixpoint logics tend to trade expressivity for simplicity and
readability, and in fact often live within the single variable fragment of the
mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL,
and the logic of common knowledge. Extending this notion to the generic
semantic framework of coalgebraic logic enables covering a wide range of logics
beyond the standard mu-calculus including, e.g., flat fragments of the graded
mu-calculus and the alternating-time mu-calculus (such as alternating-time
temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We
give a generic proof of completeness of the Kozen-Park axiomatization for such
flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on
Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer
Science, Springer, 2010, pp. 524-53
Probabilistic modal {\mu}-calculus with independent product
The probabilistic modal {\mu}-calculus is a fixed-point logic designed for
expressing properties of probabilistic labeled transition systems (PLTS's). Two
equivalent semantics have been studied for this logic, both assigning to each
state a value in the interval [0,1] representing the probability that the
property expressed by the formula holds at the state. One semantics is
denotational and the other is a game semantics, specified in terms of
two-player stochastic parity games. A shortcoming of the probabilistic modal
{\mu}-calculus is the lack of expressiveness required to encode other important
temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL).
To address this limitation we extend the logic with a new pair of operators:
independent product and coproduct. The resulting logic, called probabilistic
modal {\mu}-calculus with independent product, can encode many properties of
interest and subsumes the qualitative fragment of PCTL. The main contribution
of this paper is the definition of an appropriate game semantics for this
extended probabilistic {\mu}-calculus. This relies on the definition of a new
class of games which generalize standard two-player stochastic (parity) games
by allowing a play to be split into concurrent subplays, each continuing their
evolution independently. Our main technical result is the equivalence of the
two semantics. The proof is carried out in ZFC set theory extended with
Martin's Axiom at an uncountable cardinal
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