61 research outputs found
O-Minimal Hybrid Reachability Games
In this paper, we consider reachability games over general hybrid systems,
and distinguish between two possible observation frameworks for those games:
either the precise dynamics of the system is seen by the players (this is the
perfect observation framework), or only the starting point and the delays are
known by the players (this is the partial observation framework). In the first
more classical framework, we show that time-abstract bisimulation is not
adequate for solving this problem, although it is sufficient in the case of
timed automata . That is why we consider an other equivalence, namely the
suffix equivalence based on the encoding of trajectories through words. We show
that this suffix equivalence is in general a correct abstraction for games. We
apply this result to o-minimal hybrid systems, and get decidability and
computability results in this framework. For the second framework which assumes
a partial observation of the dynamics of the system, we propose another
abstraction, called the superword encoding, which is suitable to solve the
games under that assumption. In that framework, we also provide decidability
and computability results
Decisiveness of Stochastic Systems and its Application to Hybrid Models (Full Version)
In [ABM07], Abdulla et al. introduced the concept of decisiveness, an
interesting tool for lifting good properties of finite Markov chains to
denumerable ones. Later, this concept was extended to more general stochastic
transition systems (STSs), allowing the design of various verification
algorithms for large classes of (infinite) STSs. We further improve the
understanding and utility of decisiveness in two ways. First, we provide a
general criterion for proving decisiveness of general STSs. This criterion,
which is very natural but whose proof is rather technical, (strictly)
generalizes all known criteria from the literature. Second, we focus on
stochastic hybrid systems (SHSs), a stochastic extension of hybrid systems. We
establish the decisiveness of a large class of SHSs and, under a few classical
hypotheses from mathematical logic, we show how to decide reachability problems
in this class, even though they are undecidable for general SHSs. This provides
a decidable stochastic extension of o-minimal hybrid systems.
[ABM07] Parosh A. Abdulla, Noomene Ben Henda, and Richard Mayr. 2007.
Decisive Markov Chains. Log. Methods Comput. Sci. 3, 4 (2007).Comment: Full version of GandALF 2020 paper (arXiv:2001.04347v2), updated
version of arXiv:2001.04347v1. 30 pages, 6 figure
Finite Bisimulations for Dynamical Systems with Overlapping Trajectories
Having a finite bisimulation is a good feature for a dynamical system, since it can lead to the decidability of the verification of reachability properties. We investigate a new class of o-minimal dynamical systems with very general flows, where the classical restrictions on trajectory intersections are partly lifted. We identify conditions, that we call Finite and Uniform Crossing: When Finite Crossing holds, the time-abstract bisimulation is computable and, under the stronger Uniform Crossing assumption, this bisimulation is finite and definable
Monotonic Abstraction Techniques: from Parametric to Software Model Checking
Monotonic abstraction is a technique introduced in model checking
parameterized distributed systems in order to cope with transitions containing
global conditions within guards. The technique has been re-interpreted in a
declarative setting in previous papers of ours and applied to the verification
of fault tolerant systems under the so-called "stopping failures" model. The
declarative reinterpretation consists in logical techniques (quantifier
relativizations and, especially, quantifier instantiations) making sense in a
broader context. In fact, we recently showed that such techniques can
over-approximate array accelerations, so that they can be employed as a
meaningful (and practically effective) component of CEGAR loops in software
model checking too.Comment: In Proceedings MOD* 2014, arXiv:1411.345
An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet
We show that a special case of the Feferman-Vaught composition theorem gives
rise to a natural notion of automata for finite words over an infinite
alphabet, with good closure and decidability properties, as well as several
logical characterizations. We also consider a slight extension of the
Feferman-Vaught formalism which allows to express more relations between
component values (such as equality), and prove related decidability results.
From this result we get new classes of decidable logics for words over an
infinite alphabet.Comment: 24 page
Kleene-SchĂĽtzenberger and BĂĽchi Theorems for Weighted Timed Automata
In 1994, Alur and Dill introduced timed automata as a simple mathematical model for modelling the behaviour of real-time systems.
In this thesis, we extend timed automata with weights. More detailed, we equip both the states and transitions of a timed automaton with weights taken from an appropriate mathematical structure. The weight of a transition determines the weight for taking this transition, and the weight of a state determines the weight for letting time elapse in this state. Since the weight for staying in a state depends on time, this model, called weighted timed automata, has many interesting applications, for instance, in operations research and scheduling. We give characterizations for the behaviours of weighted timed automata in terms of rational expressions and logical formulas. These formalisms are useful for the specification of real-time systems with continuous resource consumption. We further investigate the relation between the behaviours of weighted timed automata and timed automata. Finally, we present important decidability results for weighted timed automata
Rational semimodules over the max-plus semiring and geometric approach of discrete event systems
We introduce rational semimodules over semirings whose addition is
idempotent, like the max-plus semiring, in order to extend the geometric
approach of linear control to discrete event systems. We say that a
subsemimodule of the free semimodule S^n over a semiring S is rational if it
has a generating family that is a rational subset of S^n, S^n being thought of
as a monoid under the entrywise product. We show that for various semirings of
max-plus type whose elements are integers, rational semimodules are stable
under the natural algebraic operations (union, product, direct and inverse
image, intersection, projection, etc). We show that the reachable and
observable spaces of max-plus linear dynamical systems are rational, and give
various examples.Comment: 24 pages, 9 postscript figures; example in section 4.3 expande
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