228 research outputs found
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
On the Decidability of Reachability in Linear Time-Invariant Systems
We consider the decidability of state-to-state reachability in linear
time-invariant control systems over discrete time. We analyse this problem with
respect to the allowable control sets, which in general are assumed to be
defined by boolean combinations of linear inequalities. Decidability of the
version of the reachability problem in which control sets are affine subspaces
of is a fundamental result in control theory. Our first result
is that reachability is undecidable if the set of controls is a finite union of
affine subspaces. We also consider versions of the reachability problem in
which (i)~the set of controls consists of a single affine subspace together
with the origin and (ii)~the set of controls is a convex polytope. In these two
cases we respectively show that the reachability problem is as hard as Skolem's
Problem and the Positivity Problem for linear recurrence sequences (whose
decidability has been open for several decades). Our main contribution is to
show decidability of a version of the reachability problem in which control
sets are convex polytopes, under certain spectral assumptions on the transition
matrix
Interrupt Timed Automata: verification and expressiveness
We introduce the class of Interrupt Timed Automata (ITA), a subclass of
hybrid automata well suited to the description of timed multi-task systems with
interruptions in a single processor environment. While the reachability problem
is undecidable for hybrid automata we show that it is decidable for ITA. More
precisely we prove that the untimed language of an ITA is regular, by building
a finite automaton as a generalized class graph. We then establish that the
reachability problem for ITA is in NEXPTIME and in PTIME when the number of
clocks is fixed. To prove the first result, we define a subclass ITA- of ITA,
and show that (1) any ITA can be reduced to a language-equivalent automaton in
ITA- and (2) the reachability problem in this subclass is in NEXPTIME (without
any class graph). In the next step, we investigate the verification of real
time properties over ITA. We prove that model checking SCL, a fragment of a
timed linear time logic, is undecidable. On the other hand, we give model
checking procedures for two fragments of timed branching time logic. We also
compare the expressive power of classical timed automata and ITA and prove that
the corresponding families of accepted languages are incomparable. The result
also holds for languages accepted by controlled real-time automata (CRTA), that
extend timed automata. We finally combine ITA with CRTA, in a model which
encompasses both classes and show that the reachability problem is still
decidable. Additionally we show that the languages of ITA are neither closed
under complementation nor under intersection
Decidability and Undecidability in Dynamical Systems
A computing system can be modelized in various ways: one being in analogy with transfer functions, this is a function that associates to an input and optionally some internal states, an output ; another being focused on the behaviour of the system, that is describing the sequence of states the system will follow to get from this input to produce the output. This second kind of system can be defined by dynamical systems. They indeed describe the ``local'' behaviour of a system by associating a configuration of the system to the next configuration. It is obviously interesting to get an idea of the ``global'' behaviour of such a dynamical system. The questions that it raises can be for example related to the reachability of a certain configuration or set of configurations or to the computation of the points that will be visited infinitely often. Those questions are unfortunately very complex: they are in most cases undecidable. This article will describe the fundamental problems on dynamical systems and exhibit some results on decidability and undecidability in various kinds of dynamical systems
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
Decidability and Universality in Symbolic Dynamical Systems
Many different definitions of computational universality for various types of
dynamical systems have flourished since Turing's work. We propose a general
definition of universality that applies to arbitrary discrete time symbolic
dynamical systems. Universality of a system is defined as undecidability of a
model-checking problem. For Turing machines, counter machines and tag systems,
our definition coincides with the classical one. It yields, however, a new
definition for cellular automata and subshifts. Our definition is robust with
respect to initial condition, which is a desirable feature for physical
realizability.
We derive necessary conditions for undecidability and universality. For
instance, a universal system must have a sensitive point and a proper
subsystem. We conjecture that universal systems have infinite number of
subsystems. We also discuss the thesis according to which computation should
occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2:
minor orthographic changes v3: section 5.2 (collatz functions) mathematically
improved v4: orthographic corrections, one reference added v5:27 pages.
Important modifications. The formalism is strengthened: temporal logic
replaced by finite automata. New results. Submitte
Reachability problems for PAMs
Piecewise affine maps (PAMs) are frequently used as a reference model to show
the openness of the reachability questions in other systems. The reachability
problem for one-dimentional PAM is still open even if we define it with only
two intervals. As the main contribution of this paper we introduce new
techniques for solving reachability problems based on p-adic norms and weights
as well as showing decidability for two classes of maps. Then we show the
connections between topological properties for PAM's orbits, reachability
problems and representation of numbers in a rational base system. Finally we
show a particular instance where the uniform distribution of the original orbit
may not remain uniform or even dense after making regular shifts and taking a
fractional part in that sequence.Comment: 16 page
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