39 research outputs found
The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations
We study decidability and complexity questions related to a continuous
analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of
a linear recurrent sequence. In particular, we show that the continuous version
of the nonnegativity problem is NP-hard in general and we show that the
presence of a zero is decidable for several subcases, including instances of
depth two or less, although the decidability in general is left open. The
problems may also be stated as reachability problems related to real zeros of
exponential polynomials or solutions to initial value problems of linear
differential equations, which are interesting problems in their own right.Comment: 14 pages, no figur
On Recurrent Reachability for Continuous Linear Dynamical Systems
The continuous evolution of a wide variety of systems, including
continuous-time Markov chains and linear hybrid automata, can be described in
terms of linear differential equations. In this paper we study the decision
problem of whether the solution of a system of linear
differential equations reaches a target
halfspace infinitely often. This recurrent reachability problem can
equivalently be formulated as the following Infinite Zeros Problem: does a
real-valued function satisfying a
given linear differential equation have infinitely many zeros? Our main
decidability result is that if the differential equation has order at most ,
then the Infinite Zeros Problem is decidable. On the other hand, we show that a
decision procedure for the Infinite Zeros Problem at order (and above)
would entail a major breakthrough in Diophantine Approximation, specifically an
algorithm for computing the Lagrange constants of arbitrary real algebraic
numbers to arbitrary precision.Comment: Full version of paper at LICS'1
On the Skolem Problem for Continuous Linear Dynamical Systems
The Continuous Skolem Problem asks whether a real-valued function satisfying
a linear differential equation has a zero in a given interval of real numbers.
This is a fundamental reachability problem for continuous linear dynamical
systems, such as linear hybrid automata and continuous-time Markov chains.
Decidability of the problem is currently open---indeed decidability is open
even for the sub-problem in which a zero is sought in a bounded interval. In
this paper we show decidability of the bounded problem subject to Schanuel's
Conjecture, a unifying conjecture in transcendental number theory. We
furthermore analyse the unbounded problem in terms of the frequencies of the
differential equation, that is, the imaginary parts of the characteristic
roots. We show that the unbounded problem can be reduced to the bounded problem
if there is at most one rationally linearly independent frequency, or if there
are two rationally linearly independent frequencies and all characteristic
roots are simple. We complete the picture by showing that decidability of the
unbounded problem in the case of two (or more) rationally linearly independent
frequencies would entail a major new effectiveness result in Diophantine
approximation, namely computability of the Diophantine-approximation types of
all real algebraic numbers.Comment: Full version of paper at ICALP'1
On the Polytope Escape Problem for Continuous Linear Dynamical Systems
The Polyhedral Escape Problem for continuous linear dynamical systems
consists of deciding, given an affine function and a convex polyhedron ,
whether, for some initial point in , the
trajectory of the unique solution to the differential equation
,
, is entirely contained in .
We show that this problem is decidable, by reducing it in polynomial time to
the decision version of linear programming with real algebraic coefficients,
thus placing it in , which lies between NP and PSPACE. Our
algorithm makes use of spectral techniques and relies among others on tools
from Diophantine approximation.Comment: Accepted to HSCC 201
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
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
Continuous-time orbit problems are decidable in polynomial-time
We place the continuous-time orbit problem in P , sharpening the decidability result shown by Hainry [7
On Decidability of Time-Bounded Reachability in CTMDPs
We consider the time-bounded reachability problem for continuous-time Markov
decision processes. We show that the problem is decidable subject to Schanuel's
conjecture. Our decision procedure relies on the structure of optimal policies
and the conditional decidability (under Schanuel's conjecture) of the theory of
reals extended with exponential and trigonometric functions over bounded
domains. We further show that any unconditional decidability result would imply
unconditional decidability of the bounded continuous Skolem problem, or
equivalently, the problem of checking if an exponential polynomial has a
non-tangential zero in a bounded interval. We note that the latter problems are
also decidable subject to Schanuel's conjecture but finding unconditional
decision procedures remain longstanding open problems
Invariants for Continuous Linear Dynamical Systems
Continuous linear dynamical systems are used extensively in mathematics,
computer science, physics, and engineering to model the evolution of a system
over time. A central technique for certifying safety properties of such systems
is by synthesising inductive invariants. This is the task of finding a set of
states that is closed under the dynamics of the system and is disjoint from a
given set of error states. In this paper we study the problem of synthesising
inductive invariants that are definable in o-minimal expansions of the ordered
field of real numbers. In particular, assuming Schanuel's conjecture in
transcendental number theory, we establish effective synthesis of o-minimal
invariants in the case of semi-algebraic error sets. Without using Schanuel's
conjecture, we give a procedure for synthesizing o-minimal invariants that
contain all but a bounded initial segment of the orbit and are disjoint from a
given semi-algebraic error set. We further prove that effective synthesis of
semi-algebraic invariants that contain the whole orbit, is at least as hard as
a certain open problem in transcendental number theory.Comment: Full version of a ICALP 2020 pape