1,999 research outputs found
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 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
A generalization of the problem of Mariusz Meszka
Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L
containing n positive integers not exceeding n there exists a near 1-factor in
K_p whose list of edge-lengths is L. In this paper we propose a generalization
of this problem to the case in which p is an odd integer not necessarily prime.
In particular, we give a necessary condition for the existence of such a near
1-factor for any odd integer p. We show that this condition is also sufficient
for any list L whose underlying set S has size 1, 2, or n. Then we prove that
the conjecture is true if S={1,2,t} for any positive integer t not coprime with
the order p of the complete graph. Also, we give partial results when t and p
are coprime. Finally, we present a complete solution for t<12.Comment: 15 page
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