4,758 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
Near-Optimal Complexity Bounds for Fragments of the Skolem Problem
Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial).
In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds
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
Queries with Guarded Negation (full version)
A well-established and fundamental insight in database theory is that
negation (also known as complementation) tends to make queries difficult to
process and difficult to reason about. Many basic problems are decidable and
admit practical algorithms in the case of unions of conjunctive queries, but
become difficult or even undecidable when queries are allowed to contain
negation. Inspired by recent results in finite model theory, we consider a
restricted form of negation, guarded negation. We introduce a fragment of SQL,
called GN-SQL, as well as a fragment of Datalog with stratified negation,
called GN-Datalog, that allow only guarded negation, and we show that these
query languages are computationally well behaved, in terms of testing query
containment, query evaluation, open-world query answering, and boundedness.
GN-SQL and GN-Datalog subsume a number of well known query languages and
constraint languages, such as unions of conjunctive queries, monadic Datalog,
and frontier-guarded tgds. In addition, an analysis of standard benchmark
workloads shows that most usage of negation in SQL in practice is guarded
negation
On the Positivity Problem for Simple Linear Recurrence Sequences
Given a linear recurrence sequence (LRS) over the integers, the Positivity
Problem} asks whether all terms of the sequence are positive. We show that, for
simple LRS (those whose characteristic polynomial has no repeated roots) of
order 9 or less, Positivity is decidable, with complexity in the Counting
Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277
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
Decidability of the interval temporal logic ABBar over the natural numbers
In this paper, we focus our attention on the interval temporal logic of the
Allen's relations "meets", "begins", and "begun by" (ABBar for short),
interpreted over natural numbers. We first introduce the logic and we show that
it is expressive enough to model distinctive interval properties,such as
accomplishment conditions, to capture basic modalities of point-based temporal
logic, such as the until operator, and to encode relevant metric constraints.
Then, we prove that the satisfiability problem for ABBar over natural numbers
is decidable by providing a small model theorem based on an original
contraction method. Finally, we prove the EXPSPACE-completeness of the proble
Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility
Word equations are a crucial element in the theoretical foundation of
constraint solving over strings, which have received a lot of attention in
recent years. A word equation relates two words over string variables and
constants. Its solution amounts to a function mapping variables to constant
strings that equate the left and right hand sides of the equation. While the
problem of solving word equations is decidable, the decidability of the problem
of solving a word equation with a length constraint (i.e., a constraint
relating the lengths of words in the word equation) has remained a
long-standing open problem. In this paper, we focus on the subclass of
quadratic word equations, i.e., in which each variable occurs at most twice. We
first show that the length abstractions of solutions to quadratic word
equations are in general not Presburger-definable. We then describe a class of
counter systems with Presburger transition relations which capture the length
abstraction of a quadratic word equation with regular constraints. We provide
an encoding of the effect of a simple loop of the counter systems in the theory
of existential Presburger Arithmetic with divisibility (PAD). Since PAD is
decidable, we get a decision procedure for quadratic words equations with
length constraints for which the associated counter system is \emph{flat}
(i.e., all nodes belong to at most one cycle). We show a decidability result
(in fact, also an NP algorithm with a PAD oracle) for a recently proposed
NP-complete fragment of word equations called regular-oriented word equations,
together with length constraints. Decidability holds when the constraints are
additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page
Timed Comparisons of Semi-Markov Processes
Semi-Markov processes are Markovian processes in which the firing time of the
transitions is modelled by probabilistic distributions over positive reals
interpreted as the probability of firing a transition at a certain moment in
time. In this paper we consider the trace-based semantics of semi-Markov
processes, and investigate the question of how to compare two semi-Markov
processes with respect to their time-dependent behaviour. To this end, we
introduce the relation of being "faster than" between processes and study its
algorithmic complexity. Through a connection to probabilistic automata we
obtain hardness results showing in particular that this relation is
undecidable. However, we present an additive approximation algorithm for a
time-bounded variant of the faster-than problem over semi-Markov processes with
slow residence-time functions, and a coNP algorithm for the exact faster-than
problem over unambiguous semi-Markov processes
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