358 research outputs found
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
The Polyhedron-Hitting Problem
We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC
'80 and JACM '86)---determining whether a target polyhedron V may be reached
from a starting point x under repeated applications of a linear transformation
A in an ambient vector space Q^m. In the context of program verification, very
similar reachability questions were also considered and left open by Lee and
Yannakakis in (STOC '92). We present what amounts to a complete
characterisation of the decidability landscape for the Polyhedron-Hitting
Problem, expressed as a function of the dimension m of the ambient space,
together with the dimension of the polyhedral target V: more precisely, for
each pair of dimensions, we either establish decidability, or show hardness for
longstanding number-theoretic open problems
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
Minimisation of Multiplicity Tree Automata
We consider the problem of minimising the number of states in a multiplicity
tree automaton over the field of rational numbers. We give a minimisation
algorithm that runs in polynomial time assuming unit-cost arithmetic. We also
show that a polynomial bound in the standard Turing model would require a
breakthrough in the complexity of polynomial identity testing by proving that
the latter problem is logspace equivalent to the decision version of
minimisation. The developed techniques also improve the state of the art in
multiplicity word automata: we give an NC algorithm for minimising multiplicity
word automata. Finally, we consider the minimal consistency problem: does there
exist an automaton with states that is consistent with a given finite
sample of weight-labelled words or trees? We show that this decision problem is
complete for the existential theory of the rationals, both for words and for
trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor
editing changes from previous versio
Model Checking Markov Chains Against Unambiguous Buchi Automata
We give a polynomial-time algorithm for model checking finite Markov chains
against omega-regular specifications given as unambiguous Buchi automata
On the decidability and complexity of Metric Temporal Logic over finite words
Metric Temporal Logic (MTL) is a prominent specification formalism for
real-time systems. In this paper, we show that the satisfiability problem for
MTL over finite timed words is decidable, with non-primitive recursive
complexity. We also consider the model-checking problem for MTL: whether all
words accepted by a given Alur-Dill timed automaton satisfy a given MTL
formula. We show that this problem is decidable over finite words. Over
infinite words, we show that model checking the safety fragment of MTL--which
includes invariance and time-bounded response properties--is also decidable.
These results are quite surprising in that they contradict various claims to
the contrary that have appeared in the literature
Revisiting Reachability in Timed Automata
We revisit a fundamental result in real-time verification, namely that the
binary reachability relation between configurations of a given timed automaton
is definable in linear arithmetic over the integers and reals. In this paper we
give a new and simpler proof of this result, building on the well-known
reachability analysis of timed automata involving difference bound matrices.
Using this new proof, we give an exponential-space procedure for model checking
the reachability fragment of the logic parametric TCTL. Finally we show that
the latter problem is NEXPTIME-hard
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