190 research outputs found
(Un)decidable Problems about Reachability of Quantum Systems
We study the reachability problem of a quantum system modelled by a quantum
automaton. The reachable sets are chosen to be boolean combinations of (closed)
subspaces of the state space of the quantum system. Four different reachability
properties are considered: eventually reachable, globally reachable, ultimately
forever reachable, and infinitely often reachable. The main result of this
paper is that all of the four reachability properties are undecidable in
general; however, the last three become decidable if the reachable sets are
boolean combinations without negation
Vector Reachability Problem in
The decision problems on matrices were intensively studied for many decades
as matrix products play an essential role in the representation of various
computational processes. However, many computational problems for matrix
semigroups are inherently difficult to solve even for problems in low
dimensions and most matrix semigroup problems become undecidable in general
starting from dimension three or four.
This paper solves two open problems about the decidability of the vector
reachability problem over a finitely generated semigroup of matrices from
and the point to point reachability (over rational
numbers) for fractional linear transformations, where associated matrices are
from . The approach to solving reachability problems
is based on the characterization of reachability paths between points which is
followed by the translation of numerical problems on matrices into
computational and combinatorial problems on words and formal languages. We also
give a geometric interpretation of reachability paths and extend the
decidability results to matrix products represented by arbitrary labelled
directed graphs. Finally, we will use this technique to prove that a special
case of the scalar reachability problem is decidable
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
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
Decidability of the Membership Problem for integer matrices
The main result of this paper is the decidability of the membership problem
for nonsingular integer matrices. Namely, we will construct the
first algorithm that for any nonsingular integer matrices
and decides whether belongs to the semigroup generated
by .
Our algorithm relies on a translation of the numerical problem on matrices
into combinatorial problems on words. It also makes use of some algebraical
properties of well-known subgroups of and various
new techniques and constructions that help to limit an infinite number of
possibilities by reducing them to the membership problem for regular languages
Approximated Symbolic Computations over Hybrid Automata
Hybrid automata are a natural framework for modeling and analyzing systems
which exhibit a mixed discrete continuous behaviour. However, the standard
operational semantics defined over such models implicitly assume perfect
knowledge of the real systems and infinite precision measurements. Such
assumptions are not only unrealistic, but often lead to the construction of
misleading models. For these reasons we believe that it is necessary to
introduce more flexible semantics able to manage with noise, partial
information, and finite precision instruments. In particular, in this paper we
integrate in a single framework based on approximated semantics different over
and under-approximation techniques for hybrid automata. Our framework allows to
both compare, mix, and generalize such techniques obtaining different
approximated reachability algorithms.Comment: In Proceedings HAS 2013, arXiv:1308.490
The Identity Correspondence Problem and its Applications
In this paper we study several closely related fundamental problems for words
and matrices. First, we introduce the Identity Correspondence Problem (ICP):
whether a finite set of pairs of words (over a group alphabet) can generate an
identity pair by a sequence of concatenations. We prove that ICP is undecidable
by a reduction of Post's Correspondence Problem via several new encoding
techniques.
In the second part of the paper we use ICP to answer a long standing open
problem concerning matrix semigroups: "Is it decidable for a finitely generated
semigroup S of square integral matrices whether or not the identity matrix
belongs to S?". We show that the problem is undecidable starting from dimension
four even when the number of matrices in the generator is 48. From this fact,
we can immediately derive that the fundamental problem of whether a finite set
of matrices generates a group is also undecidable. We also answer several
question for matrices over different number fields. Apart from the application
to matrix problems, we believe that the Identity Correspondence Problem will
also be useful in identifying new areas of undecidable problems in abstract
algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the
published journal version of this article, see footnote 3 on page 1
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