19 research outputs found
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
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
Decision Questions for Probabilistic Automata on Small Alphabets
We study the emptiness and ?-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and ?-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is over {0, 1}, we show they are in NP. In contrast to the Skolem-hardness of the ?-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with commuting transition matrices, we prove NP-hardness of the ?-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems
Decidability of Cutpoint Isolation for Probabilistic Finite Automata on Letter-Bounded Inputs
We show the surprising result that the cutpoint isolation problem is decidable for probabilistic finite automata where input words are taken from a letter-bounded context-free language. A context-free language ? is letter-bounded when ? ? a?^* a?^* ? a_?^* for some finite ? > 0 where each letter is distinct. A cutpoint is isolated when it cannot be approached arbitrarily closely. The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-bounded language as well as to the injectivity problem which is undecidable for PFA over letter-bounded languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, which holds even when the PFA is exponentially ambiguous. We also show that the problem is at least NP-hard and use our decision procedure to solve several related problems
Decision Questions for Probabilistic Automata on Small Alphabets
We study the emptiness and -reachability problems for unary and
binary Probabilistic Finite Automata (PFA) and characterise the complexity of
these problems in terms of the degree of ambiguity of the automaton and the
size of its alphabet. Our main result is that emptiness and
-reachability are solvable in EXPTIME for polynomially ambiguous unary
PFA and if, in addition, the transition matrix is binary, we show they are in
NP. In contrast to the Skolem-hardness of the -reachability and
emptiness problems for exponentially ambiguous unary PFA, we show that these
problems are NP-hard even for finitely ambiguous unary PFA. For binary
polynomially ambiguous PFA with fixed and commuting transition matrices, we
prove NP-hardness of the -reachability (dimension 9), nonstrict
emptiness (dimension 37) and strict emptiness (dimension 40) problems.Comment: Updated journal pre-prin
On the Identity and Group Problems for Complex Heisenberg Matrices
We study the Identity Problem, the problem of determining if a finitely
generated semigroup of matrices contains the identity matrix; see Problem 3
(Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control
Theory'' by Blondel and Megretski (2004). This fundamental problem is known to
be undecidable for and decidable for . The Identity Problem has been recently shown to be in polynomial
time by Dong for the Heisenberg group over complex numbers in any fixed
dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula.
We develop alternative proof techniques for the problem making a step forward
towards more general problems such as the Membership Problem. We extend our
techniques to show that the fundamental problem of determining if a given set
of Heisenberg matrices generates a group, can also be decided in polynomial
time
Decidability of membership problems for flat rational subsets of and singular matrices
This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove various new decidability results for matrices over . For that, we introduce the concept of flat rational sets: if is a monoid and is a submonoid, then ``flat rational sets of over '' are finite unions of the form where all 's are rational subsets of and . We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of over is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further inside . We also show a dichotomy for nontrivial group extension of in : if is a f.g. group such that , then either , for some , or contains an extension of the Baumslag-Solitar group , with , of infinite index. In the first case of the dichotomy the membership problem for is decidable but the equality problem for rational subsets of is undecidable. In the second case, decidability of the membership problem for rational subsets in is open. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable (in doubly exponential time) for flat rational subsets of over the submonoid that is generated by the matrices from with determinants in
On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond
We consider the following variant of the Mortality Problem: given matrices , does there exist nonnegative integers such that the product is equal to the zero matrix? It is known that this problem is decidable when for matrices over algebraic numbers but becomes undecidable for sufficiently large and even for integral matrices. In this paper, we prove the first decidability results for . We show as one of our central results that for this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for and for matrices over algebraic numbers and for and for matrices over real algebraic numbers. Another consequence is that the set of triples for which the equation equals the zero matrix is equal to a finite union of direct products of semilinear sets. For we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations