26 research outputs found
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
Acceptance Ambiguity for Quantum Automata
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post\u27s correspondence problem
Computational problems in matrix semigroups
This thesis deals with computational problems that are defined on matrix
semigroups, which playa pivotal role in Mathematics and Computer Science
in such areas as control theory, dynamical systems, hybrid systems, computational
geometry and both classical and quantum computing to name but
a few. Properties that researchers wish to study in such fields often turn out
to be questions regarding the structure of the underlying matrix semigroup
and thus the study of computational problems on such algebraic structures
in linear algebra is of intrinsic importance.
Many natural problems concerning matrix semigroups can be proven
to be intractable or indeed even unsolvable in a formal mathematical sense.
Thus, related problems concerning physical, chemical and biological systems
modelled by such structures have properties which are not amenable to
algorithmic procedures to determine their values.
With such recalcitrant problems we often find that there exists a tight
border between decidability and undecidability dependent upon particular
parameters of the system. Examining this border allows us to determine
which properties we can hope to derive algorithmically and those problems
which will forever be out of our reach, regardless of any future advances in
computational speed.
There are a plethora of open problems in the field related to dynamical
systems, control theory and number theory which we detail throughout
this thesis. We examine undecidability in matrix semigroups for a variety
of different problems such as membership and vector reachability problems,
semigroup intersection emptiness testing and freeness, all of which are well
known from the literature. We also formulate and survey decidability questions
for several new problems such as vector ambiguity, recurrent matrix
problems, the presence of any diagonal matrix and quaternion matrix semigroups,
all of which we feel give a broader perspective to the underlying
structure of matrix semigroups
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
On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups
In this paper we study several closely related fundamental
problems for words and matrices. First, we introduce the Identity Correspondence
Problem (ICP): whether a nite 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 nitely
generated semigroup S of integral square 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 nite set of matrices generates a
group is also undecidable. We also answer several questions for matrices
over di erent number elds. 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
On injectivity of quantum finite automata
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the injectivity problem of determining if the acceptance probability function of a MO-QFA is injective over all input words, i.e., giving a distinct probability for each input word. We show that the injectivity problem is undecidable for 8 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial state vector is real algebraic. We also show undecidability of this problem when the initial vector is rational, although with a huge increase in the number of states. We utilize properties of quaternions, free rotation groups, representations of tuples of rationals as linear sums of radicals and a reduction of the mixed modification of Post's correspondence problem, as well as a new result on rational polynomial packing functions which may be of independent interest.</div
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
Self Organising Maps for Anatomical Joint Constraint
The accurate simulation of anatomical joint models is becoming increasingly important for both realistic animation and diagnostic medical applications. Recent models have exploited unit quaternions to eliminate ingularities when
modelling orientations between limbs at a joint. This has led to
the development of quaternion based joint constraint
validation and correction methods. In this paper a novel
method for implicitly modelling unit quaternion joint
constraints using Self Organizing Maps (SOMs) is proposed
which attempts to address the limitations of current constraint validation and correction approaches. Initial results show that the resulting SOMs are capable of modelling regular spherical constraints on the orientation of the limb