229 research outputs found
Green's Relations in Finite Transformation Semigroups
We consider the complexity of Green's relations when the semigroup is given
by transformations on a finite set. Green's relations can be defined by
reachability in the (right/left/two-sided) Cayley graph. The equivalence
classes then correspond to the strongly connected components. It is not
difficult to show that, in the worst case, the number of equivalence classes is
in the same order of magnitude as the number of elements. Another important
parameter is the maximal length of a chain of components. Our main contribution
is an exponential lower bound for this parameter. There is a simple
construction for an arbitrary set of generators. However, the proof for
constant alphabet is rather involved. Our results also apply to automata and
their syntactic semigroups.Comment: Full version of a paper submitted to CSR 2017 on 2016-12-1
Effective Theories for Circuits and Automata
Abstracting an effective theory from a complicated process is central to the
study of complexity. Even when the underlying mechanisms are understood, or at
least measurable, the presence of dissipation and irreversibility in
biological, computational and social systems makes the problem harder. Here we
demonstrate the construction of effective theories in the presence of both
irreversibility and noise, in a dynamical model with underlying feedback. We
use the Krohn-Rhodes theorem to show how the composition of underlying
mechanisms can lead to innovations in the emergent effective theory. We show
how dissipation and irreversibility fundamentally limit the lifetimes of these
emergent structures, even though, on short timescales, the group properties may
be enriched compared to their noiseless counterparts.Comment: 11 pages, 9 figure
Finding All Solutions of Equations in Free Groups and Monoids with Involution
The aim of this paper is to present a PSPACE algorithm which yields a finite
graph of exponential size and which describes the set of all solutions of
equations in free groups as well as the set of all solutions of equations in
free monoids with involution in the presence of rational constraints. This
became possible due to the recently invented emph{recompression} technique of
the second author.
He successfully applied the recompression technique for pure word equations
without involution or rational constraints. In particular, his method could not
be used as a black box for free groups (even without rational constraints).
Actually, the presence of an involution (inverse elements) and rational
constraints complicates the situation and some additional analysis is
necessary. Still, the recompression technique is general enough to accommodate
both extensions. In the end, it simplifies proofs that solving word equations
is in PSPACE (Plandowski 1999) and the corresponding result for equations in
free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As
a byproduct we obtain a direct proof that it is decidable in PSPACE whether or
not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk
at CSR 2014 in Moscow, June 7 - 11, 201
The Computational Complexity of Symbolic Dynamics at the Onset of Chaos
In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
manuscrip
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
Gorenstein homological algebra and universal coefficient theorems
We study criteria for a ringâor more generally, for a small categoryâto be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neemanâs BrownâAdams representability theorem for compactly generated categories
Algebraic description of spacetime foam
A mathematical formalism for treating spacetime topology as a quantum
observable is provided. We describe spacetime foam entirely in algebraic terms.
To implement the correspondence principle we express the classical spacetime
manifold of general relativity and the commutative coordinates of its events by
means of appropriate limit constructions.Comment: 34 pages, LaTeX2e, the section concerning classical spacetimes in the
limit essentially correcte
Uniqueness of roots up to conjugacy for some affine and finite type Artin groups
Let be one of the Artin groups of finite type , and affine type and .
In this paper, we show that if and are elements of such
that for some nonzero integer , then and
are conjugate in . For the Artin group of type , this was
recently proved by J. Gonz\'alez-Meneses.
In fact, we prove a stronger theorem, from which the above result follows
easily by using descriptions of those Artin groups as subgroups of the braid
group on strands. Let be a subset of . An -braid is
said to be \emph{-pure} if its induced permutation fixes each , and
\emph{-straight} if it is -pure and it becomes trivial when we delete all
the -th strands for . Exploiting the Nielsen-Thurston
classification of braids, we show that if and are -pure
-braids such that for some nonzero integer , then
there exists a -straight -braid with
. Moreover, if , the conjugating element
can be chosen to have the first strand algebraically unlinked with the
other strands. Especially in case of , our result implies the
uniqueness of root of pure braids, which was known by V. G. Bardakov and by D.
Kim and D. Rolfsen.Comment: 15 pages, 8 figures; version published by Math.
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