214 research outputs found
Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids
There have been several attempts to extend the notion of conjugacy from
groups to monoids. The aim of this paper is study the decidability and
independence of conjugacy problems for three of these notions (which we will
denote by , , and ) in certain classes of finitely
presented monoids. We will show that in the class of polycyclic monoids,
-conjugacy is "almost" transitive, is strictly included in
, and the - and -conjugacy problems are decidable with linear
compexity. For other classes of monoids, the situation is more complicated. We
show that there exists a monoid defined by a finite complete presentation
such that the -conjugacy problem for is undecidable, and that for
finitely presented monoids, the -conjugacy problem and the word problem are
independent, as are the -conjugacy and -conjugacy problems.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1503.0091
Markov semigroups, monoids, and groups
A group is Markov if it admits a prefix-closed regular language of unique
representatives with respect to some generating set, and strongly Markov if it
admits such a language of unique minimal-length representatives over every
generating set. This paper considers the natural generalizations of these
concepts to semigroups and monoids. Two distinct potential generalizations to
monoids are shown to be equivalent. Various interesting examples are presented,
including an example of a non-Markov monoid that nevertheless admits a regular
language of unique representatives over any generating set. It is shown that
all finitely generated commutative semigroups are strongly Markov, but that
finitely generated subsemigroups of virtually abelian or polycyclic groups need
not be. Potential connections with word-hyperbolic semigroups are investigated.
A study is made of the interaction of the classes of Markov and strongly Markov
semigroups with direct products, free products, and finite-index subsemigroups
and extensions. Several questions are posed.Comment: 40 pages; 3 figure
Languages Generated by Iterated Idempotencies.
The rewrite relation with parameters m and n and with the possible length limit = k or :::; k we denote by w~, =kW~· or ::;kw~ respectively. The idempotency languages generated from a starting word w by the respective operations are wDAlso other special cases of idempotency languages besides duplication have come up in different contexts. The investigations of Ito et al. about insertion and deletion, Le., operations that are also observed in DNA molecules, have established that w5 and w~ both preserve regularity.Our investigations about idempotency relations and languages start out from the case of a uniform length bound. For these relations =kW~ the conditions for confluence are characterized completely. Also the question of regularity is -k n answered for aH the languages w- D 1 are more complicated and belong to the class of context-free languages.For a generallength bound, i.e."for the relations :"::kW~, confluence does not hold so frequently. This complicatedness of the relations results also in more complicated languages, which are often non-regular, as for example the languages WWithout any length bound, idempotency relations have a very complicated structure. Over alphabets of one or two letters we still characterize the conditions for confluence. Over three or more letters, in contrast, only a few cases are solved. We determine the combinations of parameters that result in the regularity of wDIn a second chapter sorne more involved questions are solved for the special case of duplication. First we shed sorne light on the reasons why it is so difficult to determine the context-freeness ofduplication languages. We show that they fulfiH aH pumping properties and that they are very dense. Therefore aH the standard tools to prove non-context-freness do not apply here.The concept of root in Formal Language ·Theory is frequently used to describe the reduction of a word to another one, which is in sorne sense elementary.For example, there are primitive roots, periodicity roots, etc. Elementary in connection with duplication are square-free words, Le., words that do not contain any repetition. Thus we define the duplication root of w to consist of aH the square-free words, from which w can be reached via the relation w~.Besides sorne general observations we prove the decidability of the question, whether the duplication root of a language is finite.Then we devise acode, which is robust under duplication of its code words.This would keep the result of a computation from being destroyed by dupli cations in the code words. We determine the exact conditions, under which infinite such codes exist: over an alphabet of two letters they exist for a length bound of 2, over three letters already for a length bound of 1.Also we apply duplication to entire languages rather than to single words; then it is interesting to determine, whether regular and context-free languages are closed under this operation. We show that the regular languages are closed under uniformly bounded duplication, while they are not closed under duplication with a generallength bound. The context-free languages are closed under both operations.The thesis concludes with a list of open problems related with the thesis' topics
A complete transformation rule set and a minimal equation set for CNOT-based 3-qubit quantum circuits (Draft)
We introduce a complete transformation rule set and a minimal equation set
for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum
circuits that compute the same Boolean function are reduced to the same normal
form. We can thus easily check the equivalence of circuits by comparing their
normal forms. By applying the Knuth-Bendix completion algorithm to a set of
modified 18 equations introduced by Iwama et al. 2002, we obtain a complete
transformation rule set (i.e., a set of transformation rules with the
properties of `termination' and `confluence'). Our transformation rule set
consists of 114 rules. Moreover, we discovered a minimal combination of
equations for the initial equation set
Geodesic rewriting systems and pregroups
In this paper we study rewriting systems for groups and monoids, focusing on
situations where finite convergent systems may be difficult to find or do not
exist. We consider systems which have no length increasing rules and are
confluent and then systems in which the length reducing rules lead to
geodesics. Combining these properties we arrive at our main object of study
which we call geodesically perfect rewriting systems. We show that these are
well-behaved and convenient to use, and give several examples of classes of
groups for which they can be constructed from natural presentations. We
describe a Knuth-Bendix completion process to construct such systems, show how
they may be found with the help of Stallings' pregroups and conversely may be
used to construct such pregroups.Comment: 44 pages, to appear in "Combinatorial and Geometric Group Theory,
Dortmund and Carleton Conferences". Series: Trends in Mathematics.
Bogopolski, O.; Bumagin, I.; Kharlampovich, O.; Ventura, E. (Eds.) 2009,
Approx. 350 p., Hardcover. ISBN: 978-3-7643-9910-8 Birkhause
Coherence in monoidal track categories
We introduce homotopical methods based on rewriting on higher-dimensional
categories to prove coherence results in categories with an algebraic
structure. We express the coherence problem for (symmetric) monoidal categories
as an asphericity problem for a track category and we use rewriting methods on
polygraphs to solve it. The setting is extended to more general coherence
problems, seen as 3-dimensional word problems in a track category, including
the case of braided monoidal categories.Comment: 32 page
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