187 research outputs found
Cyclic rewriting and conjugacy problems
Cyclic words are equivalence classes of cyclic permutations of ordinary
words. When a group is given by a rewriting relation, a rewriting system on
cyclic words is induced, which is used to construct algorithms to find minimal
length elements of conjugacy classes in the group. These techniques are applied
to the universal groups of Stallings pregroups and in particular to free
products with amalgamation, HNN-extensions and virtually free groups, to yield
simple and intuitive algorithms and proofs of conjugacy criteria.Comment: 37 pages, 1 figure, submitted. Changes to introductio
Partial monoids: associativity and confluence
A partial monoid is a set with a partial multiplication (and
total identity ) which satisfies some associativity axiom. The partial
monoid may be embedded in a free monoid and the product is
simulated by a string rewriting system on that consists in evaluating the
concatenation of two letters as a product in , when it is defined, and a
letter as the empty word . In this paper we study the profound
relations between confluence for such a system and associativity of the
multiplication. Moreover we develop a reduction strategy to ensure confluence
and which allows us to define a multiplication on normal forms associative up
to a given congruence of . Finally we show that this operation is
associative if, and only if, the rewriting system under consideration is
confluent
Star-Free Languages are Church-Rosser Congruential
The class of Church-Rosser congruential languages has been introduced by
McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser
congruential (belongs to CRCL), if there is a finite, confluent, and
length-reducing semi-Thue system S such that L is a finite union of congruence
classes modulo S. To date, it is still open whether every regular language is
in CRCL. In this paper, we show that every star-free language is in CRCL. In
fact, we prove a stronger statement: For every star-free language L there
exists a finite, confluent, and subword-reducing semi-Thue system S such that
the total number of congruence classes modulo S is finite and such that L is a
union of congruence classes modulo S. The construction turns out to be
effective
Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions
What is the common link, if there is any, between Church-Rosser systems,
prefix codes with bounded synchronization delay, and local Rees extensions? The
first obvious answer is that each of these notions relates to topics of
interest for WORDS: Church-Rosser systems are certain rewriting systems over
words, codes are given by sets of words which form a basis of a free submonoid
in the free monoid of all words (over a given alphabet) and local Rees
extensions provide structural insight into regular languages over words. So, it
seems to be a legitimate title for an extended abstract presented at the
conference WORDS 2017. However, this work is more ambitious, it outlines some
less obvious but much more interesting link between these topics. This link is
based on a structure theory of finite monoids with varieties of groups and the
concept of local divisors playing a prominent role. Parts of this work appeared
in a similar form in conference proceedings where proofs and further material
can be found.Comment: Extended abstract of an invited talk given at WORDS 201
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
- …