102 research outputs found
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
On the group of a rational maximal bifix code
We give necessary and sufficient conditions for the group of a rational
maximal bifix code to be isomorphic with the -group of , when
is recurrent and is rational. The case where is uniformly
recurrent, which is known to imply the finiteness of , receives
special attention.
The proofs are done by exploring the connections with the structure of the
free profinite monoid over the alphabet of
A profinite approach to complete bifix decodings of recurrent languages
We approach the study of complete bifix decodings of (uniformly) recurrent
languages with the help of the free profinite monoid. We show that the complete
bifix decoding of a uniformly recurrent language by an -charged rational
complete bifix code is uniformly recurrent. An analogous result is obtained for
recurrent languages.Comment: Original Manuscript of article to be published by De Gruyter in Forum
Mathematicum. The last section of the version in Forum Mathematicum is very
different, as there it is not proved that the Sch\"utzenberger group is an
invariant of eventual conjugacy (the argument in the Original Manuscript had
a flaw), but only that its maximal pronilpotent quotient is invariant by
eventual conjugac
Characterizing classes of regular languages using prefix codes of bounded synchronization delay
In this paper we continue a classical work of Sch\"utzenberger on codes with
bounded synchronization delay. He was interested to characterize those regular
languages where the groups in the syntactic monoid belong to a variety . He
allowed operations on the language side which are union, intersection,
concatenation and modified Kleene-star involving a mapping of a prefix code of
bounded synchronization delay to a group , but no complementation. In
our notation this leads to the language classes and
). Our main result shows that always
corresponds to the languages having syntactic monoids where all subgroups are
in . Sch\"utzenberger showed this for a variety if contains Abelian
groups, only. Our method shows the general result for all directly on
finite and infinite words. Furthermore, we introduce the notion of local Rees
products which refers to a simple type of classical Rees extensions. We give a
decomposition of a monoid in terms of its groups and local Rees products. This
gives a somewhat similar, but simpler decomposition than in Rhodes' synthesis
theorem. Moreover, we need a singly exponential number of operations, only.
Finally, our decomposition yields an answer to a question in a recent paper of
Almeida and Kl\'ima about varieties that are closed under Rees products
On Commutativity of Extractable Codes (Algebraic system, Logic, Language and Related Areas in Computer Sciences II)
Deletion and insertion are interesting and common operations which often appear in string rewriting systems. Extractable submonoids and insertable submonoids of free monoids generated by finite alphabets allow to perform deletion operations and insertion operations, respectively. A submonoid N ⊂ A* is called extractable (resp. insertable) if x, uxv ∈ N implies uv ∈ N(resp. x, uv ∈ N implies uxv E N). The code C is called extractable (resp. insertable) if the submonoid C* is extractable (resp. insertable)[7]. Both extractable and insertable codes are identical to well-known strong codes, which is deeply related to the identities of syntactic monoids of languages. This paper deals with the commutativity of extractable codes. After the preliminaries in the first section, we summarize the fundamental properties of codes above in the second section. In the last section, we deal with commutative extractable codes. At first the language operators S and Q, which make languages commutative, are introduced. We show that a commutative extractable code is finite
Random walks on semaphore codes and delay de Bruijn semigroups
We develop a new approach to random walks on de Bruijn graphs over the
alphabet through right congruences on , defined using the natural
right action of . A major role is played by special right congruences,
which correspond to semaphore codes and allow an easier computation of the
hitting time. We show how right congruences can be approximated by special
right congruences.Comment: 34 pages; 10 figures; as requested by the journal, the previous
version of this paper was divided into two; this version contains Sections
1-8 of version 1; Sections 9-12 will appear as a separate paper with extra
material adde
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