102 research outputs found

    Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

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    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

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    We give necessary and sufficient conditions for the group of a rational maximal bifix code ZZ to be isomorphic with the FF-group of ZFZ\cap F, when FF is recurrent and ZFZ\cap F is rational. The case where FF is uniformly recurrent, which is known to imply the finiteness of ZFZ\cap F, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of FF

    Inverse semigroups and varieties of finite semigroups

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    A profinite approach to complete bifix decodings of recurrent languages

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    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 FF by an FF-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

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    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 HH. 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 GHG\in H, but no complementation. In our notation this leads to the language classes SDG(A)SD_G(A^\infty) and SDH(ASD_H(A^\infty). Our main result shows that SDH(A)SD_H(A^\infty) always corresponds to the languages having syntactic monoids where all subgroups are in HH. Sch\"utzenberger showed this for a variety HH if HH contains Abelian groups, only. Our method shows the general result for all HH 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)

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    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

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    We develop a new approach to random walks on de Bruijn graphs over the alphabet AA through right congruences on AkA^k, defined using the natural right action of A+A^+. 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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