Extension of Brzozowski’s derivation calculus of rational expressions to series over the free partially commutative monoids
We introduce an extension of the derivatives of rational expressions to expressions denoting formal power series over partially commuting variables. The expressions are purely noncommutative, however they denote partially commuting power series. The derivations (which are so-called ϕ -derivations) a...
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| Vydáno v: | Theoretical computer science Ročník 400; číslo 1; s. 144 - 158 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Amsterdam
Elsevier B.V
09.06.2008
Elsevier |
| Témata: | |
| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | We introduce an extension of the derivatives of rational expressions to expressions denoting formal power series over partially commuting variables. The expressions are purely noncommutative, however they denote partially commuting power series. The derivations (which are so-called
ϕ
-derivations) are shown to satisfy the commutation relations.
Our main result states that for every so-called rigid rational expression, there exists a stable finitely generated submodule containing it. Moreover, this submodule is generated by what we call Words, that is by products of letters and of pure stars.
Consequently this submodule is free and it follows that every rigid rational expression represents a recognizable series in
K
《
A
/
C
》
. This generalizes the previously known property where the star was restricted to mono-alphabetic and connected series. |
|---|---|
| AbstractList | We introduce an extension of the derivatives of rational expressions to expressions denoting formal power series over partially commuting variables. The expressions are purely noncommutative, however they denote partially commuting power series. The derivations (which are so-called
ϕ
-derivations) are shown to satisfy the commutation relations.
Our main result states that for every so-called rigid rational expression, there exists a stable finitely generated submodule containing it. Moreover, this submodule is generated by what we call Words, that is by products of letters and of pure stars.
Consequently this submodule is free and it follows that every rigid rational expression represents a recognizable series in
K
《
A
/
C
》
. This generalizes the previously known property where the star was restricted to mono-alphabetic and connected series. |
| Author | Reutenauer, Christophe Berstel, Jean |
| Author_xml | – sequence: 1 givenname: Jean surname: Berstel fullname: Berstel, Jean organization: Institut Gaspard-Monge (IGM), Université Paris-Est, France – sequence: 2 givenname: Christophe surname: Reutenauer fullname: Reutenauer, Christophe email: Christophe.Reutenauer@uqam.ca organization: LaCIM Université du Québec Montréal, Canada |
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| Cites_doi | 10.1051/ita/1982160201391 10.1006/inco.1999.2799 10.1016/0304-3975(93)90124-C 10.1016/S0022-0000(71)80005-3 10.1016/j.tcs.2004.10.016 10.1016/S0019-9958(72)90865-0 10.1016/0304-3975(95)00182-4 10.1051/ita/1983170100031 10.3233/FI-1996-253401 10.1016/0304-3975(92)90345-G 10.1145/321239.321249 |
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| Keywords | Recognizable Free partially commutative Rational expressions Word Computer theory Formal series Product Power series Letter Commutation relation Recognizable series Derivative Free monoid By product Power |
| Language | English |
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| References_xml | – volume: 25 start-page: 217 year: 1996 end-page: 239 ident: b2 article-title: Towards an algebraic theory of context-free languages publication-title: Fund. Inform. – reference: S. Dulucq, Equations avec opérateurs: Un outil combinatoire. Thèse de doctorat, Université de Bordeaux I, 1981 – volume: 11 start-page: 481 year: 1964 end-page: 494 ident: b4 article-title: Derivatives of regular expressions publication-title: J. Assoc. Comput. Mach. – reference: G. Jacob, Représentations et substitutions matricielles dans la théorie algébrique des transductions, Thesis, University of Paris, 1975 – volume: vol. 1478 start-page: 215 year: 1998 end-page: 243 ident: b14 article-title: Expressions rationnelles sur un anneau publication-title: Topics in Invariant Theory – volume: 155 start-page: 291 year: 1996 end-page: 319 ident: b1 article-title: Partial derivatives of regular expressions publication-title: Theoret. Comput. 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| SubjectTerms | Applied sciences Computer Science Computer science; control theory; systems Data Structures and Algorithms Exact sciences and technology Free partially commutative Miscellaneous Rational expressions Recognizable Theoretical computing |
| Title | Extension of Brzozowski’s derivation calculus of rational expressions to series over the free partially commutative monoids |
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