Rational series and their languages
Rational series and their languages
On the partially commutative shuffle product
Theoretical Computer Science
The shuffle algebra and its derivations
Theoretical Computer Science
Partial derivatives of regular expressions and finite automaton constructions
Theoretical Computer Science
Towards an algebraic theory of context-free languages
Fundamenta Informaticae - Special issue on formal language theory
The Kleene-Schützenberger theorem for formal power series in partially commuting variables
Information and Computation
Derivatives of Regular Expressions
Journal of the ACM (JACM)
Derivatives of rational expressions with multiplicity
Theoretical Computer Science
Realizations by stochastic finite automata
Journal of Computer and System Sciences
Weighted and unweighted trace automata
Acta Cybernetica
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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 @f-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. This generalizes the previously known property where the star was restricted to mono-alphabetic and connected series.