Free (rational) derivation

Derivation of Rational Expressions with Multiplicity

This paper addresses the problem of turning a rational (ie regular) expression into a finite automaton. We formalize and generalize the idea of “partial derivatives” introduced in 1995 by V. Antimirov, in order to obtain a construction of an automaton with multiplicity from a rational expression describing a formal power series with coefficients in a semiring. We first define precisely what is ...

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) are shown to satisfy the commutation relations. Our main result states that for every so-called rig...

On Term Rewriting Systems Having a Rational Derivation

Several types of term rewriting systems can be distinguished by the way their rules overlap. In particular, we define the classes of prefix, suffix, bottom-up and top-down systems, which generalize similar classes on words. Our aim is to study the derivation relation of such systems (i.e. the reflexive and transitive closure of their rewriting relation) and, if possible, to provide a finite mec...

On Word Rewriting Systems Having a Rational Derivation

We define four families of word-rewriting systems: the prefix/suffix systems and the left/right systems. The rewriting of prefix systems generalizes the prefix rewriting of systems: a system is prefix (suffix) if a left hand side and a right hand side are overlapping only by prefix (suffix). The rewriting of right systems generalizes the mechanism of transducers: a system is right (left) if a l...

Free Will and Rational Coherency