The family of one-rule grid semi-Thue systems, introduced by Alfons Geser, is the family of one-rule semi-Thue systems such that there exists a letter c that occurs as often in the left-hand side as the right-hand side of the rewriting rule. We prove

Weighted Finite Automata are automata with multiplicities used to compute real functions by reading infinite words. The aim of this paper is to study what kind of functions can be computed by level automata, a particular subclass of WFA. Several results concerning the continuity and the smoothness of these functions are shown. In particular, the only smooth functions that can be obtained are the polynomials. This enables to decide whether a function computed by a level automaton is smooth or not.

We study the family S of rational cones obtained by iterated substitutions from rational cones L1, .., Ln. This family is a semi-group and to every non empty word u defined on the alphabet {L1, ..., Ln}, corresponds a rational cone U of S. We give sufficient conditions for S to be free (U = U′ implies u = u′) and to verify the subpattern property (U ⊂ U′ implies u is a subpattern of u′). We study, more particularly, the case where L1, ..., Ln are bounded rational cones.