The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature. Lifting this result to a weighted setting has been widely studied. Moreover, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalising both the free inverse monoid and graded monoids. The presence of idempotent elements in these pre-rational monoids leads in the weighted setting to consider infinite sums. To handle such sums, we will have to restrict ourselves to rationally additive semirings. Our main result is thus a generalisation of the Kleene theorem for pre-rational monoids and rationally additive semirings. As a corollary, we obtain a class of expressions equivalent to weighted two-way automata, as well as one for tree-walking automata.

This paper deals with the conversion of expressions denoting Hadamard series into weighted rotating automata. We prove that any algorithm converting rational series into one-way weighted automata can be extended to provide an algorithm which achieves our goal. We apply this to define the derivation and the follow automata of a Hadamard expression. Our method may also be used to extend algorithms which perform the inverse conversion, but it is required to enhance the algorithm to fulfill some constraints.

Two-way transducers or weighted automata are in general more powerful than one-way ones. We show that two-way automata over locally finite semirings may have undefined behaviour. We prove that it is decidable whether this behaviour is defined, and, if it is, we show that two-way automata over locally finite semirings are equivalent to one-way automata.