On the erasure of several letter-transitions
Résumé
We present here some algebraic formulas enabeling to define a $k$-automaton $\mathcal A_2$ from a given $k$-automaton $\mathcal A_1$ such that the behaviour of $\mathcal A_2$ is the behaviour of $\mathcal A_1$ after erasure of a given set of letters. This procedure contains as particular case the algebraic elimination of $\ep$-transitions. The time complexity of this process is evaluated. In the case of well-known semirings (boolean and tropical) the closure is computed in $O(n^3)$. When $k$ is a ring, the complexity can be more finely tuned.