We introduce semaphore codes associated to a Turing machine via resets. Semaphore codes provide an approximation theory for resets. In this paper we generalize the set-up of our previous paper "Random walks on semaphore codes and delay de Bruijn semigroups" to the infinite case by taking the profinite limit of $k$-resets to obtain $(-\omega)$-resets. We mention how this opens new avenues to attack the P versus NP problem.

We develop a new approach to random walks on de Bruijn graphs over the alphabet $A$ through right congruences on $A^k$, defined using the natural right action of $A^+$. A major role is played by special right congruences, which correspond to semaphore codes and allow an easier computation of the hitting time. We show how right congruences can be approximated by special right congruences.

In this paper, we provide a simple proof of the Holonomy theorem using a new Lyndon-Chiswell length function on the Karnofsky-Rhodes expansion of a semigroup. Unexpectedly, we have both a left and a right action on the Chiswell tree by elliptic maps.

We provide a simple proof of the Holonomy Theorem using a new Lyndon-Chiswell length function on the Karnofsky-Rhodes expansion of a semigroup. Unexpectedly, we have both a left and a right action on the Chiswell tree by elliptic maps.