Abstract. The paper investigates non-deterministic, probabilistic and quantum walks, from the perspective of coalgebras and monads. Nondeterministic and probabilistic walks are coalgebras of a monad (powerset and distribution), in an obvious manner. It is shown that also quantum walks are coalgebras of a new monad, involving additional control structure. This new monad is also used to describe Turing machines coalgebraically, namely as controlled ‘walks ’ on a tape. 1

We introduce the decreasing proof order. It orders a conversion above another conversion if the latter is obtained by filling any peak in the former by a decreasing diagram. The result is developed in the setting of involutive monoids.