  We study a class of finite state machines, called \defi{$w$-matching
machines}, which yield to simulate the behavior of pattern matching algorithms
while searching for a pattern $w$. They can be used to compute the asymptotic
speed, i.e. the limit of the expected ratio of the number of text accesses to
the length of the text, of algorithms while parsing an iid text to find the
pattern $w$.
  Defining the order of a matching machine or of an algorithm as the maximum
difference between the current and accessed positions during a search (standard
algorithms are generally of order $|w|$), we show that being given a pattern
$w$, an order $k$ and an iid model, there exists an optimal $w$-matching
machine, i.e. with the greatest asymptotic speed under the model among all the
machines of order $k$, of which the set of states belongs to a finite and
enumerable set.
  It shows that it is possible to determine: 1) the greatest asymptotic speed
among a large class of algorithms, with regard to a pattern and an iid model,
and 2) a $w$-matching machine, thus an algorithm, achieving this speed.
