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Toggle## McNaughton and Yamada algorithm

The algorithm of McNaughton and Yamada follows the next steps. Given a automaton To *not* states, and whose states are numbered from 1 to *not*, we give an expression for the languages composed of the words which label the paths of *i* To *j*, for any couple *i*, *j*. This expression is built by induction using a condition on the paths. This condition states that the paths only pass through certain authorized states.

Our automaton having *not* states, so we focus on building a regular expression for each of *not*×*not* languages *THE _{s, s'}* = {

*w*∈X * |

*s*.

*w*=

*s*'} formed by all the words that send a state

*s*any on another state

*s'*. If we succeed, the language recognized by the automaton will be the value of a finite union of regular expressions and therefore of a regular expression.

The idea is to proceed step by step by taking into account an increasing number of intermediate states between *s* and *s'*. The states being assumed to be numbered from 0 to *not*, we start with *no* intermediate state, then only *s _{0}*, then

*s*and

_{0}*s*, then

_{1}*s*,

_{0}*s*and

_{1}*s*, etc.

_{2}We notice *THE _{s, s'}^{(k)}* is the expression for the set of words which label paths from s to s' and whose intermediate vertices are less than or equal to k. The starting vertices s and ending s' are not intermediate, so they are not constrained to be less than or equal to k.

We build the *THE _{s, s'}^{(k)}* by induction on k, starting with k = 0, and ending with k = n. When k = n, the constraint on k is no longer a restriction, and

*THE*=

_{s, s'}^{(k)}*THE*if s ≠ s', and ε +

_{s, s' }*THE*=

_{s, s}^{(not)}*THE*.

_{s, s}For k = 0, since the vertices are numbered starting from 1, the constraint simply expresses that there is no intermediate vertex. The only paths are transitions from s to s' (we ignore a path of length 0 in state s).

We therefore have with s = p and s' = q:

For induction, we consider a path from s to s' whose intermediate vertices are smaller than k. Two cases are then possible:

- the intermediate vertices are smaller than k-1, then the label is in
*THE*_{s, s'}^{(k-1)} - the path goes through state k. We then decompose the path into parts whose intermediate vertices are smaller than k-1. For this, we consider each occurrence of the vertex k in this path: between two consecutive occurrences, the intermediate vertices are smaller than k-1. We then have the formula with s = p and s' = q: .

There are therefore n + 1 steps. Each of the steps requires the computation of n² expressions, and the size of the expressions themselves increases with k.

## Example

Consider the following automaton:

With k = 0 we have:

Then k = 1:

Then k = 2:

Then k = 3:

For the last step we just calculate the two expressions that interest us (initial states to final states):

The desired expression is therefore ab (a + b) ∗ + (b + aa) a ∗.