5

I am taking an undergraduate CS Theory course and the material on finite automata and regular languages is being taught in a non-traditional manner. Instead of using regular expressions, the closure properties of regular languages, the pumping lemma etc, to show that a language is or is not regular, all of our proofs and examples for identifying regular languages use the Myhill-Nerode theorem and boolean matrices as defined below.

Let $L = \{w_1, w_2, ...\}$ be a set of words over some alphabet $\Sigma$ and let $T_L$ be a matrix with entries $t_{ij}$ where

$$ t_{ij} = \left\{ \begin{array}{ll} 1 & \mbox{if } w_iw_j \in L \\ 0 & \mbox{otherwise} \end{array} \right. $$

Then by Myhill-Nerode, L is non-regular iff all rows of $T_L$ are distinct.

Correction: Then by Myhill-Nerode, $L$ is non-regular iff $T_L$ has an infinite number of distinct rows.

My question is, are there any readily available books, papers, or lecture notes that lean heavily on this technique and are appropriate for undergrads?

Sean Grossman
  • 51
  • 2

1 Answers1

2

I wanted to make this a comment, but it is too long.

First, your definition of the matrix is poorly written and probably wrong. There is no reason to introduce indices $i, j$. My guess is that the right definition is the following: $T$ is the infinite matrix $(T_{u,v})_{u, v \in A^*}$ defined by $$ T_{u,v} = \begin{cases} 1 &\text{if $uv \in L$}\\ 0 &\text{otherwise} \end{cases} $$ Next, your claim

$L$ is non-regular if and only if all rows of $T$ are distinct

is wrong. Take $A = \{a, b\}$ and $L = \{ u \in A^* \mid |u|_a = |u|_b\}^1$. Then $T_{1,v} = T_{ab,v}$ for all $v \in A^*$, so the rows $T_{1, -}$ and $T_{ab,-}$ are equal although $L$ is not regular. What is true however is that $L$ is regular if and only if $T$ has finitely many distinct rows.
${}_{\text{(1) $|u|_c$ denotes the number of occurrences of the letter $c$ in $u$.}}$

References. The matrix $T$ is used in the larger setting of noncommutative formal power series under the name of Hankel matrix. Here is a good reference, but I am afraid it is not really appropriate for undergraduate students.

[1] J. Berstel and C. Reutenauer, Noncommutative rational series with applications. Encyclopedia of Mathematics and its Applications, 137. Cambridge University Press, Cambridge, 2011. xiv+248 pp. ISBN: 978-0-521-19022-0

J.-E. Pin
  • 6,219
  • 21
  • 39