Star Height Problem - Families of Regular Languages With Unbounded Star Height

Families of Regular Languages With Unbounded Star Height

The first question was answered in the negative when in 1963, Eggan gave examples of regular languages of star height n for every n. Here, the star height h(L) of a regular language L is defined as the minimum star height among all regular expressions representing L. The first few languages found by Eggan (1963) are described in the following, by means of giving a regular expression for each language:

\begin{alignat}{2} e_1 &= a_1^* \\ e_2 &= \left(a_1^*a_2^*a_3\right)^*\\ e_3 &= \left(\left(a_1^*a_2^*a_3\right)^*\left(a_4^*a_5^*a_6\right)^*a_7\right)^*\\ e_4 &= \left( \left(\left(a_1^*a_2^*a_3\right)^*\left(a_4^*a_5^*a_6\right)^*a_7\right)^* \left(\left(a_8^*a_9^*a_{10}\right)^*\left(a_{11}^*a_{12}^*a_{13}\right)^*a_{14}\right)^* a_{15}\right)^* \end{alignat}

The construction principle for these expressions is that expression is obtained by concatenating two copies of, appropriately renaming the letters of the second copy using fresh alphabet symbols, concatenating the result with another fresh alphabet symbol, and then by surrounding the resulting expression with a Kleene star. The remaining, more difficult part, is to prove that for there is no equivalent regular expression of star height less than n; a proof is given in (Eggan 1963).

However, Eggan's examples use a large alphabet, of size 2n-1 for the language with star height n. He thus asked whether we can also find examples over binary alphabets. This was proved to be true shortly afterwards by Dejean & Schützenberger (1966). Their examples can be described by an inductively defined family of regular expressions over the binary alphabet as follows–cf. Salomaa (1981):

\begin{alignat}{2} e_1 & = (ab)^* \\ e_2 & = \left(aa(ab)^*bb(ab)^*\right)^* \\ e_3 & = \left(aaaa \left(aa(ab)^*bb(ab)^*\right)^* bbbb \left(aa(ab)^*bb(ab)^*\right)^*\right)^* \\ \, & \cdots \\ e_{n+1} & = (\,\underbrace{a\cdots a}_{2^n}\, \cdot \, e_n\, \cdot\, \underbrace{b\cdots b}_{2^n}\, \cdot\, e_n \,)^* \end{alignat}

Again, a rigorous proof is needed for the fact that does not admit an equivalent regular expression of lower star height. Proofs are given by (Dejean & Schützenberger 1966) and by (Salomaa 1981).

Read more about this topic:  Star Height Problem

Famous quotes containing the words families, regular, languages, star and/or height:

    “The authoritarian child-rearing style so often found in working-class families stems in part from the fact that parents see around them so many young people whose lives are touched by the pain and delinquency that so often accompanies a life of poverty. Therefore, these parents live in fear for their children’s future—fear that they’ll lose control, that the children will wind up on the streets or, worse yet, in jail.”
    —Lillian Breslow Rubin (20th century)

    “The solid and well-defined fir-tops, like sharp and regular spearheads, black against the sky, gave a peculiar, dark, and sombre look to the forest.”
    —Henry David Thoreau (1817–1862)

    “No doubt, to a man of sense, travel offers advantages. As many languages as he has, as many friends, as many arts and trades, so many times is he a man. A foreign country is a point of comparison, wherefrom to judge his own.”
    —Ralph Waldo Emerson (1803–1882)

    “Never tell me that not one star of all
    That slip from heaven at night and softly fall
    Has been picked up with stones to build a wall.”
    —Robert Frost (1874–1963)

    “I pray that, according to the riches of his glory, he may grant that you may be strengthened in your inner being with power through his Spirit, and that Christ may dwell in your hearts through faith, as you are being rooted and grounded in love. I pray that you may have the power to comprehend, with all the saints, what is the breadth and length and height and depth, and to know the love of Christ that surpasses knowledge, so that you may be filled with all the fullness of God.”
    —Bible: New Testament, Ephesians 3:16-19.