The Generalized Star-Height Problem

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Giles McKinney
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1 School of Mathematics & Statistics University of St Andrews CIRCA Lunch Seminar 9th April 2015

2 Regular Expressions Given a finite alphabet A, we define, ε (the empty word), and a in A to be regular expressions. If E and F are regular expressions then we recursively define new regular expressions by using the following operations: EF (concatenation) E F (set union) E (Kleene star) We use regular expressions to represent regular languages. For example, if A = {a, b} then A a = (a b) a represents the language in which all words end with the letter a.

3 Star-Height The star-height h(e) of a regular expression E is defined recursively by: h( ) = h(ε) = h(a) = 0, where a A; h(ef ) = h(e F ) = max{h(e), h(f )}; h(e ) = h(e) + 1. For example, over A = {a, b}, the expression (aa b ) has star-height 2.

4 Star-Height The star-height h(e) of a regular expression E is defined recursively by: h( ) = h(ε) = h(a) = 0, where a A; h(ef ) = h(e F ) = max{h(e), h(f )}; h(e ) = h(e) + 1. For example, over A = {a, b}, the expression (aa b ) has star-height 2. Then, for a language L A, we define the star-height of L by h(l) = min{h(e) : E is a regular expression for L}.

5 Generalized Star-Height Now suppose that in addition to the aforementioned operations for defining regular expressions, we also allow complementation; that is, if E is a regular expression then so is E c. Including the complement operation leads us to refer to E as a generalized regular expression.

6 Generalized Star-Height Now suppose that in addition to the aforementioned operations for defining regular expressions, we also allow complementation; that is, if E is a regular expression then so is E c. Including the complement operation leads us to refer to E as a generalized regular expression. We then define h(e c ) = h(e), and define the generalized star-height of a language L A as before. It is best to think of the (generalized) star-height of L as the nesting depth of Kleene stars in the regular expression representing L that features the fewest Kleene stars.

7 A language which has (generalized) star-height 0 is said to be star-free. We have the following result: Theorem (Schützenberger 65) A language is star-free if and only if its syntactic monoid is finite and aperiodic. This theorem gives us an algorithm for deciding whether a language has (generalized) star-height 0. However, the following is still an open question:

8 A language which has (generalized) star-height 0 is said to be star-free. We have the following result: Theorem (Schützenberger 65) A language is star-free if and only if its syntactic monoid is finite and aperiodic. This theorem gives us an algorithm for deciding whether a language has (generalized) star-height 0. However, the following is still an open question: Does there exist a language of generalized star-height greater than 1?

9 Example I For any alphabet A, the language L = A is star-free; that is, has generalized star-height 0. How?

10 Example I For any alphabet A, the language L = A is star-free; that is, has generalized star-height 0. How? The minimal automaton recognizing L is 1 A The syntactic monoid of L is the trivial monoid, which is finite and aperiodic, so L must be star-free.

11 Example I For any alphabet A, the language L = A is star-free; that is, has generalized star-height 0. How? The minimal automaton recognizing L is 1 A The syntactic monoid of L is the trivial monoid, which is finite and aperiodic, so L must be star-free. A star-free expression for L is c.

12 Example II For any alphabet A and any subset B of A, we have h(b ) = 0. The minimal automaton recognizing B is A \ B 1 2 A B The syntactic monoid of B is M(B ) = x x 2 = x, which is finite and aperiodic, so B must be star-free.

13 Example II For any alphabet A and any subset B of A, we have h(b ) = 0. The minimal automaton recognizing B is A \ B 1 2 A B The syntactic monoid of B is M(B ) = x x 2 = x, which is finite and aperiodic, so B must be star-free. A star-free expression for B is ( c (A \ B) c ) c.

14 Current Research Which languages are recognized by Rees matrix semigroups over cyclic groups?

15 Current Research Which languages are recognized by Rees matrix semigroups over cyclic groups? Potential insight could be given by an answer to the following: For any alphabet A, how can we choose B A 2 such that h(b ) = 0?

16 Current Research Which languages are recognized by Rees matrix semigroups over cyclic groups? Potential insight could be given by an answer to the following: For any alphabet A, how can we choose B A 2 such that h(b ) = 0? More generally, for any alphabet A, how can we choose B A n, where n N, such that h(b ) = 0?

Rational subsets in HNN-extensions

Rational subsets in HNN-extensions G. Sénizergues joint work with M. Lohrey ROUPE DE TRAVAIL-LGL-28/02/06 1 0- PLAN 0 INTRODUCTION 0.1 Problems 0.2 Motivations 0.3 Results 0.4 Tools 1- HNN EXTENSIONS 1.1