Star free language vs. regular language

Question

I was wondering, since $a^*$ is itself a star-free language, is there a regular language that is not a star-free language? Could you give an example?

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(from wikipdia) Lawson defines star-free languages as:

A regular language is said to be star-free if it can be described by a regular expression constructed from the letters of the alphabet, the empty set symbol, all boolean operators – including complementation – and concatenation but no Kleene star.

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Here is the proof of $a^*$ being star-free:

$\emptyset$ is star-free $\Longrightarrow$
$\Sigma^*=\bar{\emptyset}$ is star-free $\Longrightarrow$
If $A\subseteq\Sigma$ then $\Sigma^*A\Sigma^*$ is star-free $\Longrightarrow$
If $A\subseteq\Sigma$ then $A^*=\overline{\Sigma^*(\Sigma \setminus A)\Sigma^*}$ is star-free

In the last line we have $A^*=\overline{\Sigma^*(\Sigma \setminus A)\Sigma^*}$, because any word that is not of form $A^*$ contains a letter in $\Sigma \setminus A$ and vice versa.

Answer 1

Schützenberger (1965) gave an algebraic characterization of the star-free languages: a regular language is star-free if and only if its syntactic monoid is aperiodic. Contrary to the logical characterization (star-free = FO[<]), this algebraic characterization gives an algorithm to decide whether a given regular language is star-free (the regular language can be given by a finite automaton, a regular expression or a regular grammar). Using the logical characterization (due to McNaughton and Papert) one can then decide the following problem: given a WMSO formula, is there a FO formula describing the same language?

M.-P. Schützenberger, On finite monoids having only trivial subgroups, Information and Control 8 (1965), 190-194.

R.~McNaughton and S.~Papert, Counter-free automata, The M.I.T. Press, Cambridge, Mass.-London, 1971.

A full proof of Schützenberger's theorem can be found in various textbooks or survey papers. For an elementary presentation of the corresponding algorithm (without a proof), see

J.-É. Pin, Finite semigroups and recognizable languages : an introduction, in NATO Advanced Study Institute Semigroups, Formal Languages and Groups, J. Fountain (éd.), 1-32, Kluwer academic publishers, (1995).

Answer 2

Regular languages are those that can be described by weak monadic second order logic (WMSO) [1].

Star-free languages are those that can be described by first order logic with $<$ (FO[<]) [2].

The two logics are not equally powerful. One example for a language that is WMSO-definable but not FO[<]-definable is $(aa)^*$ (which is clearly regular³); this can be shown using Ehrenfeucht-Fraissé games⁴.

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1. Weak Second-Order Arithmetic and Finite Automata by Büchi (1960)
2. Counter-free automata by McNaughton and Papert (1971)

3. A WMSO-formula for $(aa)^*$ is

   $\ \begin{align} \bigl[ \forall x. P_a(x)\bigr] \land \Bigl[ \exists x. P_a(x) \to \bigl[ \exists X. X(0) &\land [\forall x,y. X(x) \land \operatorname{suc}(x,y) \to \lnot X(y)] \\ &\land [\forall x,y. \lnot X(x) \land \operatorname{suc}(x,y) \to X(y)] \\ &\land [\forall x. \operatorname{last}(x) \to \lnot X(x)] \bigr] \Bigr] \;. \end{align}$

   (If the word is not empty, $X$ is the set of all even indices.)

4. See also here.

Answer 3

Star free languages are described by regular expressions that include concatenation, complementation, union, intersection, but no Kleene-star.

Since regular languages are closed under all these operations (where complementation is the crucial point), then every star-free language is also regular.

Perhaps you mean the converse? Are all regular languages star-free? The answer to the latter is no. See this paper for details.

Answer 4

A simple separating example is (aa)*. More sophisticated: All binary strings with even (or odd) parity.