How to Prove that a Language Is Regular or Star-Free?

Abstract

This survey article presents some standard and less standard methods used to prove that a language is regular or star-free.Most books of automata theory [9,23,29,46,50] offer exercises on regular languages, including some difficult ones. Further examples can be found on the web sites math.stackexchange.com and cs.stackexchange.com. Another good source of tough questions is the recent book 200 Problems in Formal Languages and Automata Theory [37]. Surprisingly, there is hardly any exercise to prove that a language is star-free. In this paper, we present various methods to prove that a language is regular or star-free.

Well quasi-orders arising from finite ordered semigroups

“…We do not see any direct impact of the research to the work done in [1]. On the other hand, in [8], the wqo was applied to prove regularity of maximal solutions of very general language equations and inequalities (see also [12]). The theory developed in [8] may be naturaly extended to the ordered case, so our new class of ordered semigroups inducing well quasi-orders may find the application there.…”

“…Similarly, we can consider an appropriate homomorphism onto an ordered semigroup in the other mentioned examples. 1 Up to our knowledge, and also according to the survey paper [12], there is just one significant contribution to the mentioned open questions. Namely, in the paper [8] by Kunc, the questions are solved for the semigroups ordered by the equality relation.…”

“…Our paper concentrates on an application of wqos motivated by the work of Bucher, Ehrenfeucht and Haussler [1], which leads to a purely algebraic question in the realm of ordered semigroups. Notice that the topic is nicely overviewed in the recent survey paper [12] by Pin. Before we introduce the primary question, we briefly recall the role of ordered semigroups in the algebraic theory of regular languages. At first, when we talk about an ordered semigroup (S, •, ≤), we assume that the partial order ≤ is stable, i.e., compatible with the multiplication • in the sense that, for arbitrary x, y, s ∈ S, the inequality x ≤ y implies both s • x ≤ s • y and x • s ≤ y • s. The finite ordered semigroups are used to recognize regular languages similarly to unordered semigroups -see, e.g., the fundamental survey on the algebraic theory of regular languages [11] by Pin.…”

Characterization of Ordered Semigroups Generating Well Quasi-Orders of Words

“…Notice that the original paper [2] does not use terminology of ordered semigroups which was fixed in the algebraic theory of formal languages later, see, e.g., [14] by Pin, a fundamental survey in that theory. The reformulation of the question in [2] using the notion of ordered semigroups is also mentioned in the recent survey paper [15] by Pin (see Paragraph 3.5). Recall that by an ordered semigroup (S, •, ≤) we mean a semigroup (S, •) equipped with a monotone partial order ≤, i.e., with the antisymmetric quasi-order ≤ compatible with the associative multiplication • on S. In this setting, any semigroup may be also viewed as an ordered semigroup which is ordered by the equality relation.…”

Regularity Conditions for Iterated Shuffle on Commutative Regular Languages

“…We also show that the commutative star-free languages and the commutative group languages are closed under projections. For further connections on regularity conditions and closure properties, in particular for the star-free languages, see the recent survey [21].…”