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, 45, 49] 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 [36]. Surprisingly, there are very few exercises related to star-free languages. In this paper, we present various methods to prove that a language is regular or star-free.
Background
Regular and Star-Free Languages
Let’s start by reminding us what a regular language and a star-free language are.
Definition 1
The class of regular languages is the smallest class of languages containing the finite languages that is closed under finite union, finite product and star.
The definition of star-free languages follows the same pattern, with the difference that the star operation is replaced by the complement:
Definition 2
The class of star-free languages is the smallest class of languages containing the finite languages that is closed under finite union, finite product and complement.
For instance, the language 
is star-free, since 
. More generally, if B is a subset of A, then 
is star-free since
 |
On the alphabet 
, the language 
is star-free since
 |
Since regular languages are closed under complement, every star-free language is regular, but the converse is not true: one can show that the language 
is not star-free.
Early Results and Their Consequences
Kleene’s theorem [26] states that regular languages are accepted by finite automata.
Theorem 1
Let L be a language. The following conditions are equivalent:
L is regular,
L is accepted by a finite deterministic automaton,
L is accepted by a finite non-deterministic automaton.
Given a language L and a word u, the left [right] quotient of L by u are defined by 
and 
, respectively. The quotients of a regular [star-free] language are also regular [star-free].
Here is another standard result, due to Nerode.
Theorem 2
A language is regular if and only if it has finitely many left (respectively right) quotients.
Example 1
Nerode’s theorem suffices to show that if 
and 
are regular [star-free], then the language
 |
is also regular [star-free]. Indeed 
and since 
and 
are regular [star-free], this apparently infinite union can be rewritten as a finite union. Thus L is regular [star-free].
Recognition by a Monoid and Syntactic Monoid
It is often useful to have a more algebraic definition of regular languages, based on the following result.
Proposition 1
Let L be a language. The following conditions are equivalent:
L is regular,
L is recognised by a finite monoid,
the syntactic monoid of L is finite.
For readers who may have forgotten the definitions used in this proposition, here are some reminders. A language L of 
is recognised by a monoid
M if there is a surjective monoid morphism 
and a subset P of M such that 
.
The syntactic congruence of a language L of 
is the equivalence relation 
on 
defined as follows: 
if and only if, for every 
, xuy and xvy are either both in L or both outside of L. The syntactic monoid of L is the quotient monoid 
.
Moreover, the syntactic monoid of a regular language is the transition monoid of its minimal automaton, which gives a convenient algorithm to compute it. It is also the minimal monoid (in size, but also for the division ordering1) that recognises the language.
Syntactic monoids are particularly useful to show that a language is star-free. Recall that a finite monoid M is aperiodic if, for every 
, there exists 
such that 
.
Theorem 3
(Schützenberger [46]). For a language L, the following conditions are equivalent:
L is star-free,
L is recognised by a finite aperiodic monoid,
the syntactic monoid of L a finite aperiodic monoid.
Schützenberger’s theorem is considered, right after Kleene’s theorem, as the most important result of the algebraic theory of automata.
Example 2
The languages 
and 
are star-free, but the languages 
and 
are not. This is easy to prove by computing the syntactic monoid of these languages.
The following classic example is a good example of the usefulness of the monoid approach. For each language L, let 
.
Proposition 2
If L is regular [star-free], then so is 
.
Proof
Let 
be the syntactic morphism of L, let 
and let 
. Then
 |
Thus M recognises 
and the result follows.
Although the star operation is prohibited in the definition of a star-free language, some languages of the form 
are star-free. A submonoid M of 
is pure if, for all 
and 
, the condition 
implies 
. The following result is due to Restivo [43] for finite languages and to Straubing [52] for the general case.
Theorem 4
If L is star-free and 
is pure, then 
is star-free.
Here is another example, based on [51, Theorem 5]. For each language L, let
 |
Proposition 3
If L is regular [star-free], then so is 
.
Proof
Let 
be the syntactic morphism of L and let 
. Note that the conditions 
and 
are equivalent, for any 
. Setting 
and 
one gets
 |
and the result now follows easily.
Iteration Properties
The bible on this topic is the book of de Luca and Varricchio [13]. I only present here a selection of their numerous results.
Pumping
The standard pumping lemma is designed to prove that a language is non-regular, although some students try to use it to prove the opposite. In a commendable effort to comfort these poor students, several authors have proposed extensions of the pumping lemma that characterise regular languages. The first is due to Jaffe [24]:
Theorem 5
A language L is regular if and only if there is an integer m such that every word x of length 
can be written as 
, with 
, and for all words z and for all 
, 
if and only if 
.
Stronger versions were proposed by Stanat and Weiss [50] and Ehrenfeucht, Parikh and Rozenberg [15], but the most powerful version was given by Varricchio [54].
Theorem 6
A language L is regular if and only if there is an integer 
such that, for all words x, 
and y, there exist i, j with 
such that for all 
,
 |
Periodicity and Permutation
Definition 3
Let L be a language of 
.
L is periodic if, for any
, there exist integers 
such that, for all 
, 
.L is n-permutable if, for any sequence
of n words of 
, there exists a nontrivial permutation 
of 
such that, for all 
, 
.L is permutable if it is permutable for some
.
These definitions were introduced by Restivo and Reutenauer [44], who proved the following result.
Proposition 4
A language is regular if and only if it is periodic and permutable.
Iteration Properties
The book of de Luca and Varricchio [13] also contains many results about iterations properties. Here is an example of this type of results.
Proposition 5
A language L is regular if and only if there exist integers m and s such that for any 
, there exist integers h, k with 
, such that for all for all 
,
 |
1 |
for all 
.
Rewriting Systems and Well Quasi-orders
Rewriting systems and well quasi-orders are two powerful methods to prove the regularity of a language. We follow the terminology of Otto’s survey [37].
Rewriting Systems
A rewriting system is a binary relation R on 
. A pair 
from R is usually referred to as the rewrite rule or simply the rule
. A rule is special if 
, context-free if 
, inverse context-free if 
, length-reducing if 
. It is monadic if it is length-reducing and inverse context-free. A rewriting system is special (context-free, inverse context-free, length-reducing, monadic) if its rules have the corresponding properties.
The reduction relation
the reflexive and transitive closure of the single-step reduction relation
defined as follows: 
if 
and 
for some 
and some 
. For each language L, we set
 |
A rewriting system R is said to preserve regularity if, for each regular language L, the language 
is regular. The following result is well-known.
Theorem 7
Inverse context-free rewriting systems preserve regularity.
Proof
Let R be an inverse context-free rewriting system and let L be a regular language. Starting from the minimal deterministic automaton of L, construct an automaton with the same set of states, but with 1-transitions, by iterating the following process: for each rule 
and for each path 
, create a new transition 
; for each rule 
with 
and for each path 
, create a new transition 
. The automaton obtained at the end of the iteration process will accept 
.
A similar technique can be used to prove the following result [38]. If K is a regular language, then the smallest language L containing K and such that 
is regular.
Suffix Rewriting Systems
A suffix rewriting system is a binary relation S on 
. Its elements are called suffix rules. The suffix-reduction relation 
defined by S is the reflexive transitive closure of the single-step suffix-reduction relation defined as follows: 
if 
and 
for some 
and some 
. Prefix rewriting systems are defined symmetrically. For each language L, we set
 |
The following early result is due to Büchi [8].
Theorem 8
Suffix (prefix) rewriting systems preserve regularity.
Deleting Rewriting Systems
We follow Hofbauer and Waldmann [22] for the definition of deleting systems. If u is a word, the content of u is the set c(u) of all letters of u occurring in u. A precedence relation is an irreflexive and transitive binary relation. A precedence relation < on an alphabet A can be extended to a precedence relation on 
, by setting 
if 
and, for each 
, there exists 
such that 
. A rewriting system R is <-deleting if for each rule 
of R, 
.
Hofbauer and Waldmann [22] proved the following result.
Theorem 9
Every deleting string rewriting system preserves regularity.
Rules of the Form 
Rules of the form 
were studied in several papers, for instance [5, 16, 34]. The following result is due to Bovet and Varricchio [5].
Proposition 6
The rewriting systems 
and 
both preserve regularity.
This result can be used to solve the following exercise. Let L be a language such that, for all 
, 
is a semigroup. Prove that L is regular. Indeed, this condition implies that 
implies 
.
Several results were obtained by Leupold [33, 34]. Let us say that a rewriting system is k-period-expanding [k-period-reducing] if its rules are of the form 
, with 
[
] and 
. Any union of finitely many k-period-expanding and k-period reducing SRSs is called a k-periodic rewriting system.
Proposition 7
(Leupold) .
Every k-periodic rewriting system preserves regularity.
For each
, the rewriting system 
preserves regularity.For each k and for
, the rewriting system 
preserves regularity.
Well Quasi-orders
A quasi-order (or preorder) on 
is a reflexive and transitive relation. A quasi-order 
is stable (or monotone) if, for all words u, v, x, y, the condition 
implies 
. A language U is an upper set with respect to a quasi-order 
is the conditions 
and 
imply 
. The upper set generated by a language
L is the language 
A quasi-order 
on 
is a well quasi-order (wqo) if every upper set is generated by some finite language. The connection with regular languages was first established in [14] (see also [13, Theorem 6.3.1, p. 203] and [12]).
Theorem 10
A language is regular if and only if it is an upper set with respect to some stable well quasi-order on 
.
It follows that if the reduction relation defined by a rewriting system is a well quasi-order, then this rewriting system preserves regularity. Actually, a stronger property holds. Following Conway [11], let us say that a rewriting system R is a total regulator if for any language L, the language 
is regular.
Theorem 11
Any rewriting system whose reduction relation is a well quasi-order is a total regulator.
The most famous example is the rewriting system 
, which defines the subword ordering. A word 
is a subword of a word v if 
. Higman’s theorem states that if A is finite, the subword relation is a well quasi-order on 
. It follows that for any language L (regular or not), the shuffle product 
is regular.
The following result extends Higman’s theorem on the subword order. Let us say that a set H of words of 
is unavoidable if the language 
is finite.
Theorem 12
(Ehrenfeucht, Haussler, Rozenberg [14, Theorem 4.8]). If H is a unavoidable finite set of words of 
, then the reduction relation of the rewriting system 
is a well quasi-order on 
.
A similar result holds for rewriting systems with rules of the form 
, where a is a letter.
Theorem 13
(Bucher, Ehrenfeucht and Haussler [6, Theorem 2.3]). Let R be a finite rewriting system with rules of the form 
with 
and 
. The following conditions are equivalent:
the relation
is a well quasi-order,The set
is unavoidable,The set
is unavoidable.
It follows for instance that the following rewriting systems are total regulators:
 |
Bucher, Ehrenfeucht and Haussler [6] considered context-free rewriting systems related to semigroup morphims. Recall that an ordered semigroup is a semigroup equipped with a stable partial order. Let 
be a finite ordered semigroup and let 
be a semigroup morphism. Consider the rewriting system
 |
Let 
be a finite set of languages of 
. Consider a (possibly infinite) system of inequations of the form
 |
2 |
where each 
is a product built from the variables 
and arbitrary constant languages and each 
is an expression built from the variables 
and constant languages belonging to the set 
, using concatenation, possibly infinite union and possibly infinite intersection. Note that the expressions 
can also use Kleene star, since it can be rewritten as an infinite union of products.
Theorem 14
(Kunc [30]). Let 
be a semigroup morphism that recognises all languages in 
. If 
is a well quasi-order on 
, then the components of every maximal solution of (2) is regular and they are star-free is S is aperiodic.
Characterising the semigroup morphisms for which 
is a well quasi-order, is an open problem. However, Kunc found a complete answer for finite semigroups 
ordered by the equality relation.
Theorem 15
(Kunc [30]). Let 
be a finite ordered semigroup ordered by the equality relation and let 
be a surjective semigroup morphism. Then the relation 
is a well quasi-order on 
if and only if S is a chain of simple semigroups.
In particular any finite group is a simple semigroup. It follows that if L is a language recognised by a finite group, then, for any subset S of 
, the language 
is regular.
Example 3
The following example is given by Kunc [30, Example 19]. Let L be the language consisting of those words 
which contain some occurrence of b and where the difference between the length of u and the number of blocks of occurrences of b in u is even. Here is the minimal automaton of this language.
The syntactic semigroup of L is defined by the relations 
, 
, 
, 
and 
. It is a chain of two simple semigroups whose elements are represented by the words a, 
and b, 
, ab, 
, ba, 
, aba, 
, respectively.
Let us consider the inequality 
with one variable X. It is easy to verify that this inequality has a largest solution, namely the regular language 
.
Equations and Inequalities
Inequations in languages in which the right hand side is a constant language were first considered by Conway [11], see also Bala [1]. In Chap. 21 of the forthcoming Handbook of Automata Theory, Kunc and Okhotin [32] give the following remarkable result. Consider a finite system of inequations of the form
 |
3 |
where each 
is a product of arbitrary constant languages and variables, each 
is a constant regular language and each index set 
is possibly infinite.
Theorem 16
(Kunc and Okhotin [32]). Every system of the form (3) has only finitely many maximal solutions and every maximal solution has all components regular. If all 
are star-free, then the maximal solutions are star-free. Furthermore, the result still holds if any inequalities are replaced by equations.
Proof
Let 
be the simultaneous syntactic monoid of the languages 
. If 
is a solution, then so is 
. It follows that every solution is contained in a solution in which all components are recognised by h and the result follows.
Inequations of the form 
were considered by Kunc [30].
Theorem 17
(Kunc [30]). Let K be an arbitrary language and let L be a regular language. Then the greatest solution of the inequality 
is regular.
The situation is totally different for equations of the type 
. Indeed Kunc [31] has shown that there exists a finite language L such that the greatest solution of the equation 
is co-recursively enumerable complete.
Logic
Logic can be used in various ways to characterise regular languages. We consider successively logic on words, linear temporal logic and logic on trees.
Logic on Words
Let 
be a nonempty word on the alphabet A. The domainDomain of u, denoted by 
, is the set 
. For each letter 
, let 
be a unary predicate symbol, where 
is interpreted as “the letter in position x is an a”. We also use the binary predicate symbols < and S, interpreted as the usual order relation and the successor relation on 
, respectively. The language defined by a sentence 
is the set
 |
We let 
and 
denote the set of first-order and monadic second-order formulas of signature 
, respectively. Similarly, we let 
and 
denote the same sets of formulas of signature 
.
Let us say that a syntactic fragment of logic F
captures a class of languages 
if every sentence of the fragment F defines a language of 
and every language of 
can be defined by a sentence of F.
Two famous results are a natural ingredient of this survey. The first one is due to Buchi [7] and was independently obtained by Elgot [20] and Trakhtenbrot [53].
Theorem 18
(Buchi [7]).
captures the class of regular languages.
The second one relates first order logic and star-free languages.
Theorem 19
(McNaughton [35]).
captures the class of star-free languages.
Second order logic 
is much more expressive than monadic second order, but two successive results led to a complete characterisation of the syntactic fragments of 
— in the signature 
— that capture the regular languages.
A quantifier prefix is any word on the alphabet 
. A quantifier prefix class is any set of quantifier prefixes. For any quantifier prefix Q, let 
(resp. 
be the set of all formulas of the shape 
(resp. 
) where 
is a list of relations and 
is quantifier free. For every 
, let 
(resp., 
) be the set of all formulas of the form 
(resp. 
) where 
is a 
(resp. 
) formula. Finally, for every quantifier prefix class 
, let 
.
The fragment 
, also known as existential second order and frequently denoted by 
, was first explored by Eiter, Gottlob and Gurevich [17].
Theorem 20
(Eiter, Gottlob and Gurevich [17]). A syntactic fragment 
captures the regular languages if and only if 
is a quantifier prefix class contained in 
whose intersection with 
is nonempty.
The proof of this result is very difficult. It relies on combinatorial methods related to hypergraph transversals for the fragment 
and on more logical techniques for the fragment 
. Eiter, Gottlob and Gurevich further proved the following dichotomy theorem: a class 
either expresses only regular languages or it expresses some NP-complete languages.
The fragments 
, with 
, were explored by Eiter, Gottlob and Schwentick [18].
Theorem 21
(Eiter, Gottlob and Schwentick [18]). The fragments 
and 
capture the class of regular languages. Furthermore, for each 
, the fragments 
and 
only define regular languages.
For more information on this topic, the reader is invited to read the beautiful survey of Eiter, Gottlob and Schwentick [19].
Linear Temporal Logic
Linear temporal logic (LTL for short) on an alphabet A is defined as follows. The vocabulary consists of an atomic proposition 
(for each letter 
), the usual connectives 
, 
and 
and the temporal operators 
(next), 
(eventually) and 
(until). The formulas are constructed according to the following rules:
for every
, 
is a formula,if
and 
are formulas, so are 
, 
, 
, 
, 
and 
.
Semantics are defined by induction on the formation rules. Given a word 
, and 
, we define the expression “w satisfies 
at the instant n” (denoted 
) as follows:
if the n-th letter of w is an a.
(resp. 
, 
) if 
or 
(resp. if 
and 
, if (w, n) does not satisfy 
).
if 
satisfies 
.
if there exists m such that 
and 
.
if there exists m such that 
, 
and, for every k such that 
, 
.
Note that, if 
, 
only depends on the word 
.
Example 4
Let 
. Then 
since the fourth letter of w is an a, 
since the fifth letter of w is a b and 
since cb is a factor of babcba.
If 
is a temporal formula, we say that w satisfies 
if 
. The language defined by a LTL formula 
is the set 
of all words of 
that satisfy 
.
A famous result of Kamp [25] states that LTL is equivalent to the first-order logic of order. As a consequence, one gets the following result.
Theorem 22
A language of 
is star-free if and only if it is LTL-definable.
We just defined future temporal formulas but one can define in the same way past temporal formulas by reversing time: it suffices to replace next by previous, eventually by sometimes and until by since. The expressive power of this extended temporal logic remains the same: it still captures the class of star-free languages.
Rabin’s Tree Theorem
We now consider the structure 
, where each 
is a binary relation symbol, interpreted on 
as follows: 
if and only if 
. Let 
be a monadic second order formula with a free set-variable X. We write 
as a short hand for the formula 
. A language L is said to be definable in 
if there exists a monadic second order formula 
such that L satisfies 
.
The following result is a consequence of Rabin’s tree theorem [42].
Theorem 23
A language of 
is regular if and only if it is definable in 
.
Transductions
Transductions proved to be a powerful tool to study regular languages. Let us first recall some useful facts about rational and recognisable sets.
Rational and Recognisable Sets
Let M be a monoid. A subset P of M is recognisable if there exists a finite monoid F, and a monoid morphism 
such that 
. It is well known that the class 
of recognisable subsets of M is closed under Boolean operations, left and right quotients and under inverses of monoid morphisms. The recognisable subsets of a product of monoids were described by Mezei (unpublished).
Theorem 24
Let 
be monoids. A subset of 
is recognisable if and only if it is a finite union of subsets of the form 
, where 
.
Furthermore, the following property holds:
Proposition 8
Let 
be finite alphabets. Then 
is closed under product.
The class 
of rational subsets of M is the smallest set 
of subsets of M containing the finite subsets and closed under finite union, product and star (where 
is the submonoid of M generated by X). Rational sets are closed under monoid morphisms. Kleene’s theorem shows that 
, but this result does not extend to arbitrary monoids.
Matrix Representations of Transductions
Let M be a monoid. We denote by 
the semiring of subsets of M with union as addition and the usual product of subsets as multiplication. Note that both 
and 
are subsemirings of 
. Let also 
denote the semiring of 
-matrices with entries in 
.
Let M and N be two monoids. A transduction
is a relation on M and N, viewed as a function from M to 
. One extends 
to a function 
by setting 
. The inverse transduction 
is defined by 
. The transduction is rational if the set 
is a rational subset of 
.
A transduction 
admits a linear matrix representation
of degree
n if there exist 
, a monoid morphism 
, a row vector 
and a column vector 
such that, for all 
, 
.
A substitution from 
to a monoid M is a monoid morphism from 
to 
. Thus a substitution has linear matrix representation of degree 1.
Kleene-Schützenberger’s theorem (see [2]) states that a transduction 
is rational if and only if it admits a linear matrix representation with entries in 
.
The following result already suffices for most of the applications we have in mind. It relies on the fact that every monoid morphism 
can be extended to a semiring morphism 
and, for each 
, to a semiring morphism 
.
Theorem 25
Let 
be a transduction that admits a linear matrix representation 
of degree n and let P be a subset of M recognised by a morphism 
. Then the language 
is recognised by the submonoid 
of the monoid of matrices 
.
This result was generalised in [39, 40]. Let us say that a transduction 
admits a matrix representation
of degree
n if there exist a morphism 
and an expression 
, where S is a possibly infinite union of products involving arbitrary languages and the variables 
, such that, for all 
, 
. Theorem 25 can now be generalized as follows.
Theorem 26
Let 
be a transduction that admits a matrix representation 
of degree n and let P be a subset of M recognised by a morphism 
. Then the language 
is recognised by the submonoid 
of the monoid of matrices 
.
Example 5
Let us come back to the example 
. Observe that 
where 
. Clearly 
admits the matrix representation 
where 
and 
.
Example 6
Let us show that if L is a regular language and S is a subset of 
then the language
 |
is also regular. It suffices to observe that 
where the transduction 
admits the matrix representation 
, where
 |
Example 7
Finally the reader who likes more complicated examples may prove by the same method that if 
is regular, then the following language is also regular (
is the Dyck language):
 |
Decompositions of Languages
For each 
, consider the transduction 
defined by
 |
Theorem 27
Let L be a language of 
. The following conditions are equivalent:
L is rational,
for some
, 
is a recognisable subset of 
,for all
, 
is a recognisable subset of 
.
Proof
(1) implies (3). Let 
be the minimal automaton of L. For each state p, q of 
, let 
be the language accepted by 
with p as initial state and q as unique final state. Let 
. We claim that
 |
4 |
Let R be the right hand side of (4). Let 
. Let 
, 
, ..., 
. Since 
, one has 
and hence 
. Moreover, by construction, 
and hence 
.
Let now 
. Then, for some 
, one has 
, ..., 
. It follows that 
, ..., 
and thus 
and hence 
.
Profinite Topology
Let M be a monoid. A monoid morphism 
separates two elements u and v of M if 
. By extension, we say that a monoid N
separates two elements of M if there exists a morphism 
which separates them. A monoid is residually finite if any pair of distinct elements of M can be separated by a finite monoid.
Let us consider the class 
of monoids that are finitely generated and residually finite. This class include finite monoids, free monoids, free groups, free commutative monoids and many others. It is closed under direct products and thus monoids of the form 
are also in 
.
Each monoid M of 
can be equipped with the profinite metric, defined as follows. Let, for each 
,
 |
Then we set 
, with the usual conventions 
and 
. One can show that d is an ultrametric and that the product on M is uniformly continuous for this metric.
Uniformly Continuous Functions and Recognisable sets
The connection with recognisable sets is given by the following result:
Proposition 9
Let 
and let 
be a function. Then the following conditions are equivalent:
for every
, one has 
,the function f is uniformly continuous for the profinite metric.
Here is an interesting example [41].
Proposition 10
The function 
defined by 
is uniformly continuous.
Example 8
As an application, let us show that if L is a regular language of 
, then the language
 |
is also regular. Indeed, 
, where h is the function defined by 
. Observe that 
, where 
is the monoid morphism defined by 
and g is the function defined in Proposition 10. Now since 
, one gets 
by Proposition 10 and 
since f is a monoid morphism. Thus K is regular.
Uniformly continuous functions from 
to 
are of special interest. A function 
is residually ultimately periodic (rup) if, for each monoid morphism h from 
to a finite monoid F, the sequence h(f(n)) is ultimately periodic. It is cyclically ultimately periodic if, for every 
, there exist two integers 
and 
such that, for each 
, 
. It is ultimately periodic threshold t if the function 
is ultimately periodic.
For instance, the functions 
and n! are residually ultimately periodic. The function 
is not cyclically ultimately periodic. Indeed, it is known that 
if and only if n is a power of 2. It is shown in [48] that the sequence 
is not cyclically ultimately periodic.
Let us mention a last example, first given in [10]. Let 
be a non-ultimately periodic sequence of 0 and 1. The function 
is residually ultimately periodic. It follows that the function 
is not residually ultimately periodic since 
.
The following result was proved in [3].
Proposition 11
For a function 
, the following conditions are equivalent:
f is uniformly continuous,
f is residually ultimately periodic,
f is cyclically ultimately periodic and ultimately periodic threshold t for all
.
The class of cyclically ultimately periodic functions has been studied by Siefkes [48], who gave in particular a recursion scheme for producing such functions. The class of residually ultimately periodic sequences was also thoroughly studied in [10, 55] (see also [27, 28, 47]). Their properties are summarized in the next proposition.
Theorem 28
Let g and g be rup functions. Then the following functions are also rup: 
, 
, fg, 
, 
, 
. Furthermore, if 
for all n and 
, then 
is also rup.
In particular, the functions 
and 
(for a fixed k), are rup. The tetration function
(exponential stack of 2’s of height n), considered in [47], is also rup, according to the following result: if k is a positive integer, then the function f(n) defined by 
and 
is rup.
The existence of non-recursive rup functions was established in [47]: if f is a strictly increasing, non-recursive function, then the function 
is non-recursive but is rup.
Coming back to regular languages, Seiferas and McNaughton [47] proved the following result.
Theorem 29
Let 
be a rup function. If L is regular, then so is the language
 |
Here is another application of rup functions. A filter is a strictly increasing function 
. Filtering a word
by f consists in deleting the letters 
such that i is not in the range of f. For each language L, let L[f] denote the set of all words of L filtered by f. A filter is said to preserve regular languages if, for every regular language L, the language L[f] is also regular. The following result was proved in [3].
Theorem 30
A filter f preserves regular languages if and only if the function 
defined by 
is rup.
Transductions and Recognisable Sets
Some further topological results are required to extend Proposition 9 to transductions.
The completion of the metric space (M, d), denoted by 
, is called the profinite completion of M. Since multiplication on M is uniformly continuous, it extends, in a unique way, to a multiplication on 
, which is again uniformly continuous. One can show that 
is a metric compact monoid.
Let 
be the monoid of compact subsets of 
. The Hausdorff metric on 
is defined as follows. For 
, let
 |
By a standard result of topology, 
, equipped with this metric, is compact.
Let now 
be a transduction. Define a map 
by setting, for each 
, 
, the topological closure of 
. The following extension of Proposition 9 was proved in [41].
Theorem 31
Let 
and let 
be a transduction. Then the following conditions are equivalent:
for every
, one has 
,the function
is uniformly continuous.
Let us say that a transduction 
is uniformly continuous, if 
is uniformly continuous. Uniformly continuous transductions are closed under composition and they are also closed under direct product.
Proposition 12
Let 
and 
be uniformly continuous transductions. Then the transduction 
defined by 
is uniformly continuous.
Proposition 13
For every 
, the transduction 
defined by 
is uniformly continuous.
Further Examples and Conclusion
Here are a few results relating regular languages and Turing machines.
Theorem 32
([9, Theorem 3.84, p. 185]). The language accepted by a one-tape Turing machine that never writes on its input is regular.
Theorem 33
(Hartmanis [21]). The language accepted by a one-tape Turing machine that works in time 
is regular.
The following result is proposed as an exercise in [9, Exercise 4.16, p. 243].
Theorem 34
The language accepted by a Turing machine that works in space 
is regular.
Let me also mention a result related to formal power series.
Theorem 35
(Restivo and Reutenauer [44]). If a language and its complement are support of a rational series, then it is a regular language.
Many other examples could not be included in this survey, notably the work of Bertoni, Mereghetti and Palano [4, Theorem 3, p. 8] on 1-way quantum automata and the large literature on splicing systems.
I would be very grateful to any reader providing me new interesting examples to enrich this survey.
Acknowledgements
I would like to thank Olivier Carton for his useful suggestions.
Footnotes
Let M and N be monoids. We say that M divides N if there is a submonoid R of N and a monoid morphism that maps R onto M.
J.-É. Pin—Work supported by the DeLTA project (ANR-16-CE40-0007).
Contributor Information
Alberto Leporati, Email: alberto.leporati@unimib.it.
Carlos Martín-Vide, Email: carlos.martin@urv.cat.
Dana Shapira, Email: shapird@g.ariel.ac.il.
Claudio Zandron, Email: zandron@disco.unimib.it.
Jean-Éric Pin, Email: Jean-Eric.Pin@irif.fr.
References
- 1.Bala S. Complexity of regular language matching and other decidable cases of the satisfiability problem for constraints between regular open terms. Theory Comput. Syst. 2006;39(1):137–163. doi: 10.1007/s00224-005-1262-y. [DOI] [Google Scholar]
- 2.Berstel, J.: Transductions and Context-Free Languages. Teubner (1979)
- 3.Berstel J, Boasson L, Carton O, Petazzoni B, Pin J-É. Operations preserving recognizable languages. Theor. Comput. Sci. 2006;354:405–420. doi: 10.1016/j.tcs.2005.11.034. [DOI] [Google Scholar]
- 4.Bertoni A, Mereghetti C, Palano B. Quantum computing: 1-way quantum automata. In: Ésik Z, Fülöp Z, editors. Developments in Language Theory; Heidelberg: Springer; 2003. pp. 1–20. [Google Scholar]
- 5.Bovet DP, Varricchio S. On the regularity of languages on a binary alphabet generated by copying systems. Inform. Process. Lett. 1992;44(3):119–123. doi: 10.1016/0020-0190(92)90050-6. [DOI] [Google Scholar]
- 6.Bucher W, Ehrenfeucht A, Haussler D. On total regulators generated by derivation relations. Theor. Comput. Sci. 1985;40(2–3):131–148. doi: 10.1016/0304-3975(85)90162-8. [DOI] [Google Scholar]
- 7.Büchi JR. Weak second-order arithmetic and finite automata. Z. Math. Logik und Grundl. Math. 1960;6:66–92. doi: 10.1002/malq.19600060105. [DOI] [Google Scholar]
- 8.Büchi, J.R.: Regular canonical systems. Arch. Math. Logik Grundlagenforsch. 6, 91–111 (1964) (1964)
- 9.Carton, O.: Langages formels, calculabilité et complexité. Vuibert (2008)
- 10.Carton O, Thomas W. The monadic theory of morphic infinite words and generalizations. Inform. Comput. 2002;176:51–76. doi: 10.1006/inco.2001.3139. [DOI] [Google Scholar]
- 11.Conway JH. Regular Algebra and Finite Machines. London: Chapman and Hall; 1971. [Google Scholar]
- 12.D’Alessandro F, Varricchio S. Well quasi-orders in formal language theory. In: Ito M, Toyama M, editors. Developments in Language Theory; Heidelberg: Springer; 2008. pp. 84–95. [Google Scholar]
- 13.De Luca A, Varricchio S. Finiteness and Regularity in Semigroups and Formal Languages. Monographs in Theoretical Computer Science. An EATCS Series. Heidelberg: Springer; 1999. [Google Scholar]
- 14.Ehrenfeucht A, Haussler D, Rozenberg G. On regularity of context-free languages. Theor. Comput. Sci. 1983;27(3):311–332. doi: 10.1016/0304-3975(82)90124-4. [DOI] [Google Scholar]
- 15.Ehrenfeucht A, Parikh R, Rozenberg G. Pumping lemmas for regular sets. SIAM J. Comput. 1981;10(3):536–541. doi: 10.1137/0210039. [DOI] [Google Scholar]
- 16.Ehrenfeucht A, Rozenberg G. On regularity of languages generated by copying systems. Discrete Appl. Math. 1984;8(3):313–317. doi: 10.1016/0166-218X(84)90129-X. [DOI] [Google Scholar]
- 17.Eiter T, Gottlob G, Gurevich Y. Existential second-order logic over strings. J. ACM. 2000;47(1):77–131. doi: 10.1145/331605.331609. [DOI] [Google Scholar]
- 18.Eiter T, Gottlob G, Schwentick T. Second-order logic over strings: regular and non-regular fragments. In: Kuich W, Rozenberg G, Salomaa A, editors. Developments in Language Theory; Heidelberg: Springer; 2002. pp. 37–56. [Google Scholar]
- 19.Eiter T, Gottlob G, Schwentick T. The model checking problem for prefix classes of second-order logic: a survey. In: Blass A, Dershowitz N, Reisig W, editors. Fields of Logic and Computation; Heidelberg: Springer; 2010. pp. 227–250. [Google Scholar]
- 20.Elgot CC. Decision problems of finite automata design and related arithmetics. Trans. Am. Math. Soc. 1961;98:21–51. doi: 10.1090/S0002-9947-1961-0139530-9. [DOI] [Google Scholar]
- 21.Hartmanis J. Computational complexity of one-tape Turing machine computations. J. Assoc. Comput. Mach. 1968;15:325–339. doi: 10.1145/321450.321464. [DOI] [Google Scholar]
- 22.Hofbauer D, Waldmann J. Deleting string rewriting systems preserve regularity. Theor. Comput. Sci. 2004;327(3):301–317. doi: 10.1016/j.tcs.2004.04.009. [DOI] [Google Scholar]
- 23.Hopcroft, J.E., Ullman, J.D.: Introduction To Automata Theory, Languages, And Computation. Addison-Wesley Publishing Co., Reading (1979). Addison-Wesley Series in Computer Science
- 24.Jaffe J. A necessary and sufficient pumping lemma for regular languages. SIGACT News. 1978;10(2):48–49. doi: 10.1145/990524.990528. [DOI] [Google Scholar]
- 25.Kamp, J.: Tense Logic and the Theory of Linear Order. Ph.D. thesis, University of California, Los Angeles (1968)
- 26.Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Automata Studies, pp. 3–41. Princeton University Press, Princeton (1956). Ann. Math. Stud. 34
- 27.Kosaraju SR. Regularity preserving functions. SIGACT News. 1974;6(2):16–17. doi: 10.1145/1008304.1008306. [DOI] [Google Scholar]
- 28.Kozen D. On regularity-preserving functions. Bull. Europ. Assoc. Theor. Comput. Sci. 1996;58:131–138. [Google Scholar]
- 29.Kozen DC. Automata and computability. Undergraduate Texts in Computer Science. New York: Springer; 1997. [Google Scholar]
- 30.Kunc M. Regular solutions of language inequalities and well quasi-orders. Theor. Comput. Sci. 2005;348(2–3):277–293. doi: 10.1016/j.tcs.2005.09.018. [DOI] [Google Scholar]
- 31.Kunc M. The power of commuting with finite sets of words. Theory Comput. Syst. 2007;40(4):521–551. doi: 10.1007/s00224-006-1321-z. [DOI] [Google Scholar]
- 32.Kunc, M., Okhotin, A.: Language equations. In: Pin, J.E. (ed.) Handbook of Automata Theory, vol. II, chap. 21. European Mathematical Society, Zürich (2020, To appear)
- 33.Leupold P. Languages generated by iterated idempotency. Theor. Comput. Sci. 2007;370(1–3):170–185. doi: 10.1016/j.tcs.2006.10.021. [DOI] [Google Scholar]
- 34.Leupold P. On regularity-preservation by string-rewriting systems. In: Martín-Vide C, Otto F, Fernau H, editors. Language and Automata Theory and Applications; Heidelberg: Springer; 2008. pp. 345–356. [Google Scholar]
- 35.McNaughton R, Papert S. Counter-Free Automata. Cambridge: The M.I.T. Press; 1971. [Google Scholar]
- 36.Niwinśki, D., Rytter, W.: 200 Problems in Formal Languages and Automata Theory. University of Warsaw (2017)
- 37.Otto F. On the connections between rewriting and formal language theory. In: Narendran P, Rusinowitch M, editors. Rewriting Techniques and Applications; Heidelberg: Springer; 1999. pp. 332–355. [Google Scholar]
- 38.Pin J-É. Topologies for the free monoid. J. Algebra. 1991;137:297–337. doi: 10.1016/0021-8693(91)90094-O. [DOI] [Google Scholar]
- 39.Pin J-É, Sakarovitch J. Some operations and transductions that preserve rationality. In: Cremers AB, Kriegel H-P, editors. Theoretical Computer Science; Heidelberg: Springer; 1982. pp. 277–288. [Google Scholar]
- 40.Pin J-É, Sakarovitch J. Une application de la représentation matricielle des transductions. Theor. Comput. Sci. 1985;35:271–293. doi: 10.1016/0304-3975(85)90019-2. [DOI] [Google Scholar]
- 41.Pin J-É, Silva PV. A topological approach to transductions. Theor. Comput. Sci. 2005;340:443–456. doi: 10.1016/j.tcs.2005.03.029. [DOI] [Google Scholar]
- 42.Rabin MO. Decidability of second-order theories and automata on infinite trees. Trans. Am. Math. Soc. 1969;141:1–35. [Google Scholar]
- 43.Restivo, A.: Codes and aperiodic languages. In: Erste Fachtagung der Gesellschaft für Informatik über Automatentheorie und Formale Sprachen (Bonn, 1973), LNCS, vol. 2, pp. 175–181. Springer, Berlin (1973)
- 44.Restivo A, Reutenauer C. On cancellation properties of languages which are supports of rational power series. J. Comput. Syst. Sci. 1984;29(2):153–159. doi: 10.1016/0022-0000(84)90026-6. [DOI] [Google Scholar]
- 45.Sakarovitch J. Elements of Automata Theory. Cambridge: Cambridge University Press; 2009. [Google Scholar]
- 46.Schützenberger MP. On finite monoids having only trivial subgroups. Inf. Control. 1965;8:190–194. doi: 10.1016/S0019-9958(65)90108-7. [DOI] [Google Scholar]
- 47.Seiferas JI, McNaughton R. Regularity-preserving relations. Theor. Comput. Sci. 1976;2(2):147–154. doi: 10.1016/0304-3975(76)90030-X. [DOI] [Google Scholar]
- 48.Siefkes, D.: Decidable extensions of monadic second order successor arithmetic. In: Automatentheorie und formale Sprachen (Tagung, Math. Forschungsinst., Oberwolfach, 1969), pp. 441–472. Bibliographisches Inst., Mannheim (1970)
- 49.Sipser, M.: Introduction to the Theory of Computation. 3rd edn. Cengage Learning (2012)
- 50.Stanat DF, Weiss SF. A pumping theorem for regular languages. SIGACT News. 1982;14(1):36–37. doi: 10.1145/1008892.1008895. [DOI] [Google Scholar]
- 51.Stearns RE, Hartmanis J. Regularity preserving modifications of regular expressions. Inf. Control. 1963;6:55–69. doi: 10.1016/S0019-9958(63)90110-4. [DOI] [Google Scholar]
- 52.Straubing H. Relational morphisms and operations on recognizable sets. RAIRO Inf. Theor. 1981;15:149–159. doi: 10.1051/ita/1981150201491. [DOI] [Google Scholar]
-
53.Trakhtenbrot, B.A.: Barzdin
, Y.M.: Finite Automata, Behavior and Synthesis. North-Holland Publishing Co., Amsterdam (1973). Translated from the Russian by D. Louvish, English translation edited by E. Shamir and L. H. Landweber, Fundamental Studies in Computer Science, vol. 1
- 54.Varricchio S. A pumping condition for regular sets. SIAM J. Comput. 1997;26(3):764–771. doi: 10.1137/S0097539790179944. [DOI] [Google Scholar]
- 55.Zhang GQ. Automata, boolean matrices, and ultimate periodicity. Inform. Comput. 1999;152(1):138–154. doi: 10.1006/inco.1998.2787. [DOI] [Google Scholar]