Geometrically Closed Positive Varieties of Star-Free Languages

Abstract

A recently introduced operation of geometrical closure on formal languages is investigated. It is proved that the geometrical closure of a language from the positive variety , the level 3/2 of the Straubing-Thérien hierarchy of star-free languages, always falls into the variety , which is a new variety consisting of specific R-trivial languages. As a consequence, each class of regular languages lying between and is geometrically closed.

Introduction

A geometrical closure is an operation on formal languages introduced recently by Dubernard, Guaiana, and Mignot [8]. It is defined as follows: Take any language L over some k-letter alphabet and consider the set called the figure of L in [8], which consists of all elements of corresponding to Parikh vectors of prefixes of words from L. The geometrical closure of L is the language of all words w such that the Parikh vectors of all the prefixes of w lie in the figure of L. This closure operator was inspired by the previous works of Blanpain, Champarnaud, and Dubernard [4] and Béal et al. [3], in which geometrical languages are studied – using the terminology from later paper [8], these can be described as languages whose prefix closure is equal to their geometrical closure. Note that this terminology was motivated by the fact that a geometrical language is completely determined by its (geometrical) figure. In the particular case of binary alphabets, these (geometrical) figures were illustrated by plane diagrams in [8].

The class of all regular languages can be easily observed not to be geometrically closed – that is, one can find a regular language such that its geometrical closure is not regular [8] (see also the end of Sect. 2). One possible research aim could be to characterise regular languages L for which is regular, or to describe some robust classes of languages with this property. Another problem posed in [8] is to find some subclasses of regular languages that are geometrically closed. As we explain in Sect. 3, non-empty group languages have their geometrical closure equal to the universal language . For this reason, it makes sense to look for more interesting geometrically closed subclasses among star-free languages, which are known to be “group-free”. More precisely, a language L is star-free if and only if the syntactic monoid of L is aperiodic, that is, if does not contain non-trivial groups as subsemigroups.

It is well known that the star-free languages are classified into the Straubing-Thérien hierarchy based on polynomial and Boolean operations. In particular, the variety (i.e., the variety of languages of level 1) is formed by piecewise testable languages and the positive variety is formed by polynomials built from languages of level 1. We refer to the survey paper by Pin [12] for an introduction to the Straubing-Thérien hierarchy of star-free languages and the algebraic theory of regular languages in general. This theory is based on Eilenberg correspondence between varieties of regular languages and pseudovarieties of finite monoids. Note that one well-known instance of Eilenberg correspondence, which plays an essential role in our contribution, is given by the pseudovariety of finite R-trivial monoids, for which the corresponding variety of languages is denoted by . Nevertheless, we emphasise that our contribution is rather elementary, and it does not use sophisticated tools developed in the algebraic theory of regular languages.

It was proved by Dubernard, Guaiana, and Mignot [8] that the class of all binary languages from the positive variety is geometrically closed. They have obtained this result by decomposing the plane diagram of the figure of a given language into specific types of basic subdiagrams, and using this decomposition to construct a regular expression for the language .

We prove a generalisation of the above mentioned result in this contribution. Our approach is to concentrate on the form of languages that may arise as  for L taken from . In other words, we do not construct a concrete regular expression for , but we determine what kind of expression exists for such a language. In particular, we introduce a new variety of languages , which is a subvariety of the variety . Note that there is a transparent description of languages from and also an effective characterisation via the so-called acyclic automata (both are recalled in Sect. 4). The variety of languages is then characterised in the same manner: a precise description by specific regular expressions and also an automata-based characterisation are given. The letters LT in the notation refer to a characteristic property of acyclic automata in which “loops are transferred” along paths.

We show that the geometrical closure of a language from the positive variety always falls into the variety . As a consequence, each class of regular languages lying between and is geometrically closed. In particular, the positive variety is geometrically closed regardless of the alphabet, as well as is the variety .

Preliminaries

All automata considered in this paper are understood to be deterministic and finite. An automaton is thus a five-tuple , where Q is a finite set of states, is a non-empty finite alphabet, is a complete transition function, is the unique initial state, and is the set of final states. The minimal automaton of a given language L is denoted by .

By a (positive) variety of languages, we always understand what is called a (positive) -variety in [12]. We recall this notion for a reader’s convenience briefly. A class of languages is an operator, which determines, for each finite non-empty alphabet , a set of languages over . A positive variety is a class of regular languages such that is closed under quotients, finite unions and intersections, and the whole class is closed under preimages in homomorphisms. A positive variety is a variety if each is closed under complementation. Note that an alphabet could be fixed in our contribution, so homomorphisms among different alphabets play no role, and we could consider lattices of languages [9] instead of varieties of languages. However, we prefer to stay in the frame of the theory of (positive) varieties of languages as a primary aim of this paper is to describe robust classes closed under geometrical closure.

Given words u, v over an alphabet , we write if u is a prefix of v. We also write, for each ,

We call these languages the prefix closure and the prefix reduction of L, respectively. Both are prefix-closed, while and .

Proposition 1

Each positive variety is closed under the operator .

Proof

It is well known that each regular language has finitely many right quotients by words. Thus, for each alphabet and each , the language

is a finite union of right quotients of L, and its membership to follows.    

Let be a linearly ordered alphabet. The Parikh vector of a word w in  is then given by , where denotes the number of occurrences of the letter a in w. This notation extends naturally to languages: we write for . We denote by [w] the equivalence class of the kernel relation of , i.e. . Then we also write, for each language ,

and we call [L] the commutative closure of L. A language L such that is called commutative. A class of languages  is said to be closed under commutation if for each alphabet , the language [L] belongs to whenever .

In the previous paragraph we consider the mapping , where is the set of all non-negative integers. Following the ideas of [8], we introduce some technical notations concerning , whose elements are called vectors. We denote by the null vector of . Let and be vectors and be an index. We write if and, at the same time, for all . Moreover, means that for some index s. A path in is a finite sequence of vectors from  such that and for ; more specifically, we say that is a path leading to . This means that a path always begins in and each other vector of the path is obtained from the previous one by incrementing exactly one of its coordinates by one. If in addition all belong to a set , we say that is a path in F and write .

Given a word in , we write for the unique path in  such that . Conversely, for each path in , there is a unique word w such that . We denote this unique word w by . For each , we denote the set . Note that the language is prefix-closed.

Moreover, we put for each . The set is a connex figure in the sense of [8], i.e., for each , there is a path leading to such that .

Finally, the geometrical closure of L is a language . A class of languages is said to be geometrically closed if the language belongs to whenever L does, for each alphabet .

Note that the class of all regular languages is not geometrically closed, as observed in [8]. For instance, the language is regular, while its geometrical closure is the prefix closure of the Dyck language.

A Characterisation of the Geometrical Closure

We now characterise the operation of geometrical closure via three simpler operations: the prefix closure, the commutative closure, and the prefix reduction. This characterisation is a key to our later considerations.

Proposition 2

If L is a language over , then .

Proof

By definition,

If , then there is a path such that . For an arbitrary prefix u of w, we have for some . It follows that belongs to . Hence and w belongs to .

On the other hand, if w belongs to , then all prefixes u of w belong to . Thus is in for each , and is a path in , implying that w is in .    

As a direct consequence of Propositions 1 and 2, we obtain the following sufficient condition, under which a positive variety of languages is geometrically closed.

Corollary 3

Each positive variety of regular languages closed under prefix reduction and commutation is geometrically closed.

Some positive varieties of languages are geometrically closed for trivial reasons – for instance all such that for all non-empty . Let us observe that this is the case for L whenever . The proof of the following lemma is easy to see. We just note that by an absorbing state we mean a state p satisfying for every .

Lemma 4

Let L be a regular language over an alphabet and be the minimal automaton of L. Then the following conditions are equivalent:

• (i);
• (ii)for each state p in , there exists a final state reachable from p;
• (iii)every absorbing state p in is final.

The conditions of Lemma 4 are satisfied in particular for all non-empty group languages. The variety , consisting of all languages L such that the syntactic monoid is a group, is geometrically closed as a consequence. This result can be extended to languages of the form , where each is a letter, and each is a non-empty group language. Indeed, for every , there is some such that , and one can find at least one for every . Then u is a prefix of the word . This implies that . We may thus conclude that the variety , consisting of languages of level 1/2 in the group hierarchy, is geometrically closed. (The reader not familiar with the group hierarchy is referred to [12]).

In the rest of the paper, we move our attention to star-free languages.

Languages Recognised by LT-acyclic Automata

We now introduce the class of languages , which plays a central role in our main result. For every alphabet , the set consists of languages which are finite unions of languages of the form

1

The previous definition is similar to definitions of other classes of languages that have already been studied in literature. First of all, if we omit the condition , we get a definition of languages from the variety corresponding to R-trivial monoids, which we recall in more detail later. Let us conclude here just that . Secondly, if we also require in (1) for , then we obtain a variety of languages considered by Pin, Straubing, and Thérien [13] and corresponding to a pseudovariety of finite monoids denoted . Finally, if we drop in (1) the condition and then we generate a variety, then we obtain the variety of languages corresponding to the pseudovariety considered by Almeida [1, p. 236].

Since we want to characterise languages from in terms of automata, we recall the characterisation of languages from first. An automaton is acyclic if every cycle in is a loop. This means that if for some and , then also for every letter a occurring in w. The defining condition means that one can number the states in Q as in such a way that the state , with and , is always greater than or equal to p. For this reason, these automata are called extensive in [11, p. 93]. It is known that they recognise precisely R-trivial languages [6].

We say that an acyclic automaton has a loop transfer property, if implies for every and . We then call an LT-acyclic automaton for short. This means that if there is an a-labelled loop in a state p in an LT-acyclic automaton, then there is also an a-labelled loop in each state reachable from p. We may thus equivalently take in the previous definition. The first aim of this section is to show that languages recognised by LT-acyclic automata are precisely those from . We do so via a series of elementary lemmas.

Lemma 5

For a language L of the form (1), the automaton is LT-acyclic.

Proof

Let L be a language of the form (1). For every , we denote and we also put . Then it is an easy exercise to show that the automaton in Fig. 1 is the minimal automaton of L and that it is an LT-acyclic automaton.    

Fig. 1.

An LT-acyclic automaton for the language of the form (1).

Lemma 6

Let L, K be languages over an alphabet recognised by LT-acyclic automata. Then is also recognised by an LT-acyclic automaton.

Proof

The language can be recognised by the direct product of a pair of automata that recognise the languages L and K. It is a routine to check that a finite direct product of LT-acyclic automata is an LT-acyclic automaton.    

The previous two lemmas show that every language from is recognised by an LT-acyclic automaton. The following lemma strengthens this observation by implying that the minimal automaton of a language from is LT-acyclic.

Lemma 7

Let L be a language recognised by an LT-acyclic automaton. Then the minimal automaton of L is also LT-acyclic.

Proof

Let be an LT-acyclic automaton such that . The minimal automaton is a homomorphic image of some subautomaton of  [14]. It is clear that a subautomaton of an LT-acyclic automaton is LT-acyclic. Thus we may assume that has all states reachable from the initial state .

Let be a surjective mapping, which is a homomorphism from the automaton onto an automaton . We claim that is acyclic. To prove this claim, let and be such that . Then we choose some state from . For that , we have for every natural number m. Since the sequence contains only finitely many states, there are natural numbers n and m such that . Since is acyclic, we have for every letter a occurring in w. Consequently, . We showed that is acyclic.

Now let and be such that . It follows from the previous paragraph that there is such that . Since is LT-acyclic, we see that for every . Thus . We showed that is an LT-acyclic automaton. In particular, it is true for .    

Let us also prove a converse to the statements established above.

Lemma 8

Let be an LT-acyclic automaton over an alphabet . Then belongs to .

Proof

Let and let R be the set of all valid runs in the automaton , which do not use loops:

We see that the set R is finite. Moreover, for each q in Q, let denote the alphabet . Then

is a language of the form (1) for each in R and

Hence the language belongs to .    

The following theorem provides a summary of the previous lemmas.

Theorem 9

For a language , the following statements are equivalent:

• (i)L belongs to .
• (ii)L is recognised by an LT-acyclic automaton.
• (iii)The minimal automaton of L is LT-acyclic.

Proof

The statement (i) implies (ii) by Lemmas 5 and 6. The statement (ii) implies (iii) by Lemma 7. Finally, (iii) implies (i) by Lemma 8.    

One may prove that is a variety of languages in several different ways. It is possible to prove directly that the class is closed under basic language operations. It is also possible to prove that the class of LT-acyclic automata forms a variety of actions in the sense of [7]. Here we complete the previous characterisation by showing the algebraic counterpart of the class ; namely, we characterise the corresponding pseudovariety of finite monoids by pseudoidentities. We do not want to recall the notion of pseudoidentities in general. Let us only recall the implicit operation here. If we substitute for x some element s in a finite monoid M, then the image of is , which is a unique idempotent in the subsemigroup of M generated by s. It could be useful to know that, for a fixed finite monoid M, there is a natural number m such that for each .

Theorem 10

Let be an alphabet, , and the syntactic monoid of L. The following statements are equivalent:

• (i)L belongs to .
• (ii) satisfies the pseudoidentities and .
• (iii) satisfies the pseudoidentity .

Proof

Let be the minimal automaton of the language L. Then  can be viewed as the transition monoid of (see [12, p. 692]). Elements of are thus transitions of determined by words from . More formally, for , we denote by the transition given by the rule for each . Let m be a natural number such that for each s in .

Let us prove that (i) implies (ii). Suppose that L belongs to . Then is an LT-acyclic automaton by Theorem 9. In particular, the language L is R-trivial as we already mentioned. Hence, the monoid is R-trivial, i.e., satisfies the pseudoidentity . Next, let x, y be mapped to elements in which are given by words . We now need to check that . Since is acyclic, we have for every and occurring in v. Since is an LT-acyclic automaton, the loop labelled by a in state is transferred to every state reachable from . In particular, for every letter a occurring in v, there is a loop labelled by a in the state . The equality follows.

Next, let us show that the pseudoidentity is a consequence of pseudoidentities from item (ii). We may interpret x, y, z as arbitrary elements of any finite monoid M satisfying these pseudoidentities. Let m be such that for each . Then we use the second pseudoidentity from (ii) repetitively, and we get

2

By the first pseudoidentity from (ii), we get . Then we obtain using the equality (2). Thus we get .

Finally, in order to prove that (iii) implies (i), suppose that satisfies the pseudoidentity . Taking , it follows that satisfies the pseudoidentity . Hence, L is R-trivial and is acyclic. Moreover, let and be such that , and take arbitrary . Then in maps p to . Similarly, in maps p to . However, taking , , and in gives us . Therefore, . So, we see that there is a loop labelled by a in the state . We proved that is an LT-acyclic automaton and L belongs to by Theorem 9.    

Corollary 11

The class is a variety of languages corresponding to the pseudovariety of finite monoids given by

Let us also note that is known to describe the pseudovariety of finite monoids ; cf. Almeida [1, p. 212], who attributes this result to Pin. Therefore, .

The Main Result

Let us now return to the geometrical closure and prove the main result of this paper: each class of languages lying between the variety of languages and the positive variety is geometrically closed. This strengthens the result from [8] mentioned in the Introduction.

The route that we take to this result (Theorem 16) consists of three steps:

1. We recall that the class is closed under commutation [5, 10]. Although it is not necessary to obtain our main result, we refine this observation by proving that a commutative closure of a -language is piecewise testable.

2. We prove that each commutative -language belongs to .

3. We observe that the variety is closed under prefix reduction.

These three observations imply that the geometrical closure of a -language belongs to , from which our main result follows easily.

Recall the result of Arfi [2], according to which a language belongs to if and only if it is given by a finite union of languages , where are letters from and are subalphabets of . It follows by a more general result of Guaiana, Restivo, and Salemi [10], or of Bouajjani, Muscholl, and Touili [5] that is closed under commutation, and this observation is a first step to Theorem 16.

Let us show that a commutative closure of a -language is in fact piecewise testable.

Lemma 12

A commutative closure of a -language is piecewise testable.

Proof

Let an alphabet be fixed. It is clear that if are languages, then

As a result, it is enough to prove piecewise testability of [L] for all languages , with and .

Let L be of this form. Denote , and . We claim that

3

Indeed, if w is in [L], then for some , while clearly for each a in , and for each b in . Conversely, let w in be such that for each a in , and for each b in . Then for v in given by , where () is given as follows: if , then

The word v is in L by construction, hence w belongs to [L].

It remains to observe that the language [L] given by (3) is piecewise testable. However, this language is equal to

4

The language on the right-hand side of (4) is piecewise testable.    

We now proceed to prove that the geometrical closure of each language from belongs to .

Lemma 13

Every commutative language L from belongs to .

Proof

If we take into account the proof of Lemma 12 and the fact that is closed under finite unions, it is enough to prove that every language of the form (3) belongs to . We may also use the expression (4) for that language. For each letter and a natural number m, we may write . This shows that the language belongs to . Since is a variety, we see that also the language belongs to . Altogether, the language (4) belongs to the variety .    

Finally, let us observe that the variety is closed under prefix reduction.

Lemma 14

Let L be a language from for some alphabet . Then belongs to as well.

Proof

Let L be recognised by some LT-acyclic automaton . If , then L does not contain the empty word, and consequently , which belongs to . So we may assume that .

Now, simply saying, we claim that the language is recognised by the automaton constructed from by replacing all non-final states with a single absorbing non-final state . More precisely, we construct an automaton , where is a new state, for which we define for each . Furthermore, for each and , we put if , and otherwise. As contains no cycle other than a loop, the constructed automaton has the same property. Moreover, any state of reachable in  from some p in is either reachable from p in , or equal to . As for each c in , this implies that is an LT-acyclic automaton and belongs to by Theorem 9.    

Theorem 15

Let be an alphabet and . Then .

Proof

We have by Proposition 2. As is a positive variety of languages, belongs to whenever L belongs to this set by Proposition 1. The language is thus a commutative -language by [5, 10]. (Note that the language is actually commutative piecewise testable, by Lemma 12.) It follows by Lemma 13 that belongs to , and by Lemma 14 that the language belongs to as well.    

We are now prepared to state the main result of this article merely as an alternative formulation of the theorem above.

Theorem 16

Let be a class of languages containing , which is contained in . Then is geometrically closed.

There are many important (positive) varieties studied in the literature for which the main result can be applied.

Corollary 17

The following classes are geometrically closed: the positive variety , the variety , the variety , the variety of all -recognisable languages, the variety of all -recognisable languages.

The variety of all -recognisable languages coincides with the intersection of and its dual. This class has a natural interpretation in terms of logical descriptions of levels in Straubing-Thérien hierarchy (see Section 5 in [15]).

Conclusions

We have introduced a new variety of languages and we have proved that geometrical closures of languages from fall into . As a consequence, we have seen that many natural classes of star-free languages are geometrically closed, namely those between the variety and the positive variety . On the contrary, the variety of all piecewise testable languages is not geometrically closed. The example is not included in the paper due to space limitations.

There are some interesting questions in connection to the paper. First of all, one may ask how to effectively construct a regular expression for the geometrical closure for a given language L from . Note that it is effectively testable, for a given deterministic finite automaton , whether the language belongs to (see [12, p. 725]). It is not clear to us whether a regular expression for can be effectively computed from .

Nevertheless, the main open question related to the topic is to clarify the behaviour of the geometrical closure outside the class .