Computing probability of Borelian languages

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Computing probability of Borelian languages

Mustapha Arfi, Carla Selmi

To cite this version:

Mustapha Arfi, Carla Selmi. Computing probability of Borelian languages. 2012. �hal-00834017�

Computing probability of Borelian languages

Arfi M. and Selmi C., ^∗

Abstract: The Kolmogorov extension theorem shows that, for any probability lawπ onA^∗, there exists one and only one probability measure, namely Pπ, on the family of Borelian languages ofA^ωsuch thatP_π(wA^ω) =π(w). We give in this paper a method to compute the probability mesure, given by the Kolmogorov extension theorem, of Borelian languages of infinite words on a finite alphabetA. This method becomes effective in the case of rational Borelian languages when the probability law is computable, as in the case of probability law defined by an automaton.

Keywords: Words, infinite words, formal languages, probability, topology.

Introduction

Probability laws (distributions) play an important role in language theory and, in particular, in that of codes [4]. Also called cylindric measures, in the litterature de- voted to the subject, they are in fact defined on finite words over a certain alphabet.

According to Kolmogorov axtension theorem, there exists a one to one correspon- dence between the set of probability distributions and that of probability measures on infinite words over the same alphabet. They can be used in order to study mixed strategies in game theory [1, 2], which will be done in a future paper.

The purpose of this article consists in computing the probability measure of Borelian languages of infinite words. It provides an explicit method allowing to evaluate that of ω-rational languages, by decomposing them in more elementary sets in the Borelian hierarchy.

Our paper is organized in the following way: In Section 1, we recall the basic notions and properties on finite and infinite words, on the topological space of infinite words, on Borelian sets, on measures and probability laws. We state, in Section 2, the Kolmogorov extension theorem to the space of infinite words. In Section 3, we show how to obtain the probability of open and closed sets when the probability law is computable, then we apply these results, in Section 4, to compute the probability of any Borelian language. Finally, in Section 5, we give an effective method to obtain the probability ofω-rational languages.

∗Normandie Univ, France, UR LITIS F-76821 Mont Saint Aignan, France {mustapha.arfi, carla.selmi}@litislab.eu

1 Preliminaries

We recall in this section the basic notions and properties on finite and infinite words, on topological space of infinite words, on Borelian sets, on measure and on proba- bility laws.

1.1 Finite and infinite words

A wordis a finite sequence of elements of an alphabet A. We denote by A^∗ the set of all words on A. The length of a word w ∈ A^∗, denoted by |w|, is the number of letters of A composing w. The empty word ǫ is the only word of length zero.

We denote byA^k the set of all words in A^∗ length of which isk. Given two words u, v∈A^∗, we say that u is a prefix of v ifv∈uA^∗

An infinite word onAis an infinite sequencehof elements ofA, which we will write h = h0h1· · ·ht· · ·. We denote by A^ω the set of all infinite words on A. Given a word w ∈ A^∗ and an infinite word h ∈ A^ω, we say that w is a prefix of h if there exists an infinite word h^′ ∈A^ω such thath=wh^′.

IfL is a subset of A^ω, we denote by P refk(L) the set of all words that are prefixes of length k of words of L. We set P ref(L) =S

i≥0P refi(L). In the sequel, all the alphabets are supposed to be finite.

1.2 Topology of A^ω

We consider on the setA^ω the distance ddefined as follows:

d(x, y) = (1 +max{|w| |w∈P ref({x})∩P ref({y})})⁻¹

with the convention 1/∞ = 0. Equipped with this distance, A^ω is a compact and complete metric space, since A is finite. The reader is referred to [5] for more topological basic definitions. In this reference, a new operator is introduced as follows: for each subsetL⊂A^∗, let

−

→L ={u∈A^ω|u has infinitely many prefixes inL}.

In the same reference, it is shown the following proposition, which we will use in the sequel.

Proposition 1.1 Let L be a subset of A^ω

1. L is a closed set if and only if L=−−−−−−→

P ref(L).

2. L is an open set if and only if it is of the form L=XA^ω, X ⊂A^∗. Further- more, L= (X\XA⁺)A^ω.

3. L is a clopen set if and only if it is of the form L=XA^ω where X is a finite subset ofA^∗.

1.3 Borelian sets

LetE be a set. Denote by X^c the complement of a subsetX of E. A family F of subsets ofE is said a σ-algebra if it satisfies the following conditions:

• ∅ ∈ F,

• X^c ∈ F, ∀X∈ F,

• S

n≥0Xn∈ F, for all sequence (Xn)n≥0 of elements ofF.

For each collectionCof subsets ofE, there exists a smallestσ-algebra onEcontainig C which is denoted by σ(C). One says that σ(C) is the σ-algebra generated by C.

IfE is a topological space, it is usual to note B(E) the σ-algebra generated by the collection of open subsets of E. B(E) is called the Borelian σ-algebra of E. Its elements constitute the Borelian sets ofE.

Proposition 1.2 In the case of a metric spaceE, the following statements hold:

• σ(C) =B(E), where C denotes the family of closed subsets of E.

• σ(C) =B(E), where C denotes the family of clopen subsets of E.

1.4 Measures and probability laws

LetE be a set and F be a σ-algebra of subsets of F. We will say that (E,F) is a measureable space. One defines a measure on (E,F) as a mapµ:F −→IR+∪{+∞}

such that : - µ(∅) = 0, - µ(S

n≥0Xn) = P

n≥0µ(Xn), for every pairwise disjoint sequence (Xn)n≥0 of elements ofF.

The second condition is calledσ-additivity.

A probability measure on (E,F) is a measureµ:F −→[0,1] such thatµ(E) = 1.

2 Kolmogorov extension theorem

LetA be a finite alphabet. A probability law on A^∗ is a map π :A^∗ −→[0,1] such that:

- π(ǫ) = 1, - P

a∈Aπ(wa) =π(w), ∀w∈A^∗.

The second part of the definition is called coherence condition.

Example 2.1 Let A = {a, b} and α ∈ [0,1]. Consider the map π : A^∗ −→ [0,1]

given by:

π(ǫ) = 1, π(a) =α, π(b) = 1−α andπ(wa) =π(w)π(a),∀a∈A,∀w∈A^∗. The mapπ determines a probability law on A^∗, called Bernoulli probability law.

Example 2.2 [4] A probability law on words can be given by a finite automaton in the following way. Let A = hQ, Ai be a finite deterministic automaton on the alphabetA. Letπs be a probability distribution onQ,the set of states ofA. For each state q∈Q, we define a probability distributionπq on the set of edges starting in q.

Thus,

P

q∈Qπs(q) = 1 and P

a∈Aπq(q, a) = 1, ∀q∈Q.

This allows to define a probability law πγ on the set of paths in A. Let c = (q₀, a₀, q₁)(q₁, a₁, q₂). . .(qn, an, q_n+1), ai ∈A,∀0 ≤i≤n, qj ∈Q,∀0≤j ≤n+ 1, be a path in A. We set

πγ(c) =πs(q₀)πq0(q₀, a₀)πq1(q₁, a₁). . . πqn(qn, an).

The probability of empty path is one.

This in turn defines a probability lawπ on the set of words on A. Let w∈A^∗. π(w) is the sum ofπγ(c) on all paths c in A with label w.

The probability given by a finite automaton is effectively computable.

Remark 2.3 The Bernoulli law of the Example 2.1 is given by the automaton A= h{1},{(1, a),(1, b)}i withπ_s(1) = 1, π₁(1, a) =α and π₁(1, b) = 1−α.

In the sequel, all probability laws will be supposed computable.

Let nowµbe a probability measure onB(A^ω). Consider the mapπ:A^∗−→[0,1]

given by π(w) = µ(wA^ω) forw ∈A^∗. Then we have π(ǫ) = µ(A^ω) = 1. Moreover, π satisfies the coherence condition, since:

X

a∈A

π(wa) =X

a∈A

µ(waA^ω) =µ([

a∈A

waA^ω) =µ(wA^ω) =π(w),∀w∈A.

Hence,π is a probability law onA^∗, as defined at the beginning of this section. The converse statement, known as the Kolmogorov extension theorem [4], is given by Theorem 2.4 For any probability law π onA^∗, there exists one and only one prob- ability mesure Pπ on the family of Borelian subsets of A^ω such that Pπ(wA^ω) = π(w), ∀w∈A^∗.

We will say that Pπ is the probability measure associated with the probability law π defined onA^∗.

3 Computing probability of open and closed sets of A

Letπ:A^∗−→[0,1] be a probability law onA^∗. For each languageL ofA^∗, set π(L) =X

w∈L

π(w).

π defines a map from P(A^∗) into [0,1]. In the sequel, we denote by π a probability law onA^∗ and byP_π the probability measure associated with the probability lawπ 3.1 Probability of open sets

We will show how to compute the probability of open sets. We begin with the following proposition.

Proposition 3.1 Let XA^ω, X ⊂ A^∗, be a an open set of A^ω. Then Pπ(XA^ω) = π(X\XA⁺).

Proof. Proposition 1.1 implies L = XA^ω = S

x∈(X\XA⁺)xA^ω where the union is disjoint sinceX\XA⁺is a prefix set. So, we havePπ(L) =P

w∈X\XA⁺Pπ(wA^ω). By Theorem 2.4,Pπ(wA^ω) =π(w) for allw∈A^ω. We obtain,Pπ(L) =P

w∈X\XA⁺π(w)

=π(X\XA⁺).

We obtain as a corollary of the previuos proposition the following statement:

Corollary 3.2 LetXA^ω, X⊂A^∗,be a an open set ofA^ω where X⊂A^∗ is a prefix set. Then we haveP_π(XA^ω) =π(X).

We remark that the probability of a clopen setL=XA^ω is effectively computable since the set X ⊂ A^∗ is finite. In Proposition 3.1 we proved that Pπ(XA^ω) = π(X\XA⁺). In the following proposition we show how to computeπ(X\XA⁺) for each X⊂A^∗. We denote byX⁽ⁿ⁾ =X∩Aⁿ for each X⊂A^∗ and for eachn≥0.

Proposition 3.3 Let XA^ω, X ⊂A^∗, be an open set of A^ω. We have:

Pπ(XA^ω) = limn→∞π(Sn

k=0X^(k)A^n−k).

Proof. We can write XA^ω =S

n≥0X⁽ⁿ⁾A^ω =S

n≥0(Sn

k=0X^(k)A^ω). Since the first union is increasing, we havePπ(XA^ω) = lim_n→∞Pπ(Sn

k=0X^(k)A^ω).

Now, we have Sn

k=0X^(k)A^ω =Sn

k=0(X^(k)A^n−k)A^ω = (Sn

k=0X^(k)A^n−k)A^ω. So, we obtain P_π(XA^ω) = lim_n→∞P_π((Sn

k=0X^(k)A^n−k)A^ω). Since (Sn

k=0X^(k)A^n−k) is a prefix set, Corollary 3.2 implies thatPπ(XA^ω) = limn→∞π(Sn

k=0(X^(k)A^n−k)).

3.2 Probability of intersection of open sets

Now, we are able to compute also the probability of the intersection of open sets. We will use this result in the sequel to compute the probability of rationalω-languages.

Proposition 3.4 Let XA^ω, Y A^ω, X, Y ⊂A^∗ be open sets ofA^ω. We have:

Pπ(XA^ω∩Y A^ω) = lim

n→∞π( [

p+q≤n

(X^(p)A^q∩Y^(q)A^p)A^n−p−q)).

Proof.We can write:

XA^ω∩Y A^ω = (S

p≥0X^(p)A^ω)∩(S

q≥0Y^(q)A^ω)

= S

p,q≥0(X^(p)A^ω∩Y^(q)A^ω)

= S

p,q≥0(X^(p)A^qA^ω∩Y^(q)A^pA^ω)

= S

p,q≥0(X^(p)A^q∩Y^(q)A^p)A^ω

= (S

p,q≥0X^(p)A^q∩Y^(q)A^p)A^ω

= T A^ω

It follows by Proposition 3.3 that Pπ(XA^ω∩Y A^ω) = lim_n→∞π(Sn

k=0(T^(k)A^n−k)) SinceT^(k)=S

p+q=k(X^(p)A^q∩Y^(q)A^p), we have successively:

Sn

k=0T^(k)A^n−k = Sn k=0(S

p+q=k(X^(p)A^q∩Y^(q)A^p))A^n−k

= Sn k=0

S

p+q=k(X^(p)A^q∩Y^(q)A^p)A^n−k

= Sn k=0

S

p+q=k(X^(p)A^q∩Y^(q)A^p)A^n−p−q

= S

p+q≤n(X^(p)A^q∩Y^(q)A^p)A^n−p−q We obtainPπ(XA^ω∩Y A^ω) = limn→∞π(S

p+q≤n((X^(p)A^q∩Y^(q)A^p)A^n−p−q)).

3.3 Probability of closed sets

We will show in the sequel how to compute the probability Pπ for a closed set of A^ω. We begin by proving the following statement.

Lemma 3.5 For every closed set F of A^ω, we have F =T

t≥0P reft(F)A^ω. Proof.It is clear thatF ⊂T

t≥0P reft(F)A^ω. To establish the opposite inclusion, consider a word h∈T

t≥0P ref_t(F)A^ω. For each t≥0,h admits a prefix of length t in P reft(F). So, h has infinitely many prefixes in P ref(F), then h ∈ −−−−−−→

P ref(F).

SinceF is closed, it verifiesF =−−−−−−→

P ref(F) and thus,h∈F. This Lemma leads to the next result:

Proposition 3.6 For each closed setF of A^ω, we have:

Pπ(F) = lim

t→∞π(P reft(F)).

Proof. By applying the previous lemma, we can writeF =T

t≥0P reft(F)A^ω. Since this intersection is a decreasing one, it implies: π(F) = lim_t→∞Pπ(P reft(F)A^ω).

Remark now that P ref_t(F)A^ω is a clopen set and that P ref_t(F) is a prefix set, for each t ≥ 0. So Pπ(P reft(F)A^ω) = π(P reft(F)). Finally, we obtain: π(F) =

lim_t→∞π(P reft(F)).

Example 3.7 Given a real number α ∈ [0,1], consider the probability law π on A={a, b}defined by: π(ǫ) = 1, π(wa) =α, π(wb) = 1−α, ∀w∈A^∗.LetL=a^∗b^ω+ a^ω ⊂A^ω. Since L=−−−−−−→

P ref(L), L is a closed set. The computation of the languages L_t = P ref_t(L), for t ≥ 0, leads to the relation: L₀ = {ǫ} and L_t+1 = aL_t+b^t+1 (the proof is left to the reader). So, we obtain on the probability side: π(L0) = 1 and π(L_t+1) =απ(Lt) + (1−α)^t+1. A simple proof by induction gives immediately π(L_t) =Pt

i=0α^t−i(1−α)ⁱ=Pt

i=0αⁱ(1−α)^t−i. Suppose nowα≤(1−α), the other case is symetric. It implies thatαⁱ ≤(1−α)ⁱ for all i≥0. So, Pt

i=0αⁱ(1−α)^t−i≤ Pt

i=0(1−α)^t = (t+ 1)(1−α)^t. Thus, we have 0≤π(Lt)≤(t+ 1)(1−α)^t, which allows us to conclude that π(L) = lim_t→∞π(L_t) = 0, except for α = 1 where this probability is equal to one.

4 Computing probabilities on B( A

)

The Borelian sets of A^ω, as those of any metric space, are organized in the Borel hierarchy defined as follows:

Π₀ = Σ₀={L⊂A^ω |L clopen},∆₀ = Σ₀ and for alln≥0:

Σ_n+1 = {S

i≥0Li |Li ∈Πn} Π_n+1 = {T

i≥0Li |Li ∈Σn}

∆_n+1 = Σ_n+1T Π_n+1. Moreover, one can also remark that for alln≥0

Π_n = {L^c|L∈Σ_n}.

In the particular case of metric spaces, the Borel hierarchy verifies the following property:

Proposition 4.1 In a metric space, the following inclusions hold:

∆₁ ⊂Σ₁ ⊂∆₂ ⊂Σ₂ ⊂. . .

∆₁⊂Π₁ ⊂∆₂ ⊂Π₂⊂. . .

We have already seen how to compute the probability of a language belonging to the levels 0 and 1 of this hierarchy, To proceed to the computation of the probability for the upper hierarchy levels, we need to establish the following proposition.

Proposition 4.2 For any n≥0,Πn (resp. Σn) is closed under finite union (resp.

finite intersection).

Proof. We will make it by induction onn only for Πn, the case of Σn is the same if one considers the intersection. We know that any finite union of closed sets is a closed one. Thus, the property holds for n = 0. Suppose now that the property is true at the rank n, Let L, K ∈ Πn. By definition, we obtain: L =T

i≥0Li and K=T

j≥0Kj withLi, Kj ∈Πn, then L∪K = (T

i≥0L_i)∪(T

j≥0L_j) =T

i,j≥0(L_i∪K_j), L_i, K_j ∈Π_n.

Since, by induction hypothesis, Π_nis closed by finite union, it follows thatL_i∪K_j ∈

Πn. So, L∪K∈Π_n+1.

The following statement represents an immediate consequence of the previous propo- sition.

Corollary 4.3 Every Borelian languageL of level n >0 can be written as increas- ing countable union or as a decreasing countable intersection of Borelian sets all belonging to the previous level. That is

L=S

i≥0Li or L=T

i≥0Li,

where Li is a Borelian language of level n−1, ∀i≥0. In both cases, we obtain, P_π(L) = lim_i→∞P_π(L_i).

4.1 Probability of F_σ and of G_δ

In a topological space, the class Fσ (resp. Gδ) is usually defined as that consisting of all countable unions (resp. intersections) of closed (resp. open) sets. We can even take countable increasing (resp. decreasing) unions (resp. intersections) to build the set of Fσ (resp. Gδ). One can easily notice that the levels Σ₂ and Π₂ coincide exactly with the families Fσ and G_δ. These classes of Borelian languages are very important beacause, as we see later, one can show that every ω-rational Borelian language can be obtained as a boolean combination of sets belonging to FσS

Gδ. We begin with the following remark:

Remark 4.4 In a metric space, which is the case of A^ω, the open (resp. closed) sets are elements ofFσ (resp. Gδ).

Given a probability lawπ onA^∗ and a languageLbelonging to F_σ, it becomes now possible to obtain the probability of L. We know that there exists an increasing sequence (Fn)_n≥0 of closed sets such that L=S

n≥0Fn. So we have:

P_π(L) = lim_n→∞P_π(F_n), sinceP_π is a probability measure onB(A^ω).

If we suppose nowL inGδ, there exists a decreasing sequence (On)_n≥0 of open sets such thatL=T

n≥0On. The probability of Lis then given by the formula:

P_π(L) = lim_n→∞(1−P_π(O_n^c)) = 1−lim_n→∞P_π(O^c_n).

We give some examples of computation of probability of sets belonging toFσ and G_δ.

Example 4.5 Consider on the alphabetA={a, b}, the languageL=A^∗a^ω. Notice that this language is not closed for the usual topology onA^ωbecauseb^ω∈−−−−−−−→

P ref(A^∗a)\

A^∗a^ω. The language L is exactly the set of infinite words having a finite number of b. The language L belongs to Fσ, since we can write L = S

n≥0Aⁿa^ω, where Aⁿa^ω is closed. So: Pπ(L) = lim_n→∞Pπ(Aⁿa^ω). The real numbers Pπ(Aⁿa^ω) = limt→∞π(P reft(Aⁿa^ω)), for each n≥0, depend on the chosen probability law.

We denote by X_(n)=S

p≥nX^(p) for each X⊂A^∗ and for each n≥0.

Let X ⊂ A^∗. We have −→

X ∈ Gδ since X can be written as an intersection of open sets,

−

→X =T

n≥0X_(n)A^ω.

Using Proposition 3.3, we are able to compute the probability of a set of the form

−

→X , X ⊂A^∗.

Proposition 4.6 Let −→

X ⊂A^ω, X ⊂A^∗. The probability of −→

X is given by:

Pπ(−→

X) = lim_p→∞lim_q→∞π(Sq

k=p(X^(k)A^q−k)).

Proof. We can write −→ X = T

p≥0X_(p)A^ω and since this intersection is decreasing, we have Pπ(−→

X) = lim_p→∞Pπ(X_(p)A^ω). By Proposition 3.3, we obtain:

limp→∞Pπ(X_(p)A^ω) = limp→∞limq→∞π(Sq

k=0(X_(p)^(k)A^q−k))

= lim_p→∞lim_q→∞π(Sq

k=p(X^(k)A^q−k)),

carX_(p)^(k)=X^k ifk≥p and X_(p)^(k) =∅otherwise.

We denote byR^qp =Sq

k=pX^(k)A^q−k, q≥p≥0. So,P_π(−→

X) = lim_p→∞lim_q→∞π(R^qp).

We can compute the setRp^q by induction in the following way:

- R^pp =X^(p), - R^qp =Sq

k=pX^(k)A^q−k=Sq−1

k=pX^(k)A^q−kS

X^(q)=R^q−1p AS X^(q). Example 4.7 In the particular case of the Borelian language −−→

A^∗a, where A = {a, b}, we have:

- R^pp =X^(p) =A^p−1a,

- R^qp =A^p−1(A^q−p+1\ {b^q−p+1}).

Letπ be a multiplicative probability law onA^∗ and letPπ be the Kolmogorov measure associated to π. The probability of −−→

A^∗a is given by:

Pπ(−−→

A^∗a) = limp→∞limq→∞π(A^p−1(A^q−p+1\{b^q−p+1})) = 1−limp→∞limq→∞π(b^q−p+1).

We note that 1−lim_p→∞lim_q→∞π(b^q−p+1) is equal to1 if π(b) = 0and it is equal to0 otherwise.

5 Case of rational Borelian languages

On obtain, as a consequence of the Theorem of MacNaughton [5], the following proposition:

Proposition 5.1 Anyω-rational subset L of A^ω is a disjoint union of the form L=S

1≤i≤n(−→ Xi\−→

Yi), where Xi, Yi ⊂A^∗ for any 1≤i≤n.

Remark 5.2 The family of ω-rational languages belongs to level three of the Borel hierarchy. In fact, let−→

X ,−→

Y ⊂A^ω, X, Y ⊂A^∗. We can write−→ X\−→

Y =−→ X∩(−→

Y)^c. As we have −→

X =T

i≥0L_i, where L_i is an open set for all i≥0 and (−→ Y)^c =S

j≥0M_j, where M_i is a closed set for all j≥0,−→

X\−→

Y is a boolean combination of languages in Fσ∪G_δ, that is: −→

X \−→ Y = (T

i≥0Li)T (S

j≥0Mj). Applying Proposition 4.1 and Proposition 4.2, one can see that every ω-rational subset ofA^ω is included in Σ₃. We computed, in Proposition 4.6, the probability of a set inA^ω of the form −→

X with X ⊂A^∗. The following Proposition shows how to compute the probability of a set in A^ω of the form −→

X \−→

Y with X, Y ⊂ A^∗. So the probability of any ω-rational subsetL=S

1≤i≤n(−→ Xi\−→

Yi) is given by Pn

i=1Pπ(−→ Xi\−→

Yi).

Proposition 5.3 Let X, Y ⊂A^∗. We have:

Pπ(−→ X\−→

Y) = limp→∞limq→∞(π(Sq

k=pX^(k)A^q−k))−

lim_p→∞lim_q→∞(S

s,t≥p,s+t≤q(X^(s)A^t∩Y^(t)A^s)A^q−s−t)).

Proof. We can write−→ X = (−→

X\−→ Y)∪(−→

X∩−→

Y) and, since this union is disjoint, the probability of −→

X is given by Pπ(−→

X) = Pπ(−→ X \−→

Y) +Pπ(−→ X ∩−→

Y). By Proposition 4.6, we obtain:

Pπ(−→ X\−→

Y) = lim_p→∞lim_q→∞π(Sq

k=pX^(k)A^q−k)−Pπ(−→ X∩−→

Y).

Now, we write:

−

→X∩−→ Y = (T

t≥0X_(t)A^ω)∩(T

t≥0Y_(t)A^ω) =T

p≥0(X_(p)A^ω∩Y_(p)A^ω).

Since the intersection onp is decreasing, we obtain:

Pπ(−→ X∩−→

Y) = limp→∞Pπ(X_(p)A^ω∩Y_(p)A^ω).

Applying Proposition 3.4, we have:

Pπ(−→ X∩−→

Y) = limp→∞limq→∞π(Sq

k=p(X^(k)A^q−k))

= lim_p→∞lim_q→∞π(S

s,t≥p,s+t≤q(X^(s)A^t∩Y^(t)A^s)A^q−s−t).

Previous computations give:

Pπ(−→ X\−→

Y) = limp→∞limq→∞((π(Sq

k=pX^(k)A^q−k))−

lim_p→∞lim_q→∞(π(S

s,t≥p,s+t≤q(X^(s)A^t∩Y^(t)A^s)A^q−s−t))).

We denote bySp^q =S

s,t≥p,s+t≤q(X^(s)A^t∩Y^(t)A^s)A^q−s−t), q≥p≥0, q >0. We can compute the setSp^q by induction in the following way:

- Sp^2p =X^(p)A^p∩Y^(p)A^p, - Sp^q=Sp^q−1∪(S

s,t≥p,s+t=q(X^(s)A^t∩Y^(t)A^s)A^q−s−t).

Example 5.4 We compute, in this example, the probability of the Borelian language of infinite words on the alphabet A = {a, b} having infinitely many occurrences of the letterabut only a finite number of occurrences of the letterb. It is easy to verify that this language can be write as −−→

A^∗a\−−→

A^∗b.

First, we compute the set Sp^q. We have:

- Sp^2p =A^p−1aA^p∩A^p−1bA^p=∅, - Sp^q=∅ ∩(S

s,t≥p,s+t=q(A^s−1aA^t∩Y^t−1bA^s)) =A^p−1T A,

where T = {w ∈ A^∗||w| = q −p + 2,|w|a 6= 0,|w|b 6= 0} = A^q−p+2 \ {a^q−p+2, b^q−p+2}.

By Proposition 5.3, the probability of −−→

A^∗a\−−→

A^∗b is given by:

Pπ(−−→

A^∗a\−−→

A^∗b) = lim_p→∞lim_q→∞(π(R^qp)−π(Sp^q))

= lim_p→∞lim_q→∞(R^qp)−

= limp→∞limq→∞(π(A^p−1{A^q−p+2\ {a^q−p+2, b^q−p+2}}A)).

Letπ be a probability multiplicative law onA^∗ and letPπ be the Kolmogorov measure associated toπ. Applying the result of the Example 4.7, the probability of −−→

A^∗a\−−→

A^∗b is given by:

P_π(−−→

A^∗a\−−→

A^∗b) = lim_p→∞lim_q→∞(−π(b^q−p+1) +π(a^q−p+2) +π(b^q−p+2))

= limp→∞limq→∞((π(b)−1)π(b^q−p+1) +π(a^q−p+2))

= lim_p→∞lim_q→∞(−π(a)π(b^q−p+1) +π(a^q−p+2))

= π(a) lim_p→∞lim_q→∞((π(a^q−p+1)−π(b^q−p+1))).

References

[1] Arfi M., B. Ould M Lemine and C. Selmi, Strategical languages of infinite words, Information Processing Letters,109, (2009), 749-753.

[2] Arfi M., B. Ould M Lemine and C. Selmi, Mixed strategies and closed sets of infinite words, In Proceedings of the Conference Words 2009, 14-18 Septembre 2009, Salerne, Italy.

[4] M. Lothaire, Applied Combinatorics on Words, Encyclopedia of Mathe- matics and its Applications,105, Cambridge University Press, 2005.

[4] Perrin D., J. Berstel and C. Reutenauer, Codes and automata, Encyclo- pedia of Mathematics and its Applications, 129, Cambridge University Press, 2009.

[5] Perrin D. and J.-´E. Pin, Infinite words, Automata, Semigroups, Logic and Games, Elsevier, Academic Press, 2004.