Weighted First-Order Logics over Semirings

Weighted First-Order Logics over Semirings ^∗

Eleni Mandrali

and George Rahonis

Dedicated to the memory of Ferenc G´ecseg Abstract

We consider a first-order logic, a linear temporal logic, star-free expres- sions and counter-free B¨uchi automata, with weights, over idempotent, zero- divisor free and totally commutative complete semirings. We show the expres- sive equivalence (of fragments) of these concepts, generalizing in the quanti- tative setup, the corresponding folklore result of formal language theory.

1 Introduction

The expressive equivalence of monadic second-order logic and finite automata over finite words was established in [5, 16] and over infinite words in [6]. Droste and Gastin, in [8] (cf. also [9]), introduced a weighted monadic second-order logic over semirings and showed that sentences from a fragment of this logic, interpreted over finite words, are equivalent to weighted automata. A corresponding result for infi- nite words was stated in [13]. Recently in [12], the authors extended the expressive equivalence of monadic second-order logic and automata over more general struc- tures, namely valuation monoids. On the other hand, first-order (FO for short) logic (i.e., the logic obtained from monadic second-order one by relaxing second- order quantifiers) is equivalent to linear temporal logic (LTL for short), star-free expressions and counter-free B¨uchi automata (cf. for instance [7]). More interest- ingly,LTL and its alternatives serve as specification languages in model checking for real world applications [3, 22, 31]. The last few years there is also an increas- ing interest in establishing FO logic and its equivalent objects in the quantitative framework. This is motivated by the need to create model checking tools which incorporate quantitative features. In [14], the aforementioned equivalence was es- tablished in the weighted setup of arbitrary bounded lattices. Recently, in [26] (cf.

also [24]), we introduced a weighted FO logic, a weightedLTL, ω-star-free series

∗Research of the first author has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program:

Heracleitus II. Investing in knowledge society through the European Social Fund.

†Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece.

E-mail:elemandr@gmail.com

‡E-mail:grahonis@math.auth.gr

DOI: 10.14232/actacyb.22.2.2015.13

and counter-free weighted B¨uchi automata over the max-plus semiring with dis- counting and investigated fragments of them satisfying an expressive equivalence.

The convergence of infinite sums over nonnegative real numbers was ensured by the existence of discounting parameters.

In this paper, we consider a weighted FO logic, a weighted LTL, ω-star-free series and counter-free weighted B¨uchi automata over idempotent, zero-divisor free and totally commutative complete semirings. We show that there are suitable fragments of our objects so that the classes of infinitary series, derived by them, coincide. Our results can be proved for series over finite words as well, though we skip any technical detail.

The structure of our paper is as follows. Except of this introductory section, in Section 2 we recall the notion of totally commutative complete semirings and present notations used in the paper. The underlying structure for all weighted objects considered in the paper will be an arbitrary idempotent, zero-divisor free and totally commutative complete semiring.

In Section 3 we introduce the weighted LTL and define the semantics of LTL formulas interpreted as infinitary series. We consider a fragment of ourLTLnamely the fragment ofU-nesting formulas. We should note that a quantitativeLTLover De Morgan algebras was introduced for the first time in [21].

In Section 4 we consider the weightedFO logic which is in fact the one induced by the weightedMSO logic of [8, 9]. Its semantics is interpreted by infinitary series as induced by the semantics of the corresponding weightedMSO logic of [13]. We consider the fragment of weakly quantifiedFO logic formulas and in our first main result, in Section 5, we show that every series which is definable by a U-nesting LTLformula is definable also by a weakly quantifiedFO logic sentence.

In Section 6 we deal with star-free and ω-star-free series. We recall that the class of star-free languages over an alphabet A is the smallest class of languages over A which contains ∅, the singleton {a} for every a ∈ A, and which is closed under finite union, complementation and concatenation. Furthermore, the class of ω-star-free languages overAis the closure of the empty set under the operations of union, complement and concatenation with star-free languages on the left (cf. for instance [7, 23, 27, 29]). It is worth noting that the application of the star-operation (whenever it is permitted) to star-free languages is implemented by the other oper- ations. However, in the setup of series (over semirings) the complement operation is not ”too strong”. Therefore, we defined the classω-star-free series as the least class of infinitary series generated by the monomials (overAand our semiring) by applying finitely many times the operations of sum, Hadamard product, comple- ment, Cauchy product, and iteration andω-iteration restricted to series of the form P

a∈A(ka)_a where, for every a∈A, ka is an element of our semiring. The second main result of the paper, in Section 7, states that the class of definable series by weakly quantifiedFO logic sentences is contained in the class ofω-star-free series.

In Section 8 we introduce counter-free weighted automata and counter-free weighted B¨uchi automata and investigate closure properties of the classes of their behaviors. We define a fragment of the class of series accepted by counter-free weighted B¨uchi automata, namely the class of almost simpleω-counter-free series

and we show, in Section 9, that this contains the class ofω-star-free series.

Finally, in Section 10 we show that the class of almost simple ω-counter-free series is contained in the class of series which are definable byU-nestingLTLfor- mulas. In fact this last inclusion concludes the coincidence of the classes of series definable by U-nesting formulas of the weighted LTL and weakly quantified FO logic sentences, ω-star-free series and almost simple ω-counter-free series. In the Conclusion we refer to some interesting problems for further research. A prelimi- nary version of this paper appeared in [25].

2 Preliminaries

LetAbe an alphabet, i.e., a finite nonempty set. As usually, we denote byA^∗the set of all finite words overAandA⁺=A^∗\{ε}, whereεis the empty word. The set of all infinite sequences with elements inA, i.e., the set of all infinite words overA, is denoted byA^ω. A finite wordw=a₀. . . a_n−1, wherea₀, . . . , a_n−1∈A(n≥1), is written also as w=w(0). . . w(n−1) wherew(i) = a_i for every 0≤i≤n−1.

For every 0≤i≤n−1, we denote byw<i (resp. w_≤i) the prefixw(0). . . w(i−1) (resp. w(0). . . w(i)) of w and byw>i (resp. w_≥i) the suffix w(i+ 1). . . w(n−1) (resp. w(i). . . w(n−1)) ofw. For every infinite wordw=a0a1. . .which is written also asw=w(0)w(1). . ., the wordsw<i, w_≤i, w>i, w_≥iare defined in the same way, with the suffixesw>i, w_≥i being infinite words.

Throughout the paper Awill denote an alphabet.

A semiring (K,+,·,0,1) consists of a set K, two binary operations + and · and two constant elements 0 and 1 such that hK,+,0iis a commutative monoid, hK,·,1iis a monoid, multiplication distributes over addition, and 0·k=k·0 = 0 for everyk ∈K. The semiring is denoted simply byK if the operations and the constant elements are understood.

The semiringK is calledcommutative ifk·k⁰=k⁰·kfor every k, k⁰∈K. It is calledadditively idempotent (or simplyidempotent), ifk+k=k for everyk∈K.

Moreover, the semiringK is zero-sum free (resp. zero-divisor free) if k+k⁰ = 0 impliesk=k⁰= 0 (resp. k·k⁰ = 0 impliesk= 0 ork⁰ = 0) for everyk, k⁰∈K.It is well known that every idempotent semiring is necessarily zero-sum free (cf. [1]).

Next, assume that the semiring K is equipped, for every index set I, with infinitary sum operationsP

I :K^I →K,such that for every family (ki |i∈I) of elements ofK andk∈K we have

X

i∈∅

k_i= 0, X

i∈{j}

k_i =k_j, X

i∈{j,l}

k_i=k_j+k_l forj6=l, X

j∈J

X

i∈Ij

k_i

=X

i∈I

k_i, ifS

j∈JI_j =I andI_j∩I_j⁰ =∅ forj6=j⁰, X

i∈I

(k·ki) =k·X

i∈I

ki

, X

i∈I

(ki·k) =X

i∈I

ki

·k.

Then the semiringK together with the operationsP

I is calledcomplete [15, 19].

A complete semiring is said to betotally complete [18],if it is endowed with a countably infinite product operation satisfying for every sequence (ki | i ≥ 0) of elements ofK the subsequent conditions:

Y

i≥0

1 = 1, Y

i≥0

k_i=Y

i≥0

k_i⁰

k₀·Y

i≥0

k_i+1=Y

i≥0

k_i, Y

j≥1

X

i∈Ij

k_i= X

(i1,i2,...)∈I1×I2×...

Y

j≥1

k_ij,

where in the second equation k⁰₀ = k0·. . .·kn₁, k₁⁰ = kn₁+1 ·. . .·kn₂, . . . for an increasing sequence 0 < n1 < n2 < . . . , and in the last equation I1, I2, . . . are arbitrary index sets.

Furthermore, we will call a totally complete semiring K totally commutative complete if it satisfies the statement:

Y

i≥0

(ki·k_i⁰) =



 Y

i≥0

ki



·



 Y

i≥0

k_i⁰



.

Obviously a totally commutative complete semiring is commutative. For our theory, we shall also need that a totally commutative complete semiring K satisfies the property

k6= 0 =⇒ Y

i≥0

k6= 0

for everyk∈K. Therefore in the sequel, by abusing terminology, when we refer to totally commutative complete semirings we assume that they additionally satisfy the above property.

Example 1. The following semirings are totally commutative complete, and all but the second one are idempotent. Moreover, by excluding the arbitrary completely distributive complete lattices, the remaining ones are zero-divisor free.

• theboolean semiring B= ({0,1},+,·,0,1),

• the semiring (N∪ {∞},+,·,0,1) ofextended natural numbers [17],

• thearctical semiring ormax-plus semiring (R+∪ {±∞},max,+,−∞,0),

• each completely distributive complete lattice (cf. [2]) with the operations supremum and infimum, in particular each complete chain [20].

Lemma 1. Let K be an idempotent totally complete semiring andI an index set of size at most continuum. Then, the following statements hold.

(i) [10, Chap. 5, Lm. 7.3]P

I

1 = 1.

(ii) P

I

k=k for everyk∈K.

(iii) P

i∈I

ki= P

k∈K

∃i∈I,ki=k

kfor every family (ki)_i∈I inK.

Proof. (ii) By (i) and distributivity we getP

I

k=k·P

I

1 =k·1 =k.

(iii) For every k∈K we letIk ={i∈I|ki=k}. Then we get X

i∈I

ki= X

k∈K

∃i∈I,ki=k

X

I_k

k= X

k∈K

∃i∈I,ki=k

k

where the second equality follows by (ii).

In the rest of the paper K will denote a totally commutative complete, idempotent and zero-divisor free semiring.

Let Q be a set. A formal power series (or simply series)over Q and K is a mapping s:Q→K. For every v ∈Q we write (s, v) for the value s(v) and refer to it as the coefficient of s on v. The support of s is the set supp(s) = {v ∈ Q | (s, v) 6= 0}. The constant series ek (k ∈ K ) is defined, for every v ∈ Q, by

ek, v

=k. The characteristic series 1P of a set P ⊆Qis given by (1P, v) = 1 if v∈P, and (1_P, v) = 0 otherwise. We denote byKhhQiithe class of all series over QandK.

Let s, r ∈KhhQiiand k∈K. Thesum s+r, thescalar products ks and sk as well as the Hadamard product sr are defined elementwise by (s+r, v) = (s, v) + (r, v),(ks, v) =k·(s, v), (sk, v) = (s, v)·k, and (sr, v) = (s, v)·(r, v) for everyv∈Q.Abusing notations, ifP ⊆Q, then we shall identify the restriction s|P of s on P with the series s1P. Moreover, if supp (s) ⊆ P, sometimes in the sequel we shall identify s|P with s. It is a folklore result that the structure

KhhQii,+,,e0,e1

is a commutative semiring. In our paper, we work with the semirings KhhA^∗ii and KhhA^ωii of finitary and infinitary series over A and K, respectively.

LetB be another alphabet andh:A^∗→B^∗ be a nondeleting homomorphism, i.e.,h(a)6=εfor eacha∈A.Then hcan be extended to a mapping h:A^ω→B^ω by letting h(w) = (h(w(i)))i≥0 for every w ∈ A^ω. Moreover, his extended to a mapping h : KhhA^∗ii → KhhB^∗iias follows. For every s ∈KhhA^∗ii the series h(s)∈ KhhB^∗iiis given by (h(s), u) =P

w∈h⁻¹(u)(s, w) for everyu∈B^∗. Since K is complete, his also extended to a mapping h: KhhA^ωii →KhhB^ωiiwhich is defined for every seriess ∈ KhhA^ωii by (h(s), u) = P

w∈h⁻¹(u)(s, w) for every u∈B^ω. If r∈KhhB^∗ii(resp. r∈KhhB^ωii), then the series h⁻¹(r)∈KhhA^∗ii (resp. h⁻¹(r) ∈ KhhA^ωii) is determined by (h⁻¹(r), w) = (r, h(w)) for every w∈A^∗ (resp. w∈A^ω).

3 Weighted linear temporal logic

For every lettera∈Awe consider a propositionpa and we let AP ={pa |a∈A}.

As usually, for everyp∈AP we identify¬¬pwithp.

Definition 1. The syntax of formulas of the weighted linear temporal logic (weighted LTL for short)over Aand K is given by the grammar

ϕ::=k|pa| ¬ϕ|ϕ∨ϕ|ϕ∧ϕ| ϕ|ϕU ϕ|ϕ wherek∈K andpa ∈AP.

We denote by LT L(K, A) the set of all such weighted LTL formulas ϕ. We represent the semantics kϕk of formulas ϕ ∈ LT L(K, A) as infinitary series in KhhA^ωii.

Definition 2. Letϕ∈LT L(K, A).The semanticsofϕis a serieskϕk ∈KhhA^ωii which is defined inductively as follows. For everyw∈A^ω we set

- (kkk, w) =k, - (kpak, w) =

1 ifw(0) =a 0 otherwise , - (k¬ϕk, w) =

1 if (kϕk, w) = 0 0 otherwise , - (kϕ∨ψk, w) = (kϕk, w) + (kψk, w), - (kϕ∧ψk, w) = (kϕk, w)·(kψk, w), - (kϕk, w) = (kϕk, w_≥1),

- (kϕU ψk, w) =X

i≥0







 Y

0≤j<i

(kϕk, w_≥j)



·(kψk, w_≥i)



,

- (kϕk, w) =Y

i≥0

(kϕk, w_≥i).

Theeventually operator is defined as in the classicalLTL, i.e., by♦ϕ:= 1U ϕ, hence we have (k♦ϕk, w) =X

i≥0

(kϕk, w_≥i) for everyw∈A^ω.

The syntactic boolean fragmentbLT L(K, A) ofLT L(K, A) is given by the gram- mar

ϕ::= 0|1|pa| ¬ϕ|ϕ∨ϕ| ϕ|ϕU ϕ

wherepa∈AP. For every formulaϕ∈bLT L(K, A) it is easily obtained, by struc- tural induction onϕand using idempotency, thatkϕkgets only values in{0,1}. By identifying 0 with0and 1 with1it is trivially concluded that kϕk coincides with

the semantics in the boolean semiring B. The conjunction and always operators are defined, respectively, by the macros ϕ∧ψ := ¬(¬ϕ∨ ¬ψ) and ϕ := ¬♦¬ϕ.

Clearly, the application of the operators ∧ and in bLT L(K, A) formulas ϕ, ψ coincides semantically with the application of the classical operators ∧ and in ϕ, ψconsidered as classical formulas.

We aim to define a further fragment ofLT L(K, A). For this we need some pre- liminary matter. More precisely, anatomic-step formula is anLT L(K, A) formula of the formW

a∈A(k_a∧p_a) wherek_a∈K andp_a ∈AP for every a∈A. AnLTL- step formula is anLT L(K, A) formula of the form W

1≤i≤n(k_i∧ϕ_i) wherek_i∈K and ϕ_i ∈ bLT L(K, A) for every 1 ≤ i ≤ n. We shall denote by stLT L(K, A) the class of LTL-step formulas over A and K. Furthermore, we shall denote by abLT L(K, A) the class ofalmost boolean LTLformulas overAandK, i.e., formu- las of the formV

1≤i≤nϕ_iwithϕ_i∈bLT L(K, A) orϕ_i=W

a∈A(k_a∧p_a), for every 1≤i≤n.

Definition 3. The fragment U LT L(K, A)ofU-nestingLTLformulas overAand K is the least class of formulas in LT L(K, A) which is defined inductively in the following way.

• k∈U LT L(K, A)for everyk∈K.

• abLT L(K, A)⊆U LT L(K, A).

• If ϕ∈U LT L(K, A), then¬ϕ∈U LT L(K, A).

• If ϕ, ψ∈U LT L(K, A), thenϕ∧ψ, ϕ∨ψ∈U LT L(K, A).

• If ϕ∈U LT L(K, A), thenϕ∈U LT L(K, A).

• Ifϕ∈bLT L(K, A)orϕis an atomic-step formula, thenϕ∈U LT L(K, A).

• If ϕ∈abLT L(K, A) andψ∈U LT L(K, A), thenϕU ψ∈U LT L(K, A).

A series r ∈ KhhA^ωii is called ω-ULTL-definable if there is a formula ϕ ∈ U LT L(K, A) such thatr=kϕk. We shall denote by ω-U LT L(K, A) the class of ω-ULTL-definable series overAandK.

4 Weighted first-order logic

In this section, we define the weighted first-order logic (weighted FO logic, for short) and consider a syntactic fragment of it. We aim to show that the class of semantics of sentences in this fragment contains the classω-U LT L(K, A).

Definition 4. The syntax of formulas of the weighted FO logic over Aand K is given by the grammar

ϕ::=k|Pa(x)|x≤y| ¬ϕ|ϕ∨ϕ|ϕ∧ϕ| ∃xϕ| ∀xϕ wherek∈K anda∈A.

We shall denote by F O(K, A) the set of all weighted FO logic formulas over A and K. In order to define the semantics of F O(K, A) formulas, we recall the notions of extended alphabet and valid assignment (cf. for instance [30]). LetV be a finite set of first-order variables. For an infinite wordw∈A^ωwe letdom(w) =ω.

A (V, w)-assignment σis a mapping associating variables fromV to elements ofω.

For every x ∈ V and i ∈ ω, we denote by σ[x → i] the (V, w)-assignment which associatesitoxand acts asσonV \ {x}.We encode pairs (w, σ) for everyw∈A^ω and (V, w)-assignmentσ, by using the extended alphabetA_V =A× {0,1}^V. Each word inA^ω_V can be considered as a pair (w, σ) where w is the projection over A and σ is the projection over {0,1}^V. Then,σ is called avalid (V, w)-assignment whenever for every x ∈ V the x-row contains exactly one 1. In this case, we identify σwith the (V, w)-assignment so that for every first-order variablex∈ V, σ(x) is the position of the 1 on the x-row. It is well-known (cf. [7]) that the set N_V={(w, σ)|w∈A^ω, σis a valid (V, w) -assignment}is anω-star-free language overA_V. The set f ree(ϕ) of free variables in a formulaϕ∈ F O(K, A) is defined as usual.

Definition 5. Letϕ∈F O(K, A)andV be a finite set of variables withf ree(ϕ)⊆ V. The semantics ofϕis a serieskϕk_V∈KhhA^ω_Vii. Consider an element(w, σ)∈ A^ω_V. If σis not a valid assignment, then we put(kϕk_V,(w, σ)) = 0. Otherwise, we inductively define (kϕk_V,(w, σ))∈K as follows.

- (kkk_V,(w, σ)) =k, - (kP_a(x)k_V,(w, σ)) =

1 if w(σ(x)) =a 0 otherwise , - (kx≤yk_V,(w, σ)) =

1 ifσ(x)≤σ(y) 0 otherwise , - (k¬ϕk_V,(w, σ)) =

1 if (kϕk_V,(w, σ)) = 0

0 otherwise ,

- (kϕ∨ψk_V,(w, σ)) = (kϕk_V,(w, σ)) + (kψk_V,(w, σ)), - (kϕ∧ψk_V,(w, σ)) = (kϕk_V,(w, σ))·(kψk_V,(w, σ)), - (k∃xϕk_V,(w, σ)) =X

i≥0

kϕk_V∪{x},(w, σ[x→i]) ,

- (k∀xϕk_V,(w, σ)) =Y

i≥0

kϕk_V∪{x},(w, σ[x→i]) .

IfV=f ree(ϕ), then we simply writekϕkforkϕk_{f ree(ϕ)}. Moreover, by Prop. 5 in [13], it holds

(kϕk_V,(w, σ)) = kϕk, w, σ|_{f ree(ϕ)} for every (w, σ)∈ N_V.

The syntactic boolean fragmentbF O(K, A) ofF O(K, A) is defined by the gram- mar

ϕ::= 0|1|Pa(x)|x≤y| ¬ϕ|ϕ∨ϕ| ∃xϕ.

For every formulaϕ∈bF O(K, A) it is easily obtained, by structural induction on ϕand using idempotency, thatkϕkgets only values in{0,1}. By identifying 0 with 0and 1 with 1 it is trivially concluded that kϕk coincides with the semantics in the boolean semiringB. Theconjunction anduniversal quantification are defined, respectively, by the macrosϕ∧ψ:=¬(¬ϕ∨ ¬ψ) and∀xϕ:=¬∃x¬ϕ. Clearly, the application of the operators∧and∀inbF O(K, A) formulasϕ, ψcoincides seman- tically with the application of the classical operators∧and∀inϕ, ψ considered as classical formulas.

Next, we define a fragment of our logic. For this, we recall the notion of an FO-step formula from [4]. More precisely, a formula ϕ ∈ F O(K, A) is an FO- step formula ifϕ=W

1≤i≤n(ki∧ϕi) withϕi ∈ bF O(K, A) andki ∈ K for every 1 ≤ i ≤ n. Moreover, a formula ϕ ∈ F O(K, A) is called a letter-step formula wheneverϕ=W

a∈A(ka∧Pa(x)) withka∈K for everya∈A. We shall need also the following macros:

-f irst(x) :=∀yx≤y, -x=y:=x≤y∧y≤x, -x < y:=x≤y∧ ¬(x=y), -z≤x < y:=z≤x∧x < y, -ϕ→ψ:=¬ϕ∨(ϕ∧ψ).

Definition 6. A formula ϕ∈F O(K, A)will be called weakly quantified if when- ever ϕ contains a subformula of the form ∀xψ, then ψ is either a boolean or a letter-step formula with free variable x or a formula of the form y ≤ x→ ψ⁰ or z≤x < y→ψ⁰ whereψ⁰ is a letter-step formula with free variablex.

We denote byW QF O(K, A) the set of all weakly quantifiedF O(K, A) formulas over A and K. A series s ∈ KhhA^ωii is called ω-wqFO-definable if there is a sentence ϕ∈W QF O(K, A) such that s=kϕk. We write ω-wqF O(K, A) for the class ofω-wqFO-definable series inKhhA^ωii.

5 ω-U LT L-definable series are ω-wqF O-definable

In this section we show that everyω-ULTL-definable series overA and K is also ω-wqFO-definable. For this, we will prove that for everyϕ∈U LT L(K, A) there exists a sentenceϕ⁰ ∈W QF O(K, A) such that kϕk=kϕ⁰k, using the subsequent technical results.

Lemma 2. Let ϕ ∈ U LT L(K, A) such that there exists ϕ⁰(y) ∈ W QF O(K, A) with

(kϕ⁰(y)k,(w,[y→i])) = (kϕk, w_≥i)for every w∈A^ω, i≥0.

Then(k¬ϕ⁰(y)k,(w,[y→i])) = (k¬ϕk, w_≥i)for everyw∈A^ω, i≥0.

Lemma 3. Let ϕ, ψ ∈ U LT L(K, A) such that there exist ϕ⁰(y), ψ⁰(x) ∈ W QF O(K, A) with (kϕ⁰(y)k,(w,[y→i])) = (kϕk, w_≥i) and (kψ⁰(x)k,(w,[x→i])) = (kψk, w_≥i) for every w ∈ A^ω, i ≥ 0. Then, there ex- istξ1(x), ξ2(x)∈W QF O(K, A)with

(kξ1(x)k,(w,[x→i])) = (kϕ∧ψk, w_≥i) and

(kξ2(x)k,(w,[x→i])) = (kϕ∨ψk, w_≥i) for everyw∈A^ω, i≥0.

Proof. Without any loss, we assume that the variablexdoes not occur inϕ⁰ (oth- erwise we apply a renaming). We replace every occurrence ofy with xinϕ⁰, and we letξ1(x) =ϕ⁰(x)∧ψ⁰(x) andξ2(x) =ϕ⁰(x)∨ψ⁰(x) which trivially satisfy our claim.

Lemma 4. Letϕ∈K∪abLT L(K, A). Then, there existsϕ⁰(x)∈W QF O(K, A) such that(kϕ⁰(x)k,(w,[x→i])) = (kϕk, w_≥i)for everyw∈A^ω, i≥0.

Proof. Let ϕ = k ∈ K. Then we set ϕ⁰(x) = k. Next, let ϕ ∈ abLT L(K, A), i.e., ϕ = V

1≤j≤nψj with ψj ∈ bLT L(K, A) or ψj = W

a∈A(ka∧pa), for every 1 ≤ j ≤ n. If ψj ∈ bLT L(K, A), then it is well-known that there exists a for- mula ψ⁰_j(xj) ∈ bF O(K, A) with one free variable xj, such that (kψjk, w≥i) =

ψ⁰_j(xj)

,(w,[xj →i])

for every w ∈ A^ω, i ≥ 0. Without any loss, we can assume that the variable xj (1 ≤ j ≤ n) does not occur in any ψ_k⁰ (whenever ψ⁰_k ∈ bLT L(K, A)) with k 6= j (if this is not the case, then we apply a re- naming of variables). Therefore, we can replace x_j in ψ_j⁰ with a new variable x. In case ψ_j = W

a∈A(k_a∧p_a) we consider the W QF O(K, A) letter-step for- mula ψ_j⁰(x) = W

a∈A(k_a∧P_a(x)). Now it is a routine matter to show that the W QF O(K, A) formulaϕ⁰(x) =V

1≤j≤nψ_j⁰(x) satisfies our claim.

Lemma 5. Let ϕ ∈ U LT L(K, A) such that there exists a formula ϕ⁰(y) ∈ W QF O(K, A) with (kϕ⁰(y)k,(w,[y→i])) = (kϕk, w_≥i) for every w ∈ A^ω, i ≥ 0. Then, there exists a W QF O(K, A) formula ψ(x) such that (kψ(x)k,(w,[x→i])) = (kϕk, w_≥i)for everyw∈A^ω, i≥0.

Proof. We letψ(x) =∃y.(y=x+ 1∧ϕ⁰(y)) and we have (kψ(x)k,(w,[x→i])) = (k∃y.(y=x+ 1∧ϕ⁰(y))k,(w,[x→i]))

=X

j≥0

(ky=x+ 1∧ϕ⁰(y)k,(w,[x→i, y→j]))

= (ky=x+ 1∧ϕ⁰(y)k,(w,[x→i, y→i+ 1]))

+ X

j≥0,j6=i+1

(ky=x+ 1∧ϕ⁰(y)k,(w,[x→i, y→j]))

= (ky=x+ 1∧ϕ⁰(y)k,(w,[x→i, y→i+ 1]))

= (kϕ⁰(y)k,(w,[y→i+ 1]))

= (kϕk, w_≥i+1) = (kϕk, w_≥i).

for everyw∈A^ω, i≥0, where the fourth equality holds by Lemma 1(ii).

Lemma 6. Let ϕ ∈ bLT L(K, A) or ϕ be an atomic-step formula. Then, there exists ψ(y) ∈ W QF O(K, A) such that (kψ(y)k,(w,[y→i])) = (kϕk, w_≥i) for everyw∈A^ω, i≥0.

Proof. Ifϕ∈bLT L(K, A), thenϕ∈bLT L(K, A),and thus there exists a formula ψ(x) ∈ bF O(K, A) with one free variable x, such that (kψ(x)k,(w,[x→i])) = (kϕk, w≥i) for every w∈A^ω, i≥0. Ifϕ=W

a∈A(ka∧pa), then we consider the W QF O(K, A) letter-step formulaϕ⁰(x) =W

a∈A(k_a∧P_a(x)). We also consider the W QF O(K, A) formulaψ(y) =∀x.(y≤x→ϕ⁰(x)). Then, for everyw∈A^ω, i≥0 we have

(kψ(y)k,(w,[y→i])) =Y

j≥0

(ky≤x→ϕ⁰(x)k,(w,[y→i, x→j]))

=Y

j≥i

(ky≤x∧ϕ⁰(x)k,(w,[y→i, x→j]))

=Y

j≥i

(kϕ⁰(x)k,(w,[x→j]))

=Y

j≥i

(kϕk, w_≥j)

= (kϕk, w_≥i) where the fourth equality holds by Lemma 4.

Lemma 7. Let ϕ ∈ abLT L(K, A) and ψ ∈ U LT L(K, A) such that there exists ψ⁰(y) ∈ W QF O(K, A) with (kψ⁰(y)k,(w,[y→i])) = (kψk, w_≥i) for every w ∈ A^ω, i ≥ 0. Then, there exists ξ(z) ∈ W QF O(K, A) such that (kξ(z)k,(w,[z→i])) = (kϕU ψk, w_≥i)for everyw∈A^ω, i≥0.

Proof. Letϕ=V

1≤l≤mϕl. Then, by the proof of Lemma 4, there exists a formula ϕ⁰(x) =V

1≤l≤mϕ⁰_l(x) where for every 1 ≤l ≤m, ϕ⁰_l(x)∈bF O(K, A) or it is a letter-step formula with (kϕ⁰_l(x)k(w,[x→i])) = (kϕlk, w_≥i) for everyw∈A^ω, i≥

0. Moreover, we have

(kϕ⁰(x)k,(w,[x→i])) = (kϕk, w≥i) for every w ∈ A^ω, i ≥ 0. We consider the F O(K, A) formulaξ⁰(z) =∃y.(∀x.((z≤x < y)→ϕ⁰(x))∧(z≤y)∧ψ⁰(y)). For everyw∈A^ω, i≥0 we compute

(kξ⁰(z)k,(w,[z→i]))

=X

j≥0

(k∀x.((z≤x < y)→ϕ⁰(x))∧(z≤y)∧ψ⁰(y)k,(w,[z→i, y→j]))

=X

j≥0

(k∀x.((z≤x < y)→ϕ⁰(x))∧ψ⁰(y)k,(w,[z→i, y→i+j]))

=X

j≥0







 Y

0≤k<j

(kϕ⁰(x)k,(w,[x→i+k]))



·(kψ⁰(y)k,(w,[y→i+j]))





=X

j≥0







 Y

0≤k<j

(kϕk, w_≥i+k)



·(kψk, w_≥i+j)





= (kϕU ψk, w_≥i).

Now, we consider the formula ξ(z) =∃y.^

1≤l≤m(∀x.((z≤x < y)→ϕ⁰_l(x)))∧(z≤y)∧ψ⁰(y) and for everyw∈A^ω, i≥0 we get (kξ(z)k,(w,[z→i])) = (kξ⁰(z)k,(w,[z→i])) = (kϕU ψk, w_≥i). Sinceξ(z)∈W QF O(K, A), we conclude our proof.

Lemma 8. For everyU LT L(K, A)formula ϕwe can construct a W QF O(K, A) formulaϕ⁰(x)such that(kϕ⁰(x)k,(w,[x→i])) = (kϕk, w_≥i)for everyw∈A^ω, i≥ 0.

Proof. We use Lemmas 2, 3, 4, 5, 6, and 7.

Proposition 1. For every ϕ ∈ U LT L(K, A) we can construct a W QF O(K, A) sentenceϕ⁰ with kϕ⁰k=kϕk.

Proof. Letϕ∈U LT L(K, A). By the previous lemma, there exists aW QF O(K, A) formulaψ(x) such that (kψ(x)k,(w,[x→i])) = (kϕk, w_≥i), for everyw∈A^ω, i≥ 0.We consider theW QF O(K, A) sentence ϕ⁰ =∃x.(f irst(x)∧ψ(x)) and we get

(kϕ⁰k, w) = (kψ(x)k,(w,[x→0]))

= (kϕk, w) for every w∈A^ω, i.e.,kϕ⁰k=kϕk, as required.

By the above proposition, we get the main result of this section.

Theorem 1. ω-U LT L(K, A)⊆ω-wqF O(K, A).

The result of the next corollary, which is trivially obtained by the constructive proofs of this section’s lemmas and propositions, in fact generalizes the correspond- ing result that relates booleanLTLandFO logic.

Corollary 1. For everyϕ∈U LT L(K, A)we can construct aW QF O(K, A)sen- tenceϕ⁰,that uses at most three different names of variables, such thatkϕ⁰k=kϕk.

6 Star-free series

In this section, we introduce the notions of star-free and ω-star-free series over A andK. LetL⊆A^∗(resp. L⊆A^ω). As usually, we denote by 1Lthe characteristic series of L. If L is a singleton, i.e., L= {w}, then we simply write 1w for 1_{w}. Furthermore, we simply denote by kL the seriesk1L for k∈ K. The monomials over Aand K are series of the form (ka)_a fora∈Aand ka ∈K. For simplicity, we shall consider also the series of the formkε withk∈K as monomials. A series s∈KhhA^∗iiis called aletter-step series ifs=P

a∈A(ka)_a whereka∈Kfor every a∈ A. The complement sof a series sis given by (s, w) = 1 if (s, w) = 0, and (s, w) = 0 otherwise. Letr, s∈KhhA^∗ii. The (Cauchy)product of rand sis the seriesr·s∈KhhA^∗iidefined for everyw∈A^∗ by

(r·s, w) =X

{(r, u)·(s, v)|u, v ∈A^∗, w=uv}.

The nth-iteration rⁿ ∈ KhhA^∗ii (n ≥ 0) of a series r ∈ KhhA^∗ii is defined inductively by

r⁰= 1ε and rⁿ⁺¹=r·rⁿ forn≥0.

Then, we have (rⁿ, w) =Pn Q

1≤i≤n(r, u_i)|u_i∈A^∗, w=u₁. . . u_no

for every w∈A^∗. A seriesr∈KhhA^∗iiis calledproper if (r, ε) = 0. Ifris proper, then for every w∈A^∗ and n >|w|we have (rⁿ, w) = 0. Theiteration r⁺ ∈KhhA^∗iiof a proper series r ∈KhhA^∗iiis defined by r⁺ =P

n>0rⁿ. Thus, for every w∈A⁺ we have (r⁺, w) = X

1≤n≤|w|

(rⁿ, w) and (r⁺, ε) = 0.

Definition 7. The class of star-free series over Aand K, denoted bySF(K, A), is the least class of series containing the monomials (over A and K) and being closed under sum, Hadamard product, complement, Cauchy product, and iteration restricted to letter-step series.

Next, letr∈KhhA^∗iibe a finitary ands∈KhhA^ωiian infinitary series. Then, theCauchy product of rand sis the infinitary seriesr·s∈KhhA^ωiidefined for everyw∈A^ω by

(r·s, w) =X

{(r, u)·(s, v)|u∈A^∗, v∈A^ω, w=uv}¹.

1Since the semiringKis idempotent (resp. By Lemma 1(ii)), the notation of the sum in the definition of Cauchy product of two finitary series (resp. of a finitary and an infinitary series), is consistent with the standard definition.

The ω-iteration of a proper finitary series r ∈ KhhA^∗ii is the infinitary series r^ω∈KhhA^ωiiwhich is defined by

(r^ω, w) =Pn Q

i≥1(r, u_i)|u_i∈A^∗, w=u₁u₂. . .o for everyw∈A^ω.

Example 2. Letr=P

a∈A(k_a)_a∈KhhA^∗iibe a letter-step series. We will show that (r⁺)⁺=r⁺. Moreover, for everyw∈A^ω we have (r^ω, w) =Q

i≥0(r, w(i)).

Letw=w(0). . . w(n−1)∈A⁺. Then r⁺, w

=X





 Y

1≤j≤k

(r, u_j)|w=u₁. . . u_k,1≤k≤n







= Y

0≤j≤n−1

(r, w(j)). Furthermore, we get

r⁺+

, w

=X





 Y

1≤j≤k

r⁺, uj

|w=u1. . . uk,1≤k≤n







=X





 Y

1≤j≤k



 Y

0≤ij≤|uj|−1

(r, uj(ij))



|w=u1. . . uk,1≤k≤n







= Y

0≤j≤n−1

(r, w(j)) = r⁺, w .

Similarly, we can show that (r^ω, w) =Q

i≥0(r, w(i)), for everyw∈A^ω.

Definition 8. The class ofω-star-free series overAandK, denoted byω-SF(K, A), is the least class of infinitary series generated by the monomials (over A and K) by applying finitely many times the operations of sum, Hadamard product, com- plement, Cauchy product, iteration restricted to letter-step series, and ω-iteration restricted to letter-step series.

The next result is trivially proved by Definitions 7, 8 and standard arguments.

Lemma 9. Let r∈SF(K, A)(resp. r∈ω-SF(K, A)) and B ⊆A. Then r|_B^∗ ∈ SF(K, B)(resp. r|_Bω∈ω-SF(K, B)).

In the sequel, we state properties of the classes SF(K, A) and ω-SF(K, A).

More precisely, we prove a splitting lemma and the closure of the classes under inverse strict alphabetic epimorphisms and bijections.

Lemma 10. If r ∈ SF(K, A) (resp. r ∈ ω-SF(K, A)) and k ∈ K, then kr ∈ SF(K, A)(resp. kr∈ω-SF(K, A)).

Proof. We havekr=kε·r, hence we get the proof of our claim.

Lemma 11. Let L, L⁰⊆A^∗ andK, K⁰⊆A^ω. Then - 1_L∪L⁰ = 1L+ 1L⁰, 1_K∪K⁰ = 1K+ 1K⁰

- 1L∩L⁰ = 1L1L⁰, 1K∩K⁰ = 1K1K⁰

- 1_LL⁰ = 1_L·1_L⁰, 1_LK = 1_L·1_K - 1_L+= (1L)⁺ wheneverε /∈L - 1L^ω= (1L)^ω wheneverε /∈L.

Proof. We use standard arguments and the idempotency property of the semiring K. In particular, for the last statement we use Lemma 1(i).

The two subsequent results are shown by induction on the structure of star-free (resp. ω-star-free) languages and series using Lemma 11.

Lemma 12. For every L⊆A^∗ the following statements are equivalent.

(i) L is a star-free language.

(ii) 1_L∈SF(K, A).

Lemma 13. For every L⊆A^ω the following statements are equivalent.

(i) L is anω-star-free language.

(ii) 1L∈ω-SF(K, A).

Since for every L ⊆A^∗ (resp. L ⊆A^ω) and k∈ K we have kL =kε·1L, by Lemmas 12 and 13, we get Lemma 14 below.

Lemma 14. Let L ⊆ A^∗ (resp. L ⊆ A^ω) and k ∈ K. If L is star-free (resp.

ω-star-free), thenk_L∈SF(K, A)(resp. k_L∈ω-SF(K, A)).

Lemma 15. If s∈SF(K, A)(resp. s∈ω-SF(K, A)), thensupp(s)is a star-free language (resp. anω-star-free) language over A.

Proof. Using standard arguments, we state the proof by induction on the structure ofs.

Lemma 16.

(i) Let L ⊆ A^∗ be a star-free language and B,Γ ⊆ A with B∩Γ = ∅. Then 1L|B^∗ΓB^∗ =P

1≤i≤n 1M_i· 1γ_i·1_M⁰

i

where for every1≤i≤n, Mi, M_i⁰⊆ B^∗ are star-free languages, and γi∈Γ.

(ii) Let L⊆A^ω be anω-star-free language and B,Γ⊆Awith B∩Γ =∅. Then 1L|B^∗ΓB^ω =P

1≤i≤n 1M_i· 1γ_i·1_M⁰

i

where for every1 ≤i≤n, Mi ⊆B^∗ is star-free, M_i⁰⊆B^ω isω-star-free, and γi∈Γ.

Proof. We prove only (ii); Statement (i) is shown with the same arguments. By the splitting lemma forω-star-free languages (cf. Lm. 3.2. in [7]), we getL∩B^∗ΓB^ω= S

1≤i≤nM_iγ_iM_i⁰ where for every 1 ≤ i ≤ n, M_i ⊆ B^∗ is star-free, γ_i ∈ Γ, and M_i⁰⊆B^ωisω-star-free. Since 1_L|_B^∗_ΓBω = 1_L∩B^∗_ΓBω, we complete our proof using Lemma 11.

Proposition 2(Splitting lemma for finitary series). Lets∈SF(K, A)andB,Γ⊆ A with B∩Γ = ∅. Then s|B^∗ΓB^∗ = P

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃

where for every 1≤i≤n,s⁽ⁱ⁾₁ , s⁽ⁱ⁾₃ ∈SF(K, B) ands⁽ⁱ⁾₂ = (ki)_γ

i with γi∈Γ, ki∈K.

Proof. We use induction on the structure ofs. Lets= (k_a)_a, a∈A, be a monomial.

Then, if a ∈ Γ, we have s|B^∗ΓB^∗ = 1ε·((ka)_a·1ε), otherwise s|B^∗ΓB^∗ = 1_∅ ·

(k_γ)_γ·1_∅

for an arbitraryγ∈Γ. Ifs=k_ε, then agains|_B^∗_ΓB^∗ = 1_∅·

(k_γ)_γ·1_∅ for an arbitraryγ∈Γ.

Let s, r ∈ SF(K, A) satisfying the induction hypothesis. This means that s|_B^∗_ΓB^∗=P

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃

andr|_B^∗_ΓB^∗=P

1≤j≤m

r₁^(j)·

r^(j)₂ ·r₃^(j) where for every 1≤i≤nand 1≤j≤m, we haves⁽ⁱ⁾₁ , s⁽ⁱ⁾₃ , r^(j)₁ , r₃^(j)∈SF(K, B), s⁽ⁱ⁾₂ = (ki)_γ

i, r^(j)₂ = (lj)_γ0 j

, γi, γ_j⁰ ∈Γ, ki, lj ∈K. Obviously, (s+r)|B^∗ΓB^∗ has the required form.

Next letw∈B^∗ΓB^∗ and 0≤k≤ |w| −1 withw(k)∈Γ. Thenw<k, w>k∈B^∗ and we have

(s|B^∗ΓB^∗, w) =



 X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃ , w





= X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃ , w

= X

1≤i≤n

s⁽ⁱ⁾₁ , w_<k

·

s⁽ⁱ⁾₂ , w(k)

·

s⁽ⁱ⁾₃ , w_>k

where the third equality holds since for every 1≤i≤nand every decomposition w=u1u2u3 withu26=w(k) we have

s⁽ⁱ⁾₂ , u2

= 0.

Similarly

(r|_B^∗_ΓB^∗, w) =



 X

1≤j≤m

r₁^(j)·

r^(j)₂ ·r₃^(j) , w





= X

1≤j≤m

r^(j)₁ , w<k

·

r₂^(j), w(k)

·

r^(j)₃ , w>k

.

Hence,

((sr)|B^∗ΓB^∗, w) = (s|B^∗ΓB^∗, w)·(r|B^∗ΓB^∗, w)

= X

1≤i≤n

s⁽ⁱ⁾₁ , w_<k

·

s⁽ⁱ⁾₂ , w(k)

·

s⁽ⁱ⁾₃ , w_>k

· X

1≤j≤m

r^(j)₁ , w<k

·

r₂^(j), w(k)

·

r₃^(j), w>k

= X

1≤i≤n 1≤j≤m

s⁽ⁱ⁾₁ r₁^(j), w_<k

·

s⁽ⁱ⁾₂ r₂^(j), w(k)

·

s⁽ⁱ⁾₃ r^(j)₃ , w_>k

=







 X

1≤i≤n 1≤j≤m

s⁽ⁱ⁾₁ r₁^(j)

·

s⁽ⁱ⁾₂ r₂^(j)

·

s⁽ⁱ⁾₃ r₃^(j) , w







 .

Sinces⁽ⁱ⁾₁ r^(j)₁ , s⁽ⁱ⁾₃ r₃^(j)∈SF(K, B),ands⁽ⁱ⁾₂ r^(j)₂ = (ki·lj)_γ

i ifγi=γ⁰_j, and s⁽ⁱ⁾₂ r₂^(j)= 0γ for an arbitraryγ∈Γ otherwise, our claim is true for the Hadamard product.

Furthermore,

((s·r)|B^∗ΓB^∗, w) =X

{(s|B^∗ΓB^∗, u)·(r, v)|u∈B^∗ΓB^∗, v∈B^∗, w=uv}

+X

{(s, u)·(r|B^∗ΓB^∗, v)|u∈B^∗, v∈B^∗ΓB^∗, w=uv}

with

X{(s|B^∗ΓB^∗, u)·(r, v)|u∈B^∗ΓB^∗, v∈B^∗, w=uv}

=X









 X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃ , u



·(r, v)|u∈B^∗ΓB^∗, v∈B^∗, w=uv







=X









 X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃ , u



·(r|B^∗, v)|u, v∈A^∗, w=uv







=







 X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃



·r|_B^∗, w





=



 X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃

·r|B^∗

, w





=



 X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·

s⁽ⁱ⁾₃ ·r|B^∗

, w





wherer|B^∗ =r1B^∗∈SF(K, B), and the fourth equality holds since the Cauchy product distributes over the sum of series. Similarly

X{(s, u)·(r|B^∗ΓB^∗, v)|u∈B^∗, v∈B^∗ΓB^∗, w=uv}

=X





 (s, u)·



 X

1≤j≤m

r^(j)₁ ·

r₂^(j)·r^(j)₃ , v



|u∈B^∗, v∈B^∗ΓB^∗, w=uv







=X







(s|B^∗, u)·



 X

1≤j≤m

r₁^(j)·

r^(j)₂ ·r₃^(j) , v



|u, v∈A^∗, w=uv







=



s|B^∗· X

1≤j≤m

r₁^(j)·

r^(j)₂ ·r₃^(j) , w





=



 X

1≤j≤m

s|B^∗· r₁^(j)·

r^(j)₂ ·r₃^(j) , w





=



 X

1≤j≤m

s|B^∗·r₁^(j)

·

r^(j)₂ ·r₃^(j) , w



.

Thus,

((s·r)|B^∗ΓB^∗, w) =



 X

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·

s⁽ⁱ⁾₃ ·r|B^∗

, w





+



 X

1≤j≤m

s|B^∗·r₁^(j)

·

r^(j)₂ ·r₃^(j) , w



.

Therefore, the series (s·r)|B^∗ΓB^∗ has the required form.

Now, let s be a letter-step series. Then, s|B^∗ΓB^∗ = s|Γ = P

γ∈Γ(k_γ)_γ. Let w∈supp(s⁺)∩B^∗ΓB^∗, which implies that there is an index 0≤k≤ |w| −1 such

thatw<k, w>k∈B^∗ andw(k)∈Γ. Then

s⁺

|_B^∗_ΓB^∗, w

=X

{(s^m|B^∗ΓB^∗, w)|1≤m≤ |w|}=

s^|w||B^∗ΓB^∗, w

= Y

0≤j≤|w|−1

(s, w(j))

=



 Y

0≤j≤k−1

(s, w(j))



·(s, w(k))·



 Y

k<j≤|w|−1

(s, w(j))





=

(s|B)⁺·

s|Γ·(s|B)⁺ , w

=



 X

γ∈Γ

(s|_B)⁺·

(k_γ)_γ·(s|_B)⁺ , w





and this concludes the induction for letter-step series.

Finally, let s ∈ SF(K, A). Then s = 1_supp(s). Since supp (s) is a star-free language, we get that supp (s) is also star-free. Hence, by Lemma 16(i) we conclude our proof.

Proposition 3 (Splitting lemma for infinitary series). Let s ∈ω-SF(K, A) and B,Γ⊆A with B∩Γ = ∅. Then s|B^∗ΓB^ω =P

1≤i≤n

s⁽ⁱ⁾₁ ·

s⁽ⁱ⁾₂ ·s⁽ⁱ⁾₃

where for every 1 ≤ i ≤ n, s⁽ⁱ⁾₁ ∈ SF(K, B), s⁽ⁱ⁾₃ ∈ ω-SF(K, B), and s⁽ⁱ⁾₂ = (k_i)_γ

i with γ_i∈Γ, k_i∈K.

Proof. Taking into account the definition ofω-star-free series, firstly we embed the proof of Lemma 2. Furthermore, we use arguments of that proof as follows. For the operations of sum and Hadamard product we lets, r ∈ ω-SF(K, A), and for Cauchy product we lets∈SF(K, A) and r∈ω-SF(K, A). For the complement operation, we lets∈ω-SF(K, A) and we use the corresponding argument forω- star-free languages and Lemma 16(ii). Finally, letsbe a letter-step series. Then, s|B^∗ΓB^∗ = s|Γ = P

γ∈Γ(kγ)_γ. Let w ∈ supp(s^ω)∩B^∗ΓB^ω, i.e., there exists an

indexk≥0 such thatw<k∈B^∗, w>k∈B^ω, and w(k)∈Γ. Then we get ((s^ω)|B^∗ΓB^ω, w)

=X





 Y

i≥1

(s, ui)|ui∈A^∗, w=u1u2. . .







=Y

j≥0

(s, w(j))

=



 Y

0≤j≤k−1

(s, w(j))



·(s, w(k))·



 Y

j>k

(s, w(j))





=

(s|B)⁺·(s|Γ·(s|B)^ω), w

=



 X

γ∈Γ

(s|B)⁺·

(k_γ)_γ·(s|B)^ω , w





i.e.,

(s^ω)|B^∗ΓB^ω=X

γ∈Γ

(s|B)⁺·

(kγ)_γ·(s|B)^ω and this completes our proof.

Proposition 4. Let A, B be two alphabets and h : A → B a bijection. Then s ∈ SF(K, A) (resp. s ∈ ω-SF(K, A)) implies that h(s) ∈ SF(K, B) (resp.

h(s)∈ω-SF(K, B)).

Proof. There is an one-to-one correspondence between the words of A^∗ and B^∗ (resp. the words ofA^ωandB^ω) derived byh. Then, we can easily state our proof by induction on the structure of star-free (resp. ω-star-free) series.

Proposition 5. Let A, B be alphabets and h : A → B a strict alphabetic epi- morphism. Then s∈ SF(K, B) (resp. s ∈ ω-SF(K, B)) implies that h⁻¹(s) ∈ SF(K, A)(resp. h⁻¹(s)∈ω-SF(K, A)).

Proof. We prove our claim by induction on the structure of star-free (resp. ω-star- free) series. Lets= (k_b)_b be a monomial overB andK. Then,h⁻¹(s) is a letter- step series and thus a star-free series overA and K.If s=kε, then h⁻¹(s) =kε

sincehis strict. Next lets1, s2∈SF(K, B) (resp. s1, s2∈ω-SF(K, B)) such that h⁻¹(s1), h⁻¹(s2)∈SF(K, A) (resp. h⁻¹(s1), h⁻¹(s2)∈ω-SF(K, A)). Trivially h⁻¹(s1s2) =h⁻¹(s1)h⁻¹(s2) andh⁻¹(s1+s2) =h⁻¹(s1) +h⁻¹(s2).