De las dependencias coordinadoras de sector, respecto de lo siguiente:

The Hadamard product of two series σ and τ is dened as: σ Hτ ∶= ∑

w∈Σ∗

⟨σ,w⟩ ⋅ ⟨τ,w⟩w

In particular, the support of σ Hτ is the intersection of both supports of σ and τ. Hence, if one of σ or τ is a polynomial (resp. a proper series), so is σ Hτ. The Hadamard product

admits a neutral element: the series 1H that maps any word to 1 i.e., the series dened

as:

1H ∶= ∑

w∈Σ∗

1w

Example 8. We x Σ and K = Rat(Σ∗_{). Consider the series Id, which maps every}

word to (the singleton containing) itself, i.e., Id∶= ∑

w∈Σ∗

{w}w

It is rational because it is equal to the polynomial series mapping each one-symbol word on Σ into itself. Now, consider the Hadamard product of Id by itself:

Square_{∶= ∑}

w∈Σ∗

{ww}w

By the projection on the second component of Square is not rational, providing that Σ has at least two symbols.

Observe that when K is commutative, the Hadamard product commutes, i.e., σ Hτ = τ Hσ. Furthermore, it is proved in [66, Theorem III. 3.1] that in this case, the family Rat is closed under Hadamard product.

Theorem 14. If K is commutative then the family Rat is closed under Hadamard prod- uct.

As usual, we dene the successive powers of a series under Hadamard product: σHk∶= { 1H if k= 0;

σHσH(k−1) otherwise.

As expected, for any σ we have σH0= 1

H and σH1 = σ. Then, we can dene the Hadamard

star of σ as: σ⋆∶= ∑ k∈N σHk= ∑ w∈Σ∗(∑k∈N ⟨σHk_{, w}⟩w)

when the innite sum is dened.

Over rationally additive semirings (Section 2.2.2) the Hadamard star is always dened and it takes a simpler expression.

Proposition 8. If K is a rationally additive semiring and Σ is an alphabet, then for every series σ in K ⟨⟨Σ∗_{⟩⟩ we have:}

σ⋆= ∑

w∈Σ∗

The families Pol and Prop are not closed under Hadamard star as 0⋆ = 1H. The

Hadamard star is idempotent, i.e., (σ⋆)⋆ = σ⋆.

Example 9. Working on series in Rat(a∗_{) ⟨⟨a}∗_{⟩⟩, we dene the series uMult as being}

the Hadamard star of Id (see Example 8): uMult_{∶= Id⋆ = ∑}

n∈N

{akn∣ k ∈ N}an

By identifying the monoids a∗_×a∗ _{with N×N, this series denes the relation being a mul-}

tiple of. However rational series of N are rst-order denable in Presburger arithmetics, i.e., arithmetics with addition only. Hence, uMult is not rational. That implies that the family of rational series is not closed under Hadamard star, even in the K commutative and Σ unary case (observe the contrast with Theorem 14).

The Hadamard operations were introduced because they are well-suited to the behav- ior of 2-way K-fas. Indeed, this family of series is closed under Hadamard operations. Proposition 9. If σ and τ are series respectively accepted by 2-way, sweeping, rotating K-fas, so are the series the σHτ and σ⋆. Moreover, if the K-fas are deterministic, so is the resulting K-fa accepting σ Hτ.

Proof. The formal proof of the construction is left to the reader. The ideas are given in Figure 4.1, where Kσ and Kτ boxes stand for direct simulations of the K-fa accepting σ

and τ respectively and where − and + are new states that are used to cross the entire input up to the left and the right endmarker respectively. Observe that sweepingness and rotatingness are preserved. For the Hadamard product, even determinism is preserved. Example 10. A direct application of the previous proposition, is that the series uMult dened in Example 9 is accepted by a rotating transducer. We give one in Figure 4.2 (observe the similarity with the construction presented in Figure 4.1b).

The Hadamard product and the Hadamard star are called Hadamard operations. As shown in Example 8 and Example 9, the following denes a super family of the rational series.

Denition 13. The family of Hadamard series, denoted Had(K⟨⟨Σ∗_{⟩⟩) or simply Had}

if K and Σ are understood, is the closure of Rat(K ⟨⟨Σ∗_{⟩⟩) under Hadamard operations}

and sum, i.e.:

Had= [Rat,+,H,⋆]

Natural examples of Hadamard relations are Square (Example 8) and uMult (Ex- ample 9).

Kσ Kτ − ◁,−1 ∶ 1K ▷,0 ∶1K a,−1 ∶ 1K (a) Kσ − + ▷,0 ∶1K ◁,₋₁ ∶ 1 K a,−1 ∶ 1K ▷, +1 ∶ 1K a,+1 ∶ 1K (b)

Figure 4.1: Construction of transducers accepting σ Hτ (a) and σ⋆ (b) respectively.

←Ð_q Ð→_q q₊ ▷, +1 ∣  ▷, +1 ∣  a,−1 ∣  ◁, −1 ∣  a,+1 ∣ a a,+1 ∣ 

Figure 4.2: A rotating transducer accepting the relation uMult (an edge (q,q′_{) is}

labeled (s,d ∣ w) if φ maps the transition (q,s, d,q′_{) to w).}

We are able to prove that, under the assumption that K is rationally additive (see Section 2.2.2), the family Had with sum, Hadamard product and Hadamard star, is also a rationally additive semiring.

Proposition 10. If K is rationally additive, then ⟨Had(K ⟨⟨Σ∗_{⟩⟩), +,H,⋆ , 0, 1}

H⟩ is a

rationally additive semiring.

Proof. This is a direct consequence of the general following observation: consider a class Γ of functions from a set X to a set Y provided with a some internal operations ω1, ω2, . . .of

any arity. Extend each operation ω of arity r to Γ by setting, for f1, . . . fr∈ Γ and x ∈ X,

Ω(f1, . . . , fr)(x) = ω(f1(x),..., fr(x)). Every identity on X transfers to Γ.

Since ⟨Had,+,H, 0, 1_H⟩ is a semiring, we can dene the matrix product with the specic operations of sum and Hadamard product. Therefore we can inductively dene the successive powers of a matrix M, by setting MH0 is equal to the matrix which has

The transitive closure can be interpreted in the present case: let σi,j the coecient

in position (i, j) of the matrix M, the entry (k, `) of M∗ _{is the power series σ dened as}

follows:

⟨σ, w⟩ = {⟨σi1,i2, w⟩ ⋅ ⟨σi2,i3, w⟩ ⋅ . . . ⋅ ⟨σir−1,ir, w⟩ ∣ i1, . . . , ir∈ {1, ... , n}, i1= k, ir= `, r ≥ 0}

The following extends the Kleene-Scützenberger Theorem (Theorem 5) to rotating K-fas.

Corollary 3. The family of series recognized by rotating K-fas is equal to Had.

Proof. The inclusion of Had in the family of series accepted by rotating K-fas is a consequence of Theorem 5 and Proposition 9.

Now we prove the opposite direction. Let (A, φ) be a rotating K-fa with A = (Q, Σ, ▷,◁, I, F, δ). With the notation of Denition 10, let δ+1 be the intersection

δ∩ (Q × Σ_▷◁× {0,1} × Q₊₁). Observe that, for q,q′∈ Q the K-fa Kq,q′ = ((Q,Σ,▷,◁, {q},{q′},δ+1), φ)

recognizes a rational series thanks to Theorem 5.

Now, we dene for each pair (q, q′′_{) the rational series σ}

q,q′′ as the sum of the se-

ries ∣∣Kq,q′∣∣ such that (q′,◁, −1,q′′) ∈ δ.

Consider the matrix M ∈ RatQ×Q _whose (q,q_′)-entry is the series σ

q,q′. It describes

the behavior of (A,φ) between two visits of the left endmarker. Since the semiring is rationally additive, we may consider the matrix M∗_{, which describes the behavior of the}

automaton in an arbitrary number of hits, starting and ending at the left endmarker. The series recognized by (A,φ) is thus

σ= ⋃

q∈I, q′∈Q, q′′∈F(M ∗

q,q′ H∣∣Kq′,q′′∣∣)