Algebraic characterizations ofrecognizable formal power series

Ioannis Kafetzis

Seminar of Theoretical Informatics

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Recognizable Series

DefinitionA series S is called recognizable if there exists an integern ≥ 1, a morphism of monoids

μ : A ∗ → Kn×n

and two matrices λ ∈ K1×n and γ ∈ Kn×1 such that∀w ∈ A ∗

(S,w) = λμ(w)γ

The tripe (λ, μ, γ) is called a linear representation of S,and n is its dimention.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

K-Module

DefinitionSuppose K is a commutative ring.A left K-module consists of an abelian group (M,+) andan operation · : K ×M → M such that

r · (x + y) = r · x + r · y(r + s) · x = r · x + s · x

(rs) · x = r · (s · x)1K · x = x

∀r , s ∈ K ∀x, y ∈ M.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Free K-Module

DefinitionLet K be a commutative ring with 1K and S a set. A freeK-Module M on generators S is an K-Module M and a setmap i : S → M such that for any K-module N and any setmap f : S → N there is a unique K-modulehomomorphism f̃ : M → N :

f̃ ◦ i = f

The elements of i(S) in M are an K-basis for M .

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

PropositionK <A> is a free K-module having as a basis the freemonoid A ∗.We shall denote K <A> the set of polynomials over A andK and K�A� the set of formal power series over A andK.It holds that formal power series is the dual ofpolynomials.

K�A�= (K <A>)∗.

SinceS =

∑w∈A ∗

(S,w)w ∀S ∈ K <<A>>

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Each formal series S defines a linear form

S : K <A>→ K

P 7→ (S,P) =∑

w∈A ∗(S,w)(P,w)

This has a finite support since P is a polynomial.The kernel of S is denoted by KerS and is the set

KerS ={P ∈ K<A>

∣∣∣ (S,P) = 0}

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Any multiplicative morphism μ : A ∗ →M whereM is aK-algebra can be extended uniquely to a morphism ofalgebras

K <A>→M

This extension will also be denoted by μ.It holds

μ(P) =∑

w∈A ∗(P,w)μ(w)

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Syntactic Ideal

DefinitionThe syntactic ideal of a formal series S ∈ K <<A>> is thegreatest two-sided ideal of K <A> contained in KerS. It isdenoted by IS .The syntactic ideal always exists since

IS =∑

I⊂KerS

I

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Quotient Algebra

DefinitionA quotient algebra is obtained by partitioning theelements of an algebra into equivalence classes given bya congruence relation, that is an equivalence relationcompatible with all the operations of the algebra.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Syntactic Algebra

DefinitionThe syntactic algebra of a formal series S ∈ K�A�,denoted byMS , is the quotient algebra of K <A> by thesyntactic ideal of S,

MS = K <A>/ IS

The canonical morphism K <A>→MS is denoted by μS .Since Ker(μS) = IS ⊂ KerS, the series S induces a linearform onMS denoted by φS .

S = φS ◦ μS

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Reutenauer’s Theorem

TheoremA formal series is rational iff its syntactic algebra is afinitely generated module over K.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

TheoremLetM be a finitely generated right K-module, φ be aK-linear form onM, m0 be an element ofM and v be amorphism A ∗ → End(M). Then the formal series

S =∑

w∈A ∗φ(vw(m0))w

is recognizable. More precisely, ifM has a generatingsystem of n elements, then S admits a linearrepresentation of dimension n.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Proof

Let m1,m2, . . . ,mn be generators ofM. Then for eachletter α ∈ A and each j ∈ {1,2, . . . ,n} there existcoefficients ααi,j such that

vα(mj) =∑

i

miααi,j

The matrices (ααi,j)i,j define a function μ : A → Kn×n whichextends to μ : A ∗ → Kn×n.By induction we have that ∀w ∈ A ∗

vw(mj) =

n∑i=1

miμ(w)i,j

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Proof

Let λ ∈ K1×n and γ ∈ Kn×1 be given by λi = φ(mi) and

m0 =

n∑j=1

mjγj

Then

vw(m0) = vw(

n∑j=1

mjγj) =∑

j

∑i

miμ(w)i,jγj

thusφ(vw(m0)) =

∑i,j

λiμ(w)i,jγj = λμ(w)γ

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Syntactic Right Ideal

DefinitionThe Syntactic Right Ideal of a formal series S ∈ K�A� isthe greatest right ideal of K <A> contained in KerS and isdenoted by IrS .

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Since K�A�= (K <A>)∗ each f ∈ End(K <A>) definesan endomorphism of the K-module K�A� called theadjoint morphism, defined by

(S, f(P)) = (t f(S),P)

for every series S and polynomial P.t(f ◦ g) =tg◦ t f

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

For any polynomial P, Q 7→ PQ is an endomorphism ofK <A> and its adjoint morphism is denoted by S 7→ S ◦ P.Thus

(S,PQ) = (S ◦ P,Q)

In particular

(S, xy) = (S ◦ x, y)⇒ S ◦ x = x−1S

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Using the definitions above it holds that

(S ◦ P) ◦Q = S ◦ (PQ).

Thus K�A� is a K <A>- module.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Proposition

PropositionThe syntactic right ideal of a series S is

IrS = {P ∈ K <A> | S ◦ P = 0}

ThusK < A >/ IrS � S ◦ K <A>

as a right K<A>-module.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

DefinitionThe rank of a formal series S is the dimention of thespace S ◦ K <A>.

DefinitionThe Hankel matrix of a formal series S is the matrix Hindexed by A ∗ × A ∗ defined by

H(x, y) = (S, xy)

for all words x, y ∈ A ∗.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

Carlyle’s Theorem

TheoremThe rank of a formal series S equals to the codimensionof its syntactic right ideal, and also equals to the rank ofits Hankel matrix. The series S is rational if and only if thisrank is finite and in this case, its rank equals to theminimum of the dimension of the linear representation ofS.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

DefinitionA reduced linear representation of a rational series S is alinear representation of S with the minimal dimentionamong all its representations.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

ExampleLet S be a series of rank 1. It admits a representation(λ, μ, γ) with μ : K <A>→ K a morphism of algebras andλ, μ ∈ K. Set aa = μ(a) for every a ∈ A.For w = a1a2 . . .an

μ(w) = aa1aa2 . . . aan =∏a∈A

a |w |aa

Such a series is called geometric. It follows that

S = λγ

∑a∈A

aaa

∗ = λγ

ε −∑a∈A

aaa

−1

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series

References

Rational Series and Their Languages, J. Berstel, C.ReutenauerDroste, Manfred, Werner Kuich, and Heiko Vogler,eds. Handbook of weighted automata. SpringerScience & Business Media, 2009.

Ioannis Kafetzis Algebraic characterizations of recognizable formal power series