Algebraic characterizations of recognizable formal power series Ioannis Kafetzis Seminar of Theoretical Informatics Ioannis Kafetzis Algebraic characterizations of recognizable formal power series
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