Integer Weighted Finite Automata,
Matrices and Formal Power Series
over Laurent Polynomials
Vesa Halava
Department of Mathematics, University of Turku
FIN-20014 Turku, Finland
and TUCS - Turku Centre for Computer Science
E-mail: [email protected]
Turku Centre for Computer Science
TUCS Technical Report No 608
May 2004
ISBN 952-12-1349-3
ISSN 1239-1891
Abstract
It is well known that the family of regular languages (over alphabet A),
accepted by finite automata, coincides with the set of supports of the rational
and recognizable formal power series over N with the set of variables A. Here
we prove that there is a corresponding presentation for languages accepted
by integer weighted finite automata, where the weights are from the additive
group of integers, via the matrices over Laurent polynomials with integer
coefficients.
Keywords: formal power series, linear representation, Laurent polynomials, matrix, integer weighted finite automata
TUCS Laboratory
Discrete Mathematics for Information Technology
1
Introduction
It is well known that the family of languages accepted by a finite automata
(over alphabet A), can be defined also with the set of recognizable formal
power series over N, which on the other hand is equal with the set of rational
formal power series over N, where A is considered as a noncommutative set
of variables. This connection is proved by using the matrix representation
of the finite automata.
Here we give a similar representation for the family of languages accepted
with the integer weighted finite automata, see [4, 5]. In these automata the
weights are from the additive group of integers and a word is accepted, if it
has a successful path in the underlying automaton and the weight of the path
adds up to zero. We show that there is a connection between these languages
and the recognizable and rational formal power series with coefficients from
the ring of the Laurent polynomials with integer coefficients. The proof uses
the representation of the integer weighted finite automata with matrices over
the Laurent polynomials. The difference between these two constructions is
in the definition of the language defined with the series.
Next we give the basic definitions on words and languages. Let A be a
finite set of symbols, called an alphabet. A word over A is a finite sequence
of symbols in A. We denote by A∗ the set of all words over A. Note that
also the empty word, denoted by ε, is in A ∗ .
Let u = u1 . . . un and v = v1 . . . vm be two words in A∗ , where each ui
and vj are in A for 1 ≤ i ≤ n and 1 ≤ j ≤ m. The concatenation of u and v
is the word u · v = uv = u1 . . . un v1 . . . vm . The operation of concatenation
is associative on A∗ , and thus A∗ is a semigroup (containing an identity
element ε). Let A+ = A∗ \ {ε} be the semigroup of all nonempty words
over A. A subset L of A∗ is called a language.
2
Formal power series
Here we give the needed definitions and notations on formal power series.
As a general reference and for the details, we give [2, 9, 11] .
Let K be a semiring and A an alphabet. A formal power series S is a
function
A∗ → K.
Note that here A is considered as a (noncommutative) set of variables. The
image of a word w under S is denoted by (S, w) and it is called the coefficient
of w in S. The support of S is the language
supp(S) = {w ∈ A∗ | (S, w) 6= 0}.
The set of formal series over A with coefficients in K is denoted by KhhAii.
A formal series with a finite support is called a polynomial. The set of
1
polynomials is denoted by KhAi.
Let S and T be two formal series in KhhAii. Then their sum is given by
(S + T, w) = (S, w) + (T, w)
and their product by
(ST, w) =
X
(S, u)(T, v).
uv=w
We also define two external operations of K in KhhAii. Assume that a
is in K and S in KhhAii, then the series aS and Sa are defined by
(aS, w) = a(S, w)
and
(Sa, w) = (S, w)a.
P
A formal series S can also be written in the sum form S =
aw w over
∗
all w ∈ A such that aw is the coefficient of w in K, i.e. (S, w) = a w .
A formal series S in KhhAii is called proper if the coefficient of the empty
word vanishes, that is (S, ) = 0. Let S be proper formal series. Then the
family (S n )n≥0 is locally finite (see [2]), and we can define the sum of this
family, denoted by
X
S∗ =
Sn
n≥0
and it is called the star of S. Note that S 0 = 1, S 1 = S and S n = SS n−1 ,
where 1 is the identity of K under product.
The rational operations in KhhAii are the sum, the product and the star.
A formal series is called rational if it is an element of the rational closure of
KhAi, i.e. it can be defined using the polynomials KhAi and the rational
operations. The family of rational series is denoted by K rat hhAii.
As usual, we denote by K m×n the set of the m × n matrices over K.
A formal series S ∈ KhhAii is called recognizable if there exists an integer
n ≥ 1, and a monoid morphism µ : A∗ → K n×n , into the multiplicative
structure of K n×n , and two vectors ı, ρ ∈ K n such that for all words w,
(S, w) = ıµ(w)ρT .
The triple (ı, µ, ρ) is called a linear representation of S with dimension n.
The set of recognizable series over K is denoted by K rec hhAii
The next theorem is fundamental in the theory of rational series. It
was first proved by Kleene in 1956 for languages that are those series with
coefficients in the Boolean semiring. It was later extented by Schützenberger
to arbitrary semirings. For details, see [2, 9, 11].
Theorem 2.1. A formal series is recognizable if and only if it is rational.
2
3
Finite automaton
A (nondeterministic) finite automaton is a quintuple A = (Q, A, δ, q A , F ),
where Q is a finite set of states, A is a finite input alphabet, δ : Q × A → 2 Q
is a transition function, qA ∈ Q is an initial state and F is the set of final
states. A transition p ∈ δ(q, a), where p, q ∈ Q and a ∈ A, will also be
written as (q, a, p), in which case δ ⊆ Q × A × Q is regarded as a relation
(and sometimes also as an alphabet). Without loss of generality, we can
assume that
Q = {1, 2, . . . , n} for some n ≥ 1, and qA = 1 .
Indeed, renaming of the states will not change the accepted language.
A path π of A (from q1 to qn+1 ) is a sequence
π = t1 t2 . . . tk where ti = (qi , ai , qi+1 ) ∈ δ
(1)
for i = 1, 2, . . . , k. If we consider δ as an alphabet, then we can write π ∈ δ ∗ .
The label of the path π in (1) is the word kπk = a 1 a2 . . . ak . Let
A(w : p → q) = {π | π a path from p to q with kπk = w} .
Moreover, a path π ∈ A(w : p → q) is successful (for w), if p = 1 and q ∈ F .
The language accepted by A is the subset L(A) ⊆ A ∗ consisting of the labels
of the successful paths of A:
L(A) = {w ∈ A∗ | π ∈ A(w : 1 → q) for some q ∈ F } .
It is well-known that each finite automata has a matrix representation
obtained as in the following. Let A = (Q, A, δ, 1, F ) be a finite automaton
with n states, i.e., Q = {1, 2, . . . , n}. Define for all a ∈ A, the matrix
Ma ∈ Nn×n by
(
1, if j ∈ δ(i, a),
(Ma )ij =
(2)
0, otherwise.
We define a monoid morphism µ : A∗ → Nn×n by setting µ(a) = Ma , where
the operation in Nn×n is the usual matrix multiplication.
Let ı = (1, 0, . . . , 0), where only the first term is nonzero, and let ρ =
(ρ1 , ρ2 , . . . , ρn ) in Nn where
(
1, if qi ∈ F,
ρi =
(3)
0, otherwise.
The triple (ı, µ, ρ) is then called the linear representation of A.
For the proof for the following theorem, see [2, 9, 11].
3
Theorem 3.1. A language L is accepted by a finite automaton if and only
if there exists a linear representation (ı, µ, ρ) such that
w ∈ L ⇐⇒ ıµ(w)ρT 6= 0.
Note that we could have defined the matrices over the boolean semiring B
instead of the semiring N, and then replacing ıµ(w)ρ T 6= 0 by ıµ(w)ρT = 1.
But using the ring N, we achieve the following advantage.
Theorem 3.2. For a finite automaton A having a linear representation
(ı, µ, ρ), the value ıµ(w)ρT equals the number of different successful paths in
A for w.
By Theorem 3.1 and the fundamental theorem, Theorem2.1, we get the
following corollary.
Corollary 3.3. L ⊆ A∗ is a regular language if and only if there exists a
formal series SL ∈ Nrat hhAii = Nrec hhAii such that L = supp(SL ).
Note that it follows that the regular languages are closed under the
rational operations, since Nrat hhAii is.
4
Laurent polynomials and weighted automata
In this section we give a corresponding representation for the languages
accepted by the integer weighted finite automata. We begin with some
definitions.
A Laurent polynomial p ∈ Z[x, x−1 ] with coefficients in Z is a series
p(x) = . . . a−2 x−2 + a−1 x−1 + a0 + a1 x + a2 x2 + . . . ,
where there are only finitely many nonzero coefficients a i ∈ Z. The constant
term of the Laurent polynomial p ∈ Z[x, x −1 ] is a0 . The family of Laurent
polynomials with coefficients in Z forms a ring with respect to the operations
of sum and multiplication, that are defined in the usual way. Indeed, the
sum is defined componentwise and the multiplication is the Cauchy product
of the polynomials:
!
!
∞
∞
∞
X
X
X
X
i
i
ai x
bi x =
(
aj bk )xi .
i=−∞
i=−∞
i=−∞ j+k=i
Note that in the definition of Laurent polynomials we could have used also
arbitrary ring instead of Z, but here we need only the integer case. Actually, we concentrate on matrices over Laurent polynomials with integer
coefficients, that is, the elements of Z[x, x −1 ]n×n for n ≥ 1. A Laurent
polynomial matrix
M = (cij )n×n ∈ Z[x, x−1 ]n×n
4
is a n × n-square matrix the entries of which are Laurent polynomials from
Z[x, x−1 ]. For these matrices, multiplication is defined in the usual way
using the multiplication of the ring Z[x, x −1 ]. Indeed, if M1 = (cij )n×n and
M2 = (dij )n×n , then
M1 · M2 = (eij )n×n ,
where
eij =
n
X
cik dkj ∈ Z[x, x−1 ].
k=1
Also the sum for these matrices can be defined, but we are interested in
the semigroups generated by a finite number of Laurent polynomials under
multiplication.
Next we consider a generalization of finite automata where the transitions have integer weights. The type of automata we consider is closely
related to the 1-turn counter automata as considered by Baker and Book
[1], Greibach [3], and especially by Ibarra [8]. Also, regular valence grammars are related to these automata, see [7]. Moreover, the extended finite
automata of Mitrana and Stiebe [10] are generalizations of these automata.
Consider the additive group of Z of integers. A (Z-)weighted finite automaton Aγ consists of a finite automaton A = (Q, A, δ, 1, F ) as above,
except that here δ may be a finite multiset of transitions in Q × A × Q, and
a weight function γ : δ → Z. We let δ be a multiset in order to be able to define (finitely) many different weights for each transition of A. For example,
it is possible that for t1 , t2 ∈ δ, t1 = (i, a, j) = t2 and γ(t1 ) 6= γ(t2 ).
Let π = t1 t2 . . . tk be a path of A, where ti = (qi , ai , qi+1 ) for i =
1, 2, . . . , k. The weight of π is the element
γ(π) = γ(t1 ) + γ(t2 ) + · · · + γ(tk ).
Furthermore, we let
L(Aγ ) = {w ∈ A∗ | γ(π) = 0, π ∈ A(w : 1 → q) for some q ∈ F } ,
be the language of Aγ . In other words, a word is accepted by A γ if and only
if there is a successful path of weight 0 in A γ .
Next we shall introduce a matrix representation of integer weighted finite
automata with the matrices over the Laurent polynomials Z[x, x −1 ].
Let Aγ be a weighted finite automaton, where A = (Q, A, δ, 1, F ) and
γ : δ → Z. Let again Q = {1, 2, . . . , n}. Define for each element a ∈ A and
a pair of states i, j ∈ Q the Laurent polynomial
X
paij =
xγ(t) .
t=(i,a,j)∈δ
Moreover, define the Laurent polynomial matrix M a ∈ Z[x, x−1 ]n×n for all
a ∈ A by
(Ma )ij = paij .
(4)
5
Let µ : A∗ → Z[x, x−1 ]n×n be the morphism defined by µ(a) = Ma . Let
ı and ρ be the vectors as in (3). The triple (ı, µ, ρ) is called a Laurent
representation of Aγ . For completeness sake, we give here the proof of the
following result of [6].
Lemma 4.1. Let (ı, µ, ρ) be a Laurent representation of A γ , and let w ∈
A∗ . Then the coefficient of xz in µ(w)ij is equal to the number of paths
π ∈ A(w : i → j) of weight z.
Proof. We write Mu = µ(w) for each word w. We prove the claim by
induction on the length of the words. The claim is trivial, if w ∈ A. Assume
then
the claim holds for the words u, v ∈ A + , and let (Mu )ij = puij =
P zthat
z
z
z αij x , where αrs is the number
P zof zpaths fromz A(u : i → j) of weight z.
x , where βrs is the number of paths
Similarly, let (Mv )ij = pvij = z βij
from A(v : i → j) of weight z. Now,
(Mu Mv )ij =
n
X
puik pvkj
k=1
=
n X
X
=
n X
X
k=1
αzik1 xz1
z1
z2 z1 +z2
αzik1 βkj
x
=
k=1 z1 ,z2
X
z2 z2
βkj
x
z2
n
XX
z2 z1 +z2
αzik1 βkj
x
.
z1 ,z2 k=1
In other words, the coefficient of xz is equal to
wherefrom the claim easily follows.
P
z1 +z2 =z
Pn
z1 z2
k=1 αik βkj ,
The following result is an immediate corollary to Lemma 4.1.
Theorem 4.2. Let (ı, µ, ρ) be a Laurent representation of A γ , and let w ∈
A∗ Then the constant term c of ıµ(w)ρT equals the number of different
successful paths of w in Aγ . In particular, w ∈ L(Aγ ) if and only if c > 0.
Since Z[x, x−1 ] is a ring we can also study the formal power series
Z[x, x−1 ]hhAii. Note that the zero element of Z[x, x −1 ] is the zero polynomial, where all the coefficients are 0. By Theorem 2.1, we get the following
corollary.
Corollary 4.3. A language L ⊆ A∗ accepted with Aγ if and only if there
exists a formal power series SL ∈ Z[x, x−1 ]rat hhAii = Z[x, x−1 ]rec hhAii such
that
n
X
w ∈ L ⇐⇒ (SL , w) =
ai xi and a0 6= 0.
i=m
Note that the this corollary does not give any closure properties on the
family of languages accepted with integer weighted finite automata. The
closure properties of these languages were studied in [4]. For example, the
family is not closed under star.
6
Note also that the undecidability result in [5] gives undecidability result
for matrices over Laurent polynomials, see [6].
Actually, for the power series SL ∈ Z[x, x−1 ]hhAii in Corollary 4.3,
supp(SL ) = L(A), i.e., the support of SL is the regular language accepted
by the underlying automaton of Aγ . By reordering the terms according to
powers of the variable x, we get
SL =
n
X
Lz xz ,
(5)
z=m
where Lz is the sum of words of the language
{w ∈ A∗ | π ∈ A(w : 1 → q) for some q ∈ F, γ(π) = z} ⊆ A ∗ ,
with multiplicities from Z. We denote these languages simply by L z . Now
L0 = L(Aγ ) = L and L(A) = ∪z∈Z Lz . Note that this union can be infinite, since the sum (5) can be infinite, even for both directions. It follows
also by the rationality of SL that the languages Lz are in the family of
languages accepted with integer weighted finite automata, since x −z SL ∈
Z[x, x−1 ]rat hhAii.
Acknowledgements
I want to thank Dr. Tero Harju for the several comments and suggestion
for this work during our coffee breaks.
References
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[3] S. A. Greibach, An infinite hierarchy of context-free languages, J. Assoc.
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[4] V. Halava and T. Harju, Languages accepted by integer weighted finite
automata, Jewels are forever, Springer, Berlin, 1999, pp. 123–134.
[5] V. Halava and T. Harju, Undecidability in integer weighted finite automata, Fund. Inform. 38 (1999), no. 1-2, 189–200.
[6] V. Halava and T. Harju, Undecidability in matrices over Laurent polynomials, Tech. Report 600, TUCS, 2004, to appear in Adv. in Appl.
Math.
7
[7] V. Halava, T. Harju, H. J. Hoogeboom, and M. Latteux, Valence languages generated by generalized equality sets, Tech. Report 502, TUCS,
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[8] O. H. Ibarra, Restricted one-counter machines with undecidable universe problems, Math. Systems Theory 13 (1979), 181–186.
[9] W. Kuich and A. Salomaa, Semirings, automata, languages, SpringerVerlag, 1986.
[10] V. Mitrana and R. Stiebe, The accepting power of finite automata over
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[11] A. Salomaa and M. Soittola, Automata–theoretic aspects of formal
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8
Turku Centre for Computer Science
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FIN-20520 Turku
Finland
http://www.tucs.fi
University of Turku
• Department of Information Technology
• Department of Mathematics
Åbo Akademi University
• Department of Computer Science
• Institute for Advanced Management Systems Research
Turku School of Economics and Business Administration
• Institute of Information Systems Science