4. Rational functions 55
Theorem 4.7.1 Let Σ be an alphabet and M be a cancellative monoid with unicity of gcd. A functionf : Σ∗→M is subsequential if and only if the quotient Σ∗/∼f is finite.
Proof. It suffices to consider the case f 6=∅.
Assume that f is subsequential and let T = (Q, i, λ, ρ) be a trim prefix subsequential Σ-M-transducer that realizes f. By usingT, we define a right congruence∼T on Σ∗ by
s∼T t if and only if is=it, ∀s, t∈Σ∗.
This definition considers the case is=it=∅. Observe that the number of classes of the quotient Σ∗/∼T is at most card (Q) + 1.
We claim that
(4.7.3) ifs∼T t then ˆf·s= ˆf·t.
Indeed, letx∈Σ∗. AsT is prefix, we can write
(4.7.4) x( ˆf·s) = gcd( ˆf ·s, x)(gcd( ˆf·s, x))−1(x( ˆf·s))
(4.6.8b)
= gcd( ˆf ·s, x)(1(( ˆf·s)·x)) = gcd( ˆf·s, x)(1( ˆf·(sx)))(4.6.10)= (is∗x)((i(sx))ρ).
Equally,x( ˆf ·t) = (it∗x)((i(tx))ρ). Asis=itandx is arbitrary, we get (4.7.3).
Property (4.7.3) shows that Σ∗/∼T is a refinement of Σ∗/∼f. Thus, Σ∗/∼f is finite.
Conversely, assume that Σ∗/∼f is finite. From this quotient, we construct a subsequential Σ-M-transducer T as follows. The state set of T is Σ∗/∼f, except the class (if it exists). The initial state is [1], and [s] is a final state if and only if s ∈domf. The transitions and emissions are defined by
[1]λ= gcdf [s]·σ = [sσ]
[s]∗σ = gcd( ˆf·s, σ) [s]ρ= 1( ˆf ·s)
for all state [s] and all σ∈Σ such that [sσ]6=.
It follows directly from this construction that the functionϕdefined in (4.7.2) is an isomorphism from the transducer T to the minimal transducer Tf. Therefore, T is a subsequential transducer that realizes f. So,f is subsequential.
The congruence ∼f yields a procedure to construct the minimal transducer Tf from a trim prefix one T = (Q, i, λ, ρ). First, we define an equivalence relation ≡on Q as follows. For each stateq, let q−1ρ: Σ∗ →M be the function defined by
(4.7.5) x(q−1ρ) = (q∗x)((qx)ρ), ∀x∈Σ∗. Then,
(4.7.6) p≡q if and only if p−1ρ=q−1ρ.
56 4.7. Congruence of a function and reduction
The reduction of T is the construction of the quotient transducer T/≡= (Q≡, i≡, λ≡, ρ≡) defined as follows:
Q≡=Q/≡
(4.7.7a)
i≡= [i]
(4.7.7b)
i≡λ≡=iλ (4.7.7c)
[q]σ= [qσ]
(4.7.7d)
[q]∗σ =q∗σ (4.7.7e)
[q]ρ≡ =qρ (4.7.7f)
where q∈Qand σ∈Σ.
The following properties imply that (4.7.7d) does not depend on the choice of representatives.
For all word x,
px≡qx (4.7.8a)
[p]∗x= [q]∗x(4.6.10b)= gcd( ˆf·s, x) (4.7.8b)
[p]ρ≡ = [q]ρ≡
(4.6.10c)
= 1( ˆf·s) (4.7.8c)
Theorem 4.7.2 Let T = (Q, i, λ, ρ) be a trim prefix subsequential Σ-M-transducer that realizes a functionf : Σ∗→M. Then,
T/≡ ∼= Tf.
Proof. It suffices to notice that the functionϕ:Q≡→Qf defined by [is]ϕ= ˆf ·s, ∀s∈Σ∗
is an isomorphism.
We say that a subsequential transducer T isreduced if each pair of distinct states is not equiv-alent by≡. From Theorem 4.7.2, it follows that
Corolary 4.7.1 A subsequential Σ-M-transducer is minimal if and only if is trim, prefix and re-duced.
In the sequel, assume that M is a cancellative monoid with gcd, but that do not necessarily satisfy the property of the unicity of gcd. We close this section by defining a right congruence on Σ∗ for a function f : Σ∗ → M, and proving that this congruence characterizes the subsequential functions Σ∗ →M.
This congruence will be denoted again by ∼f. For each pair of wordsu, v∈Σ∗, we setu∼f v if and only if there existu, v∈M and a functionh: Σ∗→M such that
(4.7.9) (ux)f =u(xh) and (vx)f =v(xh), ∀ x∈Σ∗.
4. Rational functions 57
A more algebraic way to define this congruence can be achieved with two notations. The symbol
·will represent them from now on. One is a left action of M on the setX of all functions Σ∗ →M.
Givenh∈X and α∈M, we define the function α·h by
x(α·h) =α(xh), ∀ x∈Σ∗.
The other is a right action of Σ∗ on X. Given a function h∈ X and s∈Σ∗,h·s is the function defined by
x(h·s) = (sx)h, ∀ x∈Σ∗. Notice that these actions satisfy
α·(h·s) = (α·h)·s.
We can now rewrite (4.7.9) as
(4.7.10) f·u=u·h and f ·v=v·h.
A more appropriate version of this definition depends on the following property (see [dS04] for a proof):
Proposition 4.7.1 Every nonempty function r : Σ∗ → M can be write as α·h, where h is a function such that 1 is a gcdof imh.
We say that a function h such that 1 is a gcd of imh is normalized. In view of Proposition 4.7.1, we can rewrite (4.7.10) as
(4.7.11) f ·u= (uα)·h0 and f ·v= (vα)·h0, whereh0 is a normalized function, α∈M and h=α·h0.
Clearly, ∼f is reflexive and symmetric. The essence of the proof that it is transitive is the following proposition (see [dS04] for a proof):
Proposition 4.7.2 Let r and s be normalized functions Σ∗ → M. If there exist α, β ∈ M such thatα·r=β·s, then there exists an invertible element t∈Msuch that r=t·s.
Let us show that∼f is transitive. Letu,vandwbe words such thatu∼f vandv ∼f w. Then there exist u, v∈M and a normalized function r: Σ∗ →M such that
f·u=u·r and f·v=v·r, and ˆv, wˆ ∈M and a normalized function s: Σ∗ →M such that
f·v= ˆv·s and f·w= ˆw·s.
As v·r = ˆv·s, we can apply Proposition 4.7.2to these functions. Let t be an invertible element such thatr =t·s. Then, f·u= (ut)·s. Therefore,u∼f w.
It remains to show that ∼f is a right congruence. Letz∈Σ∗. Then, f ·(uz) = (f ·u)·z= (u·r)·z=u·(r·z).
Equally,
f·(vz) =v·(r·z).
So,uz ∼f vz.
Again, ∼f has at most one class [u] such that (ux)f =∅for all x. We denote it by . The following theorem characterizes the subsequential functions Σ∗ →M.
58 4.7. Congruence of a function and reduction
Theorem 4.7.3 Let Σ be an alphabet and M be a cancellative monoid with gcd. A function f : Σ∗→M is subsequential if and only if Σ∗/∼f is finite.
Proof. We may assume that f 6=∅.
Suppose that f is subsequential, and let T = (Q, i, λ, ρ) be a trim subsequential Σ-M-transducer that realizes f. From T, we define a right congruence∼T on Σ∗ by
s∼T t if and only if is=it, ∀s, t∈Σ∗. As in Theorem4.7.1, card (Σ/∼T) is at most card (Q) + 1.
We claim that
if s∼T t, then s∼f t.
To proof this claim, it suffices to consider the case whereisanditare defined. Lets= (iλ)(i∗s), t= (iλ)(i∗t), andh: Σ∗ →M be the function defined by
xh= ((is)∗x)((i(sx))ρ), ∀ x∈Σ∗. Ifs∼T t, then
f·s=s·h and f·t=t·h, that is,s∼f t.
So, Σ∗/∼T is a refinement of Σ∗/∼f, and we conclude that Σ∗/∼f is finite.
Conversely, suppose that Σ∗/∼f is finite. Enumerate the elements of this quotient asc1, . . . , cn. For each classci different from , fix a representativeui, a normalized functionhi : Σ∗ → M and an elementui∈M satisfying
f ·ui =ui·hi.
From these data, we shall construct a subsequential transducer T that realizes f.
At first, we show that, for every word sequivalent to ui, there existss∈M such that
(4.7.12) f ·s=s·hi.
Indeed, as s∼f ui, there exist α, β∈ M and a normalized functionr such that f ·s= α·r and f·ui=β·r. Asui·hi =β·r, Proposition4.7.2yields an invertible elementt such thatr=t·hi. So,f ·s=s·hi, where s=αt.
The state set of T isQ= (Σ∗/∼f)−. The transition function is defined by [ui]σ = [uiσ], ∀[ui]∈Q, ∀σ ∈Σ such that [uiσ]6=.
The initial state is i= [1], and a state [ui] is final if and only if ui ∈ domf. This property is equivalent to 1(f ·ui)6=∅, or equally 1hi 6=∅.
Before we describe the emissions, let us study some properties of T. At first, observe that is6=∅ if and only if [s]6=
(4.7.13a)
is= [s], ∀s∈Σ∗ such that [s]6=
(4.7.13b)
These properties follows from induction on|s|. By the definition of the final states, we have dom|T |= domf.
4. Rational functions 59
From (4.7.13b) and the definition ofQ, it follows thatT is accessible. In order to proof that it is coaccessible, let is be a state. Then, f ·s6=∅. Let x be a word such that x(f ·s) 6=∅. Then, (sx)f 6= ∅, that is, 1(f ·(sx)) 6= ∅. Therefore, [sx]6= , and [sx] is a final state. It remains to observe that (4.7.13b) implies [s]x= (is)x=i(sx) = [sx]. We then conclude that T is trim.
Let ci = [s] be a class different from , σ be a letter such that [sσ]6=∅, and cj be the class [sσ]. The definition of the emission [s]∗σ requires some manipulations. At first, writef ·(sσ) as
f·(sσ) = (f·s)·σ = (s·hi)·σ=s·(hi·σ).
Then,
(4.7.14) s·(hi·σ) =sσ·hj.
Letαi, σ ∈M andhi, σ be a normalized function such thathi·σ =αi, σ·hi, σ. Then, (4.7.14) becomes (4.7.15) (sαi, σ)·hi, σ =sσ·hj.
By Proposition 4.7.2, there exists an invertible ti, σ ∈M such that
(4.7.16) hi, σ =ti, σ·hj.
We define then
(4.7.17) [s]∗σ =αi, σti, σ.
The initial emission is 1, and the emission of a final state ci = [s] is 1hi. We claim that
(4.7.18) 1(i∗t) =t, ∀ t∈domf.
The proof is by induction on|t|:
• |t|= 0: evident.
• |t| > 0: factorizes t as sσ, where σ is a letter. Then, 1(i∗t) = 1(i∗s)((is)∗σ). By the induction hypothesis, 1(i∗s) =s. Letci = [s], and cj = [sσ]. Then, (is)∗σ =αi, σti, σ. So, 1(i∗t) =sαi, σti, σ.
By replacing (4.7.16) in (4.7.15), we get
(sαi, σti, σ)·hj =sσ·hj. By cancelling, sαi, σti, σ =sσ. So, 1(i∗t) =t.
Therefore, for allt∈domf,
t|T |= 1(i∗t)([t]ρ) =t(1hi) = 1(t·hi) = 1(f ·t) =tf, whereci = [t]. Then, we conclude that|T |=f, that is,f subsequential.
60 4.7. Congruence of a function and reduction
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Index
S´ımbolos 2A,1
<l,3 [k],1
#r,2 Rac(M),5 Rec(M),5 Σ≤l,3 dom , 2 gcdq,48 gcd(f, s), 51 fˆ,51
im , 2
N,Z,Q,R,R+,1 card,1
∼f,54
∅,1 ϕ,49 r|X,2 r(∞),2 r(k),2 r−1,2 u[i . . . j],3 u[i],3 Tf,52 Tf0,52
|A|,7
A action, 4 alphabet,3
associated elements,47 autˆomato
normalizado,7 automaton,6
accessible,7 behaviour,7
coaccessible, 7 deterministic, 7 normalized,7 trim,7 B
bijection, 2 bimorphism, 4
of a transducer, 10 C
cardinality,1 cartesian product,1 Choffrut’s Theorem, 34 congruence,5
copy of an alphabet, 4 D
decidability
of the equivalence ofk-valued rational re-lations,32
whether a rational relation isk-valued for a givenk,16
whether a rational relation is finite val-ued,15
determinization of a transducer,46 divisor, 45
E
Elgot and Mezei’s composition theorem,12 Elgot and Mezei’s decomposition theorem, 33 emission function,34
empty function, 1 empty word,3 end, 6
essential state, 36,39 F
final emission, 34 63
64 Index
fit,17
free monoid,3 function, 1
bijective,2 injective,2 surjective, 2 total, 1 G
gcd, 47
of a state, 48 graph of a relation,2 greatest common divisor,47 group,3
I
image, 1
initial emission,34 input monoid,9 invertible, 3 isomorphism,50 L
language, 3 left action,5
lexicographical order, 3 local language,7 M
matrix representation, 10 normalized,10 of a transducer, 10 trim,10
monoid, 3
cancellative, 47 commutative,3 monoid with gcd,47
monoid with unicity of gcd,47 morphism,4
fine,4
of monoids, 4 of semigroups,4
of subsequential transducers, 49 very fine,4
N
normalized function,57
O
output monoid, 9 P
partial function, 1 path, 6
input,9 label of, 6 output,9 successful, 7 prefix, 3
prefix form,49
prefix of a transducer, 49 prefix transducer, 49 product, 3
product monoid,4 proper morphism, 50 R
rational relation, 9 rational function, 13 rational set,5 recognizable set, 5 reduced transducer, 56
reduction of a subsequential transducer, 56 relation, 2
k-valued,13 composition,2 domain, 2 finite valued, 13 image, 2
inverse, 2 restriction, 2 union,2 right action,4 right congruence, 5
right congruence of a function,54 S
semigroup, 3 semiring,4 special state, 37
squaring of a transducer,35 star, 3
start, 6