Properties of some classes of rational relations
Rodrigo Nonamor Pereira Mariano de Souza
Dissertation presented to the Mathematics and Statistics
Institute of
University of S˜ao Paulo as part of the requirements for the degree of Master of Sciences
in Computer Science
(reduced version)
Advisor: Dr. Nami Kobayashi
— Sao Paulo, August 20, 2004—
Abstract
We study theoretical and algorithmic aspects of some classes of rational relations: the finite valued rational relations, thek-valued rational relations, for every positive integerk, the sequential functions, and the subsequential functions.
At first, we study some classical results concerning the rational relations, the representations of rational relations by transducers and matrices, and some closure properties.
Weber proved that everyk-valued rational relation can be decomposed into a union ofkrational functions. We present a proof for this result that employskapplications of Eilengerg’s Cross-section Theorem, and seems to be simpler than that of Weber.
By using this decomposition, we give a proof for the decidability of the equivalence of k-valued rational relations, first proved by Culik and Karhumaki and independently by Weber. On the other hand, Griffiths proved that the equivalence of finitely valued rational relations is undecidable. We also include this result of Griffiths.
We extend a Sch¨utzenberger’s property for rational functions to thek-valued rational relations.
As a consequence, we obtain a (non-polynomial) algorithm for deciding whether a transducer realizes ak-valued rational relation, for a given positive integer k.
We describe two efficient algorithms of B´eal, Carton, Prieur and Sakarovitch: an algorithm for deciding whether a rational relation is a function and another for deciding whether a rational func- tion is subsequential. Our description employs a simple symmetry property that leaded to slightly more efficient algorithms in terms of time and space (the improvement is in the multiplicative constant).
We present the Choffrut’s characterization of subsequential functions, and an algorithm for the determinization of a transducer based on this characterization.
We study the minimization of subsequential transducers by using a class of monoids that we call monoids with gcd (where gcd stands for greatest common divisor). We prove the existence of a minimal subsequential transducer for a subsequential function Σ∗→ M, where M is a cancellative monoid such that every non-empty subset has a single gcd. This result includes the minimization theory for functions whose images are in a free monoid or in the additive monoid of the non-negative real numbers.
Finally, we show a characterization of the subsequential functions Σ∗ → M, where M is a cancellative monoid with gcd, by using the right congruence of a function.
Contents
Main results ii
1 Preliminaries 1
1.1 Sets and relations . . . 1
1.2 Algebraic structures . . . 3
1.3 Rational sets . . . 5
1.4 Automata . . . 6
2 Rational Relations 9 2.1 Rational relations, transducers and matrix representations . . . 9
2.2 Properties of rational relations . . . 11
2.3 Closure properties . . . 12
3 Finite valued rational relations 13 3.1 Definitions and basic properties . . . 13
3.2 Decidability whether a finite valued rational relation is k-valued for a givenk . . . . 16
3.3 Metrics and differences . . . 19
3.4 Decomposition of a k-valued rational relation intokrational functions . . . 20
3.5 Equivalence of k-valued rational relations . . . 30
4 Rational functions 33 4.1 Subsequential functions . . . 33
4.2 Two decidability algorithms . . . 34
4.3 Determinization of a transducer . . . 43
4.4 Monoids with gcd. . . 45
4.5 Prefix form and morphisms of transducers . . . 48
4.6 Minimal subsequential transducers . . . 51
4.7 Congruence of a function and reduction . . . 54
Bibliography 61
Index 63
i
Main results
This work includes both a discussion of classical properties of rational relations, and a study of some more recent developments. In the latter, different proofs and even new properties take part.
The main results studied are listed below, accompanied by an indication of the corresponding location.
• Rational relations: Nivat’s factorization of rational relations (Theorem 2.2.1); Elgot and Mezei’s composition theorem (Theorem2.3.1).
• Finite valued rational relations: equivalence between finite valued rational relations and real- time transducers (Theorem3.1.2); decidability in polynomial time whether a rational relation is finite valued (Theorem 3.1.3); undecidability of the equivalence of finite valued rational relations, obtained by Griffiths (Theorem3.1.5).
• k-valued rational relations: extension of a Sch¨utzenberger’s characterization of the rational functions to the k-valued rational relations, for every positive integer k (Theorem 3.2.2); a proof of the decomposition of ak-valued rational relation intokrational functions, first proved by Weber (Theorem3.4.1); a proof of the decidability of the equivalence of k-valued rational relations, first proved by Culik and Karhumaki and independently by Weber (Theorem3.5.1).
• Rational functions: Elgot and Mezei’s decomposition of a rational function as a left sequential function and a right sequential function (Theorem4.1.1).
• Subsequential functions: Choffrut’s characterization of the subsequential functions (Theo- rem 4.1.2); algorithm of Sakarovitch et al. for deciding whether a rational function is subse- quential (Section4.2); algorithm for the determinization of a real-time transducer that realizes a rational function (Section4.3); study of the minimal transducer for a subsequential function Σ∗→M, where M is a cancelative monoid with unicity of gcd (Section 4.6); characterization of the rational functions Σ∗ →M, where M is a cancelative monoid with gcd (Theorem4.7.3).
iii
Chapter 1
Preliminaries
1.1 Sets and relations
We denote the set of the integers by Z, the set of the non-negative integers byN, the set of the rational numbers by Q, and the set of the real numbers byR.
The empty set will be denoted by∅. Given a positive integerk, we denote the set of the integers from 1 tok by [k].
We often identify a singleton with its only element. That is, we may write uin the place of the singleton{u}.
The cardinality of a set A will be denoted by card (A). If A is infinite, we set card (A) =∞.
The set of all subsets of A will be denoted by 2A.
The cartesian product of setsA and B is the setA×B of the pairs (a, b) such thata∈A and b∈B. A partial function fromA toB, orfunction for short, is a subset
f ⊆A×B
such that for alla∈A, if
(a, b),(a, b0)∈f, then
b=b0. We write
f :A→B to denote thatf is a function from A toB.
Given a pair (a, b)∈f, theimage of abyf is af =b.
Givena∈A, if there is no pair of the form (a, b)∈f, then we setaf =∅. Ifaf 6=∅for alla∈A, we say that f is atotal function. Iff =∅, we say that f is the empty function.
If
af 6=bf, ∀ a, b∈A such thata6=b, 1
2 1.1. Sets and relations
we say that f is injective. If
for all b∈B, there existsa∈A such thataf =b,
we say thatf is surjective. If f is total, injective and surjective, we say that it is bijective, or is a bijection.
We follow the terminology for relations between sets of [Eil74]. Given sets Aand B, a relation r from A toB is a function
r:A→2B.
However, a subset of A×B may also be called a relation fromA toB. IfA=B, then we call r a relationover A. The symbol r will also be used to denote the extension
r: 2A→2B define by
Xr = [
x∈X
xr, ∀ X∈2A. We write
A→r B or r:A→B to denote thatr is a relation from A toB.
The graph of r is the subset of A×B defined by
#r={(a, b)∈A×B:b∈ar}.
The domain and theimage of r are the sets
domr ={a∈A:ar6=∅} and imr=Ar= (domr)r, respectively.
The inverse of r is the relation r−1 :B →A define by br−1 ={a:b∈ar}, ∀ b∈B.
The restriction ofr to a subset X ⊆A is the relationr|X:A→B defined by a(r|X) =
(ar ifa∈X;
∅ otherwise.
The union of relationsr:A→B and s:A→B is the relation r∪s:A→B define by a(r∪s) =ar∪as, ∀a∈A.
The composition of relations r:A→B and s:B→C is the relationrs:A→C defined by a(rs) = (ar)s, ∀a∈A.
We also denote a composition by.
A−→r B −→s C.
Given a relation r over a set A, we write r(k), k ≥ 0, to denote the relation r . . . r over A resulting fromkcompositions ofr. The symbol r(0) represents the identity relation. Thetransitive closure ofr is the reflexive and transitive relation overA defined by
r(∞)= [
k≥0
r(k).
1. Preliminaries 3
1.2 Algebraic structures
A semigroup is a set M endowed with an associative operation, often called product. If this operation has an identity, that we denote by 1M or simply 1, then M is amonoid.
Let M be a monoid. An element m ∈ M is said to be invertible if there exists an element m−1 ∈M, called the inverse ofm, such that
mm−1=m−1m= 1.
A group is a monoid whose all elements are invertible.
If
mm0 =mm0, ∀ m, m0∈M, we say that M is commutative.
We can extend the product of M to subsets A, B⊆M as follows:
A·B ={a·b:a∈A, b∈B}.
With this extension, 2M is a monoid with identity{1M}.
Given a subset A⊆M, we define the subsetA+⊆M as A+= [
k≥1
Ak,
whereAk stands forkproducts of A byA. Thestar ofA is the set A∗=A+∪1.
We say that a M is generated by a subsetA⊆M if M =A∗. A submonoid of M is a subset N⊆M satisfying
1M∈N and N2 ⊆N.
Thefree monoid generated by a finite and nonempty set Σ, calledalphabet, is the monoid of all finite sequencesσ1. . . σn of elements of Σ, calledwords, with the operation of concatenation. The identity is the sequence of length 0, or empty word, that is denoted by 1. The length of a word u is denoted by |u|. A subset of Σ∗ is called alanguage. All alphabets considered will be supposed to be finite and nonempty.
Given a positive integer l, we denote by Σ≤l the set of words of Σ∗ with length bounded by l.
The subsequence of a word u, or subword, that starts at a position i and ends at a positionj will be denoted byu[i . . . j]. Ifi=j, this subword is the letter at the positioniof u, and it can be write as u[i]. If i > j, we set u[i . . . j] = 1. If i= 1, then the subword is a prefix of u. We write xuto denote that a word x is a prefix ofu. The relation is a partial order on Σ∗.
Letσ1, . . . , σn be an ordering of the letters of Σ. Thelexicographical order on Σ∗ is the relation
<l on Σ∗ defined as follows. For all u, v∈Σ∗,u <lv if
• uv, or
• u and v can be factorized as u=xσiu0 and v=xσjv0, where σi and σj are letters such that i < j.
4 1.2. Algebraic structures
This relation is a total order on Σ∗. Given words sand tsuch that
t=st0, we define
s−1t=t0.
The copy of an alphabet Σ is a bijectionb: Σ→Σ0, where Σ0 is an alphabet disjoint from Σ.
Let M and N semigroups. A semigroup morphism from M to N is a function f : M→N
satisfying
(ab)f = (af)(bf), ∀a, b∈M.
If M and N are monoids and 1f = 1, thenf is amonoid morphism, or simplymorphism.
We say that a morphism f : Σ∗ → Γ∗ between free monoids Σ∗ and Γ∗ is fine if Σf ⊆Γ∪1, and very fine if Σf ⊆Γ.
The following property of free monoids is fundamental:
Proposition 1.2.1 Let Σ an alphabet and M a monoid. A function f : Σ→ M admits a unique extension to a morphism Σ∗ →M.
The cartesian product of monoids M×N equipped with the operation (a, b)·(c, d) = (a·Mc, b·Nd), ∀(a, b),(c, d)∈M×N,
is a monoid with identity (1M,1N), called product monoid. A pair of morphisms f : S → M and g: S→N defines a morphism (f, g) : S→M×N given by
m(f, g) = (mf, mg), ∀ m∈M, that we call bimorphism.
A substitution is a morphism h: Σ∗→2M, where M is a monoid.
A semiring is a set K endowed with two operations: an addiction + and a multiplication ·, subject to the following laws:
• the addiction is associative, commutative and has an identity 0;
• the multiplication is associative and has an identity 1;
• these operations satisfy the distributive lawsa(b+c) =ab+acand (b+c)a=ba+ca;
• for all a∈K,a·0 = 0·a= 0.
A right action or simplyaction of a monoid M on a set Qis a partial function δ :Q×M→Q
subject to the following axioms, whereq·m stands for (q, m)δ:
q·1 =q;
(1.2.1a)
q·(mn) = (q·m)·n.
(1.2.1b)
1. Preliminaries 5
We often omit the symbol·in the expressionq·m. The symmetric axioms for a function M×Q→Q leads to the concept of left action.
A partial function Q×Σ→Q can be inductively extended to an action of the free monoid Σ∗ on Qas follows:
q·1 =q;
(1.2.2a)
q·(uσ) = (q·u)·σ, ∀ u∈Σ∗, ∀σ ∈Σ.
(1.2.2b)
A congruence on a monoid M is an equivalence relation ∼ on M that is consistent with the product of M. Formally, for all m1, m2∈M,
(1.2.3) if m1 ∼m2, then mm1 ∼mm2 and m1m∼m2m, ∀ m∈M.
The quotient M/∼ is a monoid with the operation
[m][n] = [mn], ∀m, n∈M.
We say that ∼is aright congruence if it satisfy the sole property if m1∼m2, then m1m∼m2m, ∀m∈M.
1.3 Rational sets
Definition 1.3.1 A subsetRof a monoidMis said to be recognizableif there exist a finite monoid N, a morphism f : M→N and a subsetP ⊆N such that R=P f−1. We denote the family of the recognizable subsets of M by Rec(M).
Definition 1.3.2 Let Mbe a monoid. The familyRac(M) of the rational subsets of M is the least family of subsets ofMthat includes the empty set and the singletons, and is closed by the operations of product, union and+.
A subset R⊆M is said to be rational ifR∈Rac(M).
Theorem 1.3.1 (Kleene 1956) Let Σ be an alphabet. Then, Rec(Σ∗) =Rac(Σ∗).
We list below some fundamental properties of rational and recognizable sets. Although not all of them are referenced throughout the text, we include them because they serve as fundamental pieces of several proofs in the complete version of this work [dS04]. Proofs are given in [Eil74,Ber79,dS04].
Proposition 1.3.1 Letf : M→N be a monoid morphism andA∈Rac(M). Then,Af ∈Rac(N).
Proposition 1.3.2 Let A and B be recognizable subsets of a monoid M, C a recognizable subset of a monoid N, and f : M→N a morphism. Then,
Cf−1 ∈Rec(M), A∪B∈Rec(M), A∩B ∈Rec(M) and M−A∈Rec(M).
6 1.4. Automata
Proposition 1.3.3 Let R a subset of a monoid M. Then, R is rational if and only if there exist an alphabet Σ, a rational language L⊆Σ∗ and a morphism f : Σ∗ →M such that R=Lf.
Proposition 1.3.4 Letf : M→Na surjective monoid morphism. Then, for every rational subset A of N, there exists a rational subset B of M such that A=Bf.
Proposition 1.3.5 Let M be a monoid and S1,S2 ⊆ M. Then, for every A1 ∈ Rac(S∗1) and A2 ∈Rac(S∗2),
(1.3.1) A1∪A2 ∈Rac((S1∪S2)∗), A1A2 ∈Rac((S1∪S2)∗) and A+1 ∈Rac((S1∪S2)∗).
Proposition 1.3.6 Let M be a monoid, S⊆M, R∈Rac(S∗), and x∈M. Then, Rx∈Rac((S∪Sx)∗) and xR∈Rac((S∪xS)∗).
Proposition 1.3.7 Let R be a rational subset of a monoid M. Then, there exists a finite subset S⊆M such thatR ∈Rac(S∗).
1.4 Automata
Definition 1.4.1 Let M be a monoid. An automaton over M, or M-automaton, is a machine A= (Q, E, I, T) composed of
• a finite set Q of states;
• subsets I and T of Q of initialand final states, respectively;
• a finite set E⊆Q×M×Qof transitions.
We prefer to represent a transition
e= (p, m, q) as
e:p−→m q.
A path is a sequence
c:p0 −→m1 p1 −→ · · ·m2 −→mn pn
of consecutive transitions. Its start isp0 and itsend ispn. It may be represented as c:p0 −−−−−→m1...mn pn.
The label of cis the product
|c|=m1. . . mn.
1. Preliminaries 7
The label of the empty path is 1M.
We say thatc is asuccessful path ifp0 ∈I andpn∈T. The behaviour of A is the set|A| ⊆M of the labels of its successful paths. We say that Arealizes |A|.
An state q ∈ A is accessible (coaccessible) if there exists a path from an initial state (q) to q (a final state). If all states are accessible (coaccessible), we say that Ais accessible (coaccessible).
We say that Ais trim if it is both accessible and coaccessible.
We say thatAisnormalized ifI andT are distinct singletonsiandt, respectively, no transition ends ini, and no transition starts int.
The following theorem is a particular version of Theorem VII.10.1 of [Eil74].
Theorem 1.4.1 A subset of a monoid M is rational if and only if it is the behaviour of a M- automaton.
Corolary 1.4.1 Every nonempty rational subset of a monoid M is the behaviour of a normalized trim M-automaton.
Definition 1.4.2 Let Σ an alphabet. An automaton over Σ, or Σ-automaton, is an automaton over the free monoid Σ∗, the transitions of which are labelled by letters ofΣ.
Definition 1.4.3 Let Σ an alphabet. A deterministic Σ-automaton is a machine A = (Q, i, T) composed of
• a finite set Q of states ;
• an initial state i;
• a subset T of Qof final states;
• a partial function ·:Q×Σ→Q.
We call ·the transition function of A. This function can be extended to an action of Σ∗ over Q.
Sometimes, it is useful to consider a set of paths as a language. This technique leads to the concept of local language:
Definition 1.4.4 We say that a subsetLof a free monoidΣ∗ is localif there exist subsetsA, B⊆ Σ andC ⊆Σ2 satisfying
L=AΣ∗∩Σ∗B−Σ∗CΣ∗, or
L= 1∪AΣ∗∩Σ∗B−Σ∗CΣ∗.
Clearly, every local language is rational. Proposition1.3.3can be strengthened as follows:
Theorem 1.4.2 Let R a subset of a monoid M. Then, R is rational if and only if there exist an alphabet Σ, a local language L ⊆ Σ∗ and a monoid morphism f : Σ∗ → M such that R = Lf. Furthermore, ifM is a free monoid, we may assume that f is a very fine morphism.
Chapter 2
Rational Relations
This chapter consists of a list of classical properties of rational relations, transducers and matrix representations. The main references for it are [Ber79,Eil74]
Throughout the chapter, M and N represent monoids, and Σ and Γ alphabets.
2.1 Rational relations, transducers and matrix representations
Definition 2.1.1 We say that a relation τ : M→ N is rationalif its graphic is a rational subset of the product monoid M×N.
We call M the input monoid and N theoutput monoid of τ.
Definition 2.1.2 A transducer over M and N, or M-N-transducer, is an automaton over the product monoidM×N.
We prefer to denote a pair (x, y) ∈ M×N as x/y when dealing with paths in a transducer.
Given a path
c:p−−→x/y q we callx theinput of c, and y theoutput oremission of c.
The behaviour of a M-N-transducer T is a rational subset R of the product monoid M×N, that will be viewed as a rational relation|T |: M→N. Thus,R= #|T |.
Theorem 1.4.1implies in
Theorem 2.1.1 A relation τ : M→ N is rational if and only if its graphic is the behaviour of a M-N-transducer.
Let T = (Q, E, I, T) be a M-N-transducer,L be the local language of the successful paths of T, and α:E∗ →M andβ :E∗→N be the morphisms defined by
(p, m, n, q)α=m and (p, m, n, q)β =n, ∀ (p, m, n, q)∈E.
Then,
#|T |=L(α, β).
9
10 2.1. Rational relations, transducers and matrix representations
We can also proof that
(2.1.1) |T |=α−1(∩L)β.
We call the pair (α, β) the bimorphism of T, and α andβ theinput and the output morphisms of T, respectively.
Definition 2.1.3 Let n be a positive integer. A matrix representation of dimension n over an alphabet Σ and a monoid Mis a tuple M= (Q, λ, µ, ρ) where
• Q is a set of cardinality n;
• λis a row vector of dimension n over Rac(M);
• ρ is a column vector of dimension nover Rac(M);
• µ: Σ∗ → Rac(M)Q×Q is a semigroup morphism from Σ∗ to the multiplicative monoid of the square matrices of dimensionn over the semiring Rac(M).
The behaviourof M is the relation |M|: Σ∗ →M defined by u|M|=λ(uµ)ρ, or, equivalently,
u|M|= [
p, q∈Q
λp(uµpq)ρq, for all u∈Σ∗.
In this definition, if M is a free monoid Γ∗, we say that M is a matrix representation over the alphabets Σ and Γ.
Definition 2.1.4 We say that a matrix representationM= (Q, λ, µ, ρ) is trim if
• the entries of the vectors λand ρ are ∅ or 1;
• for everyq ∈Q, there exist words u and v and statesi and tsuch that λi=, ρt= 1, uµiq 6=∅ and vµqt6=∅.
We say that M is normalized if λ has exactly one entry not null i, with λi = 1, and ρ has exactly one entry not null t, distinct from i, withρt= 1.
Theorem 2.1.2 A relation τ : Σ∗ → M is rational if and only if it can be realized by a matrix representation. Furthermore, every rational relation can be realized by a trim and normalized matrix representation.
Part of the proof of Theorem 2.1.2 is a construction of a matrix representation MT from a given transducer T. We call MT thematrix representation of T. The morphism µof MT has the following useful property:
uµpq ={x: there exists a pathp−−→u/x q inT }, ∀ u∈Σ∗, ∀p, q∈Q.
2. Rational Relations 11
2.2 Properties of rational relations
Theorem 2.2.1 (Nivat 1968 [Niv68]) A relationτ : M→N is rational if and only if there exist an alphabet Ω, a local language L⊆Ω∗ and morphisms f : Ω∗ →M and g: Ω∗ →N such that
(2.2.1) τ = M f
−1
−−→Ω∗ ∩−−→L Ω∗ −→g N.
Furthermore, if M is a free monoid, then f can be chosen as a fine morphism, the same holding for N and g; if M and N are free monoids, then f and g can be chosen as fine morphisms.
Theorem 2.2.2 (Theorem IX.5.1 of [Eil74]) Let Σ be an alphabet, and τ : Σ∗ → M be a relation such that 1τ =∅or 1τ = 1. Then, τ is rational if and only if there exist an alphabet Ω, a local language L⊆Ω∗, a very fine morphism f : Ω∗ →Σ∗ and a rational substitution h: Ω∗ →M such that
τ = Σ∗ f
−1
−−→Ω∗ ∩−−→L Ω∗ −→h M.
Nivat’s Theorem has several applications. Some of them are in the proofs of the properties listed below. It is useful also in the proof of some of the closure properties stated in next section.
Proposition 2.2.1 Let τ : M→N be a rational relation. Then, domτ is a rational subset of M, and imτ is a rational subset ofN.
Proposition 2.2.2 Let Σ be an alphabet, τ : Σ∗ → N be a rational relation, and A ∈ Rac(Σ∗).
Then, Aτ ∈Rac(N).
Theorem 2.2.3 Let τ : Σ∗ → Γ∗ be a rational relation. Then, there exists a positive integer n such that every pair (u, v)∈#r with|u|+|v| ≥n admits a factorization
(u1, u2)(w1, w2)(v1, v2) satisfying
i. (w1, w2)6= (1,1),
ii. |u1|+|u2|+|w1|+|w2| ≤nand iii. (u1, u2)(w1, w2)∗(v1, v2)⊆#τ.
Proposition 2.2.3 Let Σ andΓ be alphabets.
• Every rational relation Σ∗ → M is the behaviour of a trim normalized Σ-M-transducer with transitions labelled by pairs in (Σ∪1)×M.
• Every rational relation M → Σ∗ is the behaviour of a trim normalized M-Σ-transducer with transitions labelled by pairs in M×(Σ∪1).
• Every rational relation Σ∗ → Γ∗ is the behaviour of a trim normalized Σ-Γ-transducer with transitions labelled by pairs in (Σ∪1)×(Γ∪1).
We say that a M-N-transducer is fine if M is a free monoid and the input of every transition is a letter or the empty word.
12 2.3. Closure properties
2.3 Closure properties
Proposition 2.3.1 Let Σ andΓ be alphabets that are not both singletons. Then, Rac(Σ∗×Γ∗) is not closed neither by intersection nor by complementation.
Proposition 2.3.2 Let τ : M → N be a rational relation. Then, the inverse τ−1 : N → M is a rational relation.
Proposition 2.3.3 Let Σ and Γ be alphabets, and τ : Σ∗ → Γ∗ be a rational relation. Then, the relation τ%: Σ∗→Γ∗ defined by
#τ%={(u%, v%) : (u, v)∈#τ} is rational.
Proposition 2.3.4 Let τ : M → N be a rational relation and A be a recognizable subset of M.
Then, the restriction τ|A is a rational relation.
Proposition 2.3.5 Let Σ be an alphabet with at least two letters. Then, Rac(Σ∗×Σ∗) is not closed by transitive closure.
Theorem 2.3.1 (Elgot and Mezei 1965) Let Σ be an alphabet, M and N be monoids, andτ1: M→Σ∗ and τ2 : Σ∗ →N be rational relations. Then, the composition τ1τ2 : M→N is a rational relation.
Chapter 3
Finite valued rational relations
In this chapter, we study the rational relations whose image of every input is a finite set. This restriction brings new interesting structural properties.
The main result is a proof of the decomposition of a k-valued rational relation into k rational functions, first established by Weber.
Throughout the chapter, M and N represent monoids, and Σ and Γ alphabets.
3.1 Definitions and basic properties
Definition 3.1.1 We say that a relation τ : M→N is finite valued if card (mτ)<∞, ∀ m∈M
and k-valued, for a positive integer k, if
card (mτ)≤k, ∀ m∈M.
A 1-valued rational relation is called a rational function.
Some basic closure properties are stated below.
Proposition 3.1.1 Let τ : M→N and υ: M→N be rational relations. Then,
• if τ andυ are finite valued relations, then τ∪υ is a finite valued rational relation;
• if τ is a k-valued relation andυ is al-valued relation, then τ ∪υ is a (k+l)-valued rational relation.
Proposition 3.1.2 Let τ : M→Σ∗ andυ: Σ∗ →N be rational relations. Then,
• if τ and υ are finite valued relations, then the composition τ υ : M → N is a finite valued rational relation;
• if τ is ak-valued relation and υ is al-valued relation, then the composition τ υ: M→N is a kl-valued rational relation.
13
14 3.1. Definitions and basic properties
Proposition 3.1.3 Let τ : Σ∗ →M andυ: Σ∗ →M be finite valued rational relations. Then,
• the product τ ·υ is a finite valued rational relation;
• if 1τ =∅or 1τ = 1, then τ+ is a finite valued rational relation.
Let Σ ={σ}, Γ ={a, b}and τ : Σ+ →Γ+ be the rational function defined by σnτ =
(an ifnis even;
bn ifnis odd.
Then,τ2 is notk-valued for anyk. This counter-example shows that the class of k-valued rational relations is not closed neither by product nor by star.
The following theorem is a particular version of Proposition 1.3.7 for finite valued rational relations.
Theorem 3.1.1 Let τ : Σ∗ → Γ∗ be a rational relation such that 1τ = ∅ or 1τ = 1. Then, τ is finite valued if and only if there exists a finite subset X of Σ×Γ∗ such that#τ ∈Rac(X∗).
Next, we state three properties of matrix representations of finite valued and k-valued rational relations.
Proposition 3.1.4 Let M = (Q, λ, µ, ρ) be a trim matrix representation over Σ and Γ. Then,
|M| is finite valued if and only if
(3.1.1) card (uµpq)<∞, ∀u∈Σ∗, ∀ p, q∈Q.
Corolary 3.1.1 Let M= (Q, λ, µ, ρ) be a trim matrix representation over Σ and Γ. Then, |M|
is finite valued if and only if
(3.1.2) card (uµpq)<∞, ∀ u∈Σ∪1, ∀ p, q∈Q.
Proposition 3.1.5 Let M= (Q, λ, µ, ρ) be a trim normalized matrix representation over Σ and Γ and k be a positive integer. Then, |M|is k-valued if and only if
(3.1.3) card (uµpq)≤k, ∀ u∈Σ∗, ∀p, q∈Q.
Definition 3.1.2 We say that a Σ-Γ-transducer T = (Q, E, I, T) is real-time if E ⊆ Q×Σ× Γ∗×Q.
The following theorem relates real-time transducers to finite valued rational relations. The third condition may be viewed as a version of Nivat’s Theorem for finite valued rational relations.
3. Finite valued rational relations 15
Theorem 3.1.2 Let τ : Σ∗ →Γ∗ be a rational relation. The following conditions are equivalent:
i. τ is finite valued and 1τ =∅;
ii. τ is the behaviour of a normalized real-time transducer;
iii. τ can be factorized as Σ∗ f
−1
−−→ Ω∗ −−→∩L Ω∗ −→g Γ∗, where Ω is an alphabet, f is a very fine morphism, g is a morphism, and L is a local language over Ω.
The proof of the next property consists of two constructions: one way is a construction of a matrix representation from a real-time transducer, and the other way is the inverse construction.
Proposition 3.1.6 Let τ : Σ∗ → Γ∗ be a relation such that 1τ = 1 or 1τ = ∅. The following conditions are equivalent:
i. τ is a finite valued rational relation;
ii. τ is the behaviour of a matrix representation M = (Q, λ, µ, ρ) over Σ and Γ such that the entries of the vectorsλ and ρ are ∅or 1, andµ is a monoid morphism that satisfy
card (uµpq)<∞, ∀ u∈Σ∗, p, q ∈Q.
The proof of the next property is a normalization procedure for real-time transducers.
Proposition 3.1.7 Let τ : Σ∗ → Γ∗ be a finite valued rational relation with 1τ =∅. Then, there exists a real-time trim normalized Σ-Γ-transducer T such that|T |=τ.
The following proposition states a structural property of transducers that realizes a finite valued rational relation. This property is useful in the proofs of theorems3.1.3and 3.1.4.
Proposition 3.1.8 Let T = (Q, E, I, T) be a fine trimΣ-Γ-transducer Then,|T | is finite valued if and only if
(3.1.4) card (1µpq)<∞, ∀p, q∈Q, where µ is the morphism of the matrix representation of T.
Theorem 3.1.3 It is decidable whether a rational relationτ : Σ∗→Γ∗ is finite valued.
Theorem 3.1.4 LetT be a fine trim normalizedΣ-Γ-transducer that realizes a finite valued ratio- nal relation. Then, T can be effectively transformed into an union
T1∪ T2,
where T1 is a trim normalized real-time Σ-Γ-transducer such that
|T1|=|T ||Σ+ and T2 is aΣ-Γ-transducer such that
|T2|=|T ||1.
We conclude this section with an old and interesting decidability result.
Theorem 3.1.5 (Griffiths 1968 [Gri68]) The equivalence of rational relations Σ∗ → Γ∗ is un- decidable.
16 3.2. Decidability whether a finite valued rational relation is k-valued for a given k
3.2 Decidability whether a finite valued rational relation is k- valued for a given k
Gurari and Ibarra proved in 1983 that it is decidable in polynomial time whether a finite valued rational relation isk-valued for a given k [GI83]. In this section, we obtain this result as a consequence of a combinatorial property of real-time transducers that realize a k-valued rational relation. However, our algorithm is not polynomial.
This property is a generalization of the following Sch¨utzenberger theorem:
Theorem 3.2.1 (Sch¨utzenberger 1975 [Sch75]) Let T = (Q, E, I, T) be a real-time Σ-Γ- transducer, µ be the morphism of its matrix representation, andn= card (Q). If the condition
card (uµpq)≤1 holds for every p, q∈Q and every word u∈Σ∗ such that
|u| ≤1 + 2n(n−1), then this condition holds for every word u∈Σ∗.
The generalization is
Theorem 3.2.2 Let k > 0 be an integer, T = (Q, E, I, T) be a real-time Σ-Γ-transducer, µ be the morphism of its matrix representation, and n= card (Q). If the condition
(3.2.1) card (uµpq)≤k
holds for every p, q∈Q and every word u∈Σ∗ such that
(3.2.2) |u| ≤nk+1T(k(k+ 1)/2) + 1,
where T is the function defined by
(3.2.3) T(l) =l!
1 + X
1≤i≤l
1 i!
, ∀ l∈N, l >0, then this condition holds for every word u∈Σ∗.
Corolary 3.2.1 Let k be positive integer, and τ : Σ∗→Γ∗ be a rational relation. It is decidable if τ isk-valued.
Observe that if we takek= 1 in Theorem3.2.2, the resulting expression is not the same as that of Theorem 3.2.1. The reason is a particularity of the case k= 1 that allows a better bound.
In order to prove Theorem 3.2.2, we need three lemmas. Two of them are simple:
Lemma 3.2.1 LetΣbe an alphabet, s= card (Σ), and l >0 be an integer. For every wordu∈Σ∗ with |u| ≥sl−s+ 1, there exists some letter σ∈Σthat occurs at least l times in u.
3. Finite valued rational relations 17
Lemma 3.2.2 (Sch¨utzenberger) Let x1, x2, x3, x4, y1, y2, y3, y4 be words over an alphabet Σ such that
i. x1x4=y1y4, ii. x1x2x4 =y1y2y4, iii. x1x3x4 =y1y3y4. Then
x1x2x3x4 =y1y2y3y4.
Before we present the third lemma, let us observe that (3.2.3) is the solution of the recurrence T(1) = 2
T(l) =lT(l−1) + 1, ∀l >1
We also introduce a definition. Given a positive integerθ and an alphabet Ω, let (3.2.4) Cθ={(r, s) :r, s∈[θ] and r≤s}.
Given a functionf :Cθ →Ω, afit of f is a triple
((i, j),(i, k),(k+ 1, j)), (i, j),(i, k),(k+ 1, j)∈ Cθ satisfying
i≤k < j and (i, j)f = (i, k)f = (k+ 1, j)f.
Lemma 3.2.3 Let θ be a positive integer, Ω be an alphabet, and f : Cθ → Ω be a function. If θ≥T(card (Ω)), then f has a fit.
Proof. We shall proceed by induction ons= card (Ω).
• s= 1: evident.
• s >1: let x∈Ω∗ be the word of lengthθ defined as
x[i] = (1, i)f, ∀ i,1≤i≤θ.
AsT(s) =s(T(s−1) + 1)−s+ 1, it follows from Lemma3.2.1that there exists a letterσ ∈Ω that occurs at leastT(s−1) + 1 times inx. Then we can factorize x aswx0 where
w=x0σx1σ . . . xT(s−1)σ and
xi ∈(Ω−σ)∗, ∀i, 0≤i≤T(s−1).
For eachi, 0≤i≤T(s−1), we denote bywi a prefix of wof the form x0. . . xi.
18 3.2. Decidability whether a finite valued rational relation is k-valued for a given k
Suppose that (|wiσ|+ 1,|wjσ|)f =σ for some indices iand j withi ≥0 andi < j. Then, ((1,|wjσ|),(1,|wiσ|),(|wiσ|+ 1,|wjσ|)) is a fit of f. Otherwise, consider the function g : CT(s−1) →Ω−σ defined as
(i, j)g= (|wi−1σ|+ 1,|wjσ|)f, ∀ (i, j)∈ CT(s−1). By the induction hypothesis, g has a fit ((i, j),(i, k),(k+ 1, j)). So,
((|wi−1σ|+ 1,|wjσ|),(|wi−1σ|+ 1,|wkσ|),(|wkσ|+ 1,|wjσ|)) is a fit of f.
Proof of Theorem3.2.2. Assume that (3.2.1) is true for every word with length limited by (3.2.2), but not for every word in Σ∗.
Consider states p and q and a word u =σ1. . . σl, σi ∈Σ for every index i, of minimal length such that card (uµpq)> k. Then, there exist k+ 1 paths c1, . . . , ck+1 of the form
cj :p−−−−→σ1/vj1 qj1. . . qjl−1 −σ−−−l/vjl→q, ∀j, 1≤j≤k+ 1, with pairwise distinct outputs.
Let ξ = ξ1. . . ξl−1 be the word over the alphabet Qk+1 defined by ξi = (q1i, . . . , qk+1i), ∀ i, 1≤i < l. By the choice ofu,
l−1≥nk+1((T(k(k+ 1)/2) + 1)−1) + 1.
As card (Qk+1) =nk+1, it follows from Lemma3.2.1that someξioccurs at leastT(k(k+1)/2)+1 times in ξ. Then, we can factorizec1, . . . , ck+1 as
c1: p−−−−→u0/x10 q1i −−−−→u1/x11 . . .−−−−→uθ/x1θ q1i −−−−−−−→uθ+1/x1θ+1 q, ...
ck+1: p−−−−−−→u0/xk+10 qk+1i −−−−−−→u1/xk+11 . . .−−−−−−→uθ/xk+1θ qk+1i −−−−−−−−−→uθ+1/xk+1θ+1 q, whereθ=T(k(k+ 1)/2).
Let Ω ={{i, j}:i, j ∈[k+ 1], i6=j} and f :Cθ → Ω be the function defined as follows. For each (a, b)∈ Cθ, consider the paths
c(a, b)1 : p−−−−−−−−−−−−−→u0...ua−1/x10...x1a−1 q1i −−−−−−−−−−−−−−−−→ub+1...uθ+1/x1b+1...x1θ+1 q, ...
c(a, b)k+1 : p−u−−−−−−−−−−−−−−−−0...ua−1/xk+10...xk+1a−1→qk+1i −−−−−−−−−−−−−−−−−−−→ub+1...uθ+1/xk+1b+1...xk+1θ+1 q,
obtained by removing the closed path of the form q1j −−−−−−−−−−→ua...ub/xja...xjb q1j from cj, for 1≤j≤k+ 1.
By the minimality of u, we can choose two distinct paths c(a, b)r and c(a, b)s with the same output.
Then we define (a, b)f ={r, s}.
3. Finite valued rational relations 19
As card (Ω) = k(k+ 1)/2, θ = T(card (Ω)). Then it follows from Lemma 3.2.3 that f has a fit ((a, b),(a, e),(e+ 1, b)), that is, there exist indices r and s such that (a, b)f = (a, e)f = (e+ 1, b)f ={r, s}. By the definition off, we have
(xr0. . . xra−1)(xrb+1. . . xrθ+1) = (xs0. . . xsa−1)(xsb+1. . . xsθ+1),
(xr0. . . xra−1)(xre+1. . . xrb)(xrb+1. . . xrθ+1) = (xs0. . . xsa−1)(xse+1. . . xsb)(xsb+1. . . xsθ+1), (xr0. . . xra−1)(xra. . . xre)(xrb+1. . . xrθ+1) = (xs0. . . xsa−1)(xsa. . . xse)(xsb+1. . . xsθ+1).
By Lemma3.2.2, we conclude
crβ =xr0. . . xrθ+1 =xs0. . . xsθ+1 =csβ,
a contradiction to the hypothesis over u. So, card (uµpq) ≤ k for every pair of states p e q and every word u∈Σ∗.
3.3 Metrics and differences
Int this section we present some tools that will be used extensively in the proof of the de- composition theorem. Most of this material appeared in the works on subsequential functions of Choffrut [Cho77] and Sakarovitch et al. [BCPS03].
At first, we define two metrics on a given free monoid Σ∗. Both are functions from Σ∗×Σ∗ to the set Nof non-negative integers.
The first one maps a pair of words (s, t) to the integer
||s, t||=|s|+|t| −2|s∧t|.
The second metric supposes fixed a morphismg: Σ∗ →Γ∗, and maps a pair of words (s, t) to the integer
||s, t||g = max{||xg, yg||:x4s, y4t, and |x|=|y|}.
We call the latterg-metric. It is easy to verify the metric properties for these functions.
By using the g-metric, we define the sparsity of a set A⊆Σ∗ with card (A)>2 as follows:
||A||g = min{||u, v||g:u, v ∈A, u6=v and ug =vg}.
If there are no wordsu and v inA withug=vg, we set ||A||g=∞.
Now, we define a notation to deal with pairs of words without the common prefix. Thedifference of a pair in Σ∗×Σ∗ is the function ∆ : Σ∗×Σ∗ →Σ∗×Σ∗ given by
(s, t)∆ = ((s∧t)−1s, (s∧t)−1t),∀(s, t)∈Σ∗×Σ∗.
Let ∆Σ∗ be the set of pairs (s, t)∈Σ∗×Σ∗ such that (s, t)∆ = 1. We also use a right action
·: ∆Σ∗×(Σ∗×Σ∗)→∆Σ∗
of the product monoid Σ∗×Σ∗ on ∆Σ∗ given by
(x, y)·(s, t) = (xs, yt)∆,∀(x, y)∈∆Σ∗,∀(s, t)∈Σ∗×Σ∗.
20 3.4. Decomposition of a k-valued rational relation into k rational functions
It is easy to check that ·indeed is an action, that is:
(u, v)·(1,1) = (u, v) and
(u, v)·((x, y)(z, w)) = ((u, v)·(x, y))·(z, w), for all (u, v)∈∆Σ∗ and (x, y),(z, w)∈(Σ∗×Σ∗).
We conclude this section stating some properties of these concepts:
if (s, t)∆ = (u, v)∆, then ||s, t||=||u, v||
(3.3.1)
(s, t)∆ = (1,1) if and only if s=t (3.3.2)
(st, uv)∆ = (s, u)∆·(t, v) (3.3.3)
Proposition 3.3.1 Let(s, t)and(u, v)be pairs in∆Σ∗ andΣ∗×Σ∗, respectively. If(s, t)·(u, v)6=
(s, t), then, for every pair of positive integers k andl,
(s, t)·(u, v)k6= (s, t)·(u, v)l.
3.4 Decomposition of a k-valued rational relation into k rational functions
In this section, we prove Weber’s Theorem, wich can be stated as follows:
Theorem 3.4.1 (Weber) Let τ : Σ∗ → Γ∗ be a k-valued rational relation. Then, τ can be decomposed into a union
τ =f1∪ · · · ∪fk, where each fi, 1≤i≤k, is a rational function.
It should be clear that when we compare two relations by inclusion, we mean to compare the graphs of these relations.
To explain our proof, at first we outline a technique that fails to yield a decomposition of τ. Our proof will be a modification of this technique.
Let us consider 1τ =∅. The proof for 1τ 6=∅is a trivial consequence of this case. With this hypothesis, we apply Theorem3.1.2 to get a factorization
(3.4.1) τ : Σ∗ f
−1
−→Ω∗ ∩−→L Ω∗ −→g Γ∗,
where f is a very fine morphism, Ω is an alphabet, L is a recognizable subset of Ω∗, and g is a morphism.
The technique uses the following classical result due to Eilenberg [Eil74]:
Theorem 3.4.2 (Cross-Section Theorem) Given a morphism g : Σ∗ → Γ∗ and a recognizable subset A of Σ∗, there exists a recognizable subset B of A such that g maps B bijectively ontoAg.
3. Finite valued rational relations 21
The setB is called across-section of A withf.
The first step is to use Cross-Section Theorem iteratively in subsets of L, starting fromL, in order to getkpairwise disjoint cross-sectionsA1, . . . , Ak. For eachi, 1≤i≤k,Aiis a cross-section of L−S
1≤l<iAl withf. Next, we definek rational relations by fi = Σ∗ f
−1
−→Ω∗ ∩−→Ai Ω∗−→g Γ∗. Asf restricted to Ai is bijective, these relations are indeed functions.
A simple check shows that f1∪ · · · ∪fk ⊆τ. However, the converse is not true in general. A counter-example can be seen in Example 3.4.2.
The problem of this technique is the possibility of to include redundant words in the cross- sections, but to leave out some essential ones. To explain this point, we turn our attention to the fact that the graph #τ is the image of L by the bimorphism (f, g), that is, every pair (s, t) in this relation is the image u(f, g) of some word u in L. We say that two words u and v in L are redundantifu(f, g) =v(f, g). Thus, redundant words contribute to the same pair in #τ, and there is no need to have more than one in the cross-sections A1, . . . , Ak. However, in the construction of these sets, some redundant words can be included instead of some essential ones.
A way to remedy this problem could be to eliminate all redundant words. To do this, let ≡be the relation given byu≡vif, and only if,u(f, g) =v(f, g), and consider the setL/≡of the classes of redundant words. In order to construct a set L0 without redundant words, one could select a word in each one of these classes. However, this can produce a set L0 that is not recognizable, regardless of the way that the words are choosen. This situation is ilustrated in Example 3.4.1.
Our idea is to eliminate just some special redundant words, in such a way that it will be possible to get a decomposition by using Cross-Section Theorem.
To do this, let us define a familyLν,ν ≥0, of recognizable subsets ofL. The definition supposes fixed a lexicographical order<l in Ω∗.
Definition 3.4.1 For each integer ν ≥0, we denote by Lν the subset of L defined as follows. A word s ∈L is in Lν if, and only if, for every word t in L such that s(f, g) = t(f, g) and t <l s,
||s, t||g ≥ν.
Intuitively, a pair of words of Lν with the same image by the bimorphism (f, g) are “far” by the function || ||g, and a lower bound for this distance is ν. It follows from the definition that, as we increase the indexν, more redundant words are eliminated. So, these sets satisfy
L=L0⊇L1⊇ · · · ⊇L∞.
We denote byL∞the set of the minimal words by<l from each class in the quotientL/≡. Clearly, (f, g) restricted toL∞ is injective, but this is not always true for the other sets.
Next, we define the compositions τν = Σ∗ f
−1
−−→Ω∗ ∩−−→Lν Ω∗ −→g Γ∗, ∀ν ≥1.
Then, we prove that there exists a ν such that k applications of Cross-Section Theorem starting from Lν yield a decomposition of τ.
We begin with two fundamental properties of the sets Lν.
22 3.4. Decomposition of a k-valued rational relation into k rational functions
Proposition 3.4.1 For all ν∈N∪ ∞, τ =τν. Proof. Let sbe a word in Σ∗.
As Lν ⊆L,s(f−1(∩Lν))⊆s(f−1(∩L)), for all s∈Σ∗. So,sτν ⊆sτ.
Given a wordtinsτ, letx be the minimal word of the set (s, t)(f, g)−1 by the order<l. Thus, by the definition ofLν,x∈Lν, and t∈sτν. So,sτ ⊆sτν.
We prove that the setsLν are recognizable by showing an effective construction of an automaton that recognizes Lν.
Proposition 3.4.2 For every integer ν≥1, the setLν is recognizable.
Proof. LetA= (Q, i, T) be a trim deterministic Ω-automaton that recognizesL. For each integer ν ≥1, we define a deterministic Ω-automatonAν = (Qν, iν, Tν) as follows:
• Qν ={q∪X:q ∈Qand X ⊆(Q×Ω≤ν−1)∪(Ω≤ν−1×Q)}.
• iν ={i}.
• Tν ={q∪X ∈Qν :q∈T and X∩(T×1) =∅}.
• The transitions are defined as follows. Let
P ={q, (p1, u1), . . . ,(pa, ua),(v1, q1), . . . ,(vb, qb)}
be a state in Qν, where (pj, uj)∈(Q×Ω≤ν−1) for allj, and (vl, ql)∈(Ω≤ν−1×Q) for alll.
For eachσ ∈Ω, ifqσ =∅, thenP σ =∅; otherwise,P σis the union of qσ with the following pairs:
– for each γ ∈Ω such that qγ 6=∅,γ <lσ and σf =γf,
∗ ifσg4γg e |γg| − |σg|< ν, then (qγ, (σg)−1(γg))∈P σ;
∗ ifγg≺σg e |σg| − |γg|< ν, then ((γg)−1(σg), qγ)∈P σ;
– for each (pj, uj)∈P and each γ ∈Ω such that pjγ 6=∅and σf =γf,
∗ ifσg4uj(γg) and|uj(γg)| − |σg|< ν, then (pjγ,(σg)−1(uj(γg)))∈P σ;
∗ ifuj(γg)≺σg and|σg| − |uj(γg)|< ν, then ((uj(γg))−1(σg), pjγ)∈P σ;
– for each (vl, ql)∈P and each γ ∈Ω such that qlγ 6=∅and σf =γf,
∗ ifvl(σg)4γg and |γg| − |vl(σg)|< ν, then (qlγ, (vl(σg))−1(γg))∈P σ;
∗ ifγg≺vl(σg) and |vl(σg)| − |γg|< ν, then ((γg)−1(vl(σg)), qlγ)∈P σ.
In order to explain how Aν operates on a given input u, we describe the elements in the state iνu,u∈Ω∗. Induction on |u|shows that indeediν has these elements.
The pairs of (Q×Ω≤ν−1)∪(Ω≤ν−1×Q) represent the differencesbetween ug and the images by g of some words in Ω∗ whose images byf areuf. More precisely,iνu is the union of the state iuwith the following pairs. For every word v∈Ω∗ satisfying
iv6=∅, v <lu, uf =vf, ug4vg orvg ≺ug, and ||u, v||g < ν,
3. Finite valued rational relations 23
ifug4vg, theniνuhas the pair (iv,(ug)−1vg), and ifvg≺ug, theniνuhas the pair ((vg)−1(ug), iv).
From the definition ofTν, it follows thatiνu is a final state ofAν if, and only if, uis in L, and there is not a word v such that
v∈L, v <lu, u(f, g) =v(f, g) and ||u, v||g < ν.
From these four properties and the definition of Lν, we get|Aν|=Lν. We prove two technical lemmas before the main result.
Lemma 3.4.1 Let g: Ω∗ →Γ∗ be a morphism. For every pair of words u and v in Ω∗ and every factorizations xyz of u and rst of v such that|x|=|r|and |y|=|s|,
i) ifug =vg, and (xg, rg)∆ = (xg, rg)∆·(yg, sg), then, for every l≥1, (xylz)g= (rslt)g and
||xylz, rslt||g=||u, v||g;
ii) ifug 6=vg, then there exists an integerl0 such that, for every l≥l0, (xylz)g6= (rslt)g.
Proof. Assume the hypothesis in i). Then, for all l≥1,
(xg, rg)∆ = (xg, rg)∆·(yg, sg)l= (xg(yg)l, rg(sg)l)∆ = ((xyl)g,(rsl)g)∆.
As ug = vg, ((xy)g, (rs)g)∆·(zg, tg) = (1,1). Thus, ((xyl)g,(rsl)g)∆·(zg, tg) = (1,1), and ((xylz)g,(rslt)g)∆ = (1,1). Therefore, (xylz)g= (rslt)g.
We prove that ||xylz, rslt||g=||u, v||g by showing that the distances between the images byg of prefixesw14xylz andw2 4rsltsuch that|w1|=|w2|correspond to the distance of the images bygof some prefixes ofuand v. Ifw1 and w2 are prefixes ofx andr, respectively, there is nothing to do. Let us consider two other cases.
At first, take prefixesw1 =xylz0 and w2 =rslt0, where z0 4z,t0 4t and |z0|=|t0|. Then ((xylz0)g, (rslt0)g)∆(3.3.3)= (xg, rg)∆·(ylg, slg)·(z0g, t0g) =
(xg, rg)∆·(yg, sg)·(z0g, t0g)(3.3.3)= ((xyz0)g, (rst0)g)∆.
Then, by (3.3.1), ||w1g, w2g||=||(xyz0)g, (rst0)g||.
Now, take prefixes w1 = xyl0y0 and w2 = rsl0s0, where l0 < l, y0 4 y, s0 4 s and |y0| = |s0|.
By our hypothesis, ((xyl0y0)g,(rsl0s0)g)∆ = ((xy0)g, (rs0)g)∆. Then, by (3.3.1), ||w1g, w2g|| =
||(xy0)g,(rs0)g||. We then conclude that the set
{||w1g, w2g||:w14u, w24v and |w1|=|w2|}
is equal to
{||w1g, w2g||:w1 4xylz, w24rsltand |w1|=|w2|}, which implies||xylz, rslt||g=||u, v||g.
Now, we show ii).
If (xy)g and (rs)g are not comparable by 4, then (xylz)g6= (rslt)g for all l≥1.
Assume that these words are comparable by 4. If (xg, rg)∆ = (xg, rg)∆·(yg, sg), then, for all l ≥ 1, ((xyl)g, (rsl)g)∆ = ((xy)g,(rs)g)∆. As ug 6= vg, we conclude that ((xy)g,(rs)g)∆· (zg, tg)6= (1,1). Then, ((xyl)g, (rsl)g)∆·(zg, tg)6= (1,1), which implies (xylz)g6= (rslt)g.
Assume that (xg, rg)∆ 6= (xg, rg)∆·(yg, sg). Let us consider two cases for the lengths |yg|
and |sg|:
