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Continuity and Rational Functions

February 2018

DOI:10.48550/arXiv.1802.10555

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A word-to-word function is continuous for a class of languages

\mathcal{V}

if its inverse maps

\mathcal{V}

-languages to

\mathcal{V}

. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. Previous algebraic studies of transducers have focused on the structure of the underlying input automaton, disregarding the output. We propose a comparison of the two algebraic approaches through two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?

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arXiv:1802.10555v1 [cs.FL] 28 Feb 2018

CONTINUITY AND RATIONAL FUNCTIONS

MICHAËL CADILHAC, OLIVIER CARTON, AND CHARLES PAPERMAN

University of Oxford

e-mail address: michael@cadilhac.name

IRIF, Université Paris Diderot

e-mail address: olivier.carton@irif.fr

Université de Lille

e-mail address: charles.paperman@gmail.com

Abstract. A word-to-word function is continuous for a class of languages Vif its inverse

maps V-languages to V. This notion provides a basis for an algebraic study of transducers,

and was integral to the characterization of the sequential transducers computable in some

circuit complexity classes.

Here, we report on the decidability of continuity for functional transducers and some

standard classes of regular languages. Previous algebraic studies of transducers have fo-

cused on the structure of the underlying input automaton, disregarding the output. We

propose a comparison of the two algebraic approaches through two questions: When are

the automaton structure and the continuity properties related, and when does continuity

propagate to superclasses?

1. Introduction

The algebraic theory of regular languages is tightly interwoven with fundamental questions

about the computing power of Boolean circuits and logics. The most famous of these braids

revolves around A, the class of aperiodic or counter-free languages. Not only is it expressed

using the logic FO[<], but it can be seen as the basic building block of AC0, the class of

languages recognized by circuit families of polynomial size and constant depth, this class

being in turn expressed by the logic FO[arb](see [25] for a lovely account). This pervasive

interaction naturally prompts to lift this study to the functional level, hence to rational

functions. This was started in [5], where it was shown that a subsequential (i.e., input-

deterministic) transducer computes an AC0function iﬀ it preserves the regular languages of

AC0by inverse image. Buoyed by this clean, semantic characterization, we wish to further

investigate this latter property for diﬀerent classes: say that a function f:A∗→B∗is

V-continuous, for a class of languages V, if for every language L⊆B∗of V, the language

f−1(L)is also a language of V. Our main focus will be on deciding V-continuity for rational

functions; before listing our main results, we emphasize two additional motivations.

Key words and phrases: Transducers, rational functions, language varieties, continuity.

Extended version of the paper with the same title appearing in Proceedings ICALP’17.

LOGICAL METHODS

IN COMPUTER SCIENCE DOI:10.2168/LMCS-???

M. Cadilhac, O. Carton , and C. Paperman

Creative Commons

2 M. CADILHAC, O. CARTON, AND C. PAPERMAN

First, there has been some historical progression towards this goal. Noting, in [14], that

inverse rational functions provide a uniform and compelling view of a wealth of natural opera-

tions on regular languages, Pin and Sakarovitch initiated in [15] a study of regular-continuous

functions. It was already known at the time, by a result of Choﬀrut (see [3, Theorem 2.7]),

that regular-continuity together with some uniform continuity property characterize func-

tions computed by subsequential transducers. This characterization was instrumental in

the study of Reutenauer and Schützenberger [20], who already noticed the peculiar link

between uniform continuity for some distances on words and continuity for certain classes

of languages. This link was tightened by Pin and Silva [16] who formalized a topological

approach and generalized it to rational relations. More recently [17], the same authors made

precise the link unveiled by Reutenauer and Schützenberger, and developed a fascinating

and robust framework in which language continuity has a topological interpretation (see the

beginning of Section 3, as we build upon this theory). Pin and Silva [18] notably proposed

thereafter a study of functions for which continuity for a class is propagated to subclasses. In

addition, Daviaud et al. [7, 6] recently explored continuity notions in the spirit of Choﬀrut’s

characterization to study weighted automata and cost-register automata.

Second, the interweaving between languages, circuits, and logic that was alluded to

previously can in fact be formally stated (see again [25, 26]). A central property towards

this formalization is the correspondence between “cascade products” of automata, stacking of

circuits, and nesting of formulas, respectively. Strikingly, these operations can all be seen as

inverse rational functions [26]. These operations are intrinsic in the construction of complex

objects: languages, circuits, and formulas are often given as a sequence of simple objects

to be composed (see, e.g., [24, Section 5.5]). We remark that a suﬃcient condition for the

result of the composition to be in some given class (of languages, circuits, or logic formulas),

is that each rational function be continuous for that class. Hence deciding continuity allows

to give a suﬃcient condition for this membership question without computing the result of

the composition, which is subject to combinatorial blowup.

Here, we report on three questions, the ﬁrst two relating continuity to the main other al-

gebraic approach to transducers, while allowing a more gentle introduction to the evaluation

of proﬁnite words by transducers:

•When is the transducer structure (i.e., its so-called transition monoid) impacting its con-

tinuity? The results of Reutenauer and Schützenberger [20] can indeed be seen as the

starting point of two distinct algebraic theories for rational functions; on the one hand,

the study of continuity, and on the other the study of the transition monoid of the trans-

ducer (by disregarding the output). This latter endeavor was carried by [8]. Loosely

speaking, we show, in Section 4.1:

Theorem 1.1. Let Vbe a variety of languages among J,R,L,DA,A,COM,AB,Gnil,

Gsol, or G.

–The statement “Any rational function structurally in Vis continuous for V” holds for

V ∈ {A,Gsol,G} and does not otherwise;

–The statement “Any rational function continuous for Vis structurally in V” holds for

V ∈ {Gnil,Gsol,G} and does not otherwise.

•What is the impact of variety inclusion on the inclusion of the related classes of continuous

rational functions? When the focus is solely on the structure of the transducer, there is

a natural propagation to superclasses; when is it the case for continuity? We show, in

Section 4.2:

CONTINUITY AND RATIONAL FUNCTIONS 3

Theorem 1.2. Let Vand Wbe two diﬀerent varieties of languages among J,R,L,DA,

A,COM,AB,Gnil ,Gsol, or G. The statement “all rational functions continuous for Vare

continuous for W” holds only when one of these properties is veriﬁed:

–V,W ∈ {Gnil,Gsol ,G} and V ⊆ W;

–V=ABand W=COM;

–V=DA and W=A.

•When is V-continuity decidable for rational functions? We show, in Section 5:

Theorem 1.3. Let Vbe a variety of languages among J,R,L,DA,A,COM,AB,Gsol, or

G. It is decidable, given an unambiguous rational transducer, whether it realizes a function

continuous for V.

This constitutes our main contribution; note that the case Gnil is left open.

Contents

1. Introduction 1

2. Preliminaries 3

3. Continuity: The proﬁnite approach 7

3.1. The Preservation Lemma: Continuity is preserving equations 7

3.2. The proﬁnite extension of rational functions 9

3.3. The Syncing Lemma: Preservation Lemma applied to transducers 9

3.4. A proﬁnite toolbox for the aperiodic setting 10

4. Intermezzos 12

4.1. Transducer structure and continuity 12

4.2. Variety inclusion and inclusion of classes of continuous functions 14

5. Deciding continuity for transducers 16

5.1. Deciding continuity for group varieties 16

5.2. Deciding continuity for aperiodic varieties 19

5.3. Deciding Com- and Ab-continuity 26

6. Discussion 27

References 27

2. Preliminaries

We assume some familiarity with the theory of automata and transducers, and concepts

related to metric spaces (see, e.g., [3, 13] for presentations pertaining to our topic). Apart

from these prerequisites, for which the notation is ﬁrst settled, the presentation is self-

contained.

We will use Aand Bfor alphabets, and A∗for words over A, with 1the empty word.

For each word u, there is a smallest v, called the primitive root of u, such that u=vcfor

some c; if c= 1, then uis itself primitive. We write |u|for the length of a word u∈A∗and

alph(u)for the set of letters that appear in u. For a word u∈A∗and a language L⊆A∗,

we write u−1Lfor {v|u·v∈L}, and symmetrically for Lu−1, these two operations being

called the left and right quotients of Lby u, respectively. We naturally extend concatenation

4 M. CADILHAC, O. CARTON, AND C. PAPERMAN

and quotients to relations, in a component-wise fashion, e.g., for R⊆A∗×A∗and a pair

ρ∈A∗×A∗, we may use ρ−1Rand Rρ−1. We write Lcfor the complement of L. A variety is

a mapping Vwhich associates with each alphabet Aa set V(A∗)of regular languages closed

under the Boolean operations and quotient, and such that for any morphism h:A∗→B∗

and any L∈ V(B∗), it holds that h−1(L)∈ V(A∗). Reg is the variety that maps every

alphabet Ato the set Reg(A∗)of regular languages over A. Given two languages K, L ⊆A∗,

we say that they are V-separable if there is a S∈ V(A∗)such that K⊆Sand L∩S=∅.

Transducers. A transducer τis a 9-tuple (Q, A, B, δ, I , F, λ, µ, ρ)where (Q, A, δ, I, F )forms

an automaton (i.e., Qis a state set, Aan input alphabet, δ⊆Q×A×Qa transition set,

I⊆Qa set of initial states, and F⊆Qa set of ﬁnal states), and additionally, Bis an

output alphabet and λ:I→B∗, µ :δ→B∗, ρ:F→B∗are the output functions. We write

τq,q′for τwith I:= {q}and F:= {q′}, adjusting λand ρto output 1if they were undeﬁned

on these states. Similarly, τq,•is τwith I:= {q}and Funchanged, and symmetrically

for τ•,q. For q∈Qand u∈A∗, we write q.u for the set of states reached from qby reading

u. We assume that all the transducers and automata under study have no useless state, that

is, that all states appear in some accepting path.

With w∈A∗, let t1t2···t|w|∈δ∗be an accepting path for w, starting in a state q∈I

and ending in some q′∈F. The output of this path is λ(q)µ(t1)µ(t2)···µ(tn)ρ(q′), and we

write τ(w)for the set of outputs of such paths. We use τfor both the transducer and its

associated partial function from A∗to subsets of B∗. Relations of the form {(u, v)|v∈τ(u)}

are called rational relations.

The transducer τis unambiguous if there is at most one accepting path for each word.

In that case τq,q′is also an unambiguous transducer for any states q, q′. When τis unam-

biguous, it realizes a word-to-word function: the set of functions computed by unambiguous

transducers is the set of rational functions. Further restricting, if the underlying automaton

is deterministic, we say that τis subsequential. If τis a ﬁnite union of subsequential rational

functions of disjoint domains, we say that τis plurisubsequential.

Word distances, proﬁnite words. For a variety Vof regular languages, we deﬁne a distance

between words for which, intuitively, two words are close if it is hard to separate them

with Vlanguages. Deﬁne dV(u, v), for words u, v ∈A∗, to be 2−rwhere ris the size of

the smallest automaton that recognizes a language of V(A∗)that separates {u}from {v};

if no such language exists, then dV(u, v) = 0. It can be shown that this distance is a

pseudo-ultrametric [13, Section VII.2]; we make only implicit and innocuous use of this fact.

The complete metric space that is the completion of (A∗, dReg)is denoted c

A∗and is called

the free proﬁnite monoid, its elements being the proﬁnite words, and the concatenation being

naturally extended. By deﬁnition, if (un)n>0is a Cauchy sequence, it should hold that for

any regular language L, there is a Nsuch that either all unwith n > N belong to L, or

none does. For any x∈A∗, deﬁne the proﬁnite word xω= lim xn!, and more generally,

xω−c= lim xn!−c. That (xn!)n>0is a Cauchy sequence is a starting point of the proﬁnite

theory [13, Proposition VI.2.10]; it is also easily checked that xc×ω= lim xc×n!is equal to

xωfor any integer c≥1. Given a language L⊆A∗, we write L⊆c

A∗for its closure, and

we note that if Lis regular, Lc=Lcand for L′regular, L∪L′=L∪L′, and similarly for

intersection (see [13, Theorem VI.3.15]).

CONTINUITY AND RATIONAL FUNCTIONS 5

Equations. For u, v ∈c

A∗, a language L⊆A∗satisﬁes the (proﬁnite) equation u=vif for

any words s, t ∈A∗,[s·u·t∈L⇔s·v·t∈L]. Similarly, a class of languages satisﬁes an

equation if all the languages of the class satisfy it. For a variety V, we write u=Vv, and

say that uis equal to vin V, if V(A∗)satisﬁes u=v. For a partial function f,f(u) =Vf(v)

means that either both f(u)and f(v)are undeﬁned, or they are both deﬁned and equal in

Given a set Eof equations over c

A∗, the class of languages deﬁned by Eis the class

of languages over A∗that satisfy all the equations of E. Reiterman’s theorem shows in

particular that for any variety Vand any alphabet A,V(A∗)is deﬁned by a set of equations

(the precise form of which being studied in [9]).

More on varieties. Borrowing from Almeida and Costa [2], we say that a variety Vis su-

percancellative when for any alphabet A, any u, v ∈c

A∗and x, y ∈A, if u·x=Vv·yor

x·u=Vy·v, then u=Vvand x=y. This implies in particular that for any word w∈A∗,

both w·A∗and A∗·ware in V(A∗). We further say that a variety Vseparates words if for

any s, t ∈A∗,{s}and {t}are V-separable.

Our main applications revolve around some classical varieties, that we deﬁne over any

possible alphabet Aas follows, where x, y range over all of A∗, and a, b over A:

• J , def. by (xy)ω·x=y·(xy)ω= (xy)ω

• R, def. by (xy)ω·x= (xy)ω

• L, def. by y·(xy)ω= (xy)ω

• DA, def. by xω·z·xω=xωfor all z∈alph(x)∗

• A, def. by xω+1 =xω

• COM, def. by ab =ba

• AB, def. by ab =ba and aω= 1

• Gnil, the languages rec. by nilpotent groups

• Gsol, the languages rec. by solvable groups

• G, the languages rec. by groups

The varieties included in Aare called aperiodic varieties and those in Gare called group

varieties. Precise deﬁnitions, in particular for the group varieties, can be found in [25, 19];

we simply note that in group varieties, xωequals 1for all x∈A∗. All these varieties except

for ABand COM separate words, and only DA and Aare supercancellative. They verify:

J=R ∩ L

(

DA A AB=G ∩ COM (Gnil (Gsol (G

COM

(

On transducers and proﬁnite words. For a proﬁnite word uand a state qof an unambiguous

transducer τ, the set q.u is well deﬁned; indeed, with u= lim un, the set q.unis eventually

constant, as otherwise for some state q′, the domain of τq,q′would be a regular language

that separates inﬁnitely many un’s.

A transducer τ:A∗→B∗is a V-transducer,1for a variety V, if for some set of equa-

tions Edeﬁning V(A∗), for all (u=v)∈Eand all states qof τ, it holds that q.u =q.v. A

rational function is V-realizable if it is realizable by a V-transducer.

1The usual deﬁnition of V-transducer is based on the so-called transition monoid of τ, see, e.g., [20]; the

deﬁnition here is easily seen to be equivalent by [1, Lemma 3.2] and [5, Lemma 1].

6 M. CADILHAC, O. CARTON, AND C. PAPERMAN

Continuity. For a variety V, a function f:A∗→B∗is V-continuous2iﬀ for any L∈ V(B∗),

f−1(L)∈ V(A∗). We mostly restrict our attention to rational functions, and their being

computed by transducers implies that they are countably many. We note that much more

Reg-continuous functions exist, in particular uncomputable ones:

Proposition 2.1. There are uncountably many Reg-continuous functions.

Proof. Consider a strictly increasing function g:N→N. Deﬁne f:{a}∗→ {a}∗by

f(an) = ag(n)!. Recall that any regular language over a unary alphabet is a ﬁnite union of lan-

guages of the form ai(aj)∗. Moreover, we have that f−1(ai(aj)∗)is ﬁnite when i6≡ 0 mod j,

and coﬁnite otherwise, thus fis Reg-continuous. There are however uncountably many

functions g, hence uncountably many Reg-continuous functions f.

Continuity is a formal notion of “functions being compatible with a class of languages.”

An equally valid notion could be to consider classes of functions that contain the character-

istic functions of the languages, and closed under composition; it turns out that the largest

such class coincides with the class of continuous functions. Indeed, writing χL:A∗→ {0,1}

for the characteristic function of a language L⊆A∗:

Proposition 2.2. Let Vbe a variety such that {1} ∈ V({0,1}∗). Let Fbe the largest class

of functions such that:

(1) For any alphabet A,F ∩ {0,1}A∗={χL|L∈ V(A∗)};

(2) Fis closed under composition.

It holds that Fis well deﬁned and that it coincides with the class of V-continuous functions.

Proof. We say that a class of functions is good if it veriﬁes properties (1) and (2). We show

that the V-continuous functions form a good class, and that any good class is included in the

V-continuous functions. This implies that there is a largest good class, and that it coincides

with the class of V-continuous functions, as claimed.

(Continuous functions form a good class.) Clearly, the class of V-continuous functions is

closed under composition. Now consider a V-continuous function f:A∗→ {0,1}. By

continuity, L=f−1({1})is in V(A∗), since by hypothesis {1} ∈ V({0,1}∗). Hence f=χL

for some L∈ V(A∗), concluding this step.

(Functions in good classes are continuous.) Let f:A∗→B∗be in a good class, and let

L∈ V(B∗); we ought to show that f−1(L)is in V(A∗). It holds that:

f−1(L) = f−1(χ−1

L(1))

= (χL◦f)−1(1)

=g−1(1) .(with g=χL◦f)

Note that χLis by hypothesis in the good class, and it being closed under composition, g

also belongs to the good class. Since g∈ {0,1}A∗, there is a L′∈ V(A∗)such that g=χL′.

This implies that f−1(L) = L′, and it thus belongs to V(A∗).

2A note on terminology: There has been some ﬂuctuation on the use of the term “continuous” in the

literature, mostly when a possible incompatibility arises with topology. In [18], the authors use the term

“preserving” in the more general context of functions from monoids to monoids. In our study, we focus

on word to word functions, in which the natural topological context provides a solid basis for the use of

“continuous,” as used in [16, 5].

CONTINUITY AND RATIONAL FUNCTIONS 7

3. Continuity: The profinite approach

We build upon the work of Pin and Silva [16] and develop tools specialized to rational

functions. In Section 3.1, we present a lemma asserting the equivalence between V-continuity

and the “preservation” of the deﬁning equations for V. In the sections thereafter, we specialize

this approach to rational functions. As noted in [16], it often occurs that results about

rational functions can be readily applied to the larger class of Reg-continuous functions;

here, this is in particular the case for the Preservation Lemma of Section 3.1.

Our main appeal to a classical notion of continuity is given by the:

Theorem 3.1 ([17, Theorem 4.1]).Let f:A∗→B∗. It holds that fis V-continuous iﬀ f

is uniformly continuous for the distance dV.

Consequently, if fis Reg-continuous then it has a unique extension to the free proﬁnite

monoids, written b

f:c

A∗→c

B∗. The salient property of this mapping is that it is continuous

in the topological sense (see, e.g., [13]). For our speciﬁc needs, we simply mention that it

implies that for any regular language L, we have that b

f−1(L)is closed (that is, it is the

closure of some set).

3.1. The Preservation Lemma: Continuity is preserving equations. The Preserva-

tion Lemma gives us a key characterization in our study: it ties together continuity and

some notion of preservation of equations. This can be seen as a generalization for functions

of the notion of equation satisfaction for languages. We will need the following technical

lemma that extends [13, Proposition VI.3.17] from morphisms to arbitrary Reg-continuous

functions; interestingly, this relies on a quite diﬀerent proof.

Lemma 3.2. Let f:A∗→B∗be a Reg-continuous function and La regular language. It

holds that b

f−1(L) = f−1(L).

Proof. First note that f−1(L)⊆b

f−1(L), and that the right-hand side of this inclusion is

closed. Hence f−1(L)⊆b

f−1(L).

For the converse inclusion, ﬁrst write D=f−1(B∗), a regular language by hypothesis.

We have that b

f−1(L) = ( b

f−1(Lc))c∩D, and similarly, f−1(L) = (f−1(Lc))c∩D. This latter

equality implies that f−1(L) = f−1(Lc)c∩D, since f−1(Lc)and Dare regular.

Hence the inclusion to be shown, that is, b

f−1(L)⊆f−1(L), is equivalent to:

f−1(Lc))c∩D⊆f−1(Lc)c∩D ,

or equivalently,

f−1(Lc)∪Dc⊆b

f−1(Lc)∪Dc.

The inclusion to be shown is thus implied by f−1(Lc)⊆b

f−1(Lc), that is, since Lis

regular, by f−1(Lc)⊆b

f−1(Lc). As in the proof of the converse inclusion, the right-hand

side being closed, this inclusion holds.

8 M. CADILHAC, O. CARTON, AND C. PAPERMAN

Lemma 3.3 (Preservation Lemma).Let f:A∗→B∗be a Reg-continuous function and E

a set of equations that deﬁnes V(A∗). The function fis V-continuous iﬀ for all (u=v)∈E

and words s, t ∈A∗,b

f(s·u·t) =Vb

f(s·v·t).

Proof. (Only if.) Suppose fis V-continuous. Let u, v ∈c

A∗such that u=Vv, and s, t ∈A∗.

Since by V-continuity f−1(B∗)∈ V(A∗), either both s·u·tand s·v·tbelong to the closure of

this language, or they both do not. The latter case readily yields the result, hence suppose

we are in the former case.

By deﬁnition, u= lim unand v= lim vnfor some Cauchy sequences of words (un)n>0

and (vn)n>0. Since s·u·t=Vs·v·t, the hypothesis yields that dV(s·un·t, s ·vn·t)tends

to 0. By Theorem 3.1, fis uniformly continuous for dV, hence dV(f(s·un·t), f (s·vn·t))

also tends to 0(note that both f(s·un·t)and f(s·vn·t)are deﬁned for all nbig enough).

This shows that b

f(s·u·t) =Vb

f(s·v·t).

(If.) Suppose that fpreserves the equations of Eas in the statement. Let L∈ V(B∗), we

wish to verify that L′=f−1(L)∈ V(A∗), or equivalently by deﬁnition, that L′satisﬁes all

the equations of E. Let (u=v)∈Ebe one such equation, and s, t ∈A∗; we must show

that s·u·t∈L′⇔s·v·t∈L′.

Suppose s·u·t∈L′. Since fis Reg-continuous, it holds that b

f(s·u·t)∈L(observe that

f(s·u·t)is indeed deﬁned). By hypothesis, b

f(s·u·t) =Vb

f(s·v·t); now since L∈ V(B∗),

it must hold that b

f(s·v·t)∈L. Taking the inverse image of b

fon both sides, it thus holds

that s·v·t∈b

f−1(L), and Lemma 3.2 then shows that s·v·t∈L′. As the argument works

both ways, this shows that s·u·t∈L′⇔s·v·t∈L′, concluding the proof.

Continuity can be seen as preserving membership to V(by inverse image); this is where

the nomenclature “V-preserving function” of [18] stems from. Strikingly, this could also be

worded as preserving nonmembership to V:

Proposition 3.4. A Reg-continuous total 3function f:A∗→B∗is V-continuous iﬀ for all

L⊆A∗that do not belong to V(A∗),f(L)and f(Lc)are not V-separable.

Proof. We rely on a characterization due to Almeida [1, Lemma 3.2]: two languages Kand

Lare V-separable iﬀ for all u∈K, v ∈L, it holds that u6=Vv.

(Only if.) Suppose fis V-continuous, and let L⊆A∗be a language outside V(A∗). There

must be two proﬁnite words u, v ∈c

A∗such that u=Vv,u∈Land v∈Lc. By V-continuity

and the Preservation Lemma, b

f(u) =Vb

f(v), and moreover, b

f(u)∈f(L)and b

f(v)∈f(Lc).

The characterization above thus implies that f(L)and f(Lc)are not V-separable.

(If.) Let L∈ V(B∗), we show that f−1(L)∈ V(A∗). Suppose for a contradiction that

this is not the case. Then f(f−1(L)) and f(f−1(L)c)are not V-separable. The former is

included in L, and the latter equal to f(f−1(Lc)) ⊆Lc. In particular, this means that L

and Lcare not V-separable, which is a contradiction since L∈ V (B∗).

3In all the varieties we are interested in, one can easily modify any partial function into a total function

while preserving its continuity properties.

CONTINUITY AND RATIONAL FUNCTIONS 9

3.2. The proﬁnite extension of rational functions. The Preservation Lemma already

hints at our intention to see transducers as computing functions from and to the free proﬁnite

monoids. Naturally, if τis a rational function, its being Reg-continuous allows us to do so

(by Theorem 3.1). For u= lim una proﬁnite word, we will write τ(u)for bτ(u), i.e., the

limit lim τ(un), which exists by continuity. In this section, we develop a slightly more

combinatorial approach to this evaluation, and address two classes of proﬁnite words: those

expressed as s·u·tfor s, t words and ua proﬁnite word, and those expressed as xωfor xa

word.

Recall that for a transducer state qand a proﬁnite word u,q.u is well deﬁned. As a

consequence, if sand tare words and τis unambiguous, then there is at most one initial

state q0, one q∈q0.s and one q′∈q.u such that q′.t is ﬁnal, and these states exist iﬀ

τ(s·u·f)is deﬁned. Thus:

Lemma 3.5. Let τbe an unambiguous transducer from A∗to B∗,s, t ∈A∗and u∈c

A∗.

Suppose τ(s·u·f)is deﬁned, and let q0, q, q′be the unique states such that q0is initial,

q∈q0.s,q′∈q.u, and q′.t is ﬁnal. The following holds:

τ(s·u·t) = τ•,q(s)·τq,q′(u)·τq′,•(t).

Let us now turn to the evaluation of ω-terms:

Lemma 3.6. Let τbe an unambiguous transducer from A∗to B∗and x∈A∗. If τ(xω)is

deﬁned, then there are words s, y, t ∈B∗such that:

τ(xω) = s·yω−1·t .

Proof. Consider a large value n; we study the behavior of xn!on τ. There is an initial state

q0, a state q, and a ﬁnal state q1such that xn!is accepted by a path going from q0to q

reading xi, from qto qreading xkwith k < n, and from qto q1reading xj. Thus the

accepting path for any word of the form xm!, m > n is similar to the one for xn!: from q0

to q, looping (m!−n!)/k + 1 times on q, and then from qto q1. Let thus s=τq0,q(xi),

z=τq,q (xk), and t=τq,q1(xj). It then holds that τ((xk)m!) = s·zm!−(n!/k)+1 ·t. Letting

c=n!/k −1, this shows that τ((xk)ω) = s·zω−c·t. Now on the one hand, (xk)ω=xω, and

on the other hand, we similarly have that zω−c=yω−1by letting y=zc. We thus obtain

that τ(xω) = s·yω−1·t.

These constitute our main ways to eﬀectively evaluate the image of proﬁnite words

through transducers. Their use being quite ubiquitous in our study, we will rarely refer to

these lemmas nominally.

3.3. The Syncing Lemma: Preservation Lemma applied to transducers. We apply

the Preservation Lemma on transducers and deduce a slightly more combinatorial character-

ization of transducers describing continuous functions. This does not provide an immediate

decidable criterion, but our decidability results will often rely on it. The goal of the forth-

coming lemma is to decouple, when evaluating s·u·t(with the notations of the Preservation

Lemma), the behavior of the upart and that of the s, t part. This latter part will be tested

against an equalizer set:

Deﬁnition 3.7 (Equalizer set).Let u, v ∈c

A∗. The equalizer set of uand vin Vis:

EquV(u, v) = {(s, s′, t, t′)∈(A∗)4|s·u·t=Vs′·v·t′}.

10 M. CADILHAC, O. CARTON, AND C. PAPERMAN

Remark 3.8. The complexity of equalizer sets can be surprisingly high. For instance,

letting Vbe the class of languages deﬁned by {x2=x3|x∈A∗}, there is a proﬁnite word u

for which EquV(u, u)is undecidable. On the other hand, equalizer sets quickly become less

complex for common varieties; for instance, Lemma 3.12 will provide a simple form for the

equalizer sets of aperiodic supercancellative varieties.

Deﬁnition 3.9 (Input synchronization).Let R, S ⊆A∗×B∗. The input synchronization

of Rand Sis deﬁned as the relation over B∗×B∗obtained by synchronizing the ﬁrst

component of Rand S:

R ⊲⊳ S ={(u, v)|(∃s)[(s, u)∈R∧(s, v)∈S]}=S◦R−1).

Naturally, the input synchronization of two rational functions is a rational relation.

Lemma 3.10 (Syncing Lemma).Let τbe an unambiguous transducer from A∗to B∗and E

a set of equations that deﬁnes V(A∗). The function τis V-continuous iﬀ:

(1) τ−1(B∗)∈ V(A∗), and

(2) For any (u=v)∈E, any states p, q , any p′∈p.u, and any q′∈q.v, and letting

u′=τp,p′(u)and v′=τq,q′(v):

(τ•,p ⊲⊳ τ•,q)×(τp′,•⊲⊳ τq′,•)⊆EquV(u′, v′).

Proof. We rely on the Preservation Lemma, since τis Reg-continuous.

(Only if.) Suppose that τis V-continuous, and apply the Preservation Lemma. This shows

the ﬁrst point to be proven. For the second, we use the notation of the statement. Let

(s, s′, t, t′)∈(τ•,p ⊲⊳ τ•,q)×(τp′,•⊲⊳ τq′,•). This implies that there are words x, y ∈A∗such

that:

•s=τ•,p(x), s′=τ•,q (x);

•t=τp′,•(y), t′=τq′,•(y).

By Lemma 3.5, we have that τ(x·u·y) = s·u′·tand τ(x·v·y) = s′·v′·t′. The Preservation

Lemma then asserts that s·u′·t=Vs′·v′·t′, showing that (s, s′, t, t′)∈EquV(u′, v′).

(If.) Let (u=v)∈Eand x, y ∈A∗. We must show that τ(x·u·y) =Vτ(x·v·y). Since

τ−1(B∗)∈ V(A∗), either τis deﬁned on both x·u·yand x·v·y, or on neither; in this latter

case, the equality is veriﬁed by deﬁnition. We thus suppose that both values are deﬁned.

This implies that there are states p, q, p′, q′as in the statement, and using the same notation,

letting s, s′, t, t′just as above, the hypothesis yields that s·u′·t=Vs′·v′·t′, showing the

claim.

3.4. A proﬁnite toolbox for the aperiodic setting. In this section, we provide a few

lemmas pertaining to our study of aperiodic continuity. We show that the equalizer sets of

aperiodic supercancellative varieties are well behaved. Intuitively, the larger the varieties

are, the more their nonempty equalizer sets will be similar to the identity. For instance, if

s·xω=Axω, for words sand x, it should hold that sand xhave the same primitive root.

We ﬁrst note the following easy fact that will only be used in this section; it is reminiscent

of the notion of equidivisibility, studied in the proﬁnite context by Almeida and Costa [2].

Lemma 3.11. Let u, v be proﬁnite words over an alphabet Aand Vbe a supercancellative

variety. Suppose that there are s, t ∈A∗such that u·t=Vs·v, then there is a w∈c

A∗such

that u=Vs·wand v=Vw·t. If moreover u=vand Vis aperiodic, then u=Vs·u·t.

CONTINUITY AND RATIONAL FUNCTIONS 11

Proof. Let u= lim un. From u·t=Vs·v, and the fact that s·A∗∈ V(A∗)by supercan-

cellativity, we obtain that for nlarge enough, un·t∈s·A∗. Since uis nonﬁnite, |un|>|s|

for nlarge enough, in which case un=s·wnfor some sequence (wn)n>0. Let w∈c

A∗be

a limit point of this sequence, that exists by compacity (this is an essential property of the

free proﬁnite monoid, see, e.g., [13, Theorem VI.2.5]). It holds that u=s·w. Replacing u

by this value in the equation of the hypothesis, we thus have that s·w·t=Vs·v, and since

Vis supercancellative, that v=Vw·t.

For the last point, with u=v, we iterate the previous construction on w, since in that

case, u=Vw·t=Vs·w. This provides a sequence w=w1, w2, w3,... such that u=V

sn·wn=Vwn·tn. Taking a limit point xof (wn)n>0, it thus holds that u=Vsω·x=Vx·tω,

showing, by aperiodicity, that u=Vs·u=Vu·t.

Lemma 3.12. Let u, v be proﬁnite words over an alphabet Aand Vbe an aperiodic super-

cancellative variety. Suppose EquV(u, v)is nonempty. There are words x, y ∈A∗and two

pairs ρ1, ρ2∈(A∗)2such that:

EquV(u, v) = Id ·(x∗, x∗)ρ−1

1×ρ−1

2(y∗, y∗)·Id.

Proof. Let us ﬁrst establish the property for u=v. Assume that there are nonempty

primitive words x, y such that x·u·y=Vu; we show the statement of the lemma with these

xand y, and ρ1=ρ2= (1,1). Note that xω·u·yω=Vu, hence, since xω+1 =xωand

similarly for y, it holds that x·u=Vu·y=Vu. This and the fact that Vis supercancellative

show the right-to-left inclusion.

For the left-to-right inclusion, let s, s′, t, t′be such that s·u·t=Vs′·u·t′. Since V

is supercancellative, this implies that the equation also holds if common preﬁxes of sand

s′and common suﬃxes of tand t′are removed. We may thus assume that we are in two

possible situations, by symmetry:

(1) Suppose s′=t′= 1, that is, s·u·t=Vu, and that s, t are nonempty. By the same

token as above, this shows that s·u=Vu·t=Vu. In particular, this implies that:

s|x|·u·t|y|=Vx|s|·u·y|t|,

which implies, since Vis supercancellative, that s|x|=x|s|and t|y|=y|t|. As xand y

are primitive, this shows that s∈x∗and t∈y∗. (Note that this holds even if one of s

or tis empty.)

(2) Suppose s=t′= 1, that is, u·t=Vs′·u, and that s′, t are nonempty. By Lemma 3.11,

we have that u=Vs′·u·t, and we can appeal to the previous situation, showing that

s′∈x∗and t∈y∗.

(The cases where one of s, t is empty, in the ﬁrst point, or one of s′, t is empty, in the second,

are treated similarly. Note that it is not possible for both sand s′to be nonempty, since

that would imply that they start with diﬀerent letters, falsifying the assumed equation by

supercancellativity.)

We assumed that the x, y existed, we ought to show the other cases verify the claim.

The two situations above show that if EquV(u, u)is nonempty, then such x, y exist, although

without the guarantee that they be nonempty. Now if x·u=Vuand there are no nonempty

ysuch that x·u·y=Vu, this implies that there are no nonempty ysuch that u·y=Vu.

Consequently, in the above case, t=t′= 1, and the analysis stands. This concludes the

proof for the case u=v.

12 M. CADILHAC, O. CARTON, AND C. PAPERMAN

We will reduce the case u6=vto this one. Indeed, suppose that s·u·v=s′·v·t′. Again,

by stripping away common preﬁxes and suﬃxes, we are faced with two cases:

(1) Suppose s′=t′= 1, that is, s·u·t=Vv. It holds that EquV(u, v) = EquV(u, s ·u·t),

hence (m, m′, n, n′)∈EquV(u, v)iﬀ (m, m′·s, n, t ·n′)∈EquV(u, u), and the result

follows.

(2) Suppose s=t′= 1, that is, u·t=Vs′·v. By Lemma 3.11, there is a proﬁnite

word wsuch that u=Vs′·wand v=Vw·t, hence (m, m′, n, n′)∈EquV(u, v)iﬀ

(m·s′, m′, n, t ·n′)∈EquV(w, w), concluding the proof.

Lemma 3.13. Let x, y be words. For every aperiodic supercancellative variety V, it holds

that EquV(xω, yω) = EquA(xω, yω).

Proof. The inclusion from right to left is clear, since all equations true in Ahold in V.

In the other direction, let us write u=x|y|and v=y|x|; we have that uω=xωand

vω=yω. Suppose s·uωt=Vs′·vω·t′. In particular, since Vis supercancellative, this

means that s·unis a preﬁx of s′·vn, or vice-versa, depending on whether |s|>|s′|or the

opposite. This implies that u·up=vs·vfor some preﬁx upof uand suﬃx vsof v. Hence

(by, e.g., [11, Proposition 1.3.4]) uand vare conjugate. Their respective primitive roots are

thus conjugate (by [11, Proposition 1.3.3]); writing z·z′and z′·zfor them, we have that

uω= (z·z′)ωand vω= (z′·z)ω.

Thus the equation above reads: s·(z·z′)ω·t=Vs′·(z′·z)ω·t′. As in the proof of

Lemma 3.12, removing the common preﬁxes and suﬃxes (which we can do both in Vand

A), we are left with two possibilities:

•Suppose s′=t′= 1, that is, s·(z·z′)ω·t=V(z′·z)ω. The same argument as in Lemma 3.12

shows that s∈z′·(z·z′)∗and t∈(z·z′)∗·z, and hence the equation holds in Atoo;

•Suppose s=t′= 1, that is, (z·z′)ω·t=Vs′·(z′·z)ω. Similarly, as Vis supercancellative

and aperiodic, this shows that s′∈z·(z′·z)∗and t∈(z·z′)∗·z, and the equation holds

in Atoo, concluding the proof.

Remark 3.14. For two aperiodic supercancellative varieties Vand W, we could further

show that if both EquV(u, v)and EquW(u, v)are nonempty, then they are equal, for any

proﬁnite words u, v. It may however happen that one equalizer set is empty while the other

is not; for instance, with u= (ab)ωand v= (ab)ω·a·(ab)ω, the equalizer set of uand vin

DA is nonempty, while it is empty in A.

4. Intermezzos

We present a few facts of independent interest on continuous rational functions. Through

this, we develop a few examples, showing in particular how the Preservation and Syncing

Lemmas can be used to show (non)continuity. In a ﬁrst part, we study when the structure of

the transducer is relevant to continuity, and in a second, when the (non)inclusion of variety

relates to (non)inclusion of the class of continuous rational functions.

4.1. Transducer structure and continuity. As noted by Reutenauer and Schützenberger [20,

p. 231], there exist numerous natural varieties Vfor which any V-realizable rational function

is V-continuous. Indeed:

CONTINUITY AND RATIONAL FUNCTIONS 13

Proposition 4.1. Let Vbe a variety of languages closed under inverse V-realizable rational

function. Any V-realizable rational function is V-continuous. This holds in particular for

the varieties A,Gsol,and G.

Proof. This is due to a classical result of Sakarovitch [22] (see also [15]), stating, in modern

parlance, that a variety Vis closed under block product iﬀ it is closed under inverse V-re-

alizable rational functions (note that there has been some ﬂuctuation on vocabulary, since

wreath product was used at some point to mean block product). That A,Gsol,and Gare

closed under block product is folklore.

This naturally fails for all our other varieties, since they are not closed under inverse

V-realizable rational functions. For completeness, we give explicit constructions in the proof

of the:

Proposition 4.2. For V ∈ {J ,L,R,DA,AB,Gnil,COM}, there are V-realizable rational func-

tions that are not V-continuous.

Proof. We devise simple counter examples with A={a, b}.

(The J,Rand COM cases.) Recall that A∗a /∈ R(A∗)∪ COM(A∗). The minimal unam-

biguous two-state transducer τthat erases all of its input except for the last letter is a

(J ∩ COM)-transducer; indeed, aacts in the same way as band they are idempotent on the

transducer. However, τ−1(a) = A∗a.

(The Rand DA cases.) Consider the Dyck language Dover A; this is the (nonregular) lan-

guage of well-parenthesized expressions where ais the opening and bthe closing parenthesis.

Write D(k)for the Dyck language where parentheses are nested at most ktimes, for instance

D(0) = 1, D(1) = (ab)∗and D(2) = (a(ab)∗b)∗. These languages bear great importance in

algebraic language theory, as they separate each level of the dot-depth hierarchy [4]. It holds

in particular that D(1) /∈ DA(A∗).

Let τbe the rational function that removes the ﬁrst letter of each block of a’s and each

block of b’s; naturally, τis L-realizable. However, τ−1(Dk−1) = Dk, showing not only that

τis not continuous for DA, but also not continuous for any level of the dot-depth hierarchy.

(The Gnil and ABcases.) Consider the two-state transducer τwhere aloops on both states,

and a bon one state goes to the other. When ais read on the ﬁrst state, it produces a

x, while all the other productions are the identity. This is an AB-transducer. However,

τ(aba) = xba 6=baa =τ(baa), hence it is not AB-continuous by the Preservation Lemma,

since aba =ABbaa. For Gnil, let Lbe the language over {a, b, x}with a number of xcongruent

to 0modulo 3. It can be shown that τ−1(L)/∈ Gnil(A∗), intuitively since this language needs

to diﬀerentiate between those a’s that are an even number of b’s away from the beginning

of the word, and those which are not.

The converse concern, that is, whether all V-continuous rational functions are V-realiz-

able, was mentioned by Reutenauer and Schützenberger [20] for V=A.

Proposition 4.3. For V ∈ {J ,L,R,DA,A,AB,COM}, there are V-continuous rational func-

tions that are not V-realizable.

Proof. (The aperiodic cases.) Let A={a}, a unary alphabet. Consider the transducer

τthat removes every second a: its minimal transducer not being a A-transducer, it is not

14 M. CADILHAC, O. CARTON, AND C. PAPERMAN

A-realizable (this is a property of subsequential transducers [20]). However, all the unary

languages of Vare either ﬁnite or co-ﬁnite, and hence for any L∈ V(A∗),τ−1(L)is either

ﬁnite or co-ﬁnite, hence belongs to V(A∗).

(The ABand COM cases.) Over A={a, b}, deﬁne τto map words win aA∗to (ab)|w|, and

words win bA∗to (ba)|w|. Clearly, aand bcannot act commutatively on the transducer.

Now τ(ab) =COM τ(ba), and moreover τ(xω) =AB(ab)ω=AB1 = τ(1), hence τis continuous

for both ABand COM by the Preservation Lemma.

We delay the positive answers to that question, namely for Gnil,Gsol,G, to Corollary 5.6

as they constitute our main lever towards the decidability of continuity for these classes.

4.2. Variety inclusion and inclusion of classes of continuous functions. In this

section, we study the consequence of variety (non)inclusion on the inclusion of the related

classes of continuous rational functions. This is reminiscent of the notion of heredity studied

by [17], where a function is V-hereditarily continuous if it is W-continuous for each subvariety

Wof V. Variety noninclusion provides the simplest study case here:

Proposition 4.4. Let Vand Wbe two varieties. If V 6⊆ W then there are V-continuous

rational functions that are not W-continuous.

Proof. Let L∈ V(A∗)be such that L /∈ W(A∗). Deﬁne f:A∗→A∗as the identity function

with domain L. Clearly, as f−1(K) = K∩L, the function fis V-continuous. However,

f−1(A∗) = L /∈ W(A∗)and A∗∈ W(A∗), thus fis not W-continuous.

The remainder of this section focuses on a dual statement:

If V(W, are all V-continuous rational functions W-continuous?

4.2.1. The group cases. We ﬁrst focus on group varieties. Naturally, if 1. V-continuous

rational functions are V-realizable and 2. W-realizable rational functions are W-continuous,

this holds. Appealing to the forthcoming Corollary 5.6 for point 1 and Proposition 4.1 for

point 2, we then get:

Proposition 4.5. For V,W ∈ {Gnil ,Gsol ,G} with V(W, all V-continuous rational func-

tions are W-continuous. This however fails for V=ABand for any W ∈ {Gnil,Gsol,G}.

Proof. It remains to show the case V=AB. This is in fact the same example as in the proof

of Proposition 4.3, to wit, over A={a, b}, the rational function τthat maps w∈aA∗to

(ab)|w|, and words w∈bA∗to (ba)|w|. Indeed, we saw that this function is continuous for AB,

but it holds that τ(a) = ab on the one hand, and τ(bωa) = (ba)ωba =Wba, but ab 6=Wba.

The Preservation Lemma then shows that τis not continuous for W.

Proposition 4.6. All AB-continuous rational functions are COM-continuous.

Proof. Indeed, if u=ABvwith u, v words, then u=COM v, since these varieties separate the

same words. As COM is deﬁned using equations on words, this directly shows the claim by

the Preservation Lemma.

CONTINUITY AND RATIONAL FUNCTIONS 15

4.2.2. The aperiodic cases. We now turn to aperiodic varieties. For lesser expressive varieties,

the property fails:

Proposition 4.7. For V ∈ {J ,L,R} and W ∈ {L,R,DA,A} with V(W, there are

V-continuous rational functions that are not W-continuous.

Proof. Deﬁne τ:{a}∗→ {a, b}∗to be the rational function that changes every other ato

b; that is, τ(a2n) = (ab)n, and τ(a2n+1 ) = (ab)n·a. Note that naturally, over a single

letter, a·(aa)ω= (aa)ω·a=aω+1. Now τ(aω) = (ab)ωand τ(aω+1) = (ab)ω·a, and since

these two proﬁnite words are equal in Jand R, the Preservation Lemma shows that τis

continuous for both Jand R. However, these two proﬁnite words are not equal in L,DA,

and A, showing that τis continuous for none of those varieties.

The remaining case, that is, showing the existence of a J-continuous rational function

that is not R-continuous is done symmetrically, with the function mapping a2nto (ab)nand

a2n+1 to b·(ab)n.

Proposition 4.8. Any DA-continuous rational function is A-continuous.

Proof. First note that both DA and Asatisfy the hypotheses of Lemma 3.12. Consider

aDA-continuous rational function τ:A∗→B∗. By the Syncing Lemma, to show that it

is A-continuous, it is enough to show that 1. τ−1(B∗)∈ A(A∗), and 2. That some input

synchronizations of τ, based on equations of the form xω=Axω+1, belong to an equalizer

set of the form (by Lemma 3.6):

EquA(α·yω·β, α′·zω·β′) = {(s, s′, t, t′)|(s·α, s′·α′, β ·t, β′·t′)∈EquA(yω, zω)}.

Applying the Syncing Lemma on τfor the variety DA, we get that point 1 is true, since

τ−1(B∗)∈ DA(A∗). Similarly, point 2 is true since xω=xω+1 is an equation of DA, and

Lemma 3.13 implies that the equalizer set of the equation above is the same in DA and A.

However, this property does not hold beyond rational functions:

Proposition 4.9. There are nonrational functions that are continuous for both DA and Reg

but are not A-continuous.

Proof. Deﬁne f:{a}∗→ {a, b}∗by f(a2n) = (ab)nand f(a2n+1) = (ab)n·a·(ab)n. We ﬁrst

have to check that fis indeed Reg-continuous. Given a regular language L, we deﬁne a push-

down automaton over {a}∗that recognizes f−1(L); since all unary context-free languages

are regular, by Parikh’s theorem, this shows the claim. If the input is of the form a2n, then

the pushdown automaton may check that (ab)n∈L, by simulating the automaton for L. If

the input is of the form a2n+1, then the pushdown automaton can guess the middle position

of the input, and accordingly check that (ab)n·a·(ab)n∈L, again using the automaton for

L. This concludes the construction.

The function fbeing Reg-continuous, it has an extension b

fto the free proﬁnite monoids.

As in Proposition 4.7, checking that fis DA-continuous amounts to check that b

f(aω) =DA

f(aω+1). The left-hand side being (ab)ωwhile the right-hand side is (ab)ω·a·(ab)ω, this

holds. However, these two proﬁnite words are equal in DA but not in A, hence this function

is not A-continuous, again appealing to the Preservation Lemma.

16 M. CADILHAC, O. CARTON, AND C. PAPERMAN

5. Deciding continuity for transducers

5.1. Deciding continuity for group varieties. Reutenauer and Schützenberger showed

in [20] that a rational function is G-continuous iﬀ it is G-realizable. Since this is proven

eﬀectively, it leads to the decidability of G-continuity. In Proposition 4.1, we saw that the

right-to-left statement also holds for Gsol; we now show that the left-to-right statement holds

for all group varieties Vthat contain Gnil. As in [20], but with sensibly diﬀerent techniques,

we show that V-continuous transducers are plurisubsequential. The Syncing Lemma will

then imply that such transducers are V-transducers. Both properties rely on the following

normal form:

Lemma 5.1. Let τbe a transducer. An equivalent transducer τ′can be constructed by

adjoining some codeterministic automaton to τso that for any states p, q of τ′:

h(∃x, y)∅ 6= (τ′

p,•⊲⊳ τ′

q,•)⊆(x, y)·Idi⇒p=q .

Alternatively, the “dual” property can be ensured, adjoining a deterministic automaton to τ,

so that for any states p, q of τ′:

h(∃x, y)∅ 6= (τ′

•,p ⊲⊳ τ′

•,q)⊆Id ·(x, y)i⇒p=q .

Proof. We show the ﬁrst property, its dual being proved similarly. We construct the required

transducer in three steps.

(Step 1.) For any state q∈Q, write Rq={w|q.w ∈F}. Let Mq,q′be a deterministic

automaton recognizing the reverse of Rq∩Rq′(that is, the language consisting of the reverse

of each word therein), and deﬁne M′as the Cartesian product of all Mq,q′’s. Finally, let M

be the reverse of M′.

Deﬁne now τ′to be the Cartesian product of τand M, with the initial states being the

Cartesian product of that of τand M, and likewise for the ﬁnal states. The output function

of τ′is that of τ. Generalizing the notation R•to τ′and M, we have:

Fact 5.2. For any states p, p′of τ′,Rp∩Rp′6=∅ ⇒ Rp=Rp′.

Proof. Let p= (q, r)and p′= (q′, r′), and assume Rp∩Rp′6=∅. Note that Rp=Rq∩Rr

and Rp′=Rq′∩Rr′. Since their intersection is nonempty and M′is codeterministic, this

shows that r=r′. Now as Rrcontains a word of Rq∩Rq′, the state in Mcorresponding

to the automaton Mq,q′is ﬁnal in Mq,q′, showing that Rr⊆Rq∩Rq′. We thus see that

Rp=Rq∩Rr=Rr, and similarly, Rp′=Rr, showing the claim. of Fact 5.2.

(Step 2.) Write simply τfor the result of Step 1. In this step, we make sure that outputs are

produced “as soon as possible,” a process known as normalization (e.g., [12, Section 1.5.2])

that we sketch for completeness. For every state q, write πqfor the longest string such that

τq,•(A∗)⊆πq·B∗. Now deﬁne the new output function (λ′, µ′, ρ′)by letting:

µ′(1, q) = µ(1, q)·πq, µ′(q, a, q′) = π−1

qµ(q, a, q′)·πq′, µ′(q, 1) = π−1

qµ(q, 1) .

Writing τ′for τequipped with the output function µ′, it holds that for no state q, there is

a letter b∈B∗such that τq,•(A∗)⊆b·B∗.

(Step 3.) Write again τfor the result of the previous step. Naturally, τstill veriﬁes Fact 5.2.

Consider two states p, q such that there are x, y ∈B∗verifying ∅ 6= (τp,•⊲⊳ τq,•)⊆(x, y)·Id.

CONTINUITY AND RATIONAL FUNCTIONS 17

The ﬁrst part of this assumption implies that Rp∩Rq6=∅, and thus, by Fact 5.2, Rp=Rq.

In other words, τp,•and τq,•have the same domain. The second part of the assumption thus

indicates that every production of p(resp. of q) starts with x(resp. with y), and Step 2

asserts that x=y= 1. Hence τp,•and τq,•actually compute the same function. We can

thus merge them into a single state without changing the function realized. Repeating this

operation results in a transducer with the required property. of Lemma 5.1.

Lemma 5.3. Let Vbe a variety of group languages that contains Gnil. For any V-continuous

unambiguous transducer τ, the transducer obtained by applying the dual of Lemma 5.1, then

applying its ﬁrst part, is a plurisubsequential V-transducer.

Proof. Write τ′for the result of the dual part of Lemma 5.1 on τ, and τ′′ for the result of

the ﬁrst part of Lemma 5.1 on τ′. We show that τ′′ is a plurisubsequential V-transducer.

For clarity, we ﬁrst assume that it is plurisubsequential and show that it is a V-transducer,

then we show that it is indeed plurisubsequential.

(Assuming τ′′ plurisubsequential, it is a V-transducer.) Consider an equation u=Vv, a

state qof τ′′, and let p=q.u and p′=q.v. We show that p=p′, concluding this point. We

rely on the Syncing Lemma, since τ′′ is V-continuous; it ensures in particular that:

(τ′′

•,q ⊲⊳ τ′′

•,q)×(τ′′

p,•⊲⊳ τ′′

p′,•)⊆EquV(u′, v′)with u′=τ′′

q,p(u), v′=τ′′

q,p′(v).(5.1)

Let (s, s, t1, t2)be in the left-hand side. It holds that s·u′·t1=Vs·v′·t2, thus u′·t1=Vv′·t2

(here and in the following, we derive equivalent equations by appealing to the fact that the

free group is embedded, in a precise sense, in V[21, § 6.1.9]). Now consider another tuple

(s′, s′, t′

1, t′

2)again in the left-hand side of Equation (5.1). It also holds that u′·t′

1=Vv′·t′

hence we obtain that t1·t−1

2=Vt′

1·t′−1

2. This is in turn equal in Vto some α·β−1such that α

and βare words that do not share the same last letter. This shows that t1=α·tand t2=β·t

for some word t, and similarly for t′

1and t′

2. More generally: (τ′′

p,•⊲⊳ τ′′

p′,•)⊆(α, β)·Id, and

the normal form of Lemma 5.1 thus shows that p=p′.

(τ′′ is plurisubsequential.) In either τ′or τ′′, call a triple a states (p, q, q′)afork on aif

from p, the transducer can go to qand q′reading one a, and there is a path from qto p

reading only a’s:

q′

a∗

Dually, a triple (q, q′, p)is a reverse fork on aif the transducer can go from qand q′to

preading one a, and there is a path from pto qthat reads only a. In both cases, the fork

is proper if q6=q′. We rely on two facts to show plurisubsequentiality:

Fact 5.4. There are no proper forks or reverse forks in τ′′.

Fact 5.5. For any state pof τ′′ and any letter a, it holds that p∈p.aω.

Before proving the facts, we show how they imply the plurisubsequentiality of τ′′. Con-

sider a state pin τ′′ and a letter a. As p∈p.aωby Fact 5.5, there is a cycle of a’s on p.

18 M. CADILHAC, O. CARTON, AND C. PAPERMAN

Call qthe ﬁrst state of that cycle. Next, let q′be such that (p, a, q′)is a transition of τ′′ .

Clearly, (p, q, q′)forms a fork, hence by Fact 5.4, q=q′. Thus τ′′ is plurisubsequential. We

conclude with the proofs of the two facts:

Proof of Fact 5.4. We show the following statement: For any G-continuous transducer κ,

the result κ′of Lemma 5.1 has no proper reverse forks. This shows similarly that applying

the dual of Lemma 5.1 on a G-continuous transducer results in a transducer with no proper

forks. Since going from τ′to τ′′ cannot introduce new proper forks (as τ′is adjoined a

codeterministic automaton in Lemma 5.1), this shows the fact.

Consider a reverse fork (q, q′, p)on ain κ′. As pcan be reached from both qand q′

reading a, the product P=κ′

q,•⊲⊳ κ′

q′,•is nonempty. Write x=κ′

q′,p(a), y =κ′

p,q(an), z =

κ′

q,p(a), for some nsuch that yis deﬁned. We let hbe the longest common suﬃx of xand

z, and x=x′·hand z=z′·h.

As κis G-continuous, let us apply point 2 of the Syncing Lemma on κ′, the equation

(aω= 1), from the pair of states (q′, q′)to (q , q′). With Z=κ•,q′⊲⊳ κ•,q′, a nonempty subset

of the identity, it holds that Z×P⊆EquV(x(yz)ω−1y, 1). We write, in the following, ν−1for

νω−1, to convey the fact that ν−1is the inverse of νin V; that is: ν·ν−1=V1(this analogy

naturally carries further, since for instance, (νη)−1=Vη−1·ν−1). Let (s, s, u, u′)∈Z×P,

then:

s·u′=Vs·x·(yz)ω−1·y·u(By the Syncing Lemma)

=Vs·x·z−1·y−1·y·u

=Vs·x·z−1·u

=Vs·(x′h)·(z′h)−1·u=Vs·x′·z′−1·u .

By cancellation, this shows that u′=Vx′·z′−1·u. Since u′is a word and x′and z′do not

share a common suﬃx, there is a word wsuch that u=z′·w(this is true in the free group,

which is embedded, in a precise sense, in V[21, § 6.1.9]). This implies that u′=Vx′·wand

shows that P⊆(z′, x′)·Id, hence that q=q′by the construction of Lemma 5.1.

of Fact 5.4.

Proof of Fact 5.5. This relies on Fact 5.4. Let u, v be such that both τ′′

•,p(u)and τ′′

p,•(v)are

deﬁned. As τ′′(u·v)is deﬁned, the Preservation Lemma asserts that τ′′(u·aω·v)should

also be deﬁned. This implies that for nlarge enough, there is an accepting path for the

word u·an!·v. This path reaches some state p′after reading u, reads an!reaching a state

q, and accepts vfrom q. The path from p′to qcontains a cycle of a, hence if p′(resp. q)

is not in the cycle, this creates a proper reverse fork (resp. proper fork). Thus by Fact 5.4,

both p′and qare in the same cycle that reads an!; for nlarge enough, n!is a multiple of

the length of each cycle, this shows that p′=qand that p′∈p′.aω. This also implies that

τ′′

•,p′(u)and τ′′

p′,•(v)are both deﬁned, and since τ′′ is unambiguous, that p=p′, concluding

the proof. of Fact 5.5 and Lemma 5.3.

As an immediate corollary:

Corollary 5.6. For V ∈ {Gnil,Gsol,G}, any V-continuous rational function is V-realizable.

Theorem 5.7. Let Vbe a variety of group languages that includes Gnil and that is closed un-

der inverse V-realizable rational functions. It is decidable, given an unambiguous transducer,

whether it realizes a V-continuous function. This holds in particular for Gsol and G.

CONTINUITY AND RATIONAL FUNCTIONS 19

Proof. Lemma 5.3 together with Proposition 4.1 shows that a transducer is V-continuous iﬀ

its equivalent transducer eﬀectively computed by Lemma 5.1 and its dual is a V-transducer.

This latter property being testable, the result follows.

5.2. Deciding continuity for aperiodic varieties. We saw in Section 4.1 that the ap-

proach of the previous section cannot work: there is no correspondence between continuity

and realizability for aperiodic varieties. Herein, we use the Syncing Lemma to decide con-

tinuity in two main steps. First, note that all of our aperiodic varieties are deﬁned by an

inﬁnite number of equations for each alphabet. The Syncing Lemma would thus have us

check an inﬁnite number of conditions; our ﬁrst step is to reduce this to a ﬁnite number,

which we stress through the forthcoming notion of “pertaining triplet” of states. Second, we

have to show that the inclusion of the second point of the Syncing Lemma can eﬀectively

be checked. This will be done by simplifying this condition, and showing a decidability

property on rational relations.

We will need the following technical result in combinatorics on words in the proof of the

forthcoming Lemma 5.10:

Lemma 5.8. Let u, v, x, y, s, t ∈A∗be words verifying:

(1) u·v, x ·y∈s∗;

(2) v·u, y ·x∈t∗;

(3) sand tare primitive.

There exist z, z′∈A∗such that:

(1) s=z·z′and t=z′·z;

(2) u, x ∈s∗·z;

(3) v, y ∈t∗·z′.

Proof. Write u=sc·zand v=z′·sc′such that s=z·z′. It follows that v·u∈(z′·z)∗, and

since z′·zis primitive, it holds that t=z′·z. We can do the same with xand y, letting

x=s˙c·˙zand y= ˙z′·s˙c′, with s= ˙z·˙z′. The same reasoning then shows that t= ˙z′·˙z. Since

shas precisely |s|conjugates (by [11, Proposition 1.3.2]), it holds that ˙z=zand ˙z′=z′,

and the properties of the Lemma follow.

Deﬁnition 5.9. A triplet of states (p, q, q′)is pertaining if there are words s, u, t and an

integer nsuch that:

qq′

s| ·

t| ·

u|β′

un−1|β′′

un|β

where ·means “any word.” Further, a pertaining triplet is empty if, in the above picture,

β=β′β′′ = 1 and full if both words are nonempty; it is degenerate if only one of βor β′β′′

is empty.

20 M. CADILHAC, O. CARTON, AND C. PAPERMAN

It is called “pertaining” as the second point of the Syncing Lemma elaborates on prop-

erties of such a triplet, in particular, since uω=uω+1 is an equation of A. The following

characterization of A-continuity is then made without appeal to equations or proﬁnite words:

Lemma 5.10. A transducer τ:A∗→B∗is A-continuous iﬀ all of the following hold:

(1) τ−1(B∗)∈ A(A∗);

(2) For all full pertaining triplets (p, q, q′), there exist x, y ∈B∗and ρ1, ρ2∈(B∗)2such

that τ•,p ⊲⊳ τ•,q ⊆Id ·(x∗, x∗)ρ−1

1and τp,•⊲⊳ τq′,•⊆ρ−1

2(y∗, y∗)·Id;

(3) For all empty pertaining triplets (p, q, q′)it holds that (τ•,p ⊲⊳ τ•,q)·(τp,•⊲⊳ τq′,•)⊆Id;

(4) No pertaining triplet is degenerate.

Proof. (Only if.) Suppose τis A-continuous, and let us appeal to the Syncing Lemma.

Point 1 is then immediate. Point 2 is a direct consequence of the second point of the Syncing

Lemma and of Lemma 5.10.

We shall now check point 3, by contradiction. Let (p, q, q′)be an empty pertaining

triplet; we use the notations of Deﬁnition 5.9. Then by functionality of τ, it holds that

τ(s·uω·t) = x1·x2and τ(s·uω+1 ·t) = y1·y2. By the Preservation Lemma, and since

s·uω·t=As·uω+1 ·t, it should hold that x1·x2=y1·y2, proving point 3.

Point 4 is proven using similar ideas as point 3: with (p, q, q′)a degenerate pertaining

tuple, and using the same notations as above, either the production of s·uω·tgoing through

pis a not a ﬁnite word while the production of s·uω+1 ·tthrough q, q′is, or vice-versa. In

both cases, it is not possible for these productions to be equal in A, hence if such a case

happens, τcannot be A-continuous.

(If.) We again rely on the Syncing Lemma, the ﬁrst point of which being veriﬁed by hy-

pothesis. Let uω=uω+1 be an equation of Awith ua word; the set of such equations

deﬁnes A(A∗). Let p, q , p′, q′be states such that p′∈p.uωand q′∈q.uω+1, and let s, t be

words with p, q ∈q0.s and p′.t, q′.t ∈F. To conclude and apply the Syncing Lemma, we

need to show that:

τ•,p(s)·τp,p′(uω)·τp′,•(t) =Aτ•,q (s)·τq,q′(uω+1)·τq′,•(t).(5.2)

(This is a direct consequence of the way proﬁnite words are evaluated in a transducer, as

per Lemma 3.5.)

Consider a large number N=n!, so that p′∈p.uNand q′∈q.uN+1. With a large

enough N, there must be two states Pand Q, and integers i, j with i+j=N, such that:

•P∈p.ui,p′∈P.uj, and P∈P.uN(i.e., Pis “between” pand p′, and belongs to a loop);

•Q∈q.ui,q′∈Q.uj+1, and Q∈Q.uN.

(That such a pair exists can easily be seen on the product automaton of τby itself: The

path from (p, q)to (p′, q′′)with q′∈q′′ .u reading uNmust go twice through the same pair

of states (P, Q), and this pair respects the above requirements.)

Now deﬁne the following words:

•α=τ•,P (s·ui), β =τP,P (uN), γ =τP,•(uj·t),

•α′=τ•,Q(s·ui),β′=τQ,Q (uN),γ′=τQ,•(uj+1 ·t).

Using the same reasoning as Lemma 3.6, and the unambiguity of τ, Equation (5.2) is equiv-

alent to:

α·βω−N·γ=Aα′·β′ω−N·γ′.

CONTINUITY AND RATIONAL FUNCTIONS 21

Naturally, since α·βω−N·γ=Aα·βω·γ, and similarly for the right-hand side, Equation

(5.2) is equivalent to:

α·βω·γ=Aα′·β′ω·γ′.

To make use of the hypotheses of the present Lemma, deﬁne Q′to be in Q.u and such

that Q∈Q′.uN−1:(P, Q, Q′)is thus pertaining. The situation is then:

pp′

qq′

QQ′

s| ·

ui| ·

uj| ·

ui| ·

uj| ·

t| ·

u|β′

uN−1|β′′

uN|β

Since by hypothesis this triplet cannot be degenerate, either both of βand β′are empty,

or none are. Suppose they are both empty, then the hypothesis on empty triplets shows

that:

τ•,P (s·ui)·τP,•(uN+j·t) = τ•,Q(s·ui)·τQ′,•(uN+j·t).

The left-hand side evaluates to α·γ. Since τQ′,•(uN+j·t) = τQ′,Q(uN−1)·τQ,•(uj+1 ·t) = γ′,

the right-hand side evaluates to α′·γ′, and Equation (5.2) is thus veriﬁed.

Let us thus suppose that both βand β′are nonempty. We divide β′into b1b2such that

b1=τQ,Q′(u)and b2=τQ′,Q(uN−1). Now let x, y ∈B∗and ρ1, ρ2∈(B∗)2be the (pairs

of) words provided by point 2 for the triplet (P, Q, Q′). Deﬁne L=Id ·(x∗, x∗)ρ−1

1and

R=ρ−1

2(y∗, y∗)·Id. For any k≥1, and letting η=s·ui+k×Nand η′=uk×N+j·t, it

holds by hypothesis that:

•(τ•,P (η), τ•,Q(η)) = (α·βk, α′·β′k)∈L; (a)

•(τP,•(η), τQ′,•(η)) = (βk·γ, b2·β′k−1·γ′)∈R. (b)

Let us ﬁrst emphasize an easy property of Land R:

Fact 5.11. If (w·w′, w ·w′′)∈Lwith |w′|,|w′′|>|x|, then (w′, w′′)∈L. Moreover, if

(w, w′)∈L, then wis a preﬁx of w′or vice-versa.

Similarly, if (w′·w, w′′ ·w)∈Rwith |w′|,|w′′|>|y|, then (w′, w′′ )∈R. Moreover, if

(w, w′)∈R, then wis a suﬃx of w′or vice-versa.

Proof. We only show this for L, the case for Rbeing similar.

For the ﬁrst part of the statement, the hypothesis ensures the existence of a word z,

integers n′, n′′, and two preﬁxes x′, x′′ of xsuch that w·w′=z·xn′·x′and w·w′′ =z·xn′′ ·x′′.

If wis a preﬁx of z, the property is easy to verify. In the other cases, w=z·xn·χfor

some integer n < n′, n′′ (strictness coming from the hypothesis) and x=χχ′. Hence

w′=χ′·xn′−n−1·x′and w′′ =χ′·xn′′−n−1·x′, and thus both belong to L. The case of R

is similar.

For the second part of the statement, wand w′start with a common word z, then some

repetitions of x, and a preﬁx of x. Clearly, one has to be a preﬁx of the other. of Fact 5.11.

22 M. CADILHAC, O. CARTON, AND C. PAPERMAN

We ﬁrst focus on the consequences of (a). First, since either α·βkis a preﬁx of α′·β′k

or vice-versa, it holds that either αis a preﬁx of α′·β′k, or α′a preﬁx of α·βk, for some k.

Suppose for instance that α′=α·βc·βp, with β=βp·βs; the other case will be treated later.

Appealing to Fact 5.11, for kbig enough, factoring out α′yields that ((βsβp)k−c−1βs, β′k)∈L.

Hence (βsβp)∗and β′∗ share common preﬁxes of unbounded length, implying that βsβpand

β′are powers of a same primitive word z1(by, e.g., [11, Proposition 1.3.5]).

Now similarly focusing on (b), we obtain that γis a suﬃx of β′k·γ′or γ′is a suﬃx of

βk·γ, for some k. Suppose for instance that γ=β′

s·β′c′·γ′, with β′=β′

p·β′

s, again delaying

the other case. It follows, just as above, that βand β′

sβ′

pare powers of a same primitive

word z2. Noting that (ηc)ω=ηω, for any η, Equation (5.2) is thus equivalent to:

α·zω

2·β′

s·β′c′·γ′=Aα·βc·βp·zω

1·γ′.

Lemma 5.8 indicates that there exist words z, z′such that z1=z·z′,z2=z′·z, and

β′

s∈z∗

2·z′, βp∈z′·z∗

1. By eliminating αand γ′we thus obtain that there are some integers

n1, n2such that Equation (5.2) is equivalent to zω

2·z′·zn1×c′

1=Azn2×c

2·z′·zω

1, which clearly

holds as both sides evaluate to (z′·z)ω·z′.

(Remaining cases.) We made two suppositions: α′is a preﬁx of α, and γis a suﬃx of γ′.

The case where αis a preﬁx of α′and γ′a suﬃx of γis entirely symmetric. Let us keep our

supposition on α′and assume that γ′is a suﬃx of γ; the last remaining case is similar to

this one.

Let us thus write γ′=˙

βs·βc′·γ, with β=˙

βp·˙

βs. We then obtain, factoring out γ′

this time, that (βk−c−1˙

βp, β′k)∈R. This implies that (˙

βs˙

βp)and β′are powers of the same

primitive word, which can only be z1. Writing z2for the primitive root of β, Lemma 5.8

shows the existence of words z , z′such that z1=z·z′,z2=z′·z, and βp∈z∗

2·z′,˙

βs∈z·z∗

By eliminating αand γ, we similarly obtain that Equation (5.2) is equivalent, for some n1, n2,

to zω

2=Azn1×c

2·βp·zω

1·˙

βs·zn2×c′

2. Then both sides evaluate to zω

2, hence Equation (5.2)

holds. of Lemma 5.10.

Example 5.12. We show that the transducer of Proposition 4.3 is A-continuous. Let τbe:

p q

a|a

a|1

First, the function is total, hence the ﬁrst point of Lemma 5.10 is veriﬁed. Second,

there are no empty nor degenerate pertaining triplets, hence the third and fourth points

are veriﬁed. Now the full pertaining triplets are (p, p, p),(p, p, q),(q, q, q),and (q, q, p). We

check that the pertaining triplet (p, p, q)veriﬁes the second condition of Lemma 5.10, the

other cases being similar or clear. The ﬁrst half of the condition is immediate. Now τp,•⊲⊳

τq,•={(a⌊n+1/2⌋, a⌊n/2⌋)|n≥0}which veriﬁes the condition.

We now show that the property of Lemma 5.10 is indeed decidable:

Proposition 5.13. It is decidable, given a rational relation R⊆A∗×A∗, whether there is

a word x∈A∗and a pair ρ∈(A∗)2, such that R⊆Id ·(x∗, x∗)ρ−1).

Proof. We rely on the classical result that it is decidable whether a rational relation is

included in the identity [23, p. 650].

CONTINUITY AND RATIONAL FUNCTIONS 23

We ﬁrst tackle a related, simpler decision problem: Given a rational relation R⊆

(A∗×A∗)and a word x∈A∗, check whether R⊆Id ·(x∗, x∗). Write f:A∗→A∗for

the function that removes the longest suﬃx in x∗of its argument, and note that fis a

rational function. Closure under inverse and composition of rational relations implies that

R′={(f(u), f (v)) |(u, v)∈R}is a rational relation computable from R. We have that

R′⊆Id if and only if R⊆Id ·(x∗, x∗), hence the decision problem at hand is equivalent to

checking whether R′⊆Id, which is decidable.

We now reduce the main decision problem to the previous one. To do so, let us suppose

that such an xand ρexist, and search for them; if this search fails, no such xand ρexist.

First note that we may suppose that one component of ρis the empty word. Indeed,

write x′=xρ−1

1, x′′ =xρ−1

2. Assume x′is also a preﬁx of x′′ (the symmetric case being

similar). We may thus write x=x′yand x′′ =x′z, and have that R⊆Id ·(yx′)∗,(yx′)∗z,

showing that x:= yx′and ρ:= (1, z−1yx′)ﬁt the requirements.

We thus suppose that ρ2is empty (the symmetric case being similar). We ﬁrst check

that R⊆Id. If this is not the case, we are provided with a pair (u, v)∈R\Id (again by [23,

p. 650]). All the suﬃxes of uprovide us with candidates for xρ−1

1; we go through all these

candidates x′(including the empty word).

We then verify that all pairs (u, v)∈Rare such that uends with x′(this is decidable,

e.g., since R∩(A∗u)c×A∗=∅is decidable [3, Proposition 2.6, Proposition 8.2]). If this

is not the case, then x′cannot be xρ−1

1. Otherwise, let us write R′=R(x′,1)−1, a rational

relation.

We now check again that R′⊆Id; if it is the case, we are done, otherwise, we are given

a pair (u, v)∈R′\Id. Naturally, this implies that uis a preﬁx of v, or vice-versa. In the

former case, write v=u·z; it should hold that z∈x∗, and this provides us with candidates

for x: all the possible roots z′of z. We may now test that one such z′starts with x′, and

check whether R⊆Id ·(z′∗, z′∗ )using the above decision problem. If this holds, then there

do exist an xand a ρverifying R⊆Id ·(x∗, x∗)ρ−1. Moreover, if such words exist, this

procedure will ﬁnd them.

Remark 5.14. In general, the problem of deciding, given a rational relation Rand a

recognizable relation K, whether R⊆Id ·K, is undecidable. Indeed, testing R∩Id =∅is

undecidable [3], and equivalent to testing:

R⊆Id ·(A+× {1})∪({1} × A+)∪[

a6=b∈A

(a·A∗×b·A∗),

the right-hand side being of the form Id ·K.

Theorem 5.15. It is decidable, given an unambiguous transducer, whether it realizes an

A-continuous function.

Proof. This is a consequence of Lemma 5.10: Given a transducer, one can list all its pertain-

ing triplets, and whether they are empty, full, or degenerate. For full pertaining triplets, the

property of Lemma 5.10 is checked with Proposition 5.13 and the same Proposition applied

on the reverse of the transducer. The property for empty triplets can be checked since the

inclusion of a rational relation in Id is decidable.

The rest of this section focuses on conditions à la Lemma 5.10 for J,R,L, and DA. In

each of these cases, we deﬁne the proper notion of “pertaining” and rewrite the conditions

of Lemma 5.10 to match the deﬁning equations. Since the proofs are simple variants of

24 M. CADILHAC, O. CARTON, AND C. PAPERMAN

that of Lemma 5.10, we omit them; we note that in each case, the conditions are eﬀectively

veriﬁable.

5.2.1. The case of J.We use a diﬀerent set of equations to deﬁne J, that can easily be

proved to be equivalent to the one given in the preliminaries. Speciﬁcally, Jis deﬁned over

any alphabet Aby the set of equations xω=y·xω·z, with y, z ∈alph(x). The deﬁnition of

“pertaining” then reads as follows:

Deﬁnition 5.16. For two alphabets C, D, a quadruplet of states (p, q, q ′, q′′)is (C, D)-per-

taining if there are words s, u, t with alph(u) = C, words z, z′∈C∗, and an integer nsuch

that:

qq′q′′

s| ·

t| ·

z| ·

un|β

un|β′

z′| ·

∈

alph(u)∗

∈

alph(u)∗

and moreover alph(β)∪alph(β′) = D. The pertaining quadruplet is empty if D=∅; it is

full if alph(β) = alph(β′)6=∅, and degenerate otherwise.

Lemma 5.17. A transducer τ:A∗→B∗is J-continuous iﬀ all of the following hold:

(1) τ−1(B∗)∈ J (A∗);

(2) For all full (C, D)-pertaining quadruplet (p, q, q ′, q′′), it holds that:

τ•,p ⊲⊳ τ•,q ·ε, τq,q′(C∗)·τq′,q′⊆Id ·(D∗, D∗)and

τp,•⊲⊳ τq′,q′·ε, τq′,q′′ (C∗)·τq′′,•⊆(D∗, D∗)·Id ;

(3) For all empty (C, D)-pertaining quadruplet (p, q, q′, q′′), it holds that:

(τ•,p ·τp,•)⊲⊳ τ•,q ·ε, τq,q′(C∗)·τq′,q′·ε, τq′,q′′ (C∗)·τq′′ ,•⊆Id ;

(4) No pertaining quadruplet is degenerate.

5.2.2. The case of R.Again, we slightly diverge from the usual equations for R, as presented

in the Preliminaries. Indeed, Ris also deﬁned, over any alphabet A, by xω=xω·ywith

y∈alph(x). We turn to the deﬁnition of “pertaining:”

Deﬁnition 5.18. For an alphabets C, a triplet of states (p, q, q′)is C-pertaining if there

are words s, u, t with alph(u) = C, words z∈C∗, and an integer nsuch that:

CONTINUITY AND RATIONAL FUNCTIONS 25

qq′

s| ·

t| ·

z| ·

∈

alph(u)∗

un|β

un|β′

The pertaining triplet is empty if, in the above picture, β=β′= 1; it is full if none of β , β′

is empty, and degenerate otherwise.

Lemma 5.19. A transducer τ:A∗→B∗is R-continuous iﬀ all of the following hold:

(1) τ−1(B∗)∈ R(A∗);

(2) For all full C-pertaining triplets (p, q, q′), there exist x∈B∗and ρ∈(B∗)2such that

both inclusions hold:

τ•,p ⊲⊳ τ•,q ⊆Id ·(x∗, x∗)ρ−1,

τp,•⊲⊳ τq,q ·ε, τq,q′(C∗)·τq′,•⊆alph(x)∗,alph(x)∗·Id ;

(3) For all empty C-pertaining triplets (p, q, q′), it holds that:

(τ•,p ·τp,•)⊲⊳ τ•,q ·ε, τq,q′(C∗)·τq′,•⊆Id ;

(4) No pertaining triplet is degenerate.

Note that the xof Proposition 5.13 can be eﬀectively found. The case of Lcan be

simply seen as the reversal of the previous case.

5.2.3. The case of DA.Similarly, we use a slightly less standard equational deﬁnition of DA.

Indeed, DA is also deﬁned, over any alphabet A, by xω=xω·y·xωwith y∈alph(x). The

deﬁnition of “pertaining” reﬂects these equations:

Deﬁnition 5.20. For an alphabet C, a triplet of states (p, q , q′)is C-pertaining if there are

words s, u, t with alph(u) = C, a word z∈C∗, and an integer nsuch that:

qq′

s| ·

t| ·

z| ·

∈

alph(u)∗

un|β

un|β′un|β′′

26 M. CADILHAC, O. CARTON, AND C. PAPERMAN

Further, a pertaining triplet is empty if, in the above picture, β=β′=β′′ = 1; it is

left-empty if only β′is empty, right-empty if only β′′ is empty, full if none of β, β ′, β′′ is

empty, and degenerate in the other cases.

Lemma 5.21. A transducer τ:A∗→B∗is DA-continuous iﬀ all of the following hold:

(1) τ−1(B∗)∈ DA(A∗);

(2) For all full C-pertaining triplets (p, q, q′), there exist x, y ∈B∗and ρ1, ρ2∈(B∗)2such

that these three inclusions hold:

τ•,p ⊲⊳ τ•,q ⊆Id ·(x∗, x∗)ρ−1

1,

τp,•⊲⊳ τq′,•⊆ρ−1

2(y∗, y∗)·Id ,

τq,q′(C∗)⊆alph(x·y)∗;

(3) For all empty C-pertaining triplets (p, q, q′)it holds that:

(τ•,p ⊲⊳ τ•,q)·(ε, τq,q′(C∗)) ·(τp,•⊲⊳ τq′,•)⊆Id ;

(4) For all right-empty C-pertaining triplets (p, q, q′), there exist x, y ∈B∗and ρ1, ρ2∈

(B∗)2such that:

τ•,p ⊲⊳ τ•,q ⊆Id ·(x∗, x∗)ρ−1

1and τp,•⊲⊳ (ε, τq,q′(C∗)) ·τq′,•⊆ρ−1

2(y∗, y∗)·Id ;

(5) For all left-empty C-pertaining triplets (p, q, q′), there exist x, y ∈B∗and ρ1, ρ2∈(B∗)2

such that:

τ•,p ⊲⊳ τ•,q ·(ε, τq,q′(C∗))⊆Id ·(x∗, x∗)ρ−1

1and τp,•⊲⊳ τq′,•⊆ρ−1

2(y∗, y∗)·Id ;

(6) No pertaining triplet is degenerate.

5.3. Deciding Com- and Ab-continuity. The case of COM and ABis comparatively much

simpler, in particular because these varieties are deﬁned using a ﬁnite number of equations

for each alphabet. However, the argument relies on diﬀerent ideas:

Theorem 5.22. For V=COM,AB, it is decidable, given an unambiguous transducer, whether

it realizes a V-continuous function.

Proof. We apply the Syncing Lemma. Its ﬁrst point is clearly decidable. We reduce its

second point to decidable properties about semilinear sets (see, e.g., [10]). We also rely on

the notion of Parikh image, that is, the mapping Pkh :A∗→NAsuch that Pkh(w)maps

a∈Ato the number of a’s in the word w.

Since every AB-continuous function is COM-continuous (Proposition 4.6), the conditions

to test for AB-continuity are included in those for COM-continuity—this can also be seen as

a consequence of the fact that if u, v are words, EquAB(u, v) = EquCOM(u, v).

Let τ:A∗→B∗be a given transducer. Consider an equation ab =ba and four states

p, p′, q, q′of τ. Write u=τp,p′(ab)and v=τq,q′(ba). We ought to check, by the Syncing

Lemma, the inclusion in EquCOM(u, v) = {(s, s′, t, t′)|s·u·t=COM s′·v·t′}of some input

synchronization. Now this set is the set of (s, s′, t, t′)such that Pkh(s·u·t) = Pkh(s′·v·t′),

and is thus deﬁned by a simple semilinear property. The input synchronizations themselves,

e.g., τ•,p ⊲⊳ τ•,q, are rational relations, and their component-wise Parikh image is thus a

semilinear set. Since the inclusion of semilinear sets is decidable, the inclusion of the second

point of the Syncing Lemma is also decidable.

CONTINUITY AND RATIONAL FUNCTIONS 27

For AB, we should additionally check the equations aω= 1. The reasoning is similar.

Consider three states (p, p′, q), and write x·uω−1·yfor τp,p′(aω). By commutativity and

the fact that uω−1acts as an inverse of uin the equations holding in AB, we have that

(s, s′, t, t′)∈EquAB(x·uω−1·y, 1) iﬀ s·t=ABs′·u·t′. This again reduces the inclusion of

the second point of the Syncing Lemma to a decidable semilinear property.

6. Discussion

We presented a study of continuity in functional transducers, on the one hand focused on

general statements (Section 3), on the other hand on continuity for classical varieties. The

heart of this contribution resides in decidability properties (Section 5), although we also

addressed natural and related questions in a systematic way (Section 4). We single out two

main research directions.

First, there is a sharp contrast between the genericity of the Preservation and Syncing

Lemma and the technicality of the actual proofs of decidability of continuity. To which extent

can these be uniﬁed and generalized? We know of two immediate extensions: 1. the generic

results of Section 3 readily apply to Boolean algebras of languages closed under quotient,

a relaxation of the conditions imposed on varieties, and 2. the varieties Gpof languages

recognized by p-groups can also be shown to verify Proposition 4.1 and Lemma 5.3, hence

Gp-continuity is decidable for transducers. Beyond these two points, we do not know how to

show decidability for Gnil (which is the join of the Gp), and the surprising complexity of the

equalizer sets for some Burnside varieties (e.g., the one deﬁned by x2=x3, see the Remark

on page 10) leads us to conjecture that continuity may be undecidable in that case, hence

that no uniﬁed way to show the decidability of continuity exists.

Second, the notion of continuity may be extended to more general settings. For instance,

departing from regular languages, it can be noted that every recursive function is continuous

for the class of recursive languages. Another natural generalization consists in studying

(V,W)-continuity, that is, the property for a function to map W-languages to V-languages

by inverse image. This would provide more ﬂexibility for a suﬃcient condition for cascades

of languages (or stackings of circuits, or nestings of formulas) to be in a given variety.

Acknowledgment. We are deeply indebted to Shaull Almagor, Jorge Almeida (in particular

for the Remark on page 10), Luc Dartois, Bruno Guillon (in particular for the Remark on

page 23), Ismaël Jecker, and Jean-Éric Pin for their insightful comments and kind help.

The ﬁrst and third authors are partly funded by the DFG Emmy Noether program

(KR 4042/2); the second author is funded by the DeLTA project (ANR-16-CE40-0007).

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Jan 1994

Howard Straubing

Part 1 Mathematical preliminaries: words and languages automata and regular languages semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples definitions. Part 3 Finite automata: monadic second-order sentences and regular languages regular numerical predicates infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game application to FO [decreasing] application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid calculation of the syntactic monoid application to FO [decreasing] semidirect products categories and path conditions pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing] a hierarchy in FO [decreasing] another characterization of FO [+1] sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples languages in (FO + MOD(P))[decreasing] languages in (FO + MOD)[+1] languages in (FO + MOD)[Reg] summary. Part 8 Circuit complexity: examples of circuits circuits and circuit complexity classes lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1 formulas with arbitrary numerical predicates regular languages and non-regular numerical predicates special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem proofs of the category theorems.

Algebraic Combinatorics on Words

Article

Apr 2002

M. Lothaire

Combinatorics on words has arisen independently within several branches of mathematics, for instance number theory, group theory and probability, and appears frequently in problems related to theoretical computer science. The first unified treatment of the area was given in Lothaire's book Combinatorics on Words. Originally published in 2002, this book presents several more topics and provides deeper insights into subjects discussed in the previous volume. An introductory chapter provides the reader with all the necessary background material. There are numerous examples, full proofs whenever possible and a notes section discussing further developments in the area. This book is both a comprehensive introduction to the subject and a valuable reference source for researchers.

Continuity and Rational Functions

Abstract

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