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The Alternation Hierarchy of First-Order Logic on Words is Decidable

January 2025

DOI:10.48550/arXiv.2501.14899

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Authors:

Howard Straubing

Boston College

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We show that for any

i > 0

, it is decidable, given a regular language, whether it is expressible in the

\Sigma_i[<]

fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the notion of polynomial closure of a class of languages

\mathcal{V}

, that is, finite unions of languages of the form

L_0a_1L_1\cdots a_nL_n

where each

a_i

is a letter and each

L_i

a language of

\mathcal{V}

. We show that if a class

\mathcal{V}

of regular languages with some closure properties (namely, a positive variety) has a decidable separation problem, then so does its polynomial closure Pol(

\mathcal{V}

). The resulting algorithm for Pol(

\mathcal{V}

) has time complexity that is exponential in the time complexity for

\mathcal{V}

and we propose a natural conjecture that would lead to a polynomial time blowup instead. Corollaries include the decidability of half levels of the dot-depth hierarchy and the group-based concatenation hierarchy.

Commutative diagram of semigroups for the main construction. It indicates that some compositions of morphisms are equal, by following edges from the same source and destination; for instance, following paths from K + 1 to M , the diagram expresses that π ℓ • ▷ 1 = π ℓ • π.

…

Commutative diagram of semigroups for the main construction, with additional morphisms.

…

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Available via license: CC BY 4.0

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The Alternation Hierarchy of First-Order Logic on

Words is Decidable

Corentin Barloy

Ruhr-Universität Bochum

Bochum, Germany

Michaël Cadilhac

DePaul University

Chicago, USA

Charles Paperman

Université de Lille

Lille, France

Howard Straubing

Boston College

Boston, USA

Abstract—We show that for any

i > 0

, it is decidable, given a

regular language, whether it is expressible in the

Σi[<]

fragment

of ﬁrst-order logic

FO[<]

. This settles a question open since 1971.

Our main technical result relies on the notion of polynomial

closure of a class of languages

, that is, the set of languages

L0a1L1· · · anLn

where each

is a letter and each

a language

. We show that if a class

of regular languages with some

closure properties (namely, a positive variety) has a decidable

separation problem, then so does its polynomial closure

Pol(V)

The resulting algorithm for

Pol(V)

has time complexity that is

exponential in the time complexity for

and we propose a natural

conjecture that would lead to a polynomial time blowup instead.

Corollaries include the decidability of half levels of the dot-

depth hierarchy and the group-based concatenation hierarchy.

Index Terms—regular languages, ﬁrst-order logic, alternation

hierarchy, concatenation hierarchy, proﬁnite theory

I. INTRODUCTION

First-order formulas can be used to express properties of

strings. In this setting, one quantiﬁes over positions in a word

(

∃x, ∀y

), can test whether a position appears before another

(

x < y

), and whether a position contains a given letter (e.g.,

a(x)

b(x)

). For instance, over the alphabet

A={a, b, c}

the following sentence says that there is a position with an

such that after it, every position with a

is eventually followed

by a position with a c:

(∃x)ha(x)∧(∀y)(y > x ∧b(y)) →(∃z)[z > y ∧c(z)]i.

As a regular expression, the language deﬁned by this sentence

— that is, the set of strings having the stated property — can be

written as

A∗a(a+c+bA∗c)∗

. A key measure of the complexity

of such a formula is the number of times it alternates between

existential and universal quantiﬁcations. In this example, the

sentence is logically equivalent to the following sentence in

preﬁx form:

(∃x)(∀y)(∃z)ha(x)∧(y > x ∧b(y)) →[z > y ∧c(z)]i.

The resulting sentence consists of three blocks of alternating

existential and universal quantiﬁers — in this instance the

‘blocks’ each consist of a single quantiﬁer — beginning with

an existential quantiﬁer. So the language is deﬁned by a

Σ3[<]

sentence:

to indicate that it starts with an existential quantiﬁer

block,

for the number of quantiﬁer blocks,

as this is the

only predicate on positions used beside the letter predicates.

We will also say that the language itself is in Σ3[<].

Can our previous formula be “improved” under that metric?

The astute reader may have realized that the language deﬁned

can be reworded as “the string contains an

and ends either

with aor c;” this can be written as:

(∃x)(∃y)(∀z)a(x)∧(a(y)∨c(y)) ∧[¬(y < z)].

The preﬁx contains only two blocks of alternating quantiﬁers,

beginning with an existential quantiﬁer: thus the language is

Σ2[<]

. We note that this complexity measure is conjectured

to be closely related to the minimal depth of an equivalent

Boolean circuit and that depth is tied to the speed at which the

circuit can be evaluated [32] — this conjecture is known to

hold up to

Σ2[<]

[4]. It is thus of crucial importance to ﬁnd

what is the minimal number of alternations required to deﬁne

a given language.

Such an investigation ought to start with a simpler question:

What are the languages that can be deﬁned by ﬁrst-order

sentences? This was answered by Schützenberger [29] and

McNaughton and Papert [12] in the 1960’s: they are the so-

called star-free languages, and one can decide, given a regular

language, if it is star-free. This left open the ﬁner-grained

question: can the minimal number of alternations be computed?

We provide a positive answer to that question.

To state our main technical theorem, we will work with an

equivalent formulation of the languages in each

Σi[<]

, that is,

the languages deﬁnable by a sentence in preﬁx form starting

with

∃

and alternating between quantiﬁer types at most

i−1

times. The only quantiﬁer-free sentences are equivalent to

⊤

(always true) and

⊥

(always false), hence

Σ0[<]

consists of

the two languages

∅

and

A∗

. To go higher in the hierarchy,

it turns out that a simple operation sufﬁces: the polynomial

closure. For

a class of languages, let

Pol(V)

be the set

of languages that are ﬁnite unions of languages of the form

L0a1L1· · · anLn

, where each

is a letter of

and each

a language of

. We write

Πi[<]

for the dual of

Σi[<]

, that

is, the set of complements of languages in

Σi[<]

. Pin and

Weil [19], building on Perrin and Pin [13], showed that:

Theorem 1. For any i > 0,Σi[<] = Pol(Πi−1[<]).

Our previous example language can be seen to be in

Σ2[<]

using this language-theoretic construction as follows. First,

{ε}

, the language that only contains the empty word, is the

complement of

A∗aA∗∪A∗bA∗∪A∗cA∗

, so it is in

Π1[<]

Second, the language of words that contain an

and end either

in aor cis A∗a{ε} ∪ A∗aA∗c{ε}, hence it is in Σ2[<].

Beside polynomial closure, the precise statement of our main

theorem requires two more ingredients:

arXiv:2501.14899v1 [cs.FL] 24 Jan 2025

•

Some closure properties of the classes of regular languages

that we study: they have to be “positive varieties of

languages” (formally deﬁned in Section II), which entails

closure under union and intersection, but not complement;

notably, these closure properties are preserved by taking

the dual or the polynomial closure of a class;

•

The problem of

V-

separation for a class of regular

languages

: it asks, given a pair

(L, L′)

of regular

languages, whether there is a language

K∈ V

with

L⊆K

and

L′∩K=∅

. Note that this relation is not

symmetric. Naturally, a language

is in

if, and only

if,

V-

separable from its complement; thus if we

can decide the separation problem for a class of regular

languages, we can decide whether a given regular language

is a member of the class.

Theorem 2 (Main theorem).Let

be a positive variety of

regular languages. If

V-

separation is decidable, then so is

Pol(V)-separation.

Since

Σ0[<]-

separation is trivially decidable and the decid-

ability of

Πi[<]-

separation is equivalent to that of

Σi[<]-

sepa-

ration, Theorem 1 and Theorem 2 readily imply the decidability

of separation and membership for each level

Σi[<]

and

Πi[<]

of the alternation hierarchy of ﬁrst-order logic, and for their

intersections ∆i[<].

Other applications and complexity

•

is a class of regular languages with a decidable

separation problem, then we show that separation is

decidable for the levels of the hierarchy built by starting

with

and applying Boolean then polynomial closure.

These hierarchies are called concatenation hierarchies,

the most prominent of which we just saw. Another well-

studied example is the group concatenation hierarchy (see,

e.g., [16, 27, 23]) which starts from group languages.

•

For time complexity, we consider that the input regular

languages are provided as morphisms — this can be

exponentially larger than the equivalent automaton. In

that setting, we obtain that if

V-

separation is decidable

in time

c(n)

, then

Pol(V)-

separation is decidable in time

O(c(2O(n)))

. This implies that

Σi[<]-

separation can be

decided in a tower of exponentials of height

. We propose

a conjecture that would entail that our algorithm only adds

a polynomial blow-up and, consequently, that

Σi[<]-

sep-

aration and FO[<]-separation are both in PTIME.

Historical background

In 1960, McNaughton [11] raised the question of what

properties of ﬁnite-state machines could be expressed in ﬁrst-

order logic. Schützenberger [29] discovered the solution to one

of the problems raised by McNaughton, by showing that the

regular languages deﬁnable by sentences of ﬁrst-order logic are

precisely those whose syntactic monoids are aperiodic—that

is, contain no nontrivial groups. Since the syntactic monoid

of a regular language can be computed from any of the

usual representations of the language (e.g., an automaton

that recognizes the language, or a regular expression), this

provides an effective test for whether a regular language

is ﬁrst-order deﬁnable. Schützenberger wrote in terms of

operations on languages (Boolean operations and concatenation)

rather than ﬁrst-order sentences; thus the class of languages

deﬁnable in this way became known as the star-free languages,

since they can be deﬁned by extended regular expressions

in which complementation is allowed along with union and

concatenation, but in which the star operation is not used.

The book by McNaughton and Papert [12] gave an integrated

presentation in terms of logic, extended regular expressions,

automata and ﬁnite monoids.

Cohen and Brzozowski [7] introduced a complexity measure

on the star-free languages, which they called dot-depth: The

0-level of the hierarchy consists of the ﬁnite and coﬁnite

languages over a ﬁnite alphabet

k≥0

then the

k+1

level consists of products

L1· · · Lr,

where

r≥1

, and each

is a language in level

and the

k+ 1

level is the Boolean

closure of the level

k+1

Cohen and Brzozowski posed the

problem of effectively determining whether a given star-free

language belongs to a particular level of the hierarchy. In the

present paper, we settle this problem for the

k+1

levels. The

question for the Boolean-closed integer levels remains open.

Brzozowski and Knast [6] proved that the hierarchy is

strict. (See also Thomas [35] who gave a proof of this fact

using model-theoretic games.) Straubing [33] and Thérien [34]

studied a slightly different hierarchy in the star-free languages

over a given alphabet

. Set

V0={∅, A∗}

, then, inductively:

BVi=Boolean closure of Vi,and Vi+1 = Pol(BVi).

The class of star-free languages is then the union of the

Vi.

The class

is precisely the class of

Σi[<]

languages that we

consider in the present paper [19, 13]. This is closely related

to the original formulation of the dot-depth hierarchy: From an

algebraic perspective it is the simpler and more fundamental of

the two. A solution of the membership problem for any of the

levels of this hierarchy implies one for the corresponding level

of the dot-depth hierarchy [33]. The membership problem for

is fairly simple. It was solved for

BV1

in a famous theorem

of Simon [31], for

by Arﬁ [3], and, with a simpler proof,

by Pin and Weil [19]. More recently, in a major breakthrough,

Place and Zeitoun [22, 25] found solutions for

BV2,V3

and

V4.

Pin also contributed a thorough historical account in 2017,

covering more than 45 years of research on that topic [18].

Here, we provide a solution for all Vi,i≥0.

Organization of the paper

In Section II, we introduce all the concepts required to

go through our proof. We strive to keep the mathematical

objects as simple as possible, sometimes at the cost of spelling

out arguments that would have been more elegantly proved

using more powerful topological or algebraic tools. Sections III

to IX present the proof of our main theorem, starting with

an overview. In Section X, we provide some corollaries to

our main theorem. In Section XI, we close with some open

questions. All missing proofs appear in the appendix.

II. PRELIMINARIES

We assume familiarity with basic automata theory, including

the deﬁnitions of alphabet, word, language, and regular

language. Every statement singled out as a Lemma, Theorem,

or Corollary in this section has an elementary proof that is

provided in appendix.

A. Sets, languages

We will manipulate partially ordered sets

(M, ≤)

and, with

E⊆M

, we will write

↑E

for the upset

{y|(∃x∈E)[x≤y]}

and ↓Efor the downset {y|(∃x∈E)[y≤x]}.

We use

A, B

for ﬁnite alphabets and write

A∗

for the set

of words over

. We write

for the empty word and

A+

for

A∗\ {ε}

. Classes of regular languages are usually written

or W.

A language

separates a language

from another language

L′

L⊆K

and

L′∩K=∅

. With

a class of regular

languages, a language

V-

separable from

L′

if there is

K∈ V

that separates

from

L′

. We say that

V-

separation

is decidable if there is an algorithm that decides, given two

regular languages L, L′, whether Lis V-separable from L′.

For

a class of regular languages, a

V-

polynomial is a ﬁnite

union of languages of the form

L0a1L1· · · anLn

, where each

is a letter of

and each

a language of

. The polynomial

closure of V, written Pol(V), is the set of all V-polynomials.

B. Algebraic structures

In the algebraic approach to regular languages, languages live

alongside algebraic objects. Monoids and semigroups describe

equivalence classes of words within a language, similarly to

the way in which an automaton considers two words to be

equivalent if they induce the same map from states to states.

Semigroups, monoids, morphisms.

Asemigroup is a set

equipped with an associative product. For a subset

E⊆M

we write

⟨E⟩

for the subsemigroup of

generated by

i.e., the closure of

under the product of

. A monoid is

a semigroup that has a neutral element; for a monoid

, we

write that element 1M.

A semigroup element

is idempotent if

e2=e

. If

is a

ﬁnite semigroup, for any m∈Mthere is a value r > 0such

that

is idempotent, and this is the unique idempotent among

the positive powers of

(this is reminiscent of the notion of

order in ﬁnite groups). We write

mω

for the idempotent power

of m.

Throughout this article, all semigroups and monoids are

ﬁnite, with the exceptions of the free monoid

A∗

over a ﬁnite

alphabet

and the monoid

A∗

, which will be deﬁned in

Section II-C.

Asemigroup morphism is a function

µ:M→N

with

M, N

semigroups such that

µ(mm′) = µ(m)µ(m′)

for every

m, m′∈M

. It is a monoid morphism if additionally

M, N

are

monoids and µ(1M) = 1N.

A language

L⊆A∗

is recognized by a monoid

if there

is a morphism

µ:A∗→M

and a subset

E⊆M

such that

L=µ−1(E)

. A language is regular if, and only if, it is

recognized by some ﬁnite monoid. This is effective: Given a

ﬁnite automaton, one can compute a morphism

µ:A∗→M

and a set

such that

µ−1(E)

is the language of the automaton.

However, the monoid

may be exponentially larger than the

automaton: we stress that all our complexity results assume

that the morphism (and hence the monoid) is provided as input.

We will often view a monoid

as an alphabet and consider

words in

M∗

. As a prime example, we will rely on the surjective

morphism

πM:M∗→M

, for any monoid

(or

πM:M+→

when

is a semigroup): this is the canonical morphism

, which simply evaluates the product in

. For instance,

m1m2

is a 2-letter word in

M∗

and the product of

and

m2in Mis m, then πM(m1m2) = m.

An ordered semigroup is a semigroup

equipped with a

partial order ≤that is compatible with the product: for every

m, m′, n ∈M

such that

m≤m′

, we have that

mn ≤m′n

and

nm ≤nm′

. An ordered monoid is an ordered semigroup that

is a monoid. A morphism

µ:M→N

of ordered semigroups

or monoids is a morphism in the sense deﬁned above, with the

additional property

m≤m′

implies

µ(m)≤µ(m′)

for every

m, m′∈M.

We say that an ordered monoid

recognizes a language

L⊆A∗

if there is a morphism

µ:A∗→M

and a subset

E⊆Msuch that L=µ−1(↑E).

Remark 3 (Recognition by ordered vs. unordered monoids).

Note that this notion of recognition is a stronger requirement

than in the unordered case: For example, if

A={a, b}

and

is the ordered monoid

{0,1}

with the ordering

0≤1

, then

recognizes

A∗aA∗

in the unordered sense by the morphism

mapping

and

, and taking

E={0}

. However

is not upwardly closed, so this

does not recognize this

language in the ordered sense. Observe that the monoid

{0,1}

does recognize

A∗aA∗

in the ordered sense if we change the

order to

1≤0

, or to the trivial order in which 0 and 1 are

incomparable. Note additionally that every ﬁnite monoid can

be seen as an ordered monoid with the trivial order

m≤m′

if,

and only if,

and

m′

are equal. This shows that a language

is regular if, and only if, it is recognized by some ordered

monoid.

Finally, we introduce a generalization of morphisms (due

to Tilson, in Chapter XII of Eilenberg [10]). A relational

morphism

τ:M△

−→ N

between two semigroups

M, N

is a

function from

that satisﬁes, for every

m, m′∈M

τ(m)=∅

and

τ(m)τ(m′)⊆τ(mm′)

— the product of two

elements

E, F ∈2N

is deﬁned as

{ef |e∈E∧f∈F}

. For

monoids, we require additionally that 1N∈τ(1M).

Varieties.

A class of regular languages is a positive variety

if it is closed under union, intersection, inverse morphisms

(from morphisms

µ:A∗→B∗

), and quotient (the operation

that maps a language

and a word

{v|uv ∈L}

, and

Formally, a variety maps alphabets to sets of languages over that alphabet.

As is standard, we elect to sidestep the resulting notational burden by

considering varieties themselves to be sets of languages. This has no impact

on our results and their proofs.

symmetrically, the operation that maps to

{v|vu ∈L}

We note that if

is a positive variety, then so are its dual

{L|L∈ V} and its polynomial closure Pol(V)[15].

A class of ordered ﬁnite monoids is a variety (of ordered

monoids) if it is closed under taking ordered submonoids (i.e.,

subsets of a monoid closed under the product), quotients (i.e.,

images of ordered morphisms) and ﬁnite direct products.

We denote varieties of ordered monoids in boldface (

)

and the set of languages they recognize in calligraphic style

(

). The correspondence

V→ V

is a bijective correspondence

between varieties of ordered monoids and positive varieties

of regular languages; this theorem is due to Eilenberg [10]

in the unordered case and to Pin [14] for varieties of ordered

monoids. Accordingly, we will write

Pol(V)

for the variety of

ordered monoids that corresponds to

Pol(V)

. We will recall in

Theorem 8 how

Pol(V)

can be deﬁned directly from

; for this,

we require the formalism of proﬁnite words and inequalities.

C. Proﬁnite words and inequalities

Proﬁnite words allow us to speak, in a succinct way, about

inﬁnite sequences of words that are ultimately indistinguishable

by any ﬁxed monoid. Here we give a self-contained sketch of

the theory that does not require any background in topology,

and that is sufﬁcient for our purposes in this paper. For a more

thorough introduction, see the survey by Pin [17].

Convergence.

We say that an inﬁnite sequence

(mi)i>0

elements of a ﬁnite monoid

is convergent if the sequence

is ultimately constant; that is, there is some

m∈M

such

that

mi=m

for all sufﬁciently large

. We write

limimi

. The termwise product

(mini)i>0

of two convergent

sequences

(mi)i>0

and

(ni)i>0

is itself convergent, and clearly

limimini= (limimi)(limini).

We say that an inﬁnite sequence

u= (wi)i>0

of words in

A∗

is convergent if for every morphism

µ:A∗→M

, where

is a ﬁnite monoid, the sequence

(µ(wi))i>0

is convergent.

Convergent sequences embody the idea of “a sequence of words

ultimately indistinguishable by ﬁnite monoids.”

An obvious example of a convergent sequence of words

is any ultimately constant sequence. A more interesting (and

important) example is the sequence

{wi!}i>0,

where

w∈A∗

µ:A∗→M

is any morphism into a ﬁnite monoid, and

i > 0

is large enough, then

µ(wi!) = µ(w)i!=µ(w)ω

, the unique

idempotent power of µ(w), hence limiµ(wi!) = µ(w)ω.

We say that two convergent sequences

u= (u1, u2, . . .)

and

u′= (u′

1, u′

2, . . .)

are equivalent if

limiµ(ui) = limiµ(u′

for

all morphisms µfrom A∗into a ﬁnite monoid.

The free proﬁnite monoid.

What does a convergent sequence

of words converge to? Let us equip the set of inﬁnite sequences

of words with an internal operation: termwise concatenation,

i.e., with

u= (u1, u2, . . .)

and

v= (v1, v2, . . .)

, the concatena-

tion

(u1v1, u2v2, . . .)

. Clearly,

is convergent if

and

are. This turns the set of convergent sequences into a monoid

A∗

. If

µ:A∗→M

is a morphism into a ﬁnite monoid, then

setting, for every

u∈˜

A∗

˜µ(u) = limiµ(ui)

gives a morphism

from ˜

A∗into M.

Equivalence of convergent sequences is a congruence on

A∗

The quotient of

A∗

by the equivalence relation is called the

free proﬁnite monoid of

and is denoted

A∗

. Its elements,

that is, the equivalence classes of convergent sequences, are

called proﬁnite words. If

µ:A∗→M

is a morphism into a

ﬁnite monoid, then the morphism

˜µ

maps equivalent words to

the same value, and thus we obtain a morphism

bµ:c

A∗→M.

Thus if

u= (u1, u2, . . .)

is a convergent sequence of words,

we can deﬁne

limiui

to be the equivalence class of

. This

gives us the following continuity property: For every morphism

µ:A∗→M, where Mis ﬁnite, bµ(limiui) = limibµ(ui).

Everything that we do with proﬁnite words will depend

on these morphisms

bµ

. This allows us to blur the distinction

between convergent sequences and their equivalence classes.

In particular, we will often say that a convergent sequence of

words is a proﬁnite word.

Some proﬁnite words.

The monoid

A∗

embeds in

A∗

mapping each

w∈A∗

to the sequence

(w, w, . . .)

. (For any

two distinct words

w, w′

, there is a morphism into a ﬁnite

monoid that maps

and

w′

to different elements, so this map

is indeed injective.) As we discussed above, if

u∈A∗

and

µ:A∗→M

is a morphism into a ﬁnite monoid, then the

sequence

v= (ui!)i>0

is convergent, and maps under

bµ

µ(u)ω

. We thus denote

uω

, which gives us the equation

bµ(uω) = µ(u)ω.

This implies that

uω

itself is idempotent:

uω=uωuω. Abusing notation, we write uω+1 for uuω.

Convergent sequences of proﬁnite words.

The existence of

bµ

for a morphism

naturally leads to extending the notion of

convergence to inﬁnite sequences of proﬁnite words. We do not

need to introduce new elements into our algebra to capture the

limits of such sequences; any convergent sequence of proﬁnite

words behaves the same, with respect to morphisms, as a single

proﬁnite word. Let us make this more precise. A sequence

(ui)i>0

of proﬁnite words is convergent if for every morphism

µ:A∗→M

, with

a ﬁnite monoid,

(bµ(ui))i>0

is ultimately

constant. We then have

Lemma 4. For every convergent sequence

(ui)i>0

of proﬁnite

words over

, there exists a unique

w∈c

A∗

such that for

every morphism

µ:A∗→M

with

ﬁnite, we have

bµ(w) =

limibµ(ui).

We can then write

v= limiui

. In particular, we can extend

the deﬁnition of

uω

given above to the case where

is a

proﬁnite word. We will repeatedly rely on two additional

properties of convergence in

A∗

, given in the next two lemmas.

The ﬁrst allows us to extend morphisms from

A∗

into a

free proﬁnite monoid (over a possibly different alphabet) to

continuous morphisms on c

A∗:

Lemma 5. Let

A, B

be ﬁnite alphabets, and let

µ:A∗→b

B∗

be a morphism. There is a unique morphism

bµ:c

A∗→c

B∗

such that

bµ(w) = µ(w)

for every

w∈A∗

, and such that

bµ

is continuous; that is, if

(ui)i>0

is a convergent sequence of

proﬁnite words over

, then

(bµ(ui))i>0

is convergent, and

limibµ(ui) = bµ(limiui).

The second is a sequential compactness property for se-

quences of proﬁnite words:

Lemma 6. Every sequence of elements of

A∗

has a convergent

subsequence.

Inequalities.

An ordered monoid

satisﬁes the proﬁnite

inequality

u≤v

, for proﬁnite words

u, v

, if for each morphism

µ:A∗→M

, one has

bµ(u)≤bµ(v)

. In particular, if a monoid

satisﬁes a given inequality, so do all of its quotients and

submonoids. For a set

of proﬁnite inequalities, we write

JSK

for the set of ordered monoids that satisfy all the inequalities

of S.

Reiterman’s theorem (in the ordered setting) asserts that a

class

of ordered monoids is a variety if, and only if, it can

be deﬁned by a set of proﬁnite inequalities [21]. This latter set

is precisely the set of proﬁnite inequalities that are satisﬁed

by all the monoids in

. Let us write

u≤Vv

to indicate that

every monoid of

satisﬁes

u≤v

. Another way of wording

Reiterman’s theorem is thus to say that

M∈V

if, and only if,

for every inequality

u≤Vv

and every morphism

µ:A∗→M

one has

bµ(u)≤bµ(v)

. Note that

≤V

is transitive, since the

order of any ordered monoid is.

We make use of a few simple properties of the relation

≤V

Lemma 7. Let

be a variety of ordered monoids, and let

A, B be ﬁnite alphabets.

(a)

(Multiplicativity) If

u1, u2, v1, v2∈b

A∗,

with

ui≤Vvi

for

i= 1,2,then u1u2≤Vv1v2.

(b)

(Continuity) If

(ui)i>0

(vi)i>0

are convergent sequences

of proﬁnite words in

A∗,

with

ui≤Vvi

for all

i > 0,

then

limiui≤Vlimivi.

(c)

(Preservation under continuous morphisms) If

µ:A∗→

B∗

is a morphism and

u, v ∈c

A∗

with

u≤Vv,

then

bµ(u)≤bµ(v).

We are now equipped to express the polynomial closure in

purely algebraic terms. A self-contained proof is in appendix.

Theorem 8 ([19]).Let

be a variety of ordered monoids.

The following holds:

Pol(V) = Juω+1 ≤uωvuω,for any u≤VvK.

Finally, we note that if

V⊆W

are two varieties, then

u≤Wv

implies

u≤Vv

: an inequality that holds in a variety

holds in any subvariety.

D. V-pairs

Consider a variety of ordered monoids

and a ﬁnite monoid

M. The set of V-pairs of Mis:

{(dπM(u),dπM(v)) |u, v ∈d

M∗∧u≤Vv}.

Note that this set does not depend on whether

is ordered

or what its order is. Crucially, this set would be the same if

we were to use, instead of

πM

, any other surjective morphism:

Lemma 9. For any surjective morphism

µ:A∗→M

, the set

{(bµ(u),bµ(v)) |u, v ∈c

A∗∧u≤Vv}

is the set of

V-

pairs of

. If

is not surjective, then it is a

subset of all the V-pairs.

When considering a

V-

pair

(m, m′)

, we will call witnesses

a pair of proﬁnite words

u≤Vv

with

m=dπM(u)

and

m′=dπM(v)

. Witnesses of

V-

pairs always exist. A perhaps

surprising fact about

V-

pairs is that, because they “collapse”

inﬁnite properties of proﬁnite words into a ﬁnite monoid, they

are not transitive: if

(m, m′)

and

(m′, m′′)

are

V-

pairs, it does

not imply that

(m, m′′)

is a

V-

pair. Speciﬁcally,

(m, m′)

may

be witnessed by

u≤Vv

, and

(m′, m′′)

u′≤Vv′

, and

there is no reason why

and

u′

would be equal, so we cannot

deduce

u≤Vv′

, which would indeed imply that

(m, m′′)

aV-pair.

E. The semantics of V-pairs: separation

When

is ordered, our last rewording of Reiterman’s

theorem provides a salient semantics for the compatibility

of pairs and the order:

M∈V

if, and only if, for any

V-

pair

(m, n)

m≤n

. In general, even when

is not

ordered,

V-

pairs offer a ﬁne-grained characterization of how

V-

languages can separate preimages of elements of

. We

include an elementary proof both for completeness and as a

warm up to proﬁnite methods:

Lemma 10 ([1] for the unordered case).Let

be a variety of

ordered monoids. Let

µ:A∗→M

be a surjective morphism

and

m, n ∈M

, for a monoid

. The pair

(m, m′)

is a

V-

pair

if, and only if,

Lm=µ−1(m)

is not

V-

separable from

Lm′=µ−1(m′).

Proof. Left-to-right direction.

Left-to-right direction.

Let

u, v ∈c

A∗

such that

u≤V

and

bµ(u) = m, bµ(v) = m′

— they exist by Lemma 9. Let

K∈ V

such that

Lm⊆K

, we show that

contains a word

of Lm′, so that no language of Vseparates Lmfrom Lm′.

By hypothesis on

, there is a surjective morphism

ν:A∗→

, with

N∈V

, and

E⊆N

such that

K=ν−1(↑E)

. Let us

write

u, v

as convergent sequences

(ui)i>0,(vi)i>0

. For any

large enough,

µ(un) = m

, so that

un∈Lm⊆K

. This

means that

ν(un)∈ ↑E

and, ultimately, that

bν(u)∈ ↑E

. Since

N∈V

, and

u≤Vv

, this entails that

bν(v)∈ ↑E

. However,

for any

large enough, we have both that

µ(vn) = m′

, so

that vn∈Lm′, and bν(v) = ν(vn), so that vn∈K∩Lm′.

Right-to-left direction.

Assume

is not

V-

separable from

Lm′

. We need to show that there are

u, v ∈c

A∗

such that

bµ(u) = m, bµ(v) = m′

, and

u≤Vv

. To do so, we will look

at all the possible ways a language that contains

can be

recognized by a monoid in

and use these to construct a

convergent sequence of words from

and a convergent

sequence of words from

Lm′

. The limits of these two sequence

will be the proﬁnite words u, v that we require.

Let

ν1, ν2, . . .

be an enumeration of all morphisms from

A∗

to a monoid of

and write

νi:A∗→Mi

. Construct a family

(µi)i>0

of morphisms, with

µi:A∗→M1×· · ·×Mi

, by setting

µ1=ν1

and, for

i > 1

, letting

µi(w)=(µi−1(w), νi(w))

for

any

w∈A∗

. Note

M1× · · · × Mi∈ V

and that its order is

the product of the orders on

, so that if

µi(w)≤µi(w′)

then for any j≤i,νj(w)≤νj(w′).

Since

is not

V-

separable from

Lm′

, for every

the set

↑µi(Lm)∩µi(Lm′)

is nonempty; let

ui∈Lm, vi∈Lm′

such that

µi(ui)≤µi(vi)

. Let

be a convergent subsequence

(ui)i>0

and similarly

a convergent subsequence of

(vi)i>0

such that the indices taken in the two sequences are the same

for

and

(we are allowed to do that as a consequence of

compactness). We clearly have that

bµ(u) = m

and

bµ(v) = m′

We now show that

u≤Vv

by going back to the deﬁnition.

Consider any morphism from

A∗

to a monoid in

: it is

νi

for some

. There is a

j > i

such that

appears in the

sequence

appears in the sequence

, and

bνi(u) = νi(uj)

and

bνi(v) = νi(vj)

. Since

µj(uj)≤µj(vj)

and

i < j

, we

have that

νi(uj)≤νi(vj)

; consequently,

bνi(u)≤bνi(v)

. This

concludes the proof.

We say that

V-

pairs are computable if there is an algorithm

which, given an arbitrary monoid

, computes its

V-

pairs.

The previous lemma implies:

Corollary 11. Let

be a variety of ordered monoids. The

problem of

V-

separation is decidable if, and only if,

V-

pairs

are computable. Moreover, the two problems are polytime-

reducible to one another.

Remark 12. Recall that our main goal is to show that if

V-

separation is decidable, then so is

Pol(V)-

separation. A

simpler property, already used by [25], is the following: if

V-

separation is decidable, then so is

Pol(V)-

membership. This

is easily shown:

Pol(V)-

membership can be decided by testing

that the inequalities of Theorem 8 hold, that is, for a monoid

testing

dπM(uω+1)≤dπM(uωyxω)

for all

u≤Vv

. Equivalently,

this means that

mmω≤mωm′mω

for all

V-

pairs

(m, m′)

. Since the decidability of

V-

separation implies that

V-

pairs

are computable, this last property can be tested.

III. MAIN THEOREM AND STRUCTURE OF THE PROOF

In light of Corollary 11, our main theorem can be equiva-

lently worded as the following, which is the statement we will

prove:

Theorem 13 (Main theorem, reformulated).Let

be a variety

of ordered monoids and assume that

V-

pairs are computable.

There is an algorithm that computes the

Pol(V)-

pairs of any

given monoid.

We now sketch the proof structure, which is split in several

sections. Let

be a monoid,

a variety of ordered monoids,

and assume that

V-

pairs are computable; we provide an

algorithm that computes the Pol(V)-pairs of M.

•

In Section IV, we introduce the notion of relational

expansion of

. This is a ﬁnite semigroup that contains

and some extra information. We can extract from a

relational expansion, by projection, a subset of

M×M

that we call the graph of the expansion.

•

In the same Section IV, we introduce two types of rela-

tional expansions: A relational expansion is Pol(V)-cor-

rect if its graph contains only

Pol(V)-

pairs of

Pol(V)-

complete if it contains all

Pol(V)-

pairs of

. We

also introduce a property with a more technical deﬁnition:

a relational expansion is

Pol(V)-

stable if it satisﬁes a

closure condition related to the inequalities in Theorem 8.

•

In Section V, we show that if a relational expansion

Pol(V)-

stable, then it is

Pol(V)-

complete. To com-

pute

Pol(V)-

pairs, it is thus sufﬁcient to construct a

Pol(V)-

stable,

Pol(V)-

correct relational expansion of

its graph is exactly the set of

Pol(V)-

pairs of

. This is

what the algorithm for Theorem 13 does.

•

In Section VI, we provide an inﬁnite family of relational

expansions

M0, M1, M2, . . .

. These expansions are

computable provided that

V-

pairs are computable. The

algorithm for Theorem 13 will compute each of these

expansions, one by one.

•

In Section VII, we show that this sequence is eventually

constant. We call

M∞

this ﬁnal relational expansion and

note that there is a computable property that asserts that

M∞

: the algorithm for Theorem 13 will then stop

computing

’s. We show in that section that

M∞

Pol(V)-stable, implying that it is Pol(V)-complete.

•

In Section VIII, we show that

M∞

Pol(V)-

correct.

Its graph is thus the set of

Pol(V)-

pairs of

, and the

algorithm for Theorem 13 can return that set.

•

In Section IX, we recapitulate this process by explicitly

providing the algorithm for Theorem 13.

IV. COR RE CT,COMPLETE,AND STABL E RE LATIONAL

EXPANSIONS

Deﬁnition 14. Let

M, N

be ﬁnite monoids,

η:N→M

a morphism, and

ρ:N△

−→ M

be a relational morphism. The

monoid

together with the pair

(η, ρ)

is relational expansion

is a surjective morphism and for any

x∈N

η(x)∈ρ(x)

. We usually say that

itself is a relational

expansion, with

(η, ρ)

implicitly deﬁned, and, conversely, we

may speak of the relational expansion induced by (η, ρ).

The graph of Nis the subset of M×Mdeﬁned by:

[

x∈N

{(η(x), m)|m∈ρ(x)}.

Let

be a variety of ordered semigroups. The relational

expansion Nis:

•Pol(V)-

correct if its graph contains only

Pol(V)-

pairs

of M;

•Pol(V)-

complete if its graph contains all

Pol(V)-

pairs

of M;

•Pol(V)-

stable if for any

V-

pair

(x, y)

with

x2=x

and any α, β ∈ρ(x), we have that αη(y)β∈ρ(x).

We will also use these deﬁnitions for semigroup relational

expansions, where the morphisms are adjusted to be semigroup

morphisms.

Remark 15 (On the deﬁnition of stable).Our main goal

is to construct, given a monoid

, a

Pol(V)-

correct and

Pol(V)-

complete relational expansion of

. We will show in

the next section that stability implies completeness; we ﬁrst

sketch the intuition behind the deﬁnition of stability.

The deﬁnition of

Pol(V)-

stable is designed to mimic the

inequalities of Theorem 8. Let us make that more concrete.

Assume we wish to construct a relational expansion of a monoid

such that its graph is exactly the set of

Pol(V)-

pairs.

Theorem 8 provides a set of inequalities that generate

Pol(V)

so we can build our relational expansion by including those.

Speciﬁcally, consider the relational expansion of

induced

by letting

η:M→M

be the identity, and

ρ:M△

−→ M

deﬁned by setting

ρ(m)

{m}

m2=m

and, otherwise, to:

{mm′m|(m, m′)aV-pair of M}.

Disregarding the fact that

may not be a relational morphism,

this relational expansion is

Pol(V)-

correct thanks to Theo-

rem 8. Indeed, let

u≤Vv

witness the

V-

pair

(m, m′)

with

m2=m

. Theorem 8 asserts that

uω+1 ≤Pol(V)uωvuω

, and

projecting each side of this inequality with

dπM

, we conclude

that (m, mm′m)is a Pol(V)-pair of M.

So what are the

Pol(V)-

pairs that are missing for this

expansion to be

Pol(V)-

complete? First, we may need some

extra closure properties to ensure that

is indeed a relational

morphism (a property that is crucially needed in the forthcom-

ing Lemma 18) — this is not hard to ensure. Second, and much

more importantly, the operation

JSK

, as used in Theorem 8,

induces a closure under transitivity: recall that

u≤Wv

and

v≤Ww

implies

u≤Ww

. Some proﬁnite inequalities induced

by this closure do not appear as

uω+1 ≤uωvuω

. This is

why the deﬁnition of stability is not simply requiring that

η(x)η(y)η(x)∈ρ(x)

, but

αη(y)β∈ρ(x)

, for

α, β ∈ρ(x)

If the graph of the relational expansion is seen as “the

Pol(V)-

pairs already computed,” then this simple modiﬁcation

ensures that a step of transitivity is applied. We encourage

the reader to keep this subtlety in mind while going through

the proof that

Pol(V)-

stability entails

Pol(V)-

completeness

(Lemma 18). We will further discuss how

Pol(V)-

stability

and

Pol(V)-

correctness conspire to make ﬁnding a relational

expansion with both properties difﬁcult in Section VI-A.

V. Pol(V)-STABLE I MP LI ES Pol(V)-COMPLETE

To show the claimed result, we will rely on the following

lemma which appears in [2, Thm. 3.1 & Cor. 3.2] (a self-

contained proof is in the appendix):

Lemma 16. Let

be a variety of ordered monoids such that

Pol(W) = W

. Let

u, v ∈c

A∗

with

u≤Wv

. Suppose that there

are

u′, u′′ ∈c

A∗

and

ℓ∈A∪ {ε}

such that

u=u′ℓu′′

. Then

there exist

v′, v′′ ∈c

A∗

such that

v=v′ℓv′′

with

u′≤Wv′

and u′′ ≤Wv′′.

We will also rely on a proﬁnite version of the classical

factorization forest theorem of Simon:

Theorem 17. Let

µ:A∗→M

be a morphism with

a ﬁnite

monoid. There is a function

d:c

A∗→ {0,1,...,3|M|}

called

the decomposition depth such that for any u∈c

A∗:

•If d(u)=0, then u=εor u∈A;

•

d(u)>0

, then there are words

u1, u2∈c

A∗

with

d(u1), d(u2)< d(u)

such that one of the following holds:

–u=u1u2

, in which case we say that

has binary type;

–u=u1u′u2

for some

u′∈c

A∗

with

bµ(u1) = bµ(u′) =

bµ(u2) = bµ(u)

an idempotent of

, in which case we

say that uhas idempotent type.

We are now ready to show the main result of this section:

Lemma 18. Any relational expansion that is

Pol(V)-

stable is

also Pol(V)-complete.

Proof.

Let

be a relational expansion of some monoid

and let

(η, ρ)

be the morphisms that induce

. Recall that

is a surjective morphism from

and

is a relational

morphism from

, such that for any

x∈N

η(x)∈ρ(x)

We show that for any

u≤Pol(V)v

with

u, v ∈c

N∗

, we

have that

(bη(u),bη(v))

is in the graph of

(recall that

surjective, so that any

Pol(V)-

pair of

can be obtained in

that fashion). To do so, we show that

bη(v)∈bρ(u)

. This is

shown by induction on the decomposition depth of

with

respect to πN, as deﬁned in Theorem 17.

If uhas decomposition depth 0.

In that case,

is either

the empty word or a single letter.

•

u=ε

, then

cπN(u)=1N

. Since

u≤Pol(V)v

implies

that

u≤Vv

, the pair

(1N,cπN(v))

is a

V-

pair of

. In

addition, since

is a relational expansion of

, we get

that

1M∈ρ(1N)

and

Pol(V)-

stability then assert that

ρ(1N)

contains

1Mbη(v)1M=bη(v)

. Since

ρ(1N) = bρ(u)

we are done.

•

u=x∈N

, then Lemma 16, seeing

εxε

, provides

ε≤Pol(V)v1, v2

with

v=v1xv2

. The previous case

(

u=ε

) shows that

bη(vb)∈bρ(ε)

b= 1,2

. Additionally,

since

is a relational expansion of

, we have that

bη(x)∈bρ(x), and thus:

bη(v) = bη(v1)bη(x)bη(v2)∈bρ(ε)bρ(x)bρ(ε)⊆bρ(x).

Inductive step: Case uhas binary type.

This implies that

u=u1u2

with

b= 1,2

, of strictly smaller decomposition

depth. Lemma 16 provides us with a rewriting of

v1v2

with

ub≤Pol(V)vb

b= 1,2

. The induction hypothesis then

asserts that

bη(vb)∈bρ(ub)

, for

b= 1,2

. We obtain, as desired:

bη(v) = bη(v1v2) = bη(v1)bη(v2)∈

bρ(u1)bρ(u2)⊆bρ(u1u2) = bρ(u).

Inductive step: Case uhas idempotent type.

This implies

that

u=u1u′u2

with

u1, u2

of strictly smaller decomposition

depth and

cπN(u) = cπN(u1) = cπN(u′) = cπN(u2) = x

with

x∈N

idempotent. Lemma 16 shows that

v=v1v′v2

with

ub≤Pol(V)vb

b= 1,2

, and

u′≤Pol(V)v′

. The induction

hypothesis asserts that

bη(vb)∈bρ(ub)

, for

b= 1,2

. From

u′≤Pol(V)v′

, we obtain that

u′≤Vv′

, implying that

(cπN(u′),cπN(v′)) = (x, cπN(v′))

is a

V-

pair of

. From

Pol(V)-stability and since bρ(ub) = ρ(x)for b= 1,2:

bη(v) = bη(v1v′v2) = bη(v1)bη(v′)bη(v2)∈ρ(x) = bρ(u).

VI. A S EQUENCE M0, M1, M2, . . . OF RELATI ONAL

EXPANSIONS

A. The difﬁculty

Before going further, we identify the challenges presented

when one tries to construct a

Pol(V)-

stable,

Pol(V)-

correct

relational expansion.

It is fairly easy to construct a

Pol(V)-

stable relational

expansion of a monoid

: the one induced by letting

η:M→M

be the identity map and

ρ:M△

−→ M

ρ(m) = M

, for any

m∈M

, is

Pol(V)-

stable, but since

its graph is

M×M

, this does not give us any interesting

information on

Pol(V)-

pairs. Similarly, we have seen in

Remark 15 that it is not hard to construct a

Pol(V)-

correct

relational expansion. An even simpler

Pol(V)-

correct relational

expansion than the one of Remark 15 is the one induced by

setting

η:M→M

to the identity and

to map any

m∈M

{m}

. This expansion is correct for any variety, since its graph

is simply

{(m, m)|m∈M}

. Constructing a

Pol(V)-

stable,

Pol(V)-

correct relational expansion is thus a task where our

two goals seem to pull in opposite directions.

A natural strategy is to start with the trivial relational

expansion, the one with

ρ(m) = {m}

, and to add elements

ρ(m)

, iteratively, so that we reach

Pol(V)-

stability — we

simply need to ensure that if we add an element

m′

ρ(m)

we have that (m, m′)is a Pol(V)-pair.

The most natural inductive deﬁnition is as follows. Let

η:M→M

be the identity, and for every

m∈M

, deﬁne

ρ0(m) = {m}

and set

ρi(m)

to be

ρi−1(m)

m2=m

, and,

otherwise, set ρi(m)to:

ρi−1(m)∪

{αm′β|α, β ∈ρi−1(m)∧(m, m′)aV-pair of M}.

Since this follows closely the deﬁnition of

Pol(V)-

stability,

this certainly seems to converge to a

Pol(V)-

stable relational

expansion (we disregard the technicality that

ρi

may not

be a relational morphism). One would like to conclude

that the relational expansion

induced by

(η, limiρi)

Pol(V)-

correct, by inductively proving that each relational

expansion is Pol(V)-correct. Let us attempt to do so.

Assume that the relational expansion at level

i−1

Pol(V)-

correct. We want to show that the new elements added

to the graph of the relational expansion at level

are indeed

Pol(V)-pairs.

Consider

m∈M

such that

m2=m

, a

V-

pair

(m, m′)

and

α, β ∈ρi−1(m)

, we want to show that

(m, αm′β)

is a

Pol(V)-

pair. Since

(m, m′)

is a

V-

pair of

, it is witnessed

by proﬁnite words

u≤Vv

. Similarly, by hypothesis, we have

that

(m, α)

is witnessed by

uα≤Pol(V)vα

and

(m, β)

witnessed by

uβ≤Pol(V)vβ

. Assume that

uω=uα=uβ

then we can deduce:

uω+1 ≤Pol(V)uωvuω(Theorem 8)

≤Pol(V)uαvuβ.

Taking the image under

dπM

on both sides, we do ﬁnd that

(m, αm′β)

is a

V-

pair. Here lies the crux of the difﬁculty:

We cannot assume, in general, that

uω=uα=uβ

. We have

no information within this relational expansion about how the

computed Pol(V)-pairs can be witnessed. We will thus build

relational expansions that embed some information to help us

“lift” pre-computed Pol(V)-pairs in a simultaneous way.

Let us stress that we do not know whether the relational

expansion we just constructed is

Pol(V)-

correct. As far as we

know, it might be, but we simply do not know how to show

it. We will come back to this point in Remark 25, where we

present a property of our more complex expansions that would

entail that this construction is indeed correct.

B. Construction

Let

be a ﬁnite monoid. We construct a sequence of

relational expansions of

with the speciﬁc aim of obtaining, in

the limit, a monoid that is a

Pol(V)-

stable relational expansion

, that we will show is

Pol(V)-

correct in Section VIII.

Each expansion will be a subsemigroup of

M×2M

where the

projection on the left will be surjective and the projection on

the right will be a relational morphism (consistent with the

deﬁnition of relational expansion). In the limit, this expansion

will have a neutral element and thus be a monoid.

Machinery on M×2M.

We write

π: (M×2M)+→M×

for the canonical semigroup morphism of

M×2M

. For

any element

x= (m, E)∈M×2M

, we write

ℓ

(x) = m

and

πr(x) = E

and extend

ℓ

and

πr

to morphisms from

(M×2M)+

to, respectively,

and

. We have, in particular,

that

π(w) = (π

ℓ

(w), πr(w))

, and that

ℓ

◦π=π

ℓ

and

πr◦π=

πr.

We identify a partial order

⪯

over

M×2M

(m, E)⪯

(m′, E′)⇔(m=m′∧E⊆E′)

. For a set

K⊆M×2M

, we

write

↓K

for the set of elements in

M×2M

that are

⪯

-smaller

than some element in

. It should be noted that although,

strictly speaking,

M×2M

is an ordered monoid with respect

to this partial order, and the

↓

operator has the same meaning

as before, we will see

M×2M

as an unordered monoid, and

rely on ⪯as a tool in our construction.

The construction.

We construct a sequence

M0, M1, M2, . . .

of subsemigroups of M×2Mas follows:

•M0={(m, {m})|m∈M};

•

For any

, we deﬁne a semigroup morphism

▷i: (M×

2M)+→M×2M

as follows. Let

x= (m, E)∈M×2M

Let

be equal to

is a monoid, and to

Mi∪

{1Mi}

otherwise, with

1Mi

a new neutral element. We

extend

ℓ

to a monoid morphism by setting

ℓ

(1Mi) =

1M. Now let E′:= Eif x2=x, and

E′:= E∪{απ

ℓ

(y)β|(x, y)aV-pair of M1

i∧α, β ∈E};

otherwise. Then we set

▷i(x)=(m, E′)

, and extend this

to a morphism on (M×2M)+.

•

For any

, the set

is the subset of maximal elements

under

⪯

, in other words, it is the smallest subset

of Misuch that Mi⊆ ↓Ki;

•

For any

Mi+1 =⟨▷i(Ki)⟩

, that is,

Mi+1

is the

subsemigroup of

M×2M

generated by

▷i(Ki)

; in other

words, Mi+1 =▷i(K+

i).

We note these two simple properties about that construction:

Fact 1. For any i≥0and u∈d

M+

i,bπ(u)⪯b

▷i(u).

Fact 2. For any

i≥0

and any

u∈d

M+

bπ

ℓ

(u)∈bπr(u)

; this

shows that every

is indeed a semigroup that is a relational

expansion of M.

The situation.

The state of affairs is depicted in Figure 1 as

a commutative diagram of semigroups to which we will be

adding more edges in Section VIII-A.

VII. THE SEQUENCE EVENTUALLY RE ACHES A MONOID

M∞THAT IS A Pol(V)-STABLE RELATIONAL EXPANSION

Lemma 19. There is a value

such that

▷i

is the identity

over

and

Mi=Mi+1 =· · ·

. The value of

can be taken

to be the ﬁrst index such that Ki−1=Ki.

Proof.

For any

, by deﬁnition,

▷i(x, E) = (x, F )

for some

F⊇E

. This implies that

↓Ki⊆ ↓▷i(Ki)⊆ ↓Ki+1

. There is

thus a value isuch that:

↓Ki−1=↓▷i−1(Ki−1) = ↓Ki=↓▷i(Ki).

Since all the

’s only contain incomparable elements,

Ki−1=

Ki.

We now show that

▷i

is the identity over

. From

↓Ki=

↓▷i(Ki)

, we deduce that these sets have the same maximal

elements

, showing that

Ki⊆▷i(Ki)

. But

|▷i(Ki)| ≤

|Ki|

, since

▷i

is a function, so

▷i(Ki) = Ki

. Now for any

x∈Ki

, Fact 1 asserts that

x⪯▷i(x)

, and since

contains

only incomparable elements and both

and

▷i(x)

are in

▷i(x) = x

. The same argument shows that

▷i−1

is the identity

over Ki−1=Ki, and so:

Mi+1 =⟨▷i(Ki)⟩=⟨▷i−1(Ki−1)⟩=Mi.

We now show that

▷i

is the identity over

. For any

x∈Mi+1

, we have that

x=▷i(x1· · · xn)

for some

xj∈Ki

and so:

▷i(x) = ▷i(▷i(x1· · · xn))

=▷i(▷i(x1)· · · ▷i(xn))

=▷i(x1· · · xn) = x.

Since Mi=Mi+1, we are done.

Finally, since

Mi=Mi+1

, we have that

▷i=▷i+1

and

Ki=Ki+1

, showing that

Mi+2 =⟨▷i+1(Ki+1 )⟩

is the same

⟨▷i(Ki)⟩=Mi+1

. Iterating this argument shows that

Mi=

Mi+1 =· · ·.

We write

M∞, K∞,▷∞, . . .

for the objects at level

given

by the previous lemma. An immediate consequence of the

previous lemma is the following:

Corollary 20.

M∞

is a monoid that is a

Pol(V)-

stable

relational expansion of M.

Proof.

That

M∞

is a monoid is proved in appendix. We

argue that

M∞

Pol(V)-

stable. Let

(x, y)

be a

V-

pair

M∞

with

x2=x

and

α, β ∈πr(x)

; we need to show

that

απ

ℓ

(y)β∈πr(x)

. By construction,

πr(▷∞(x))

contains

απ

ℓ

(y)β

, but since

▷∞(x) = x

πr(▷∞(x)) = πr(x)

and we

are done.

VIII. M∞IS Pol(V)-CORRECT

The goal of this section is to show:

Theorem 21.

M∞

is a

Pol(V)-

correct relational expansion

of M.

Proof.

In the next section, in Lemma 22, we will show that

for every

, and

x∈Mi

, we can ﬁnd a proﬁnite word

τi(x)

whose image under

bπ

ℓ

is the same as

. Furthermore, for

every

m∈πr(x)

, there is a proﬁnite word

that maps to

through

bπ

ℓ

, with

τi(x)≤Pol(V)um

. This immediately entails

that (π

ℓ

(x), m)is a Pol(V)-pair of M, by Lemma 9.

Thus for every

w∈K+

, and every

m∈πr(▷i(w))

, the pair

(π

ℓ

(w), m)

is a

Pol(V)-

pair of

. Since

M∞=⟨K∞⟩

and

▷∞

is the identity map over

M∞

, this implies in particular that

for every

x∈M∞

and every

m∈πr(x)

, the pair

(π

ℓ

(x), m)

is a Pol(V)-pair of M.

A. Burst and Lift

Burst.

Let

i≥0

. Let us ﬁx a pair of witnesses for each

V-

pair of

; speciﬁcally, for

(x, y)

V-

pair of

, let

uxy ≤Vvxy

be proﬁnite words in

M+

such that

bπ(uxy ) = x

and bπ(vxy ) = y.

Let the continuous morphism

bursti:d

K+

i→d

M+

be deﬁned,

for any x∈Ki, by bursti(x) = xif x2=x, and otherwise:

bursti(x) = Y

(x,y)aV-pair

of Mi

uω+1

xy .

Note that the order in which the product of the

uω+1

is taken is

not important and that

bursti

is indeed a continuous morphism

(by Lemma 5). We note the following elementary fact:

Fact 3. For any iand u∈d

K+

i,bπ(bursti(u)) = bπ(u).

Lift.

Let

i≥0

. We write

lifti:

[

M+

i+1 →d

K+

for the

continuous morphism deﬁned by setting, for any

x∈Mi+1

lifti(x)

to be an arbitrary element

w∈K+

such that

▷i(w) = x

. Note that it is a morphism of words extended to the

respective proﬁnite semigroups; in particular, if

x, y ∈Mi+1

and

z=xy

is their product in

Mi+1

, it is not true that

lifti(z) =

lifti(x)lifti(y)

. However,

▷i(lifti(z)) = ▷i(lifti(x)lifti(y))

and, more generally:

Fact 4. For any

i≥0

and any

u∈

[

M+

i+1

, we have

▷i(lifti(u)) = bπ(u).

Burst and lift.

Deﬁne inductively

τi:d

K+

i→d

M+

by setting

τ0=burst0

and

τi=τi−1◦lifti−1◦bursti

. (Here,

is short

for the French word

émoin.) Combining Fact 3 and Fact 4:

Fact 5. For any

i≥0

and any

u∈d

K+

, we have that

bπ

ℓ

(u) = bπ

ℓ

(τi(u)).

M+

0K+

0M+

1K+

1M+

2K+

2· · ·

M0M1M2· · ·

ππ▷0

⊇

ππ▷1

⊇

ππ

⊇

ℓ

Figure 1. Commutative diagram of semigroups for the main construction. It indicates that some compositions of morphisms are equal, by following edges

from the same source and destination; for instance, following paths from K+

1to M, the diagram expresses that π

ℓ

◦▷1=π

ℓ

◦π.

The situation.

The current state of affairs is depicted as a

commutative diagram of semigroups in Figure 2. To include

bursti,lifti,

and

τi

in this diagram, the top row is a sequence

of proﬁnite semigroups.

B. Correctness lemma

We are now ready to prove the correctness lemma:

Lemma 22. For any

i≥0

, any

u∈d

K+

, and any

m∈

πr(b

▷i(u))

, there is a proﬁnite word

um∈d

M+

, such that

bπ

ℓ

(um) = mand τi(u)≤Pol(V)um.

Proof.

First, we note that it is enough to prove the claim for

the case where uis a single-letter word:

Fact 6. If the claim holds for any

u∈Ki

, then it holds for

any u∈d

K+

Proof.

The full proof is in appendix. It proceeds by showing

that the statement is closed under concatenation over ﬁnite

words, in the sense that if the claim is satisﬁed for two ﬁnite

words

u:= w1

and

u:= w2

then it will also be satisﬁed for

u:= w1w2

. Then, it is argued that the claim over ﬁnite words

implies that it holds over proﬁnite words, using elementary

techniques.

Equipped with the previous fact, we need only show the

claim for

an element of

that we will write

. We proceed

by induction on

. Let

m∈πr(▷i(x))

. We construct the word

um∈d

M+of the claim as follows.

Base case: i= 0.

By deﬁnition of

, we have that

(t, {t})for some t∈M. We distinguish several cases:

•

x2=x

, then

τ0(x) = burst0(x) = x

, and

▷0(x) = x

Since

m∈πr(▷0(x)) = πr(x)

, we have that

m=t

Consequently, setting um:= xsatisﬁes the claim.

•

Assume now that

x2=x

and that

can be written as

απ

ℓ

(y)β

for a

V-

pair

(x, y)

and

α, β ∈πr(x)

Since

πr(x) = {t}

, it must hold that

α=β=t

. We thus

have that

m=tπ

ℓ

(y)t

. Let

uxy ≤Vvxy

be the witnesses

in d

M+

0of the V-pair (x, y). We have that:

uω+1

xy ≤Pol(V)uω

xyvxy uω

xy.

Since there are proﬁnite words

p, s ∈d

M+

with

bπ(p) =

bπ(s) = xsuch that burst0(x) = puω+1

xy s, we have:

τ0(x) = burst0(x) = puω+1

xy s≤Pol(V)puω

xyvxy uω

xys

As xis idempotent, um:= puω

xyvxy uω

xysis such that

bπ

ℓ

(um) = π

ℓ

(bπ(puω

xy)bπ(vxy )bπ(uω

xys))

=π

ℓ

(xyx) = tπ

ℓ

(y)t=m,

so that umsatisﬁes the claim.

•

The only case left is if

x2=x

and

m∈πr(x)

. This

entails that

m=t

. However, since

(x, x)

is a

V-

pair of

, this means that

m=t=ttt

can be expressed as

απ

ℓ

(y)β

for

α, β ∈πr(x)

and

(x, y)

V-

pair of

, and

this is thus covered by the previous case.

Inductive step.

We assume the claim to be true at level

i−1and show it holds at level i. We consider two cases:

Inductive step: Case m∈πr(x).

Let

u∈

[

K+

i−1

be de-

ﬁned as

u=lifti−1(bursti(x))

. Fact 4 implies that

▷i−1(u) =

π(x)

, and we can thus apply the induction hypothesis on

and

m∈πr(b

▷i−1(u))

, giving us a proﬁnite word

um∈d

M+

such that

bπ

ℓ

(um) = m

and

τi−1(u)≤Pol(V)um

. But

τi−1(u) = τi(x)

, so we obtain

τi(x)≤Pol(V)um

and we

are done.

Inductive step: Case m /∈πr(x).

We have

x2=x

, by

construction, and so

m=απ

ℓ

(y)β

for some

V-

pair

(x, y)

, and

α, β ∈πr(x)

. Consider the pair of proﬁnite words

uxy ≤Vvxy

M+

that witness the

V-

pair

(x, y)

. We can

write

bursti(x)

puω+1

xy s

for some proﬁnite words

p, s

that

both map to

through

bπ

. Since

uxy ≤Vvxy

, we have that

uω+1

xy ≤Pol(V)uω

xyvxy uω

xy, and so:

bursti(x) = puω+1

xy s≤Pol(V)puω

xyvxy uω

xys.

In order to use our induction hypothesis, let

upxy =

lifti−1(puω

xy)

and

uxys =lifti−1(uω

xys)

, two proﬁnite words

[

K+

i−1

. By Fact 4, and since

bπ(puω

xy) = bπ(uω

xys) = x

, we

have that

▷i−1(upxy) = b

▷i−1(uxys) = x

. This shows that

α, β ∈πr(x) = πr(b

▷i−1(upxy)) = πr(b

▷i−1(uxys)).

We can thus apply the induction hypothesis at level

i−1

twice, setting ﬁrst

u:= upxy

and

m:= α

then

u:= uxys

and

m:= β

, to obtain proﬁnite words

uα, uβ∈d

M+

such

M+

K+

M+

K+

M+

K+

2· · ·

M0M1M2· · ·

bπbπb

▷0

τ0

burst0

bπ

lift0

bπb

▷1

burst1

τ1

bπ

lift1

bπ

burst2

τ2

ℓ

Figure 2. Commutative diagram of semigroups for the main construction, with additional morphisms.

that

bπ

ℓ

(uα) = α

bπ

ℓ

(uβ) = β

τi−1(upxy)≤Pol(V)uα

, and

τi−1(uxys)≤Pol(V)uβ.

We thus obtain, via several applications of Lemma 7, the

following chain, where ≤is ≤Pol(V):

τi(x) = τi−1(lifti−1(bursti(x)))

=τi−1(lifti−1(puω+1

xy s))

≤τi−1(lifti−1(puω

xyvxy uω

xys))

≤τi−1(lifti−1(puω

xy)) ·τi−1(lifti−1(vxy )) ·

τi−1(lifti−1(uω

xys))

≤τi−1(upxy)·τi−1(lifti−1(vxy )) ·τi−1(uxys )

≤uα·τi−1(lifti−1(vxy)) ·uβ.

We can now set

um:= uα·τi−1(lifti−1(vxy)) ·uβ

, and we

obtain bπ

ℓ

(um) = απ

ℓ

(y)β=m.

IX. WRAPPING UP: THE ALGORITHM

A. The algorithm

To compute the

Pol(V)-

pairs of a monoid

, we can

compute each relational expansion

one after the other

(recall that

with possibly a neutral element added

to ensure it is a monoid). These relational expansions can be

computed provided that

V-

pairs are computable. Once we reach

Ki−1=Ki

, we have that

i=M∞

(Lemma 19). Since

M∞

Pol(V)-

stable (Corollary 20) — hence

Pol(V)-

complete

(Lemma 18) — and

Pol(V)-

correct (Theorem 21), its graph

is exactly the set of

Pol(V)-

pairs of

. As an algorithm, we

obtain the pseudocode of Algorithm 1, in which

Exp

takes the

role of the successive expansions

0, M 1

1, . . .

and

MaxElts

is used for

K0, K1, . . .

An analysis of this algorithm shows

that:

Theorem 23. If

V-

pairs are computable in time

cV(n)

with

cV(n)≥n

, then

Pol(V)-

pairs are computable in time

O(cV(2O(n))).

Algorithm 1

Compute

Pol(V)-

pair of a monoid, assuming

V-pairs are computable

1: procedure POLVPAIRS(M)▷ M a ﬁnite monoid

2: Exp ← {(m, {m})|m∈M}▷ Exp =M0=M1

3: MaxElts ←Exp ▷ M axElts =K0

4: repeat

5: NewExp ← {(m, E)∈Exp |m2=m}

6: for all (m, E)∈Exp with m2=mdo

7: E′←E

8: for all ((m, E),(m′, F )) V-pair of Exp do

9: for all α, β ∈Edo

10: E′←E′∪ {αm′β}

11: end for

12: end for

13: NewExp ←NewExp ∪ {(m, E′)}

14: end for

15: OldM axElts ←M axElts

16: Exp ← ⟨N ewExp⟩1

17: MaxElts ← {(m, T )∈Exp |

18: (∀T′=T)[(m, T ′)∈Exp ⇒T⊆ T′]}

19: until OldM axElts =M axElts

20: return {(m, m′)|(∃E)[(m, E)∈Exp ∧m′∈E]}

21: end procedure

B. Tightness: Early exit and a conjecture

The halting condition of the previous algorithm is the one

provided by Lemma 19. A different condition is to test whether

Exp

itself is

Pol(V)-

stable; however, if

Pol(V)-

stable,

then

Mi=Mi+1

, hence this condition would not reduce

signiﬁcantly the number of times the main loop is taken. Rather

than testing whether

Pol(V)-

stable, we can test whether

’s graph induces a

Pol(V)-

stable relational expansion (with

as base monoid). This latter property can be worded as a

property of the graph itself, which can be tested in polynomial

time:

Lemma 24. We say that a subset

G⊆M×M

Pol(V)-

tight

if for every

V-

pair

(m, n)

with

m2=m

and for every

(m, m′),(m, m′′)∈G, we have that (m, m′nm′′ )∈G.

Consider the sequence of relational expansions

M0, M1, . . .

deﬁned earlier. If there is

i≥0

such that the graph

Miis Pol(V)-tight, then Gis the set of Pol(V)-pairs of M.

Proof. By Theorem 21, Gcontains only Pol(V)-pairs of M.

Note that

is a submonoid of

M×M

: if

(m, m′),(n, n′)∈

, then there are

x, y ∈Mi

such that

m=π

ℓ

(x), n =π

ℓ

(y)

and

m′∈πr(x), n′∈πr(y)

. Consequently,

mn =π

ℓ

(xy)

and

m′n′∈πr(x)πr(y)⊆πr(xy). Hence (mn, m′n′)∈G.

Consider now the relational expansion of

induced by

letting

η:M→M

be the identity and letting

ρ(m) = {m′|

(m, m′)∈G}

for any

m∈M

. Note that

is indeed a

relational morphism, since

is a submonoid of

M×M

, and

that, by construction, the graph of this expansion is G.

The hypothesis on

immediately implies that this expansion

Pol(V)-

stable. By Lemma 18,

thus contains all the

Pol(V)-pairs of M.

Remark 25. If any relational expansion

has a

Pol(V)-

tight

graph

, then

is the smallest submonoid

M×M

that is

Pol(V)-

tight. To show this, ﬁrst note that

P⊆G

by deﬁnition.

Next, the argument of the previous lemma showing that

contains all

Pol(V)-

pairs of

can be repeated for

, hence

G⊆P, since Gis the set of all Pol(V)-pairs.

This set

is also the graph of the construction presented

in Section

VI-A

and it is computable in polynomial time given

the monoid

, provided that

V-

pairs are also computable in

polynomial time. We conjecture:

Conjecture 26. For any variety of ordered monoid

and

any monoid

, one of the expansions

M0, M1, . . .

has a

Pol(V)-

tight graph. Consequently, if

V-

pairs are computable

in polynomial time, so are Pol(V)-pairs.

X. APPLICATION TO CONCATENATION HIERARCHIES

Deﬁnition 27. The concatenation hierarchy over a positive

variety of regular languages

is deﬁned by setting

ch0(V) = V

and, for any

i≥0

chi+1

2(V) = Pol(chi(V))

and

chi+1(V) =

Bool(chi+1

2(V))

, where

Bool

denotes the Boolean closure of

a class. We write

ch(V)

for

Sichi(V)

. We write

chi(V)

and

ch(V)

for the algebraic counterparts of these varieties, as

provided by the ordered version of Eilenberg theorem.

Remark 28. Two concatenation hierarchies are extensively

studied in the literature. The ﬁrst one was mentioned in the

introduction: it is deﬁned by setting

V:= {∅, A∗}

, and it turns

out that

chi+1

2(V)

Σi[<]

. The other one is deﬁned by setting

V:= G

, the variety of group languages, that is, languages

recognized by groups. We note that separability for

, in both

cases, can be decided in polynomial time: this is trivial for

V={∅, A∗}

and proved by van Rooijen [28], and appearing

in [26], in the case of V=G.

Theorem 29. Let

be a positive variety of regular languages

and assume

V-

separation is decidable in time

cV(n)

. Let

Tower(n, i)

be deﬁned by

Tower(n, 0) = n

and

Tower(n, i) =

2Tower(n,i−1)

. For any

i≥0

chi+1

2(V)-

separation is decidable

in time Tower(O(cV(n)), i).

Corollary 30. Assume Conjecture 26 holds. Let

be a positive

variety of regular languages and assume

V-

separation is

decidable in time

cV(n)

. For any

i≥0

chi+1

2(V)-

separation

is decidable in time polynomial in

cV(n)

. Further, so is

ch(V)-separation.2

Proof.

The ﬁrst part is immediate. As for

ch(V)-

separability,

note that Conjecture 26 implies that there is a function that

takes a set of pairs

of a monoid

, and produces another set

of pairs

E′

, such that for any variety

, if

were the

W-

pairs

, then

E′

is the

Pol(W)-

pairs of

. In particular, this

means that for any

i≥0

, if the

chi−1

2(V)-

pairs of

are

the same as the

chi+1

2(V)-

pairs, then the set of pairs is the

same for any

chj+1

2(V)

j≥i

. Since the sets of pairs only

shrink in size from

chi−1

2(V)

chi+1

2(V)

, this means that the

ch(V)-

pairs of

are the same as its

ch|M|2+1

2(V)-

pairs. We

can thus compute the

ch(V)-

pairs in polynomial time and, by

Corollary 11, decide

ch(V)-

separation in polynomial time.

XI. PERSPECTIVES AND OPEN QUESTIONS

Boolean closures.

We leave open whether membership is

decidable for the Boolean closure of

Σi[<]

. Even though the

literature has long been focused on a solution for both

Σi[<]

and its Boolean closure, the historical “dot-depth” question

was asked about the latter. However, we conjecture:

Conjecture 31. For any positive variety

of regular languages,

V-

separation is decidable, then membership in the Boolean

closure of Vis decidable.

FO with other predicates.

Common extensions of

revolve

around equipping ﬁrst-order logic with additional predicates on

the numerical value of positions to deﬁne more languages. The

logic

FO[arb]

, for instance, allows any predicate (e.g.,

x=y×

), while

FO[<, +1,mod]

only enriches

FO[<]

with predicates

x=y+1

and

x= 0 mod p

, for any constant

. A striking fact,

conjectured by McNaughton in the 1960s [11] and showed by

Barrington, et al. [5], is that the regular languages of

FO[arb]

are precisely the languages expressible in

FO[<, +1,mod]

Techniques of [20] translate the decidability of membership

to the alternation hierarchy

Σi[<]

to the alternation hierarchy

Σi[<, +1]

, while an ordered-monoid version of [9] would show

the same for

Σi[<, +1,mod]

. This leaves open the decidability

for

Σi[mod]

. If the requirement of “variety” were relaxed to

“quotienting algebra” in our main result, then a result of [24]

would entail:

Conjecture 32. Separation for Σi[mod]is decidable.

With

V={∅, A∗}

, the class

ch(V)

is equal to the class

of star-free

languages. It is known that

SF-

pairs are computable in exponential time,

hence, by Corollary 11,

SF -

separation is also decidable in exponential time.

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CONTENTS

I Introduction 1

II Preliminaries 3

II-A Sets, languages . . . . . . . . . . . . . 3

II-B Algebraic structures . . . . . . . . . . . 3

II-C Proﬁnite words and inequalities . . . . . 4

II-D V-pairs.................. 5

II-E The semantics of V-pairs: separation . 5

III Main theorem and structure of the proof 6

Correct, complete, and stable relational expan-

sions 6

VPol(V)-stable implies Pol(V)-complete 7

A sequence

M0, M1, M2, . . .

of relational expan-

sions 8

VI-A The difﬁculty . . . . . . . . . . . . . . 8

VI-B Construction . . . . . . . . . . . . . . . 8

VII

The sequence eventually reaches a monoid

M∞

that is a Pol(V)-stable relational expansion 9

VIII M∞is Pol(V)-correct 9

VIII-A Burst and Lift . . . . . . . . . . . . . . 9

VIII-B Correctness lemma . . . . . . . . . . . 10

IX Wrapping up: The algorithm 11

IX-A The algorithm . . . . . . . . . . . . . . 11

IX-B Tightness: Early exit and a conjecture . 11

X Application to Concatenation Hierarchies 12

XI Perspectives and Open Questions 12

Appendix 15

A Proof of Lemma 4 . . . . . . . . . . . . 15

B Proof of Lemma 5 . . . . . . . . . . . . 15

C Proof of Lemma 6 . . . . . . . . . . . . 15

D Proof of Lemma 7 . . . . . . . . . . . . 16

E Proof of Theorem 8 . . . . . . . . . . . 16

F Proof of Lemma 9 . . . . . . . . . . . . 17

G Proof of Corollary 11 . . . . . . . . . . 17

H Proof of Lemma 16 . . . . . . . . . . . 17

I Proof of Theorem 17 . . . . . . . . . . 18

J Proof of Fact 1 . . . . . . . . . . . . . 18

K Proof of Fact 2 . . . . . . . . . . . . . 18

L Proof of the ﬁrst part of Corollary 20 . 19

M Proof of Fact 3 . . . . . . . . . . . . . 19

N Proof of Fact 4 . . . . . . . . . . . . . 19

O Proof of Fact 5 . . . . . . . . . . . . . 19

P Proof of Fact 6 . . . . . . . . . . . . . 19

Q Proof of Theorem 23 . . . . . . . . . . 19

R Proof of Theorem 29 . . . . . . . . . . 19

APPENDIX

A. Proof of Lemma 4

Let

(ui)i>0

be a convergent sequence of proﬁnite words

over

We will show how to construct a proﬁnite word

such that

bµ(w) = limibµ(ui).

We enumerate all the morphisms

ϕi:A∗→Mi, i > 0

into ﬁnite monoids. This can be done because there are only

countably many such morphisms. We then form direct products

of these, giving a morphism

ψi=ϕ1× · · · ϕi:A∗→M1× · · · × Mi, i > 0.

Let

mi= limkb

ϕi(uk)∈Mi.

We then have

limkb

ψi(uk) = (m1, . . . , mi).

We claim that there is a convergent sequence

(wi)i>0

ﬁnite words over Asuch that for each j > 0,

limkψj(wk)=(m1, . . . , mj).

Assuming the claim for the moment, we complete the proof

as follows: If

µ:A∗→M

is a morphism with

ﬁnite, then

µ=ϕj

and

M=Mj

for some

Since

ϕj

is the composition

ψj

with projection onto the

jth

component, our claim shows

that

limkµ(wk) = mj= limkbµ(uk).

Letting

denote the proﬁnite word represented by the sequence

(wi)i≥0gives the desired result.

To prove the claim, for each

p > 0,

we choose

large

enough so that

ψp(uk) = limkc

ψp(uk)=(m1, . . . , mp)

for all

k≥kp.

We let

vp,1, vp,2, . . .

be a sequence of ﬁnite

words representing ukp.Since

limrψp(vp,r) = c

ψp(ukp)=(m1, . . . , mp),

we can choose

large enough so that

ψp(vp,rp) =

(m1, . . . , mp).Set wp=vp,rp.

Let

j > 0.

k≥j

then

ψj(wk)

is the projection of

ψk(wk)

onto the ﬁrst jcomponents; that is

ψj(wk)=(m1, . . . , mj)

for k≥j, which proves the claim.

Observe that by deﬁnition, all sequences of ﬁnite words

representing the sequence

(ui)i>0

in this sense are equivalent,

so the proﬁnite word wis unique.

B. Proof of Lemma 5

Let µ:A∗→c

B∗as in the statement of the lemma. Let

u= (ui)i≥0

be a convergent sequence of ﬁnite words over

The sequence

(µ(ui))i≥0

of proﬁnite words over

is also convergent, since if

ν:B∗→M

is a morphism into a ﬁnite monoid, the sequence

(bν(µ(ui)))i>0

is ultimately constant. It is immediate that if

u′= (u′

i)i≥0

is another convergent sequence equivalent to u,

then the sequences

(µ(ui))i≥0,(µ(u′

i))i≥0

are also equivalent,

in the sense that they have the same limit under any morphism

into a ﬁnite monoid. By Lemma 4,

(µ(ui))i≥0

is represented by

a unique proﬁnite word

over

We thus get a well-deﬁned

map bµ:u7→ wfrom c

A∗to c

B∗.

It is clear that if

bµ(u),bµ(u′)∈c

A∗

are represented by

w, w′

in this way, then

ww′

represents

bµ(uu′),

and thus

bµ

is a

morphism. We are also able to conclude, again by composing

bµ

with a morphism

ν:B∗→M

with

ﬁnite, that if

(ui)i>0

is a convergent sequence of proﬁnite words over A, then

limibµ(ui) = bµ(limiui).

Thus bµis continuous.

C. Proof of Lemma 6

(Many readers will recognize this as an application of the

König inﬁnity lemma; however we will spell out the argument

in detail.) Let

u= (ui)i≥0

be a sequence of proﬁnite words. We

will show how to extract a convergent subsequence. As in the

proof of Lemma 4, we consider an enumeration

ϕi:A∗→Mi

of the morphisms from

A∗

into ﬁnite monoids, and the products

ψi=ϕ1× · · · ϕi:A∗→M1× · · · × Mi

of these morphisms. Since

is ﬁnite, there is some

m1∈M1

such that

ϕ1(ui) = m1

for inﬁnitely many values of

thus extract the subsequence consisting of these

We denote

this subsequence by (u1,i)i≥0as well.

We next look at the elements

ψ2(u1,k)

M1×M2

for

k≥2.

The ﬁrst component of all these elements is

. Again,

since

M1×M2

is ﬁnite there is some

m2∈M2

that appears

inﬁnitely many times among these elements. We extract the

subsequence of these elements, and label the resulting terms

u2,2, u2,3,....

We continue in this fashion. After the

jth

step we have a

subsequence

u1,1, u2,2...,uj,j , uj,j+1 , uj,j+2

such that if

i≤k≤j, ψi(uk,k )=(m1, . . . , mi).

The result

is an inﬁnite subsequence

(uk,k )k>0

such that for each

the sequence

(ψi(uk,k ))k>0

is ultimately constant. Thus the

sequence

(ϕi(uk,k ))k>0

is ultimately constant as well. Since

every morphism from

A∗

into a ﬁnite monoid is one of the

ϕi,

we conclude that the subsequence (uk,k )k>0is convergent.

D. Proof of Lemma 7

(a) Let

ui≤Vvi,

for

i= 1,2,

and let

M∈V.

Let

µ:A∗→

be a morphism and let

mi=bµ(ui), ni=bµ(vi).

We then

have, by multiplicativity of the partial order on M,

bµ(u1u2) = m1m2≤n1n2=bµ(v1v2),

so u1u2≤Vv1v2.

(b) let

µ:A∗→M

be a morphism, with

M∈V.

Let

(ui)i>0, v = (vi)i>0,

be convergent sequences of proﬁnite

words over A. For isufﬁciently large, we have

bµ(u) = µ(ui)≤µ(vi) = bµ(v).

So u≤Vv.

µ:A∗→B∗

be a morphism with

u, v ∈c

A∗, u ≤Vv.

Let ν:B∗→M, where M∈V.From u≤Vvwe have

bν(bµ(u)) ≤bν(bµ(v)),

and thus bµ(u)≤Vbµ(v).

E. Proof of Theorem 8

We start with some routine lemmas on proﬁnite words.

Lemma 33. Assume that

pxωq=u1· · · un

for some proﬁnite

words

p, x, q, u1, . . . , un

. There are

i≥1

and proﬁnite words

α, β such that:

1) pxω=u1· · · ui−1α,

2) αxωβ=ui,

3) xωq=βui+1 · · · un.

Proof. This is shown by induction on n.

n= 1

, then

pxωq=u1

. We let

i:= 1

α:= pxω

β:= xωq

, and we indeed have that

αxωβ=pxω·xω·xωq=

pxωq=u1.

For

n > 1

, assume

pxωq

can be decomposed as

u1· · · un

Writing

u′

n−1=un−1un

and applying the induction hypothesis

on the decomposition

pxωq=u1· · · un−2u′

n−1

, we obtain

i′, α′, β′

with the properties of the lemma. If

i′< n −1

then

we are done, since the same

i′, α′, β′

also satisfy the properties

for the original decomposition. If i′=n−1, then we have:

α′xωβ′=u′

n−1=un−1un,

and so in particular

un−1un= (α′xω)(xωβ′)

. By equidivisi-

bility, we get that either:

•un−1=α′xωβ

for some proﬁnite word

, in which case

we set i:= n−1,α:= α′, and we are done;

•un=αxωβ′

for some proﬁnite word

, in which case

we set i:= n,β:= β′, and we are done.

Lemma 34. Assume that

pxω+1q=u1· · · un

for some

proﬁnite words

p, x, q, u1, . . . , un

. There are

i≥1

and proﬁnite

words α, β such that:

1) pxω=u1· · · ui−1α,

2) αxω+1β=ui,

3) xωq=βui+1 · · · un.

Proof.

We apply Lemma 33 on

(px)·xω·q

, that is, by

setting

p:= px

, and obtain

i′, α′, β′

such that: 1)

(px)xω=

u1· · · ui′−1α′

, 2)

α′xωβ′=ui′

, 3)

xωq=β′ui′+1 · · · un

. By

multiplying the ﬁrst equation by

xω−1

on the right, we obtain:

pxω= (px)xωxω−1=u1· · · ui′−1α′xω−1.

Let

i:= i′

α:= α′xω−1

, and

β:= β′

. We obtain, as required:

1) pxω=u1· · · ui−1α,

2) αxω+1β=α′xω−1xω+1 β′=α′xωβ′=ui′=ui,

3) xωq=β′ui′+1 · · · un=βui+1 · · · un.

The main statement is now proved as follows:

Theorem 35. Let

be a variety of ordered monoids. The

following holds:

Pol(V) = Jxω+1 ≤xωyxω,for any x≤VyK.

Proof. Left to right inclusion.

Left to right inclusion.

Let

L∈Pol(V)

, we

show that it satisﬁes the inequalities. By deﬁnition,

L0a1L1· · · anLn

for languages

. By continuity of

concatenation, we also have that:

L=L0a1L1· · · anLn.

Let

x≤Vy

and

p, q ∈c

A∗

. We need to show that

pxω+1q∈L

implies

pxωyxωq∈L

in order to conclude

this proof direction. From pxω+1q∈Lwe get that

pxω+1q=v0a1v1· · · anvn,with, for all i, vi∈Li.

x=ε

, we need to show that

pyq ∈L

. In that case,

let

i, α, β

be such that

p=v0a1v1· · · aiα

vi=αβ

, and

q=βai+1 vi+1 · · · anvn

. Since

vi∈Li

and

Li∈V

, and since

ε≤Vyby hypothesis, αyβ ∈Li, and thus pyq ∈L.

x=ε

, we apply Lemma 34 and since no letter

can be

written as αxω+1β, we have i, α, β such that:

1) pxω=v0a1v1· · · aiα,

2) αxω+1β=vi,

3) xωq=βai+1 vi+1 · · · anvn.

Since

vi∈Li

and

Li∈V

, since

x≤Vy

, and since

αxω+1β=

αxωxxωβ

, we get that

v′

i:= αxωyxωβ∈Li

. Finally, we have,

as required, that:

(pxω)·y·(xωq) = (pxω)·(xωyxω)·(xωq)

= (v0a1v1· · · aiα)(xωyxω)(βai+1vi+1 · · · anvn)

=v0a1v1· · · ai(αxωyxωβ)ai+1 vi+1 · · · anvn

=v0a1v1· · · aiv′

iai+1vi+1 · · · anvn∈L.

Right to left inclusion.

Let

be a language recognized

by a morphism

η:A∗→(M, ≤)

such that

satisﬁes the

inequalities of the statement. We show that L∈Pol(V).

We may assume that

is surjective by restraining its range

to its image, which would be a quotient of

, hence still

satisfy the inequalities of the statement.

Fix, for all

m∈M

, a language

Lm∈V

with

η−1(m)⊆

and such that

η−1(m′)∩Lm=∅

if, and only if,

(m, m′)

is a V-pair of M. Such languages exist by Lemma 10.

For any word

w∈A∗

, we build a language

by induction

on the factorization forest for w:

•

is a single letter or the empty word, then

Kw=

L1M{w}L1M,

•

In the binary case, with

w=w1w2

, we set

Kw=

Kw1Kw2,

•

In the idempotent case, with

w=w1· · · wn

and

η(w) =

η(w1) = · · · =η(wn) = m

an idempotent of

, we set

Kw=Kw1LmKwn.

Clearly, for any

Kw∈Pol(V)

. Moreover, there are only

ﬁnitely many different

since factorization forests are of

bounded depth. To conclude, we show that L=Sw∈LKw.

Fact 7. For any w∈A∗,w∈Kw.

Proof.

This is shown by induction on the forest factorization

w∈A∗

. This is true if

is a single letter, since

ε∈L1M

The binary case is immediate from the induction hypothesis.

In the idempotent case, using the notations of the construction,

it is enough to note that

w1∈Kw1, wn∈Kwn

by induction

hypothesis and w2· · · wn−1∈η−1(m)⊆Lm.

Fact 8. For any w∈A∗and any w′∈Kw,η(w)≤η(w′).

Proof.

This is shown by induction on the forest factorization

w∈A∗

. If

is a letter, then

w′=w′

1ww′

with

η(w′

1) =

η(w′

2)=1M. This implies that η(w) = η(w′).

In the binary case,

is factorized as

w1w2

, and thus

w′=

w′

1w′

with

w′

1∈Kw1

w′

2∈Kw2

. By induction hypothesis,

we have that

η(w1)≤η(w′

and

η(w2)≤η(w′

, and so

η(w) = η(w1)η(w2)≤η(w′

1)η(w′

2) = η(w′).

In the idempotent case,

w=w1w2· · · wn

with

η(wi) = m

an idempotent of

for every

. By construction,

w′=

w′

1w′

2w′

with

w′

1∈Kw1, w′

2∈Lm, w′

3∈Kwn

. From

w′

2∈

, we deduce that

(m, η(w′

2))

is a

V-

pair of

, that is,

there are proﬁnite words

u≤Vv

such that

bη(u) = m, bη(v) =

η(w′

. Since

satisﬁes the inequality

uω+1 ≤uωvuω

, and

since

is idempotent, we get that

m≤mη(w′

2)m

. The

induction hypothesis asserts that

m≤η(w′

1), η(w′

, and so

we obtain

η(w) = m≤η(w′

1)η(w′

2)η(w′

3) = η(w′)

and we

are done.

The ﬁrst fact shows that

L⊆Sw∈LKw

. For the converse

direction, let

E⊆M

such that

L=η−1(↑E)

. Let

w∈L

and w′∈Kw. The second fact shows that η(w)≤η(w′)and

since

η(w)∈ ↑E

, we have that

η(w′)∈ ↑E

, implying that

w′∈L.

F. Proof of Lemma 9

Let

u, v ∈c

A∗

with

u≤Vv,

and

µ:A∗→M

a morphism

into a ﬁnite monoid. We ﬁrst show that

(bµ(u),bµ(v))

is a

pair of

If we consider

as a ﬁnite alphabet, then the map

µ|A:A→M

extends to a morphism

θ:A∗→M∗

such that

πMθ=µ. We have b

θ(u)≤Vb

θ(v)by Lemma 7. Thus

(bµ(u),bµ(v)) = ( d

πMθ(u),d

πMθ(v)) = (dπM(b

θ(u)),dπM(b

θ(v)))

is a V-pair of M.

Conversely, suppose that

(m1, m2)

is a

-pair of

and

that

is surjective. We will show that

(m1, m2) = (bµ(u),bµ(v))

for some

u, v ∈c

A∗

with

u≤Vv.

There exist

x, y ∈M∗

such

that

x≤Vy

and

u=dπM(x), v =dπM(y).

Viewing

as a

ﬁnite monoid, we map

θ:M→A∗

by setting

θ(m) = w,

where

µ(w) = m.

(Surjectivity guarantees that such a word

exists.) Let

u=b

θ(x), v =b

θ(y).

By Lemma 7,

u≤Vv.

Since

bµ(u) = m1,bµ(v) = m2,(m1m2)is a V-pair.

G. Proof of Corollary 11

The left-to-right direction is an immediate consequence of

Lemma 10.

For the other direction, consider

L, L′⊆A∗

two regular

languages. First, we note that we can assume that

and

L′

are recognized by the same surjective morphism. Indeed, let

µ:A∗→M, µ′:A∗→M′

be surjective morphisms such that

there are sets

E, E ′

with

L=µ−1(↑E)

and

L′=µ′−1(↑E′)

We can construct the surjective morphism

θ:A∗→M×M′

by letting

θ(a)=(µ(a), µ′(a))

for any

a∈A

, and then

L=θ−1(↑E×M′)

and

L′=θ−1(M× ↑E′)

— note that

these are indeed inverse images of upsets.

Thus, we assume that

L, L′

are recognized by the surjective

morphism

µ:A∗→M

, so that

L=µ−1(↑E)

and

L′=

µ−1(↑E′)

. We show that

V-

separable from

L′

if, and only

if, for every

m∈ ↑E

and

m′∈ ↑E′

, the language

µ−1(m)

V-

separable from

µ−1(m′)

. This, together with Lemma 10,

entails that

V-

separation is decidable if

V-

pairs are computable.

Assume

K∈ V

separates

from

L′

, then certainly, for

every

m∈ ↑E

and

m′∈ ↑E′

separates

µ−1(m)

from

µ−1(m′)

. Conversely, if for any

m∈ ↑E

and

m′∈ ↑E′

there is a language

Km,m′∈ V

that separates

µ−1(m)

from

µ−1(m′)

, then

K=Sm∈↑ETm′∈↑E′Km,m′

separates

from

L′

; indeed,

Tm′∈↑E′Km,m′

separates

µ−1(m)

from every

µ−1(m′)

m′∈ ↑E′

, which means that it separates

µ−1(m)

from

Sm′∈↑E′µ−1(m′) = L′

. Consequently,

separates

from

L′

. Additionally,

is a ﬁnite union of ﬁnite intersections

of languages in V, and since Vis a positive variety, K∈ V .

H. Proof of Lemma 16

Let

be a variety of ordered monoids such that

Pol(W) =

Let

u, v ∈c

A∗

with

u≤Wv.

Suppose that there are

u′, u′′ ∈c

A∗

and

a∈A

such that

u=u′au′′

. We will show

that there exist

v, v′∈c

A∗

with

u′≤Wv′, u′′ ≤Wv′′

and

v=v′av′′.

As in previous proofs, we can enumerate all the morphisms

from

A∗

into members of

and, for each

form the direct

product of the ﬁrst

of these. The result is a sequence of

morphisms

ψi:A∗→Mi,

where each

Mi∈W,

with the following property: If

ϕ:A∗→

M∈W

is any morphism, then

factors through

ψi

for all

sufﬁciently large

(Note that we are using the fact that

closed under ﬁnite direct products.)

Since

u∈c

A∗, u = lim ui

for some convergent sequence of

ﬁnite words

ui.

Likewise, there are ﬁnite words

vi, u′

i, u′′

with

v= lim vi, u′= lim u′

i, u′′ = lim u′′

Let

k > 0.

There is some

j0,

depending on

such that if

j≥j0,then the values

ψk(uj), ψk(u′

j), ψk(u′′

j), ψk(vj)

are all constant independent of j. Now we deﬁne languages

L′

k=ψ−1

k(↑ψk(u′

j0))

L′′

k=ψ−1

k(↑ψk(u′′

j0))

Since these languages are pre-images of up-sets in

Mk,

they

belong to

. Let

Lk=L′

kaL′′

Since

Pol(W) = W, Lk∈

Let

µk:A∗→Nk∈W

be a morphism that recognizes

Lk.

We have

u′

jau′′

j∈L′

kaL′′

for

j≥j0.

There is thus some

jk> j0

such that if

j≥jk

then

µk(vj)

is constant independent

of j, and

µk(u′

jau′′

j) =

µk(u)≤

µk(v) = µk(vj).

This implies that

vj∈Lk,

and thus

vj=v′

jav′′

for some

v′

j∈L′

k, v′′

j∈L′′

We thus have (by the deﬁnitions of

L′

and

L′′

ψk(u′

j)≤ψk(v′

j), ψk(u′′

j)≤ψk(v′′

whenever j≥jk.In this manner we obtain sequences

(u′

jk)k>0,(u′′

jk)k>0,(v′

jk)k>0,(v′′

jk)k>0

of ﬁnite words. By sequential compactness, we can extract

parallel convergent subsequences from

v′

and

v′′

converging

to limits

v′, v′′

respectively. Let’s label the terms of these sub-

sequences

V′

i, V ′′

From Lemma 7 we have

u′≤Wv′, u′′ ≤W

v′′.

We also have, by Lemma 7

u=u′au′′ ≤Wv′av′′,

and from continuity of multiplication,

v′av′′ = (limiV′

i)a(limiV′′

i) = limiV′

iaV ′′

Each

V′

and

V′′

has the form

v′

j, v′′

for some sequence of

indices jthat increases with i. Since v′

jav′′

j=vj,we get

v′av′′ = limjvj=v.

This completes the proof of the lemma in the case where

ℓ

is a

letter of

We now consider the case

ℓ=ϵ,

so that

u=u′u′′.

We have

u′= lim u′

for some sequence of ﬁnite words

u′

. If

u′′ = 1

then we take

v′′ =v,

so we can assume that

u′′

is the limit of a sequence

(u′′

of nonempty ﬁnite words.

Since the alphabet

is ﬁnite, there is some letter

a∈A

such

that inﬁnitely many terms of the sequence have the form

aw′′

where

w′′

i∈A∗.

By sequential compactness, the sequence

(w′′

has a convergent subsequence

(w′′

ij)

converging to some

w∈c

A∗.So

u=u′u′′ = (limju′

ij)a(limjw′′

ij) = u′aw.

By the previous case

v=v′ax

for some proﬁnite words

v′

and

with

u′≤Wv′

and

w≤Wx.

Set

v′′ =ax.

Lemma 7, we get

u′′ =aw ≤Wax =v′′.

I. Proof of Theorem 17

Finite word case.

The existence of a decomposition depth

δ:A∗→ {0,1, . . . , D}

for ﬁnite words is known as Si-

mon’s factorization forest theorem [30] (see also Colcombet’s

survey [8, Chap. 18]). In Simon’s theorem, words

with

idempotent type can be further written

u=u1· · · un

with

µ(u) = µ(u1) = · · · =µ(un)

an idempotent of

. To

recover the types listed in the present theorem, we simply

see the case

n= 2

as a binary type, and when

n > 2

, we set

u′:= u2· · · un−1

, since

µ(u′) = µ(u2)· · · µ(un−1) = µ(u)

by idempotency.

General case.

We extend the aforementioned function

into a function

for proﬁnite words. For any

u∈c

A∗

we set

d(u)

to be the minimal value

such that there is a sequence

(ui)i≥0

of ﬁnite words that converges to

with

δ(ui)≤k

for

any

. (The existence of such a sequence is ensured by the

fact that

has ﬁnite range.) Note that the number of

’s with

δ(ui)< k

has to be ﬁnite, otherwise the sequence of these

’s would converge to

, contradicting the minimality of

d(u)

We can thus assume that all the ui’s are such that δ(ui) = k.

Let

u= (ui)i≥0∈c

A∗

with

δ(ui) = d(u)

for any

. We

show the theorem by induction on

d(u)

. If

d(u)=0

then

every

is either a letter or the empty word, hence so is

. If

d(u) = k

, we address two cases depending of the type of the

’s (note that these two cases overlap, and when this happens,

either case can be followed):

•

There are inﬁnitely many

’s with binary type. By

extraction of a subsequence, we can assume that all

’s

are of binary type. For every

we can thus write

ui=

ui,1ui,2

with

δ(ui,1), δ(ui,2)< k

. By two applications of

Lemma

we can extract a further subsequence of the

so that we can assume that both

(ui,1)i≥0

and

(ui,2)i≥0

are convergent. We write

and

for their respective

limits. This concludes the case since

d(u1), d(u2)< k

and u=u1u2by continuity of the product.

•

There are inﬁnitely many

’s with idempotent type. By

extraction of a subsequence, we can assume that all

’s

are of idempotent type and that their images in

are

equal to the same idempotent

e∈M

. For every

, we

can write

ui=ui,1u′

iui,2

with

δ(ui,1), δ(ui,2)< k

, and

µ(ui,1) = µ(u′

i) = µ(ui,2) = e

. Again, by Lemma 6 we

can assume that

(ui,1)i≥0

(u′

i)i≥0

, and

(ui,2)i≥0

are all

convergent, and we write

u′

, and

for their respective

limits. This concludes the case since

d(u1), d(u2)< k

and

u=u1u′u2

by continuity of the product. By continuity

of µ, we also have bµ(u1) = bµ(u′) = bµ(u2) = bµ(u) = e.

J. Proof of Fact 1

Since

bπ

and

▷i

are continuous morphisms, we need only

show the claim for

u=x∈Mi

, that is, we need to show

x⪯

▷i(x)

. This is clear by construction, since

ℓ

(x) = π

ℓ

(▷i(x))

and πr(x)⊆πr(▷i(x)).

K. Proof of Fact 2

Since

bπ

ℓ

and

bπr

are continuous morphisms, we need only

show the claim for u=x∈Mi.

The claim holds for

by construction. Let

i≥0

and

assume the claim holds for

. Let

x∈Mi+1

and let

w∈K+

such that

▷i(w) = x

. The claim at level

with

u:= w

asserts

that

ℓ

(w)∈πr(w)

. Fact 1 then asserts that

π(w)⪯▷i(w)

and so π

ℓ

(x) = π

ℓ

(w)∈πr(w)⊆πr(▷i(w)) = πr(x).

L. Proof of the ﬁrst part of Corollary 20

We show that

M∞

is a monoid. There is an element

(1M, E)∈K∞

, by construction. Take any other element

(m, F )∈K∞

. Their product

(m, E F )

, and since

1M∈

by Fact 2,

(m, F )⪯(m, E F )

. However,

ex ∈M∞

since

▷∞

is the identity map, and so there is an element

(m, F ′)

K∞

such that

(m, E F )⪯(m, F ′)

. As

K∞

contains only

incomparable elements,

F=EF =F′

, and so

ex =x

Similarly,

xe =x

, and thus

is a neutral element for

⟨K∞⟩

Note that

ℓ

is a monoid morphism, since

ℓ

(e)=1M

, and that

πr

is a relational morphism of monoids, since

1M∈πr(e) = E

M. Proof of Fact 3

Since

bπ

and

bursti

are continuous morphisms, we need only

show this over letters, so let us assume that

u=x∈Ki

By deﬁnition of

bursti(x)

, if

is not an idempotent then

the property holds. Otherwise,

bursti(x)

is the concatenation

of proﬁnite words that all map to

xω+1 =x

via

bπ

, and so

bπ(bursti(x)) = x.

N. Proof of Fact 4

Since

▷i◦lifti

and

bπ

are continuous morphisms, we need

only show this over letters, so let us assume that

u=x∈Mi+1

This is then immediate since

▷i(lifti(x)) = x

by construction.

O. Proof of Fact 5

We have that

bπ

ℓ

(lifti−1(bursti(u))) = bπ

ℓ

(lifti−1(u))

Fact 3. Applying

bπ

ℓ

on both sides of the equation given by

Fact 4, we get that

ℓ

▷i−1(lifti−1(u))) = bπ

ℓ

(u)

, and by

construction of

▷i−1

ℓ

▷i−1(lifti−1(u))) = bπ

ℓ

(lifti−1(u))

This entails that

bπ

ℓ

(u) = bπ

ℓ

(lifti−1(bursti(u)))

, and in turn,

iterating the argument, that bπ

ℓ

(u) = bπ

ℓ

(τi(u)).

P. Proof of Fact 6

First, we show that the statement is closed under concatena-

tion over ﬁnite words, in the sense that if the claim is satisﬁed

for two ﬁnite words

u:= w1

and

u:= w2

then it will also be

satisﬁed for

u:= w1w2

. Let

m∈πr(▷i(w1w2))

. Since

πr◦▷i

is a morphism,

πr(▷i(w1w2)) = πr(▷i(w1))πr(▷i(w2))

, and

m=m1m2

with, for

b= 1,2

mb∈πr(▷i(wb))

. By

hypothesis, there are two words

um1, um2∈d

M+

such that,

for

b= 1,2

bπ

ℓ

(umb) = mb

and

τi(wb)≤Pol(V)umb

. This

implies that bπ

ℓ

(um1um2) = m1m2=mand:

τi(u) = τi(w1w2) = τi(w1)τi(w2)≤Pol(V)um1um2,

so that

um:= um1um2

, which is not

by hypothesis, satisﬁes

the conclusion of the claim.

Next, we argue that showing the claim over ﬁnite words

implies that it holds over proﬁnite words. Let

u= (wj)j≥0∈

K+

and

m∈πr(b

▷i(u))

. Since

πr◦▷i

is a morphism into a

ﬁnite semigroup, it is eventually constant over

(wj)j≥0

; we

can thus assume that

m∈πr(▷i(wj))

for any

j≥0

(taking a

subsequence of

(wj)j≥0

as needed). For any

j≥0

, applying

the lemma with

u:= wj

and

provides a proﬁnite word

such that bπ

ℓ

(vj) = mand τi(wj)≤Pol(V)vj.

The sequence

(vj)j≥0

has a convergent subsequence, by

compactness; let

be its limit. By continuity of

bπ

ℓ

, we

have that

bπ

ℓ

(um) = m

. Since

τi

is continuous,

(τi(wj))j≥0

converges to τi(u).

We are thus considering two sequences,

(τi(wj))j≥0

and

(vj)j≥0

, with

τi(wj)≤Pol(V)vj

for any

j≥0

, such that the

former sequence converges to

τi(u)

and the latter to

. By

Lemma 7, τi(u)≤Pol(V)um.

Q. Proof of Theorem 23

Consider one pass through the main loop of Algorithm 1.

Note that

|Exp| ∈ O(2n)

. The internal loop at line

takes

time

, while the loop at line

is executed

n2×22n

times,

each time requiring to call the algorithm for

V-

pairs. All in

all, lines

take time

O(cV(2O(n)))

, since

is at

least linear. As was shown in the proof of Lemma 19, the

set

↓MaxElts

always grows, hence the main loop will be

executed at most O(2n), showing the claim.

R. Proof of Theorem 29

We ﬁrst note that Theorem 1 is an example of a more

general property, easily shown using De Morgan’s laws [24,

Lemma 33]: for any positive variety

Pol(Bool(V))

is equal

to the polynomial closure of the dual of

, that is, the set

co-Vof complements of languages in V. Hence:

chi+1

2(V) = Pol(co-chi−1

2(V)).

Consequently, if we inductively assume that we can decide

separation for

chi−1

2(V)

, we simply need to show that we

can decide separation for

co-chi−1

2(V)

with the same time

complexity in order to be able to apply Theorem 23 and

complete the proof. The time complexity bound given in the

statement holds inductively.

On the algebraic side, the ordered version of Eilenberg shows

that the variety of ordered monoids that corresponds to

co-V

for any positive variety of regular languages

, is the set

co-V

of ordered monoids of

in which the partial order is reversed.

This implies that for any proﬁnite words

u, v

, we have that

u≤Vv

if, and only if,

v≤co-Vu

. Hence for any monoid

(m, m′)

is a

co-V-

pair of

if, and only if,

(m′, m)

V-

pair of

, so that computing

co-V-

pairs is as easy as

computing

V-

pairs. By Corollary 11, this means we can decide

co-V-separation, and we are done.

ResearchGate has not been able to resolve any citations for this publication.

Generic Results for Concatenation Hierarchies

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May 2019
THEOR COMPUT SYST

In the theory of formal languages, the understanding of concatenation hierarchies of regular languages is one of the most fundamental and challenging topic. In this paper, we survey progress made in the comprehension of this problem since 1971, and we establish new generic statements regarding this problem.

Separating regular languages with two quantifier alternations

Article

Full-text available

Jul 2017
LOG METH COMPUT SCI

Thomas Place

We investigate a famous decision problem in automata theory: separation. Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes the first one and is disjoint from the second. Typically, obtaining an algorithm for separation yields a deep understanding of the investigated class C. This explains why a lot of effort has been devoted to finding algorithms for the most prominent classes. Here, we are interested in classes within concatenation hierarchies. Such hierarchies are built using a generic construction process: one starts from an initial class called the basis and builds new levels by applying generic operations. The most famous one, the dot-depth hierarchy of Brzozowski and Cohen, classifies the languages definable in first-order logic. Moreover, it was shown by Thomas that it corresponds to the quantifier alternation hierarchy of first-order logic: each level in the dot-depth corresponds to the languages that can be defined with a prescribed number of quantifier blocks. Finding separation algorithms for all levels in this hierarchy is among the most famous open problems in automata theory. Our main theorem is generic: we show that separation is decidable for the level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in the special case of the dot-depth, we push this result to the level 5/2. In logical terms, this solves separation for

\Sigma_3

: first-order sentences having at most three quantifier blocks starting with an existential one.

Group Separation Strikes Back

Conference Paper

Jun 2023

The factorisation forest theorem

Chapter

Sep 2021

Thomas Colcombet

The Regular Languages of First-Order Logic with One Alternation

Conference Paper

Aug 2022

Separation and covering for group based concatenation hierarchies

Conference Paper

Jun 2019

The dot-depth hierarchy, 45 years later

Book

Jan 2017

Jean-Eric Pin

The ω-inequality problem for concatenation hierarchies of star-free languages

Article

Jan 2017

The problem considered in this paper is whether an inequality of ω-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing–Thérien hierarchy.

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Article

Jul 2017

We investigate quantifier alternation hierarchies in first-order logic on finite words. Levels in these hierarchies are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language in the levels

\mathcal{B}{\Sigma}_2

(finite boolean combinations of formulas having only one alternation) and

{\Sigma}_3

(formulas having only two alternations and beginning with an existential block). Our proofs work by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.

An Explicit Formula for the Intersection of Two Polynomials of Regular Languages

Conference Paper

Jun 2013
Lect Notes Comput Sci

Jean-Eric Pin

Let

\mathcal{L}

be a set of regular languages of A *. An

\mathcal{L}

-polynomial is a finite union of products of the form L 0a 1L 1 ⋯ a n L n , where each a i is a letter of A and each L i is a language of

\mathcal{L}

. We give an explicit formula for computing the intersection of two

\mathcal{L}

-polynomials. Contrary to Arfi’s formula (1991) for the same purpose, our formula does not use complementation and only requires union, intersection and quotients. Our result also implies that if

\mathcal{L}

is closed under union, intersection and quotient, then its polynomial closure, its unambiguous polynomial closure and its left [right] deterministic polynomial closure are closed under the same operations.

The Alternation Hierarchy of First-Order Logic on Words is Decidable

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