Abstract
The model of programs over (finite) monoids, introduced by Barrington and Thérien, gives an interesting way to characterise the circuit complexity class 
and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in 
, a small variety of finite aperiodic monoids. First, we give a fine hierarchy within the class of languages recognised by programs over monoids from 
, based on the length of programs but also some parametrisation of 
. Second, and most importantly, we make progress in understanding what regular languages can be recognised by programs over monoids in 
. We show that those programs actually can recognise all languages from a class of restricted dot-depth one languages, using a non-trivial trick, and conjecture that this class suffices to characterise the regular languages recognised by programs over monoids in 
.
Introduction
In computational complexity theory, many hard still open questions concern relationships between complexity classes that are expected to be quite small in comparison to the mainstream complexity class 
of tractable languages. One of the smallest such classes is 
, the class of languages decided by Boolean circuits of polynomial length, logarithmic depth and bounded fan-in, a relevant and meaningful class, that has many characterisations but whose internal structure still mostly is a mystery. Indeed, among its most important subclasses, we count 
, 
and 
: all of them are conjectured to be different from each other and strictly within 
, but despite many efforts for several decades, this could only be proved for the first of those classes.
In the late eighties, Barrington and Thérien [3], building on Barrington’s celebrated theorem [2], gave an interesting viewpoint on those conjectures, relying on algebraic automata theory. They defined the notion of a program over a monoid M: a sequence of instructions (i, f), associating through function f some element of M to the letter at position i in the input of fixed length. In that way, the program outputs an element of M for every input word, by multiplying out the elements given by the instructions for that word; acceptance or rejection then depends on that outputted element. A language of words of arbitrary length is consequently recognised in a non-uniform fashion, by a sequence of programs over some fixed monoid, one for each possible input length; when that sequence is of polynomial length, it is said that the monoid p-recognises that language. Barrington and Thérien’s discovery is that 
and almost all of its significant subclasses can each be exactly characterised by p-recognition over monoids taken from some suitably chosen variety of finite monoids (a class of finite monoids closed under basic operations on monoids). For instance, 
, 
, 
and 
correspond exactly to p-recognition by, respectively, finite monoids, finite aperiodic monoids, finite solvable groups and finite solvable monoids. Understanding the internal structure of 
thus becomes a matter of understanding what finite monoids from some particular variety are able to p-recognise.
It soon became clear that regular languages play a central role in understanding p-recognition: McKenzie, Péladeau and Thérien indeed observed [12] that finite monoids from a variety 
and a variety 
p-recognise the same languages if and only if they p-recognise the same regular languages. Otherwise stated, most conjectures about the internal structure of 
can be reformulated as a statement about where one or several regular languages lie within that structure. This is why a line of previous works got interested into various notions of tameness, capturing the fact that for a given variety of finite monoids, p-recognition does not offer much more power than classical morphism-recognition when it comes to regular languages (see [8, 10, 11, 13, 14, 20–22]).
This paper is a contribution to an ongoing study of what regular languages can be p-recognised by monoids taken from “small” varieties, started with the author’s Ph.D. thesis [7]. In a previous paper by the author with McKenzie and Segoufin [8], a novel notion of tameness was introduced and shown for the “small” variety of finite aperiodic monoids 
. This allowed them to characterise the class of regular languages p-recognised by monoids from 
as those recognised by so called quasi-
morphisms and represented a first small step towards a new proof that the variety 
of finite aperiodic monoids is tame. This is a statement equivalent to Furst’s, Saxe’s, Sipser’s [6] and Ajtai’s [1] well-known lower bound result about 
. In [8], the authors also observed that, while 
“behaves well” with respect to p-recognition of regular languages, the variety 
, a subclass of 
, does, in contrast, “behave badly” in the sense that monoids from 
do p-recognise regular languages that are not recognised by quasi-
morphisms.
Now, 
is a well-studied and fundamental variety in algebraic automata theory (see, e.g., [15, 16]), corresponding through classical morphism-recognition to the class of regular languages in which membership depends on the presence or absence of a finite set of words as subwords. This paper is a contribution to the understanding of the power of programs over monoids in 
, a knowledge that certainly does not bring us closer to a new proof of the tameness of 
(as we are dealing with a strict subvariety of 
), but that is motivated by the importance of 
in algebraic automata theory and the unexpected power of programs over monoids in 
. The results we present in this article are twofold: first, we exhibit a fine hierarchy within the class of languages p-recognised by monoids from 
, depending on the length of those programs and on a parametrisation of 
; second, we show that a whole class of regular languages, that form a subclass of dot-depth one languages [15], are p-recognised by monoids from 
while, in general, they are not recognised by any quasi-
morphism. This class roughly corresponds to dot-depth one languages where detection of a given factor does work only when it does not appear too often as a subword. We actually even conjecture that this class of languages with additional positional modular counting (that is, letters can be differentiated according to their position modulo some fixed number) corresponds exactly to all those p-recognised by monoids in 
, a statement that is interesting in itself for algebraic automata theory.
Organisation of the Paper. Following the present introduction, Sect. 2 is dedicated to the necessary preliminaries. In Sect. 3, we present the results about the fine hierarchy and in Sect. 4 we expose the results concerning the regular languages p-recognised by monoids from 
. Section 5 gives a short conclusion.
Note. This article is based on unpublished parts of the author’s Ph.D. thesis [7].
Preliminaries
Various Mathematical Materials
We assume the reader is familiar with the basics of formal language theory, semigroup theory and recognition by morphisms, that we might designate by classical recognition; for those, we only specify some things and refer the reader to the two classical references of the domain by Eilenberg [4, 5] and Pin [16].
General Notations and Conventions. Let 
. We shall denote by 
the set of all 
verifying 
. We shall also denote by [i] the set 
. Given some set E, we shall denote by 
the powerset of E. All our alphabets and words will always be finite; the empty word will be denoted by 
.
Varieties and Languages. A variety of monoids is a class of finite monoids closed under submonoids, Cartesian product and morphic images. A variety of semigroups is defined similarly. When dealing with varieties, we consider only finite monoids and semigroups, each having an idempotent power, a smallest 
such that 
for any element x. To give an example, the variety of finite aperiodic monoids, denoted by 
, contains all finite monoids M such that, given 
its idempotent power, 
for all 
.
To each variety 
of monoids or semigroups we associate the class 
of languages such that, respectively, their syntactic monoid or semigroup belongs to 
. For instance, 
is well-known to be the class of star-free languages.
Quasi
Languages. If S is a semigroup we denote by 
the monoid S if S is already a monoid and 
otherwise.
The following definitions are taken from [17]. Let 
be a surjective morphism from 
to a finite monoid M. For all k consider the subset 
of M (where 
is the set of words over 
of length k). As M is finite there is a k such that 
. This implies that 
is a semigroup. The semigroup given by the smallest such k is called the stable semigroup of
. If S is the stable semigroup of 
, 
is called the stable monoid of
. If 
is a variety of monoids or semigroups, then we shall denote by 
the class of such surjective morphisms whose stable monoid or semigroup, respectively, is in 
and by 
the class of languages whose syntactic morphism is in 
.
Programs over Monoids. Programs over monoids form a non-uniform model of computation, first defined by Barrington and Thérien [3], extending Barrington’s permutation branching program model [2]. Let M be a finite monoid and 
an alphabet. A program
P
over
M
on
is a finite sequence of instructions of the form (i, f) where 
and 
; said otherwise, it is a word over 
. The length of P, denoted by 
, is the number of its instructions. The program P defines a function from 
to M as follows. On input 
, each instruction (i, f) outputs the monoid element 
. A sequence of instructions then yields a sequence of elements of M and their product is the output P(w) of the program. A language 
is consequently recognised by P whenever there exists 
such that 
.
A language L over 
is recognised by a sequence of programs 
over some finite monoid M if for each n, the program 
is on 
and recognises 
. We say 
is of length s(n) for 
whenever 
for all 
and that it is of length at most s(n) whenever there exists 
verifying 
for all 
.
For 
and 
a variety of monoids, we denote by 
the class of languages recognised by sequences of programs over monoids in 
of length at most s(n). The class 
is then the class of languages p-recognised by a monoid in 
, i.e. recognised by sequences of programs over monoids in 
of polynomial length.
The following is an important property of 
.
Proposition 1
([12, Corollary 3.5]). Let 
be a variety of monoids, then 
is closed under Boolean operations.
Given two alphabets 
and 
, a 
-program on 
for 
is defined just like a program over some finite monoid M on 
, except that instructions output letters from 
and thus that the program outputs words over 
. Let now 
and 
. We say that L
program-reduces to
K if and only if there exists a sequence 
of 
-programs (the program-reduction) such that 
is on 
and 
for each 
. The following proposition shows closure of 
also under program-reductions.
Proposition 2
([7, Proposition 3.3.12 and Corollary 3.4.3]). Let 
and 
be two alphabets. Let 
be a variety of monoids. Given 
in 
and 
from which there exists a program-reduction to K of length t(n), for 
, we have that 
. In particular, when K is recognised (classically) by a monoid in 
, we have that 
.
Tameness and the Variety 
We won’t introduce any of the proposed notions of tameness but will only state that the main consequence for a variety of monoids 
to be tame in the sense of [8] is that 
. This consequence has far-reaching implications from a computational-complexity-theoretic standpoint when 
happens to be equal to a circuit complexity class. For instance, tameness for 
implies that 
, which is equivalent to the fact that 
does not contain the language 
of words over 
containing a number of 1s not divisible by m for any 
(a central result in complexity theory [1, 6]).
Let us now define the variety of monoids 
. A finite monoid M of idempotent power 
belongs to 
if and only if 
for all 
. It is a strict subvariety of the variety 
, containing all finite monoids M of idempotent power 
such that 
for all 
, itself a strict subvariety of 
. The variety 
is a “small” one, well within 
.
We now give some specific definitions and results about 
that we will use, based essentially on [9], but also on [16, Chapter 4, Section 1].
For some alphabet 
and each 
, let us define the equivalence relation 
on 
by 
if and only if u and v have the same set of k-subwords (subwords of length at most k), for all 
. The relation 
is a congruence of finite index on 
. For an alphabet 
and a word 
, we shall write 
for the language of all words over 
having u as a subword. In the following, we consider that 
has precedence over 
and 
(but of course not over concatenation).
We define the class of piecewise testable languages
as the class of regular languages such that for every alphabet 
, we associate to 
the set 
of all languages over 
that are Boolean combinations of languages of the form 
where 
. In fact, 
is the set of languages over 
equal to a union of 
-classes for some 
(see [18]). Simon showed [18] that a language is piecewise testable if and only if its syntactic monoid is in 
, i.e. 
.
We can define a hierarchy of piecewise testable languages in a natural way. For 
, let the class of
k-piecewise testable languages
be the class of regular languages such that for every alphabet 
, we associate to 
the set 
of all languages over 
that are Boolean combinations of languages of the form 
where 
with 
. We then have that 
is the set of languages over 
equal to a union of 
-classes. Let us define 
the inclusion-wise smallest variety of monoids containing the quotients of 
by 
for any alphabet 
: we have that a language is k-piecewise testable if and only if its syntactic monoid belongs to 
, i.e. 
. (See [9, Section 3].)
Fine Hierarchy
The first part of our investigation of the computational power of programs over monoids in 
concerns the influence of the length of programs on their computational capabilities.
We say two programs over a same monoid on the same set of input words are equivalent if and only if they recognise the same languages. Tesson and Thérien proved in [23] that for any monoid M in 
, there exists some 
such that for any alphabet 
there is a constant 
verifying that any program over M on 
for 
is equivalent to a program over M on 
of length at most 
. Since 
, any monoid in 
does also have this property. However, this does not imply that there exists some 
working for all monoids in 
, i.e. that 
collapses to 
.
In this section, we show on the one hand that, as for 
, while 
collapses to 
for any super-polynomial function 
, there does not exist any 
such that 
collapses to 
; and on the other hand that 
does optimally collapse to 
for each 
.
Strict Hierarchy
Given 
, we say that 
is a k-selector over
n if 
is a function of 
that associates a subset of [n] to each vector in 
. For any sequence 
such that 
is a k-selector over n for each 
—a sequence we will call a sequence of
k-selectors—, we set 
, where for each 
, the language 
is the set of words over 
of length 
that can be decomposed into 
consecutive blocks 
of n letters where the first k blocks each contain 1 exactly once and uniquely define a vector 
in 
, where for all 
, 
is given by the position of the only 1 in 
(i.e. 
) and v is such that there exists 
verifying that 
is 1. Observe that for any k-selector 
over 0, we have 
.
We now proceed similarly to what has been done in Subsection 5.1 in [8] to show, on one hand, that for all 
, there is a monoid 
in 
such that for any sequence of k-selectors 
, the language 
is recognised by a sequence of programs over 
of length at most 
; and, on the other hand, that for all 
there is a sequence of k-selectors 
such that for any finite monoid M and any sequence of programs 
over M of length at most 
, the language 
is not recognised by 
.
We obtain the following proposition.
Proposition 3
For all 
, we have 
. More precisely, for all 
and 
, we have 
.
Collapse
Looking at Proposition 3, it looks at first glance rather strange that, for each 
, we can only prove strictness of the hierarchy inside 
up to exponent 
. We now show, in a way similar to Subsection 5.2 in [8], that in fact 
does collapse to 
for all 
, showing Proposition 3 to be optimal in some sense.
Proposition 4
Let 
. Let 
and 
be an alphabet. Then there exists a constant 
such that any program over M on 
for 
is equivalent to a program over M on 
of length at most 
.
In particular, 
for all 
.
Regular Languages in 
The second part of our investigation of the computational power of programs over monoids in 
is dedicated to understanding exactly what regular languages can be p-recognised by monoids in 
.
Non-tameness of 
It is shown in [8] that 
, thus giving an example of a well-known subvariety of 
for which p-recognition allows to do unexpected things when recognising a regular language. How far does this unexpected power go?
The first thing to notice is that, though none of them is in 
, all languages of the form 
and 
for 
an alphabet and 
are in 
. Indeed, each of them can be recognised by a sequence of constant-length programs over the syntactic monoid of 
: for every input length, just output the image, through the syntactic morphism of 
, of the word made of the 
first or last letters. So, informally stated, programs over monoids in 
can check for some constant-length beginning or ending of their input words.
But they can do much more. Indeed, the language 
does not belong to 
(compute the stable monoid), yet it is in 
. The crucial insight is that it can be program-reduced in linear length to the piecewise testable language of all words over 
having ca as a subword but not the subwords cca, caa and cb by using the following trick (that we shall call “feedback-sweeping”) for input length 
: read the input letters in the order 
, output the letters read. This has already been observed in [8, Proposition 5].
Lemma 1
.
Using variants of the “feedback-sweeping” reading technique, we can prove that the phenomenon just described is not an isolated case.
Lemma 2
The languages 
, 
, 
, 
and 
do all belong to 
.
Hence, we are tempted to say that there are “much more” regular languages in 
than just those in 
, even though it is not clear to us whether 
or not. But can we show any upper bound on 
? It turns out that we can, relying on two known results.
First, since 
, we have 
, so Theorem 6 in [8], that states 
, implies that 
.
Second, let us define an important superclass of the class of piecewise testable languages. Let 
be an alphabet and 
(
); we define 
. The class of dot-depth one languages is the class of Boolean combinations of languages of the form 
, 
and 
for 
an alphabet, 
and 
. The inclusion-wise smallest variety of semigroups containing all syntactic semigroups of dot-depth one languages is denoted by 
and verifies that 
is exactly the class of dot-depth one languages. (See [11, 15, 19].) It has been shown in [11, Corollary 8] that 
(if we extend the program-over-monoid formalism in the obvious way to finite semigroups). Now, we have 
, so that 
and hence 
.
To summarise, we have the following.
Proposition 5
.
In fact, we conjecture that the inverse inclusion does also hold.
Conjecture 1
.
Why do we think this should be true? Though, for a given alphabet 
, we cannot decide whether some word 
of length at least 2 appears as a factor of any given word w in 
with programs over monoids in 
(because 
), Lemma 2 and the possibilities offered by the “feedback-sweeping” technique give the impression that we can do it when we are guaranteed that u appears at most a fixed number of times in w, which seems somehow to be what dot-depth one languages become when restricted to belong to 
. This intuition motivates the definition of threshold dot-depth one languages.
Threshold Dot-Depth One Languages
The idea behind the definition of threshold dot-depth one languages is that we take the basic building blocks of dot-depth one languages, of the form 
for an alphabet 
, for 
and 
, and restrict them so that, given 
, membership of a word does really depend on the presence of a given word 
as a factor if and only if it appears less than l times as a subword.
Definition 1
Let 
be an alphabet. For all 
and 
, we define 
to be the language of words over 
containing 
as a subword or u as a factor, i.e. 
. Then, for all 
(
) and 
, we define 
.
Obviously, for each 
an alphabet, 
and 
, the language 
equals 
. Over 
, the language 
contains all words containing a letter c verifying that in the prefix up to that letter, ababab appears as a subword or ab appears as a factor. Finally, the language 
over 
of Lemma 1 is equal to 
.
We then define a threshold dot-depth one language as any Boolean combination of languages of the form 
, 
and 
for 
an alphabet, for 
and 
.
Confirming the intuition briefly given above, the technique of “feedback-sweeping” can indeed be pushed further to prove that the whole class of threshold dot-depth one languages is contained in 
, and we dedicate the remainder of this section to prove it. Concerning Conjecture 1, our intuition leads us to believe that, in fact, the class of threshold dot-depth one languages with additional positional modular counting is exactly 
. We simply refer the interested reader to Section 5.4 of the author’s Ph.D. thesis [7], that contains a partial result supporting this belief, too technical and long to be presented here.
Let us now move on to the proof of the following theorem.
Theorem 1
Every threshold dot-depth one language belongs to 
.
As 
is closed under Boolean operations (Proposition 1), our goal is to prove, given an alphabet 
, given 
and 
(
), that 
is in 
; the case of 
and 
for 
is easily handled (see the discussion at the beginning of Subsect. 4.1). To do this, we need to put 
in some normal form. It is readily seen that 
where the 
’s are defined thereafter.
Definition 2
Let 
be an alphabet.
For all 
, 
and 
, set 
.
Building directly a sequence of programs over a monoid in 
that decides 
for some alphabet 
and 
seems however tricky. We need to split things further by controlling precisely how many times each 
for 
appears in the right place when it does less than l times. To do this, we consider, for each 
, the language 
defined below.
Definition 3
Let 
be an alphabet.
For all 
(
), 
, 
, we set
 |
Now, for a given 
, we are interested in the words of 
such that for each 
verifying 
, the word 
indeed appears as a factor in the right place. We thus introduce a last language 
defined as follows.
Definition 4
Let 
be an alphabet.
For all 
(
), 
, 
, we set
 |
We now have the normal form we were looking for to prove Theorem 1: 
is equal to the union, over all 
, of the intersection of 
and 
. Though rather intuitive, the correctness of this decomposition is not so straightforward to prove and, actually, we can only prove it when for each 
, the letters in 
are all distinct.
Lemma 3
Let 
be an alphabet, 
and 
(
) such that for each 
, the letters in 
are all distinct. Then,
 |
Our goal now is to prove, given an alphabet 
, given 
and 
(
) such that for each 
, the letters in 
are all distinct, that for any 
, the language 
is in 
; closure of 
under union (Proposition 1) consequently entails that 
. The way 
and 
are defined allows us to reason as follows. For each 
verifying 
, let 
be the language of words w over 
containing 
as a subword but not 
and such that 
with 
and 
, where 
and 
. If we manage to prove that for each 
verifying 
we have 
, we can conclude that 
does belong to 
by closure of 
under intersection, Proposition 1. The lemma that follows, the main lemma in the proof of Theorem 1, exactly shows this. The proof crucially uses the “feedback sweeping” technique, but note that we actually don’t know how to prove it when we do not enforce that for each 
, the letters in 
are all distinct.
Lemma 4
Let 
be an alphabet and 
such that its letters are all distinct. For all 
and 
, we have
 |
Proof (Sketch)
Let 
be an alphabet and 
such that its letters are all distinct. Let 
and 
. We let
 |
If 
, the lemma follows trivially because L is piecewise testable and hence belongs to 
, so we assume 
.
For each letter 
, we shall use 
distinct decorated letters of the form 
for some 
, using the convention that 
; of course, for two distinct letters 
, we have that 
and 
are distinct for all 
. We denote by A the alphabet of these decorated letters. The main idea of the proof is, for a given input length 
, to build an A-program 
over 
such that, given an input word 
, it first ouputs the 
first letters of w and then, for each i going from 
to n, outputs 
, followed by 
(a “sweep” of 
letters backwards down to position 
, decorating the letters incrementally) and finally by 
(a “sweep” forwards up to position i, continuing the incremental decoration of the letters). The idea behind this way of rearranging and decorating letters is that, given an input word 
, as long as we make sure that w and thus 
do contain 
as a subword but not 
, then 
can be decomposed as 
where 
, 
, and 
are minimal, with z containing 
as a subword for some 
if and only if 
. This means we can check whether 
by testing whether w belongs to some fixed piecewise testable language over A.
As explained before stating the previous lemma, we can now use it to prove the result we were aiming for.
Proposition 6
Let 
be an alphabet, 
and 
(
) such that for each 
, the letters in 
are all distinct. For all 
, we have 
.
We thus derive the awaited corollary.
Corollary 1
Let 
be an alphabet, 
and 
(
) such that for each 
, the letters in 
are all distinct. Then, 
.
However, what we really want to obtain is that 
without putting any restriction on the 
’s. But, in fact, to remove the constraint that the letters must be all distinct in each of the 
’s, we simply have to decorate each of the input letters with its position minus 1 modulo a big enough 
. This finally leads to the following proposition.
Proposition 7
Let 
be an alphabet, 
and 
(
). Then 
.
This finishes to prove Theorem 1 by closure of 
under Boolean combinations (Proposition 1) and by the discussion at the beginning of Subsect. 4.1.
Conclusion
Although 
is very small compared to 
, we have shown that programs over monoids in 
are an interesting subject of study in that they allow to do quite unexpected things. The “feedback-sweeping” technique allows one to detect presence of a factor thanks to such programs as long as this factor does not appear too often as a subword: this is the basic principle behind threshold dot-depth one languages, that our article shows to belong wholly to 
.
Whether threshold dot-depth one languages with additional positional modular counting do correspond exactly to the languages in 
seems to be a challenging question, that we leave open. In his Ph.D. thesis [7], the author proved that all strongly unambiguous monomials (the basic building blocks in 
) that are imposed to belong to 
at the same time are in fact threshold dot-depth one languages. However, the proof looks much too complex and technical to be extended to, say, all languages in 
. New techniques are probably needed, and we might conclude by saying that proving (or disproving) this conjecture could be a nice research goal in algebraic automata theory.
Acknowledgements
The author thanks the anonymous referees for their helpful comments and suggestions.
Contributor Information
Alberto Leporati, Email: alberto.leporati@unimib.it.
Carlos Martín-Vide, Email: carlos.martin@urv.cat.
Dana Shapira, Email: shapird@g.ariel.ac.il.
Claudio Zandron, Email: zandron@disco.unimib.it.
Nathan Grosshans, Email: nathan.grosshans@polytechnique.edu, https://www.di.ens.fr/~ngrosshans/.
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