The theory of stabilisation monoids
and regular cost functions
?
Thomas Colcombet
Liafa/Cnrs/Universit´e Paris 7, Denis Diderot, France
Abstract. We introduce the notion of regular cost functions: a quanti-
tative extension to the standard theory of regular languages.
We provide equivalent characterisations of this notion by means of au-
tomata (extending the nested distance desert automata of Kirsten), of
history-deterministic automata (history-determinism is a weakening of
the standard notion of determinism, that replaces it in this context),
and a suitable notion of recognisability by stabilisation monoids. We
also provide closure and decidability results.
1 Introduction
When considering standard regular languages (say on finite words), some re-
sults appear as cornerstones on which the whole theory is constructed. The first
such kind of results are the equivalences between many different formalisms: non-
deterministic automata, deterministic automata, recognisability by monoids, reg-
ular expressions, etc. The second one consists in the numerous closure properties
that regular languages enjoy: union, intersection, projection (mapping under a
length-preserving morphism), complementation, etc. From these facts one can
derive a third kind of results: the equivalence with logical formalisms such as
monadic (second-order) logic. Finally, all these properties do not come at an un-
affordable price: emptiness is decidable, and hence the satisfaction of the logic
is also decidable.
In this paper, we present a quantitative extension to the standard notion of
regularity in which those cornerstone results still hold. We consider a quantitative
notion of regularity which allows to attach non-negative integer values to words,
such as the number of occurrences of a pattern, the length of segments, etc. One
also possess some freedom for combining those values, e.g., using minimum or
maximum. One can for instance describe the maximum number of occurrences of
letter a that are not separated by a letter b. Those integer values are considered
modulo an equivalence which preserves the existence of bounds, but does not
preserve exact values – as opposed to the usual way one considers quantitative
forms of automata. This is the price to pay for keeping all equivalences and
closure properties.
Originally, this work aimed at unifying and reinterpreting some recent results
from the literature. Let us review them.
?
Supported by the Anr project Jade: ‘Jeux et Automates, D´ecidabilit´e et Extensions’
First, in [9] Kirsten gives a new proof to the decidability of the (restricted)
star-height problem
1
. This problem is known to be decidable from Hashiguchi
[7], but with a very difficult proof. The first part in Kirsten’s proof consists in
reducing the star-height problem to a problem of limitedness: decide the exis-
tence of a bound for some function defined by means of a nested distance desert
automata, a new form of automata introduced for this purpose. The second part
consists in proving the decidability of this limitedness problem. This is done by
turning this automata-related question into an algebraic one: the automaton is
translated into a monoid equipped with a stabilisation operator ]. The limit-
edness problem becomes easy to decide in this presentation. Kirsten’s paper is
itself the continuation of a long line of research concerning distance automata,
tropical semiring, desert automata, etc. [6, 8, 11–17].
Second, the paper [3] provides a study of an extension of the monadic second-
order logic over infinite words with new ‘bound’ quantifiers such as: ‘there exists
a set of arbitrary large size satisfying some property’. The goal being different,
the presentation is also significantly different, and getting results comparable to
the ones in the present paper requires a translation that we cannot detail here.
However, two new forms of automata are introduced in [3] as intermediate objects
in the proofs, namely B-automata and S-automata. The class of B-automata
corresponds essentially to the non-nested variant of the nested desert distance
automata, while the class of S-automata is a new dual variant. The decidability
of limitedness can be derived from this work but with a bad complexity (non-
elementary, as opposed to [9]). Independently, B-automata were also introduced
in [1] under the name of R-automata, and the decidability of the limitedness
problem established using another technique, yielding better complexity.
Other applications of the technique have also been described. Still in this
framework, the restricted star-height problem for trees has been shown decidable
[4], and the Mostowski hierarchy problem
2
has been reduced to the corresponding
limitedness problem over infinite trees [5], which remains open. The existence
of a bound on the number of iterations necessary for reaching the fixpoint of a
monadic second-order formula over words has been also shown decidable using
distance automata [2].
Contribution. Our contribution can be roughly described as 1) a unification
of the ideas in [9] and [3], and 2) the development of a suitable mathematical
background and the establishment of new results in order to make this theory a
complete extension of the standard theory of regular languages. Let us be more
precise.
The first contribution lies in the definition of a cost function: cost functions
are mappings from words (or from any set in general) to ω + 1 quotiented by a
suitable equivalence that preserves the notion of bound (≈ in the paper). In our
1
Problem: given a regular language L of words and an integer k, is it possible to
describe L with a regular expression using at most k nesting of Kleene stars?
2
The hierarchy induced by the number of priorities used by a non-deterministic parity
automaton running on infinite trees.
framework, cost functions can be seen as a refinement of the notion of language
(each language can be seen as a cost function, while the converse is not true).
We then introduce B- and S-automata, automata that accept cost functions
rather than languages. Those are slight extensions of the automata in [3]. We
establish the equivalence of the two forms of automata, via an elementary con-
struction, as well as the equivalence with their history-deterministic form. The
new notion of history-determinism is a weakening of the classical notion of de-
terminism (deterministic automata are strictly weaker in this framework). It is
needed for the further extension of the theory to trees. Quiet naturally, we call
regular the cost functions described by one of these formalisms.
The second aspect of the theory that we develop is the algebraic formalism.
We introduce the notion of stabilisation monoids: finite monoids equipped with a
stabilisation operator, inspired from [9]. We develop a mathematical framework
– new to the knowledge of the author – in order to define the semantics of
stabilisation monoids. The key result here is the existence of unique semantics
(that we call compatible mappings) for each stabilisation monoid
3
. Building on
these notions, we introduce the notion of recognisable cost functions. As we may
expect, these happen to be exactly the regular cost functions.
While describing the above objects, we prove the closure of regular cost func-
tions under operations which correspond to union, intersection, projection and
dual of projection in the world of languages. We also provide decision procedures
subsuming the limitedness results from [9].
Structure of the paper. We present in Section 2 cost functions and the au-
tomata part of the theory. We present in Section 3 the algebraic framework, and
the equivalent notion of recognisability.
Some notations
As usual, we denote by ω the set of non-negative integers and ω + 1 the set
ω ∪ {ω}. Those are ordered by 0 < 1 < · · · < ω. The identity mapping over ω is
id. Given a set E, E
ω
is the set of sequences of ω-length of elements in E. Such
sequences will be denoted by bold letters (a, b,. . . ). We fix a finite alphabet A
consisting of letters. The set of words over A is A
∗
. The empty word is ε. The
concatenation of a word u and word v is uv. The length of word u is |u|. The
number of occurrences of a letter a in u is |u|
a
.
2 Regular cost functions
We introduce in Section 2.1 the notion of cost function. We present B and S-
automata in Section 2.2, and their history-deterministic form in Section 2.3. The
key duality result is the subject of Section 2.4.
3
This result is reminiscent of the one for infinite words stating that each finite Wilke
algebra can be uniquely extended into an ω-semigroup.
2.1 Cost functions
A correction function is a mapping from ω to ω. From now, the symbols α, α
0
, . . .
implicitly designate correction functions. Given x, y in ω + 1, x 4
α
y holds
if x ≤ α(y) in which α is the extension of α with α(ω) = ω. For every set E,
4
α
is extended to (ω + 1)
E
in a natural way by f 4
α
g if f (x) 4
α
g(x) for
all x ∈ E, or equivalently f ≤ α ◦ g. Intuitively, f is dominated by g after it has
been ‘stretched’ by α. One also writes f ≈
α
g if f 4
α
g and g 4
α
f.
Some elementary properties of 4
α
are:
Fact 1 If α ≤ α
0
and f 4
α
g, then f 4
α
0
g. If f 4
α
g 4
α
h, then f 4
α◦α
h.
Example 1. Over ω × ω, maximum and sum are equivalent for the doubling cor-
rection function (for short, (max) ≈
×2
(+)). Proof: for all x, y ∈ ω, max(x, y) ≤
x + y ≤ 2 × max(x, y).
Our second example concerns mappings from sequence of words to ω. Given
words u
1
, . . . , u
n
∈ {a, b}
∗
, we have |u
1
. . . u
n
|
a
≈
α
max(|K|, max
i=1...n
|u
i
|
a
)
in which K is the set of indices i such that |u
i
|
a
≥ 1 and α(θ) = θ
2
. Proof:
max(|K|, max
i=1...n
|u
i
|
a
) ≤ |u|
a
≤ Σ
i∈K
|u
i
|
a
≤ (max(|K|, max
i=1...n
|u
i
|
a
))
2
.
One also defines f 4 g (resp. f ≈ g) to hold if f 4
α
g (resp. f ≈
α
g) for
some α. A cost function (over a set E) is an equivalence class of ≈ (i.e., a set of
mappings from E to ω + 1). The relation 4 has other characterisations:
Proposition 1. For all f, g from E to ω + 1, the following items are equivalent:
– f 4 g,
– ∀n ∈ ω.∃m ∈ ω.∀x ∈ E.g(x) ≤ n → f (x) ≤ m , and;
– for all X ⊆ E, g|
X
is bounded implies f|
X
is bounded.
The last characterisation shows that the relation ≈ is an equivalence relation
that preserves the existence of bounds. Indeed, all this theory can be seen as an
automata theoretic method for proving the existence/non-existence of bounds.
Cost functions over some set E ordered by 4 form a lattice. Given a sub-
set X ⊆ E, one denotes by χ
X
its characteristic mapping defined by χ
X
(x) = 0
if x ∈ X, ω otherwise. It is easy to see that for all X, Y ⊆ E, χ
X
4 χ
Y
iff
Y ⊆ X. To this respect, the lattice of cost functions is a refinement of the lat-
tice of subsets of E equipped with the superset ordering. Keeping this in mind,
the notion of regular cost function developed in the paper is an extension of
the standard notion of regular language. This extension is strict as soon as E
is infinite: there are cost functions that are not equivalent to any characteristic
mapping. Consider for instance the size mapping over words, or the number of
occurrences of some letter.
2.2 Automata
We present here the automata model we use. A cost automaton (that can be
either a B-automaton or an S-automaton) is a tuple hQ, A, In, Fin, Γ, ∆i in
which Q is a finite set of states, A is the alphabet, In and Fin are respectively
the set of initial and final states, Γ is a finite set of counters, and ∆ ⊆ Q × A ×
{, i, r, c}
Γ
× Q is the set of transitions. The idea behind the letter in {, i, r, c}
Γ
(called an action) is that each counter (the value of which ranges over ω) can
either be left unchanged (), be incremented by one (i), be reset to 0 (r), or
be checked (c). A run σ of an automaton over a word a
1
. . . a
n
is defined as a
sequence q
0
, a
1
, c
1
, q
1
, . . . , q
n−1
, a
n
, c
n
, q
n
such that q
0
is initial, q
n
is final and for
all i = 1 . . . n, (q
i−1
, a
i
, c
i
, q
i
) ∈ ∆. Given a run σ, each counter ι ∈ Γ is initialized
with value 0 and evolves from left to right according to c
i
(ι): if c
i
(ι) is or c, the
value is left unchanged, if it is i, it is incremented by 1, if it is r, the counter is
reset. The set C(σ) ⊆ ω is the set of values taken by the counters when checked
(i.e., the value of counter ι when c
i
(ι) = c). The difference between B-automata
and S-automata comes from their dual semantics, [[·]]
B
and [[·]]
S
respectively:
for all u ∈ A
∗
, [[A]]
B
(u) = inf{sup C(σ) : σ run over u} ,
and, [[A]]
S
(u) = sup{inf C(σ) : σ run over u} ,
in which we use the standard convention that inf ∅ = ω and sup ∅ = 0. Remark
that if A is a non-deterministic finite automaton in the standard sense, accepting
the language L, then it can be seen as a cost automaton without counters. Seen as
a B-automaton, [[A]]
B
(u) = χ
L
, while seen as an S-automaton [[A]]
S
(u) = χ
A
∗
\L
.
Remark 1 (variants). The other similar automata known from the literature can
essentially be seen as special instances of the above formalism. The B-automata
and S-automata in [3] use only actions in {, i, cr} in which cr is an atomic
operation that checks the counter and immediately resets it. The models are
equivalent but the history-determinism (see below) cannot be achieved for S-
automata in this restricted form. The hierarchical automata correspond to the
case when Γ = {1, . . . , n} and for all transitions (p, a, c, q), if for all i ∈ Γ ,
if c(i) 6= implies c(j) = r for all j < i. The nested distance desert automata of
Kirsten corresponds to hierarchical B-automata that use actions in {, ic, r} in
which ic is an atomic operation which increments the counter and immediately
checks it. The R-automata in [1] use also actions in {, ic, r}, but without the
hierarchical constraint. All those models are equivalent, up to ≈.
We conclude the section by showing some easy closure properties. Given a
mapping f from A
∗
to ω + 1 and a length-preserving morphism h from A
∗
to B
∗
(B is another alphabet) the inf-projection and sup-projection of f with respect
to h are the mappings f
inf,h
and f
sup,h
from B
∗
to ω + 1 defined for v ∈ B
∗
by:
f
inf,h
(v) = inf{f (u) : h(u) = v} and f
sup,h
(v) = sup{f(u) : h(u) = v}.
By simply adapting the standard constructions for intersection, union, and pro-
jection of non-deterministic automata, we get:
Proposition 2. The mappings accepted by B-automata (resp. S-automata) are
closed under min and max. The mappings accepted by B-automata (resp. S-
automata) are closed under inf-projection (resp. sup-projection).