The Power of Commuting
with Finite Sets of Words
Michal Kunc
∗
Department of Mathematics, Masaryk Universi ty,
Jan´aˇckovo n´am. 2a, 602 00 Brno, Czech Republic
kunc@math.muni.cz, http://www.math.muni.cz/~kunc/
March 10, 2006
Abstract
We construct a finite language L such that the largest language
commuting with L is not recursively enumerable. This gives a negative
answer to the question raised by Conway in 1971 and also strongly dis-
proves Conway’s conjecture on context-freeness of maximal solutions
of systems of semi-linear inequalities.
1 Introduction
In this paper we address the question whether the larg est solution of any
language equation of t he form XL = LX is regular provided L is a finite
or regular language. It is known that in several algebraic structures related
to algebras of formal languages, two elements commute if and only if they
can be generated by the same element. For instance, two words commute if
and only if they are powers of the same word (due to the defect theorem),
and such a characterizatio n was proved valid also for polynomials and f ormal
series in non-commuting variables over a field (in particular, for multisets
∗
Supported by the project MSM0021622409 of the Ministry of Education of the Czech
Republic.
1
of words) by Bergman [3] and Cohn [7], respectively. But in the case of
languages the situation is completely different.
Systems of language equations and inequalities were studied mainly in
connection with context-free languages since these languages can be described
as components of smallest solutions of explicit systems of polynomial equa-
tions, i.e. equations with the operatio ns of union and concatenation. Much
less attention was devoted to systems of implicit equations. An att empt to
initiate development of a unified theory o f general language equations has
been recently made by Okhotin; in particular, he proved that recursive (re-
cursively enumerable, co-recursively enumerable) languages are exactly lan-
guages definable as unique (smallest, largest) solutions of systems of implicit
language inequalities using concatenation and all Boolean o perations [19] and
that each such language can be encoded even in a single explicit equation
and precisely defined by an explicit system consisting of two equations [20].
Known results on regularity of maximal solutions of systems of implicit
language inequalities are surveyed in [14]. Such issues were first addressed
by Conway [8], who observed that inequalities of the form E ⊆ L, where E
is a regular function of variables and L is a regular language, possess only
finitely many maximal solutions, all of them are regular and computable.
Maximal solutions are also simple in the case of inequalities where vari-
ables are concatenated only from one side. It is well known that regular
languages can be characterized as components of smallest or largest solu-
tions of explicit systems of right- linear equations. Regular solutions of more
general systems were studied for example by Leiss [17]. For systems of im-
plicit right-linear inequalities, i.e. inequalities of the form
K ∪ K
1
X
1
∪ · · · ∪ K
n
X
n
⊆ L ∪ L
1
X
1
∪ · · · ∪ L
n
X
n
,
where K, K
1
, . . . , K
n
and L, L
1
, . . . , L
n
are constant languages, it is known
that their la r gest solutions are always regular provided all constant languages
on right-hand sides are regular [16]. Moreover, if all constant languages
occurring in the system are regular, then computability of the largest solution
follows from Rabin’s results on MSO logic over infinite trees [21]; actually,
the computation of the solution is an ExpTime-complete problem [1, 5 , 2].
The problem of regularity of the largest language commuting with a g iven
regular language was formulated already by Conway [8 , p. 55 and 124]
in 197 1. There are actually two variants of the problem depending on whether
we allow the resulting language to contain the empty word or not: The
2
largest solution of the equation XL = LX is denoted C(L) and its largest
solution without the empty word is denoted C
+
(L). The languages C(L) and
C
+
(L) a r e in general different and no direct relation between them (except
for the obvious inclusion C
+
(L) ⊆ C(L)) has been found yet. The problem
was recently studied in several articles (e.g. [22, 6, 9, 10]), but a ffirmative
answers were given only for regular codes [10] a nd at most ternary sets of
words [9]. Moreover, it even remained an open problem whether the largest
language commuting with a given finite set of words is recursive. On the
other hand, the complements of languages C(L) and C
+
(L) are always re-
cursively enumerable provided the language L is recursive; this is a special
case of a general result about systems of language equations, which is proved
in [19]. A summary of known results concerning commutation of languages
and some examples can be found in t he recent survey [12].
In this paper we g ive t he most negative possible answer to Conway’s
problem by showing that there exists a finite language L such that C(L) is
not recursively enumerable. More precisely, we show that the complement of
the language computed by an a rbitra r y Minsky machine can be encoded into
a solution of a commutation equation. This contrasts with the fact that the
largest solution of the inequality XK ⊆ LX is regular provided the language
L is regular, a s demonstrated in [13]. On the other hand, negative results
analogous to those for commutation equations were proved in [15] for systems
consisting of two such inequalities.
We formulate our results for the case of languages C
+
(L) too, and further
we show that for a regular language L the difference C(L)\C
+
(L) also does not
have to be recursively enumerable, which answers a question posed in [11].
In addition, our results disprove Conway’s conjecture [8, p. 129] stating that
every maximal solution of a system of so-called semi-linear inequalities is
context-free.
Before dealing with the main result of this paper, we demonstrate the
techniques employed in its proof by proving several weaker results. In Sec-
tion 3 we consider the situation where L is only required to be a star-free
language. First, we give an example of a star-free language L such that C(L)
is non-regular, and then we describe how the construction can be improved
to show that C(L) does not even have to be recursively enumerable. Let us
mention that this is in accord with the results obta ined in [13]: complicated
cases arise for star-free languages (or equivalently, languages recognizable by
aperiodic monoids), whereas maximal solutions of such equations with con-
stant languages recognizable by finite simple semigroups (in particular, by
3
finite groups) are always regular. Based on the constructions presented in
this section, we additionally prove that it is undecidable whether two star-
free languages are conjugated via some language containing the empty word.
This result is a step t owards dealing with basic problems about conjugacy of
languages formulated in [4].
Section 4 is devoted to the case of finite languages; by encoding the
example from the b eginning of Section 3 into finitely many words, we show
that the language C(L) can be non-regular even for a finite language L.
Finally, in Section 5 we combine the techniques of Sections 3 and 4 to obtain
a finite set of words L such that C(L) is not recursively enumerable.
Basic notions employed in our considerations are recalled in the following
section. For a more comprehensive introduction to the theory of formal
languages the reader is referred to [23].
2 Preliminaries
We denote the sets of positive and non-negative integers by N and N
0
, re-
spectively. Throughout the pa per we consider a certain finite alphabet A.
As usual, we write A
+
for the set of all non-empty finite words over A, and
A
∗
for the set obtained from A
+
by adding the empty word ε. For a letter
a ∈ A and a positive integer n ∈ N, we denote the set {ε, a, . . . , a
n
} by a
≤n
.
A word u ∈ A
∗
is called a factor of v ∈ A
∗
if v = wu ˆw for some words
w, ˆw ∈ A
∗
; it is called a prefix (suffix ) of v if v = uw (v = wu, respectively)
for some w ∈ A
∗
.
Languages over the alphabet A are arbitrary subsets o f A
∗
, and we say
that a language L ⊆ A
∗
is ε-free if ε /∈ L. The reverse of a language L is
defined as
{a
n
· · · a
1
| a
1
, . . . , a
n
∈ A, a
1
· · · a
n
∈ L}
and denoted rev(L). The basic operation on languages is concatenation de-
fined by the rule K · L = {uv | u ∈ K, v ∈ L}, and we use the standard no-
tation L
+
=
S
m∈N
L
m
and L
∗
= L
+
∪ {ε}. Regular languages are languages
definable by finite automata, or equivalently, by rational expressions. The
basic tool for proving non-r egularity of languages is the well-known pumping
lemma (see e.g. [23]). A language L ⊆ A
∗
is called star-free if it can be ob-
tained fr om finite languages using the operations of union, complementation
and concatenation; in particular, for every B ⊆ A, the languages B
+
and B
∗
are star-free.
4
For every language L over A we denote by C(L) the largest language
over A which commutes with L and by C
+
(L) the largest ε-f r ee language
over A which commutes with L. Such languages C(L) and C
+
(L) certainly
exist for every language L since the union of arbitrarily many languag es
commuting with L commutes with L as well. Is is clear that we always
have C
+
(L) ⊆ C(L), L
∗
⊆ C(L) and L
+
⊆ C
+
(L). Further, the languages
C(L) a nd C
+
(L) are easily seen to be closed under concatenation and so they
form a submonoid and a subsemigroup of the free monoid A
∗
, r espectively.
Another interesting property of the languages C(L) and C
+
(L) is that they
remain unchanged when we replace L with its closure under concatenation,
i.e. C(L) = C(L
+
) and C
+
(L) = C
+
(L
+
).
Intuitively, we can view the commutation equation XL = LX as a game
of two players, the attacker and the defender. A position of the game is an
arbitrary word w ∈ A
∗
. At each step of the game, the attacker adds any
word from L to an arbitrary side of w, and the defender has to respond by
removing some word belonging to L from the opposite side of the resulting
word. The word thus obtained is a new position of the game. The attacker
wins the g ame if the defender has no move available, and the defender wins
if he manages to continue playing forever. Then it is easy to observe that the
largest solution of the equation XL = LX is exactly the set of all positions
of the game where the defender has a winning strategy.
This intuitive view is a lso reflected by the structure of proofs in this
paper, which consist of two parts: Proving that a given word w does not
belong to C(L) amounts to finding a winning strategy for the atta cker on w.
And conversely, in order to prove that w lies in C(L), we describ e a set of
positions containing w such that no matter how the attacker plays in one of
these positions, the defender is always able to return to such a position.
3 Star-Free Languages
The aim of this section is to construct a star-free language L such that the
largest solution of the equation XL = LX is not recursively enumerable.
This is achieved by encoding an arbitrary Minsky machine M into a star-
free language L in such a way that C(L)∩uv
∗
w = {uv
n
w | n /∈ L(M)}, where
u, v and w are certain words and L(M) ⊆ N
0
is the set computed by the
machine M. Because the construction is rather technical, let us first present
it in a simplified form which shows that the largest language commuting with
5