Solutions of twisted word equations, EDT0L languages, and context-free groups

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by  Volker Diekert

SOLUTIONS OF TWISTED WORD EQUATIONS, EDT0L
LANGUAGES, AND CONTEXT-FREE GROUPS
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VOLKERDIEKERTANDMURRAYELDER
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2

n
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J Abstract. We prove that the set of all solutions for twisted word equa-

2 tions with regular constraints is an EDT0L language and can be computed
1 inPSPACE. Itfollowsthatthesetofsolutionstoequationswithrationalcon-
straints in a context-free group (= finitely generated virtually free group) in

] reducednormalformsisEDT0L.Wecanalsodecide(inPSPACE)whetheror
R
not the solution set is finite, which was an open problem. Our results gen-
G eralizetheworkbyLohreyandS´enizergues (ICALP 2006) andDahmani and
Guirardel (J. of Topology 2010) with respect to complexity and withrespect
.
h to expressive power. Neither paper gave any concrete complexity bound and
t bothrelyontheexponentofperiodicity,sotheresultinthesepapersconcern
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only subsets of solutions, whereas our results concern all solutions. We do
m
more, wegive, insomesense, the “optimal” formallanguage characterization
[ ofthefullsolutionset.

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1. Introduction
v
7 In a seminal paper [24] Makanin showed that the problem WordEquations is
9
decidable. The first complexity estimation of that problem was a tower of several
2
exponentialfunctions, but this droppeddowntoPSPACE byPlandowski[29]using
3
0 compression. The insight that long solutions of word equations can be efficiently
. compressed is due to [30] which also led to the (still valid) conjecture that Word-
1
Equations is NP-complete. Until 2013 the known decidability proofs for solving
0
7 word equations were long and technical with an accompanied reputation for being
1 difficult. This changed drastically when Jez˙ applied his recompression technique:
v: hepresentedasimple NSPACE(nlogn)algorithmto solvewordequations[18]. Ac-
i tually his method achieved more: it describes all solutions, copes with rational
X
constraints (which is essential in applications), and it extends to free groups [6].
r
Builtonthe ideasin[6]Ciobanuandthepresentauthorsshowedthatthefullsolu-
a
tionsetofagivenwordequationwithrationalconstraintsisEDT0L[3]. (Thiswas
known before only for quadratic word equations by [11].) EDT0L-languages are
definedbyacertaintypeofLindenmayersystems,see[35]. ThemeaningEDT0Lis
not easy to initially digest, but fortunately there is a veryconvenientcharacteriza-
tion of an EDT0L-language: all we need is a rational set of endomorphisms over a
free monoid which applied to any word yields the language. The motivation for [3]
was to prove that the full solution set in reduced words of equations in free groups
is an indexed language, a problem which was open at that time [12, 17]. However,
the resultis strongersince EDT0L forms a strict subclass ofindexed languages[9].
2010 Mathematics Subject Classification. 03D05,20F65,20F70, 68Q25,68Q45.
Key words and phrases. Equation in a virtually free group; twisted equation; EDT0L language;
PSPACE..
ResearchsupportedbytheAustralianResearchCouncil(DiscoveryProjectDP160100486).
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Transfer results as in [15, 3] from words to free groups have a long history. In
the 1980s Makanin showedthat the existential and positive theories of free groups
are decidable [25]. In 1987 Razborov gave a description of all solutions for an
equationinafreegroupvia“Makanin-Razborov”diagrams[31,32]whichformeda
cornerstoneintheindependentworkofKharlampovichandMyasnikov[20]andSela
[37] on the positive solution of Tarski’sconjectures about the elementary theory in
free groups.
The motivation for the present paper is along this line. We show that given a
finitelygenerated(f.g.forshort)virtuallyfreegroupV thereisaPSPACE-algorithm
which produces for a given equation with rational constraints an effective descrip-
tionofEDT0Llanguageswhichdescribesthe fullsolutionsetinreducedwordover
a natural set of generators. Several remarks are in order here. First, in general
virtually free group have torsion; and this is serious obstacle to apply the known
techniques. The reason to study virtually free groups is motivated by ubiquitous
presence of word hyperbolic groups [14]. Solving equations in torsion-free hyper-
bolic groups reduces to solving equations in free groups [34], but solving equations
in word hyperbolic groups with torsion reduces to solving equations in virtually
free groups which in turn reduces to solving “twisted” word equations with ratio-
nal constraints [4]. The question of whether solving “twisted” word equations is
decidable was asked by Makanin ([26] Problem 10.26(b)) and solved in [4]. The
conclusion was that it is decidable whether a given equation over a f.g. virtually
free group is solvable. This result which was also independently shown by Lohrey
and S´enizergues [23]. (Actually, [23] proves a more general transfer result.) What
is in common: both papers use a bound on the so-called “exponent of periodicity”.
Because of this both are unable to describe all solutions. Moreover, neither paper
gives any concrete complexity bounds.
Therefore, the present paper extends [4, 23] in various aspects. For a fixed
f.g.virtuallyfreegroupV wepresentaPSPACE-algorithmwhereoninputanequa-
tion with rational constraints the output is an NFA which defines the full solution
set an EDT0L language. This is done by solving the same problem for twisted
wordequationswithrationalconstraints. So,themainnew contributionisentirely
within combinatorics on words. We must deal with twisted word equations; and
although we follow the general scheme to define a sound and complete algorithm
to produce an NFA A describing all solutions, the technical details are quite far
from previous methods. For example, for twisted equations it does not make sense
to “uncross” pairs ab where a, b are different letters because once all pairs ab are
uncrossed the twisting may produce new crossing pairs ba, uncrossing them leads
to new crossing pairs ab etc. Thus, our underlying method is quite different from
the original recompressiondue to Jez˙.
The class of f.g. virtually free groups appears in many different ways. For ex-
ample, a fundamental theorem of Muller and Schupp (relying on [8]) says that
f.g. group is virtually free if and only if it is context-free [27]. This means that,
given any set of monoid generators A, the set of words w ∈ A∗ which represent
1 ∈ V forms a context-free language. Other characterizations include: (1) funda-
mental groups of finite graphs of finite groups [19], (2) f.g. groups having a Cayley
graph with finite treewidth [22], (3) universal groups of finite pregroups [33], (4)
groups having a finite presentation by some geodesic string rewriting system [13],
2
and(5)f.g.groupshavingaCayleygraphwithdecidablemonadicsecond-orderthe-
ory[22]. Proofsfor these equivalencesare in[7]. The transformationsareeffective.
For example, starting from a context-free grammar for the word problem, we can
constructa representationas a fundamental groupof finite graphsof finite groups.
Then we can find a representation as a free-by-finite group. (We repeat a proof
for that step here because we need it for the complexity bounds.) Having that,
it is immediate to construct a (deterministic) push-down automaton for the word
problem, which yields some context-free grammar for the word problem. Thus, it
is legitimate to assume our input is a fundamental group of a finite graph of finite
groups with its natural set of generators.
2. Preliminaries
Byanalphabet wemeanafiniteset. Theelementsarecalledletters(orsymbols).
By Σ∗ we denote the free monoid overΣ. The elements of a free monoidare called
words. The length of word w is denoted by |w|, and |w| counts how often a letter
a
aappearsinw. LetM beanymonoidandu,v∈M. Wesaythatuisafactorofv,
if we can write v =xuy for some x,y ∈M. If we can write v =uy (resp. v =xu),
then we say that u is a prefix (resp. suffix). If u is a prefix of v, then we also
write u ≤ v. The prefix relation is of no interest in groups, of course. With a few
exceptionswedenotetheneutralelementinmonoidsby1. Inparticular,theempty
word in a free monoid is also denoted by 1. An involution of a set is a bijection
x 7→ x such that x = x for all x in the set. For example, the identity map is an
involution. A monoid with involution additionally has to satisfy xy = yx. If G is
a group, then it is a monoid with involution by taking g = g−1 for all g ∈ G. By
default, we identify g and g−1 for groupelements. In the following, everyalphabet
comes with an involution.
Amorphismbetweensetswithinvolutionisamappingrespectingtheinvolution.
Amorphismbetweenmonoidswithinvolutionisahomomorphismϕ:M →N such
that ϕ(x) = ϕ(x). For ∆ ⊆ M ∩N we say that it is a ∆-morphism if ϕ(x) = x
for all x ∈ ∆. A bijective morphism is called an automorphism and the set of
automorphisms on a set (or monoid) M forms the group Aut(M).
Every group homomorphism is a morphism of monoids with involution. Let G
be agroup. Itactsonaset(withinvolution)X byamappingx7→g·xif1·x=x,
f ·(g ·x) = (fg)·x) (and f ·x = f ·x) for all f,g ∈ G and x ∈ X. If G acts
on a monoid (with involution) M, then we additionally demand that every group
element acts as an automorphism: f ·(xy) = (f ·x)(f ·y). For better readability
we frequently write f(x) instead of f ·x.
Thespecificationofregularconstraintsisgivenherebyassigningtoeachconstant
andvariableanelementinfinite monoid(typicallythe finitemonoidisamonoidof
Boolean matrices and arises as the transformation monoid of a finite automaton.)
Bymakingthefinitemonoidlarger,wecanturnitintoamonoidN withinvolution
and where G acts on it, see the appendix. This allows us to represent regular
constraintsusingamorphismµ:(A∪(G×X))∗ →N whichrespectstheinvolution
andtheactionofG. InthefollowingwefixthefinitemonoidN andweassumethat
allmorphismstoN respecttheinvolutionandGaction. WesaythatM isaNG-i-
monoid ifM isamonoidwithinvolutionandaGactiontogetherwithamorphism
µ:M →N. If not explicitly stated otherwise all monoids under consideration are
3
NG-i-monoids. In particular, N is an NG-i-monoid. A morphism between NG-i-
monoids is morphism ϕ : M → M′ such that ϕ(g ·x) = g ·(ϕ(x)) and µ′ϕ = µ.
In the following, if not explicitly stated otherwise a morphism means a morphism
between NG-i-monoids.
The guiding example is given as follows. Let A be an alphabet with involution.
(Taking the identity means that our results also apply to the case of free monoids
without a predefined involution; another typical situation is a subset of a group
and g = g−1.) The involution extends to A∗: for a word w = a ···a we let
1 m
w = a ···a . The monoid A∗ is called the free monoid with involution over A.
m 1
If a = a for all a ∈ Σ then w is simply the word w read from right-to-left and
w = w means that w is a palindrome. Note that every automorphism on A∗ is
induced by some automorphism on the unique minimal generating set A. Thus,
Aut(A) = Aut(A∗) and |Aut(A∗)| ≤ |A|!. In our setting G is a finite subgroup of
Aut(A): theactionofGonwordsislengthpreservinganditrespectstheinvolution.
We allow self-involuting letters a = a. As a consequence we also allow that there
are f ∈G and a∈A such that f(a)=a. It is also clear that f(uv)=f(u)f(v) for
f ∈Gandu,v ∈A∗. Ifa∈Aisaletterandw∈A∗ isaword,thenwesaythatais
G-visible in w iff(a)appearsinw forsomef ∈G. Moreover,assumethatforeach
letter a ∈ A we are given µ(a) ∈ N such that µ(a) = µ(a) and f(µ(a)) = µ(f(a))
for all f ∈G. Then we can extend µ to the free monoid with involution such that
A∗ becomes an NG-i-monoid. We also say that A is an NG-i-alphabet.
2.1. Rational languages and EDT0L. Regular languages in finitely generated
free monoids can be defined via nondeterministic finite automata (NFA for short)
or via recognizability via homomorphisms to finite monoids, to mention just two
possible definitions. The notion of a nondeterministic finite automaton extends
to every monoid M as follows. An NFA is a directed finite graph A with initial
and final states, where the transitions are labeled with elements of the monoid M.
A transition labeled by 1 ∈ M is called an ε-transition as it is the tradition for
NFA’s over free monoids. We say that m ∈ M is accepted by the automaton A
if there exists a path from some initial to some final state such that multiplying
the edge labels together yields m. This defines the accepted language L(A) =
{m∈M | m is accepted by A}. According to [10] a subset L ⊆ M is rational if
and only if L is accepted by some NFA over M.
An NFA is called trim if every state is on some path from an initial to a final
state. For a trim A we have L(A)6=∅ if and only if A6=∅. The acronym EDT0L
refers to Extended, Deterministic, Table, 0 interaction, and Lindenmayer, see the
handbook[36]. AsubsetLinak-folddirectproductA∗×···×A∗ iscalledEDT0L
if there some (extended) alphabet C with c ,...,c ∈ C such that A ⊆ C and a
1 k
rational set R⊆End(C∗) of endomorphisms over C∗ such that
L={(h(c ),...,h(c ))| h∈R}.
1 k
ThismeansthatwehaveaneffectivedescriptionofLasfollows: thereisanNFAA
wherethetransitionsarelabeledbya“deterministictable”ofpairs(c,u )∈C×C∗
c
(encodingthe endomorphismwhichmapsc to u )andthereareletters c ,...,c ∈
c 1 k
C such that (w ,...,w ) ∈ L if and only if there is some h ∈ L(A) ⊆ End(C∗)
1 k
such that (w ,...,w ) = (h(c ),...,h(c )). The idea is that properties of L (like
1 k 1 k
emptiness or finiteness) become structural properties of A. The classical situation
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refers to k =1; and our definition uses a characterizationof EDT0L languagesdue
to Asveld [1].
2.2. Twisted variables and NG-i-monoids with types. Let B and Y be two
disjoint NG-i-alphabets. We call B the alphabet of constants and Y the set of
twisted variables. It is convenient to choose a set X with involution of minimal
size such that every Y ∈ Y has the form Y = f(X) for some X ∈ X and f ∈ G.
In the following, by a variable we mean X ∈ X and thus, every twisted variable
Y ∈Y canbe writtenasf·X forsomef ∈G. Moreover,weassumeX 6=X forall
variables. IfGactswithoutfixedpointsonY,then weidentify Y =G×X andthe
action becomes g·(f,X)=(gf,X). By M(B,X,θ,µ) we denote an NG-i-monoid
which is generated by B ∪{f(X)| f ∈G,X ∈X} together with a finite set θ of
homogeneous defining relations. That is, every (x,y) ∈ θ satisfies |x| = |y|. We
always assume that (x,y)∈θ implies µ(x)= µ(y), (x,y)∈θ, and (f(x),f(y))∈θ
for all f ∈G, even if these relations are not listed in the specification of θ.
Forcomplexityissueswerequire|x|≤2foreach(x,y)∈θ and|θ|∈O(|G|kSk2)
where kSk is specified in Theorem2. The homogeneity conditionmakes it possible
tosolvethewordproblemandallothercomputationalissuesinthequotientmonoid
M(B,X,θ,µ) = M(B,X,∅,µ)/{x=y | (x,y)∈θ}. The uniform complexity is in
nondeterministiclinearspacewhichisgoodenoughforourpurposes. Thesedetails
are easy to see and left to the interested reader. By M(B,θ,µ) we denote the
NG-i-monoid-submonoidwhich is generated by B.
3. The main result on twisted word equations
We begin with an alphabet of constants A (as always with involution) where G
is a subgroup of Aut(A) as in Section 2. Initially, the set of twisted variables is
G×V whereV denotesthe initialsetofvariables. ThegroupGactsbyf·(g,X)=
(fg,X), and hence, without fixed points on twisted variables. For a word w in
constants and f ∈ G we use the notation f(w) = (f,w); and we hence identify
(A∪(G×V))∗ = ((G×(A∗ ∪V))∗. We abbreviate (1,x) as x for x ∈ A∗ ∪V.
By µ : A∗ → N we mean a homomorphism which respects the involution and
0
the action of G. Thus A∗ is, via µ , an NG-i-monoid. Assume that µ has been
0 0
extendedtoamappingµ :A∗∪V →N suchthatµ (X)=µ (X),thenµ extends
0 0 0 0
to a morphismµ :(A∪(G×V))∗ →N of NG-i-monoidsby µ (f,X)=f·µ (X).
0 0 0
At this stage we don’t have defined types, so we work over free monoids.
A system S of twisted word equations with regular constraints is given by the
following data:
• Asetofpairs{(U ,V )| 1≤i≤s}whereU ,V ∈(A∪(G×V))∗ aretwisted
i i i i
words.
• A morphism µ :(A∪(G×V))∗ →N.
0
Asusual,atwistedequation(U ,V )isalsosimplywrittenasU =V . Asolution of
i i i i
S is givenamorphismσ :V →A∗ whichis (uniquely) extendedto anA-morphism
of NG-i-monoids σ :(A∪(G×V))∗ →A∗ such that
• σ(U )=σ(V ) for all twisted equations (U ,V ).
i i i i
• µ σ(X)=µ (X) for all variables. Hence, µ σ =µ .
0 0 0 0
Example1. LetA={a,a,b,b},V ={X,X,Y,Y,Z,Z},f,g ∈Gdefinedbyf(a)=
b,g(a) = a,g(b) = b, U = (f,X)a(g,Y), V = Z, U = (f,Y)b, V = ab(g,X),
1 1 2 2
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U =Xa, V =b(f,X) and (for simplicity) µ (x)=1 for all x∈A∪V. A solution
3 3 0
is given by σ(X)=bab,σ(Y)=baab,σ(Z)=abaabaab.
IfσisasolutionofS wealsosaythatσsolvesS,thenforV = X ,X ,...,X ,X
1 1 k k
(cid:8) (cid:9)
the full solution set Sol(S) of S is defined as
Sol(S)={(σ(X ),...,σ(X ))∈A∗×···×A∗ | σ solves S}.
1 k
Our main structural result shows that Sol(S) is effectively EDT0L. Actually, we
can compute an effective presentation in polynomial space. In order to measure
complexities we need a notion of input size. We define the size kSk by
kSk=|G|+|A|+|V|+s+ |U V |.
X i i
1≤i≤s
Convention. Thereisanimplicitbound|G|≤|A|!. SinceGcanbe muchsmaller,
our complexity bounds take kSk and |G| as parameters into account. For better
readability we don’t measureN. Therefore,we add the generalhypotheses that N
isgiveninsuchawaythatthespecificationandallnecessarycomputationsoverN,
suchas performinga multiplication, computing the involutionorthe G action,can
be done in polynomialspacewith respectto kSk. Thus, in the example of Boolean
m×m matrices, we may allow that m is polynomial in kSk although the size of
N becomes 2m2. That is, we do not ever need to actually store the entire set of
matrices N.
Theorem2. ThereisaPSPACEalgorithm whichtakesasinputasystemoftwisted
word equations with regular constraints S and V as above with input size kSk. The
output in an extended alphabet C of size O(|G|2kSk2), letters c ∈ C for each
X
X ∈ V, and a trimmed NFA A accepting a rational set of A-morphisms L(A) ⊆
End(C∗) such that
(1) Sol(S)={(h(c ),...,h(c ))∈C∗×···×C∗ | h∈L(A)}.
X1 Xk
The algorithm stores intermediate equations with a length bound in O(|G|kSk2).
Moreover, Sol(S) = ∅ if and only if L(A) = ∅; and |Sol(S)| < ∞ if and only if A
doesn’t contain any directed cycle.
Theorem 2 implies that Sol(S) is effectively EDT0L, and that we can decide
in polynomial space (resp. deterministic exponential time) whether S is solvable
and whether or not there are only finitely many solutions. The problem to decide
emptiness of Sol(S) is known to be PSPACE-hard by [21] because the intersection
problemof regularlanguagesis a specialcase. If the finite monoidN is not partof
the input, then the best known lower bound is NP-hardness.
4. Twisted conjugacy and δ-periodic words
Before we dive into the proof of Theorem 2 we show how to solve a particular
kindofatwistedequation: conjugacy. Forwordsx,y,z ∈A∗ with16=z astandard
exercise in combinatorics on words [16] shows:
(2) zy =xz ⇐⇒ ∃r,s∈A∗∃e∈N:x=rs∧y =sr∧z =(rs)er.
This fact is crucial in Makanin’s classical approach [24] to solve (untwisted) word
equations. Here, we need a variant of (2) in the twisted environment. We say that
words x,y ∈A∗ are twisted conjugate if there are f,g,h∈G and z ∈A∗ such that
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zg(y) = h(x)f(z). We also say that |x| = |y| is the offset of the conjugacy. A
twisted conjugacy equation is a twisted equation of the form
(3) Z(g,Y)=(h,X)(f,Z).
Proposition 3. Let σ be a solution of the twisted equation (3) such that the offset
|σ(X)| satisfies 1 ≤ |σ(X)| < |σ(Z)|. Then there are words r ∈ A+, s ∈ A∗ and
e,j ∈N with 0≤j <|G| such that |rs|=σ(X) and
(4) σ(Z)=((rs)f(rs)···f|G|−1(rs))ef0(rs)···fj−1(rs)fj(r).
Z Y
↓1 ↓g
↑h ↑f
X Z
u f(u) f2(u) fj(r) Y
h(v)
v u f(u) f2(u) fj−1(u) fj(r)
Figure 1. Twisted conjugacy
Proof. Let v = σ(X) and u = h(v). Since 1 ≤ |σ(X)| < |σ(Z)| the word u is a
proper nonempty prefix of σ(Z). If 2|u| ≤ |σ(Z)|, then uf(u) is a prefix of σ(Z),
and so on. Thus, σ(Z) is a prefix of the word in uf(u)f2(u)···f|σ(X)|(u). Next,
observe that f|G|(u)=f0(u)=u for every word u∈A∗. Thus,
σ(Z)=[uf(u)f2(u)···f|G|−1(u)]euf(u)···fj−1(u)fj(r)
where 0 ≤ i < |G|, u = rs and the |r| suffix of Z is where the pattern runs out,
see Figure 7. We then have σ(Y) = g−1fj(sf(r)). Hence, the nonempty word
u and the length |σ(Z)| define a unique factorization u = rs, integers 0 ≤ e and
0≤j <|G| such that σ(Z) has the desired form above. (cid:3)
Recall that a word p is primitive if it cannot be written as p = re with e ≥ 2.
Inparticular,the empty word1 is notprimitive. It is well-known(andeasyto see)
thatanonemptywordpisprimitiveifandonlyifp2 cannotbewrittenasp2 =xpy
with x6=1 and y 6=1.
Let w,p ∈ A+ be nonempty words. We say that w has period |p| if w is a
prefix of p|w|. In other words, if w = a ···a with a ∈ A, then a = a for all
1 n i i+|p|
1 ≤ i ≤ n−|p|. A word may have several periods, for example w = aabaabaa has
periods 3,6,7,8. If |p| is the least period of w, then |p|≤|w| and we can choose p
tobe primitive suchthatp≤w. Forexample,aab≤aabaabaais aprimitive prefix
and |aab|=3.
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Definition 4. We say that a word w is δ-periodic if it has some period less or
equal to than δ. A δ-periodic word w is called longδ-periodic if |w|≥3δ, and very
long δ-periodic if |w|≥10δ.
Using this terminology, Proposition 3 yields the following result.
Corollary 5. Let ε ∈ N, f,g,h ∈ G, and x,y,z ∈ A∗ be words with 1 ≤ |x| ≤ ε
and |z|≥10|G|ε. If we have zg(y)=h(x)f(z), then z is a very long |G|ε-periodic
word.
Moreover, let z = αwβ be any factorization with |w| = |x|. Then every letter b
occurring in z satisfies b=f(a) for some f ∈G and some letter a occurring in w.
Proof. By the proposition we have
z =((rs)f(rs)···f|G|−1(rs))ef0(rs)···fj−1(rs)fj(r)
where |fi(rs)|=|rs|≤ε so z has a period
|(rs)f(rs)···f|G|−1(rs)|≤|G|ε,
and |z|≥10|G|ε by hypothesis so z is a long |G|ε-periodic word.
Forthesecondclaim,ifz =αwβwith|w|=|x|thenwisafactoroffi(rs)fi+1(rs)
oflength|rs|. Ifwe writers=a ...a ,thenany letterb inz satisfiesb=fk(a ).
1 |x| ℓ
Letι∈{i,i+1}sothatfι(a )isaletterinv,thenb=fk(a )=fk(f−ι(fι(a )))=
ℓ ℓ ℓ
fk−ι(a). (cid:3)
An important property of δ-periodic words is the following.
Lemma 6. Let w be a δ-periodic word and w = per = qfs such that p,q are
primitive |p|≤|q|≤δ, 16=r ≤p, 16=s≤q, and |w|≥2δ. Then p=q, e=f ≥1,
and r =s.
Proof. The assertionis clear for |p|=|q|. Hence we may assume that p is a proper
prefix of q. Since q ≤ w we conclude q ≤ pδ. Since |w| ≥ 2δ, and |p| ≤ |q| ≤ δ we
see pq ≤ w ≤ qδ. Thus q occurs as factor inside qq: we have pqs = qq for some s.
Since 1≤|p|<|q|, this contradicts the primitivity of q. (cid:3)
Let u be a prefix (resp. factor, resp. suffix) of some nonempty word w. We say
that u is a maximal δ-periodic prefix (resp. factor, resp. suffix) in w if we cannot
extend the occurrence of the factor u inside w by any letter to the right or left, to
see a δ-periodic word.
4.1. Preprocessing to pass to a triangulated system. A twisted word equa-
tionU =V iscalledtriangulated ifU containsatmost2andV atmostonevariable.
By standard methods (using at most 2kSk more variables), it is enough to show
Theorem2inthe casewhereeacheachequationU =V equals(f,x)(g,y)=(h,z)
i i
where x,y,z ∈ A∪V. This is equivalent to (h−1f,x)(h−1g,y) = z. Moreover,
we can assume that z = Z is variable. Hence the starting point is a system of
equations(f,x)(g,y)=Z. During the processwe needa more generalform, (stan-
dard) equations appear as u(f,x)w(g,y)v = u′Zv′ where u,w,v,u′,v′ are words
over constants. Whenever such an equation with |u| = |u′| = |v| = |v′| appears,
we make a consistency check that u = u′ and v = v′; in the other case we stop
with “unsolvable”. That is, in the nondeterministic process this branch is rejected.
Finally, by adding a new zero 0 = 0 and a new neutral element 1 = 1, to N, we
canassumethatµ(w)=1impliesw =1forallwordsoverconstantsandµ(X)6=0
8
forvariables. Moreover,weassumethatforeveryu(f,X)w(g,Y)v =u′Zv′ itsdual
equation v(g,Y)w(f,X)u=v′ Z u′ is part of the system too.
Duringtheprocesswehavetoenlargethe setsofconstantsandvariables. Inthe
beginning we fix two disjoints alphabets with involution C and Ω. All constants
are drawn from C and all variables are drawn from Ω. We require |Ω| = |C| and
that |C| is large enough, but polynomial in the input size kSk. More precisely we
have |C|∈O(|G|2kSk2). Throughout we use following notation.
• A⊆B =B ⊆C, X =X, Y =G×X ⊆Ω and Σ=C∪Ω.
• GactsonB∪Y,theactionandtheinvolutionextendthoseonA∪(G×V).
• µ:Σ∗ →N satisfies µ(a)=µ (a) for a∈A.
0
• a,b,c,[p],[r,s,λ],... refer to letters in C.
• u,v,w,... refer to words in C∗.
• X,Y,Z,[X,p],... refer to variables in Ω, X 6=X for all variables.
• x,y,z,... refer to words in Σ∗.
These conventions hold everywhere unless explicitly stated otherwise. They also
apply to primed symbols such as B′, X′ etc.
4.2. The initialwordequationW . Fortechnicalreasonsweencodetheinitial
init
(triangular) system {(U =V )| 1≤i≤s} of twisted equations in variables V =
i i
X ,X 1≤i≤k as a single word. For this we assume that A contains special
i i
(cid:8) (cid:12) (cid:9)
marker s(cid:12)ymbols # and # with µ(#) = µ(#) = 0. Thus, they cannot be used
by any solution σ(X) since µ(X) 6= 0 by assumption. We also assume that for
each a ∈ A there is a trivial equation a = a. (This ensures that letters form A
will always be visible in equations.) Let U = U #···#U #U #···#U # and
1 s 1 s
V =V #···#V #V #···#V #.
1 s 1 s
The initial equation W ∈((A∪(G×V))∗ is defined as:
init
(5) W =#X #···#X #U##X #···#X #V#.
init 1 k 1 k
Note that σ(W) =σ(W) if and only if σ(U ) =σ(V ) for all i. We fix n =|W |.
i i init
Note that this implies n>|A|+|V| and kSk∈|G|+Θ(n).
Definition 7. An extended equation is a tuple (W,B,X,θ,µ), where θ is a type
and:
(1) W ∈(B∪Y)∗ (with Y =G×X) and |W|≤|C∪Ω|.
(2) W =#x #···#x #u##x #···#x #v# for some
1 k 1 k
u = u #···#u #u #···#u and v = v #···#v #v #···#v with x ,
1 s 1 s 1 s 1 s i
u , v ∈(B∪Y)∗ and µ(x u v )6=0.
i i i i i
(3) Given W as above we call u =v a local equation.
i i
(4) We say W as above is a standard equation if θ =∅ and all local equations
are triangular
(5) If a variable X appears somewhere in θ, then X is called typed. Werequire
that for a typed variable X there exists a primitive word θ(X) ∈ B∗ such
that Xθ(X)=θ(X)X is true in M(B,X,θ,µ).
(6) A solution is a morphism σ :M(B,X,θ,µ)→M(B,θ,µ) such that:
• σ(W)=σ(W). (Equivalently: σ(u )=σ(v ) for all i.)
i i
• σ(X)∈p∗, whenever X is typed and p=θ(X).
(7) An entire solution is a pair (α,σ) where α : M(B,θ,µ) → M(A,∅,µ ) is
0
an A-morphism and σ is a solution.
9
ThestatesoftheNFAAweareaimingforareextendedequationsandtransitions
are certain labeled arcs between states.
If (W,B,X,θ,µ)−h→(W′,B′,X′,θ′,µ′) is a transition, then h:M(B′,θ′,µ′)→
M(B,θ,µ) is a morphism (in the opposite direction of the arc) which is speci-
fied a mapping h : ∆ → B∗ with ∆ ∩ A = ∅. We assume that it can be ex-
tended to a morphism h : M(B′,θ′,µ′) → M(B,θ,µ) by leaving all letters in
B′\ f(d) d∈∆∪∆,f ∈G invariant. Thus, if we say that h(c) = w, then we
(cid:8) (cid:12) (cid:9)
don’t need(cid:12)to say h(c) = w or f(c) = f(w) for f ∈ G or µ′(c) = µ(w). Moreover,
for each constant a ∈ A we know its stabilizer G = {g ∈G| g(a)=a} = G .
a a
New constants appear only by compression. This means a word w is replaced a
fresh letter c by specifying h(c) = w. At this point the stabilizer of w is known:
for example if w = ab, then G = {g ∈G| g(w)=w} = G ∩G . Hence, we
w a b
define the stabilizer G = G ; and we also introduce fresh letters f(c) with the
c w
property f(c) = g(c) ⇐⇒ g−1f ∈ G . We also define µ(f(c)) = µ(f(w)) and
w
c = c ⇐⇒ w = w; and we check that h(p) = h(q) for all (p,q) ∈ θ. In this
way, we make sure that h is a morphism.(Recall that this means a morphism of
NG-i-monoids.) The morphism h induces a endomorphism of C∗ which respects
the involution assuming h(c) = c for all c ∈ C \B′. However, outside B′ neither
the action of B nor the value of µ is defined, so C∗ is not an NG-i-monoid. The
crucial observation is that whenever
(W ,B ,X ,θ ,µ )h−s→+1 ···−h→t (W ,B ,X ,θ ,µ )
s s s s s t t t t t
is a labeled path and w ∈ B∗ is word, then h = h ···h can be viewed either as
t 1 t
a morphism h : M(B ,θ ,µ ) → M(B ,θ ,µ ) or as an endomorphism of C∗. If
t t t s s s
we have w ∈B∗, then h defines a word h(w)∈B∗ and the corresponding element
t s
h(w) ∈ M(B ,θ ,µ ). By ε we denote the identity of C∗. Then ε appears as a
s s s
label transitions (W,B,X,θ,µ) −→ε (W′,B′,X′,θ′,µ′) where h : M(B′,θ′,µ′) →
M(B,θ,µ)isamorphismwithh(a)=aforalla∈B′. Forexample,wemighthave
B′ ⊆B or (B′,µ′)=(B,µ) and θ′ ⊆θ.
5. The ambient NFA F
We are ready to define an NFA F which contains the trimmed NFA A we are
aimingforasasubautomaton. WeshowthatF issound: thismeansinthenotation
of Theorem 2
(6) {(h(c ),...,h(c ))∈C∗×···×C∗ | h∈L(F)}⊆Sol(S).
X1 Xk
The states of F are extended equations and there are two types of transitions: a
substitution transforms the variables and does not affect the constants; a compres-
sion affects the constants, it may change variables only by changing types.
5.1. States. We define the states ofthe NFA F by a subsetof extendedequations
(W,B,X,θ,µ) according to Definition 7, subject to the following type restriction:
wheneverX appearsinadefiningrelation(x,y)∈θ,then(uptosymmetry)thereis
aletteraandatypedvariableY (possiblyX =Y)suchthatfirst,(x,y)=(Xa,aY)
and second, θ(X)a = aθ(Y) holds in M(B,θ,µ). Recall that for a typed variable
X there exists a primitive word θ(X) ∈ B∗ such that Xθ(X) = θ(X)X holds in
M(B,X,θ,µ). (In order to have θ(X) be defined through θ we might require that
θ(X)is the shortestprimitive wordwith Xθ(X)=θ(X)X andthat it is unique by
that property. However, this is not essential.)
10