Equations between labeled directed graphs
Existence of solutions
Garreta-Fontelles A., Miasnikov A., Ventura E.
May 2013
Motivational problem
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H1 and H2 two subgroups of the free group generated by
X ∪ A, F (X , A).
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H1 is generated by w1 , . . . , wn , and H2 is generated by
v1 , . . . , vm .
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Find F (A)-morphisms h : F (X , A) → F (A) such that
H1 h = hh(w1 ), . . . , h(wn )i = hh(v1 ), . . . , h(vm )i = H2 h
as subgroups of F (A).
F (A)-morphism means that h(a) = a for all a ∈ A.
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X is viewed as a set of variables, and A as a set of constants.
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H1 = H2 denotes the previous equation between subgroups.
Translation into an equation between graphs
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Particular example: H1 = hy , axai, H2 = hxy , ayai.
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Take two labeled directed graphs Γ1 and Γ2 as in the following
picture.
Γ1
a
a
x
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Γ2
y
y
x
a
a
y
Goal: find words h(x) and h(y ) in F (A) such that when we
substitute them for x and y , and then we reduce the graphs,
we obtain isomorphic graphs*.
* If we allow h(x) and h(y ) not to be reduced, then we need isomoprhic graphs
modulo ”hanging trees”.
Notation
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Γ- labeled directed graph with labels in X ∪ A, and h an
F (A)-morphism.
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Γh denotes the graph obtained from Γ by substituting its
edges with label x ∈ X by h(x), as in the example:
a
a
c
b
x
c
b
b
a
b
The first graph is Γ. The second is Γh , where h(x) = bab.
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red(Γh ) is the graph obtained from Γh by applying a maximal
sequence of Stallings foldings.
Definition
Definition
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Γ1 , Γ2 - labeled directed graphs with labels in X ∪ A.
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V = vi1 , . . . , vini - distinguished vertices of Γi , i = 1, 2. And,
f - map between them: f (v1 j) = v2f (j) .
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The equation Γ1 =V Γ2 has as solutions F (A)-morphisms h
such that there exists an isomorphism φ from coreV (red(Γ1 h ))
to coreV (red(Γ2 h )), with φ(v1j ) = f (v1j ).
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coreV (red(Γ1 h )) is the graph obtained from red(Γ1 h ) by
cutting all hanging trees that do not contain any vertex from
V.
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Alternatively, we can ask for red(Γ1 h ) = red(Γ2 h ). (No
removal of hanging trees).
More applications of equations between graphs
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Our goal: to decide effectively wether a system of graph
equations has a solution or not, keeping an eye on the
problem of describing these solutions (future work?).
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Solving systems of graph equations implies solving, for
example,
1. Systems of subgroup equations H1 = H2 .
2. Systems of word equations in the free group.
3. Systems of word equations on a free group with rational
constraints.
4. Systems of equations between finite deterministic automata.
Word equations seen as graph equations
Example of how to translate a word equation into a graph equation.
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w1 (X , A) = w2 (X , A) is an equation between two words w1
and w2 in F (X , A). Say w1 = axa−1 x −1 , w2 = 1. x ∈ X ,
a ∈ A.
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Set Γ1 and Γ2 to be:
Γ1
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v a
x
a
x u
w
Γ2
V = {v , u, w } are distinguished vertices. f (v ) = f (u) = w .
A solution to the above equation is h(x) = a2 . Then we have:
v =u
a
red(Γ1 h )
w
a
a
They are the same once we apply coreV .
red(Γ2 h )
Systems of word equations with rational constraints
Systems of word equations with rational constraints.
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They are systems of word equations in a free group, restricting
each variable to belong to a given regular language. A
particular case: systems of word equations, restricting that the
variables belong to given subgroups of F (A).
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The existence of solutions of systems of word equations with
rational constraints was solved by Diekert, Gutiérrez, and
Hagenah.
Our results
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Reduction of solvability of systems of graph equations to
solvability of systems of word equations with rational
constraints. (Then the method by Diekert, Gutiérrez, and
Hagenah can be applied to solvability of systems of graph
equations).
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Alternative and direct solution to the problem of solvability of
systems of graph equations, with potential applications to the
problem of giving a description of the solutions.
Tools in our direct approach
Definition (Branch folding)
We make branch foldings instead of the usual foldings. In a branch
folding we choose two paths and we fold them together:
c
b
a
c
a
b
a
c
b
d
a
a
a
b
b
d
a
d
d
c
b
Tools
Fix notation: S = Γh = ∆1 → . . . → ∆n = red(Γh ) denotes a
sequence of branch foldings, applied to Γh until it is reduced.
Definition (Bases)
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The 0-bases of (Γ, h, S) are the edges in Γ with label in X .
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The i-bases of (Γ, h, S) are the 0-bases transformed into ∆i ,
as in the example:
a
b
b
a
b
c
a
b
c
b
Following the example before, the graph on the left is ∆1 = Γh ,
and the graph on the right is ∆2 = red(Γh ).
Outline of the direct approach
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Observation: A solution to Γ1 = Γ2 induces a solution to one
among an infinite number of systems of equations, and vice
versa.
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Each of this system of equations depends on red(Γ1 h ) and
red(Γ2 h ), and the bases on them. h is any morphism.
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Problem: There are infinitely many systems as above. The
number of graphs arising from foldings, forgetting about the
bases, is ”finite” (in a sense). But the bases can go along the
graphs in infinitely many ways.
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Example.
The problem above can be overcomed in two steps:
1. Identify subwords in the elements h(x), x ∈ X , such that,
0
when removed, we have h0 , a new morphism, where red(Γ1 h )
0
and red(Γ2 h ) and its bases are essentially the same as
red(Γ1 h ), red(Γ2 h ), and its bases.
If there are no such subwords, call h a minimal morphism.
h0 is still a solution to the system.
2. There are finitely many minimal morphisms, up to Bulitko
lemma.
Thank you!