Equations between labeled directed graphs Existence of solutions - - PowerPoint PPT Presentation

Equations between labeled directed graphs

Existence of solutions Garreta-Fontelles A., Miasnikov A., Ventura E. May 2013

Motivational problem

◮ H1 and H2 two subgroups of the free group generated by

X ∪ A, F(X, A).

◮ H1 is generated by w1, . . . , wn, and H2 is generated by

v1, . . . , vm.

◮ Find F(A)-morphisms h : F(X, A) → F(A) such that

H1h = h(w1), . . . , h(wn) = h(v1), . . . , h(vm) = H2h as subgroups of F(A).

F(A)-morphism means that h(a) = a for all a ∈ A.

◮ X is viewed as a set of variables, and A as a set of constants. ◮ H1 = H2 denotes the previous equation between subgroups.

Translation into an equation between graphs

◮ Particular example: H1 = y, axa, H2 = xy, aya. ◮ Take two labeled directed graphs Γ1 and Γ2 as in the following

picture. Γ1 Γ2 y a x a x y a y a

◮ 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

◮ Γ- labeled directed graph with labels in X ∪ A, and h an

F(A)-morphism.

◮ Γh denotes the graph obtained from Γ by substituting its

edges with label x ∈ X by h(x), as in the example: a b c a b c x b a b The first graph is Γ. The second is Γh, where h(x) = bab.

◮ red(Γh) is the graph obtained from Γh by applying a maximal

sequence of Stallings foldings.

Definition

Definition

◮ Γ1, Γ2 - labeled directed graphs with labels in X ∪ A. ◮ V = vi1, . . . , vini - distinguished vertices of Γi, i = 1, 2. And,

f - map between them: f (v1j) = v2f (j).

◮ The equation Γ1 =V Γ2 has as solutions F(A)-morphisms h

such that there exists an isomorphism φ from coreV (red(Γ1h)) to coreV (red(Γ2h)), with φ(v1j) = f (v1j).

◮ coreV (red(Γ1h)) is the graph obtained from red(Γ1h) by

cutting all hanging trees that do not contain any vertex from V .

◮ Alternatively, we can ask for red(Γ1h) = red(Γ2h). (No

removal of hanging trees).

More applications of equations between graphs

◮ 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?).

◮ 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.

◮ w1(X, A) = w2(X, A) is an equation between two words w1

and w2 in F(X, A). Say w1 = axa−1x−1, w2 = 1. x ∈ X, a ∈ A.

◮ Set Γ1 and Γ2 to be:

Γ1 v u w Γ2 a x a x

◮ 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 red(Γ1h) w red(Γ2h) a a a They are the same once we apply coreV .

Systems of word equations with rational constraints

Systems of word equations with rational constraints.

◮ 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).

◮ The existence of solutions of systems of word equations with

rational constraints was solved by Diekert, Guti´ errez, and Hagenah.

Our results

◮ Reduction of solvability of systems of graph equations to

solvability of systems of word equations with rational

• constraints. (Then the method by Diekert, Guti´

errez, and Hagenah can be applied to solvability of systems of graph equations).

◮ 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: a b c a b d a b d c b c a a a b d a b c d

Tools

Fix notation: S = Γh = ∆1 → . . . → ∆n = red(Γh) denotes a sequence of branch foldings, applied to Γh until it is reduced.

Definition (Bases)

◮ The 0-bases of (Γ, h, S) are the edges in Γ with label in X. ◮ The i-bases of (Γ, h, S) are the 0-bases transformed into ∆i,

as in the example: a b c b a b b a b c 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

◮ Observation: A solution to Γ1 = Γ2 induces a solution to one

among an infinite number of systems of equations, and vice versa.

◮ Each of this system of equations depends on red(Γ1h) and

red(Γ2h), and the bases on them. h is any morphism.

◮ 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.

◮ Example.

The problem above can be overcomed in two steps:

• 1. Identify subwords in the elements h(x), x ∈ X, such that,

when removed, we have h′, a new morphism, where red(Γ1h′) and red(Γ2h′) and its bases are essentially the same as red(Γ1h), red(Γ2h), and its bases. If there are no such subwords, call h a minimal morphism. h′ is still a solution to the system.

• 2. There are finitely many minimal morphisms, up to Bulitko

lemma.

Thank you!