Equations between labeled directed graphs Existence of solutions Garreta-Fontelles A., Miasnikov A., Ventura E. May 2013
Motivational problem ◮ H 1 and H 2 two subgroups of the free group generated by X ∪ A , F ( X , A ). ◮ H 1 is generated by w 1 , . . . , w n , and H 2 is generated by v 1 , . . . , v m . ◮ Find F ( A )-morphisms h : F ( X , A ) → F ( A ) such that H 1 h = � h ( w 1 ) , . . . , h ( w n ) � = � h ( v 1 ) , . . . , h ( v m ) � = H 2 h 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. ◮ H 1 = H 2 denotes the previous equation between subgroups.
Translation into an equation between graphs ◮ Particular example: H 1 = � y , axa � , H 2 = � xy , aya � . ◮ Take two labeled directed graphs Γ 1 and Γ 2 as in the following picture. Γ 1 Γ 2 y y x a a a a y x ◮ 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. ◮ Γ h denotes the graph obtained from Γ by substituting its edges with label x ∈ X by h ( x ), as in the example: a a b c b c b b x a The first graph is Γ . The second is Γ h , where h ( x ) = bab . ◮ red (Γ h ) is the graph obtained from Γ h by applying a maximal sequence of Stallings foldings.
Definition Definition ◮ Γ 1 , Γ 2 - labeled directed graphs with labels in X ∪ A . ◮ V = v i 1 , . . . , v in i - distinguished vertices of Γ i , i = 1 , 2. And, f - map between them: f ( v 1 j ) = v 2 f ( j ) . ◮ The equation Γ 1 = V Γ 2 has as solutions F ( A )-morphisms h such that there exists an isomorphism φ from core V ( red (Γ 1 h )) to core V ( red (Γ 2 h )), with φ ( v 1 j ) = f ( v 1 j ). ◮ core V ( red (Γ 1 h )) is the graph obtained from red (Γ 1 h ) by cutting all hanging trees that do not contain any vertex from V . ◮ Alternatively, we can ask for red (Γ 1 h ) = red (Γ 2 h ). (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 H 1 = H 2 . 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. ◮ w 1 ( X , A ) = w 2 ( X , A ) is an equation between two words w 1 and w 2 in F ( X , A ). Say w 1 = axa − 1 x − 1 , w 2 = 1. x ∈ X , a ∈ A . ◮ Set Γ 1 and Γ 2 to be: Γ 1 Γ 2 v u w 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 ) = a 2 . Then we have: v = u a red (Γ 1 h ) red (Γ 2 h ) w a a They are the same once we apply core V .
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: c b c a a b d a b d a a a d a b d c c b b
Tools Fix notation: S = Γ h = ∆ 1 → . . . → ∆ n = red (Γ h ) denotes a sequence of branch foldings, applied to Γ h 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 a b c b c b b b a Following the example before, the graph on the left is ∆ 1 = Γ h , and the graph on the right is ∆ 2 = red (Γ h ) .
Outline of the direct approach ◮ Observation: A solution to Γ 1 = Γ 2 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 (Γ 1 h ) and red (Γ 2 h ), 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 (Γ 1 h ′ ) and red (Γ 2 h ′ ) and its bases are essentially the same as red (Γ 1 h ), red (Γ 2 h ), 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!
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