1 Graphs and Regular Trees Minimization

So far, trees are defined formally as “tree domains” (TDom ⊆ P(L(N))). Informally, we know that graphs can model some trees. To reason about properties of trees, we define the following functions: sub(T , i) ∈ TDom×N⇀ TDom := {l | (i :: l) ∈ T} sub(T ,p) ∈ TDom×L(N) ⇀ TDom := {l | (p@l) ∈ T} Sub(T) ∈ TDom→ P(TDom) := {sub(T ,p) | p ∈ L(N)} These functions compute the n-th direct subtree, the subtree reached on path p, and the set of all subtrees of some tree domain T . This leads us to the central definition: Definition 1.1 (Regular Tree) A tree domain T is called regular iff Sub(T) is finite, i.e. there are only finitely many different subtrees. To talk about graphs, we need to define them formally: Definition 1.2 (Graph) A graph G = (V, E) consists of a set V ⊆ N of nodes and an edge function E ∈ V × N ⇀ V such that ∀v ∈ V ∃n ∈ N : {m ∈ N | (v,m) ∈ Dom(E)} = {0, . . . , n− 1}. A graph is called finite if its node set V is finite. You might know this definition of graphs from automata theory: V is usually called the set of states, and E the transition function. Let’s connect our definitions of trees and graphs now. We can define the extension of the transition function E: . ∈ V ×L(N) → V ∪ {⊥} Let m ∈ N, p ∈ L(N). Then