Tree walking automaton

A tree walking automaton (TWA) is a type of finite automaton that deals with tree structures rather than strings. The concept was originally proposed in Template:Harvtxt.

The following article deals with tree walking automata. For a different notion of tree automaton, closely related to regular tree languages, see branching automaton.

Definition

All trees are assumed to be binary, with labels from a fixed alphabet Σ.

Informally, a tree walking automaton A (TWA) is a finite state device which walks over the tree in a sequential manner. At each moment A visits a node v in state q. Depending on the state q, the label of the node v, and whether the node is the root, a left child, a right child or a leaf, A changes its state from q to q‘ and moves to the parent of v or its left or right child. A TWA accepts a tree if it enters an accepting state, and rejects if its enters a rejecting state or makes an infinite loop. As with string automata, a TWA may be deterministic or nondeterministic.

More formally, a (nondeterministic) tree walking automaton over an alphabet Σ is a tuple A = (Q, Σ, I, F, R, δ) where Q is a finite set of states, its subset I, F, and R is the set of initial, accepting and rejecting states, respectively, and δ ⊆ (Q × { root, left, right, leaf } × Σ × { up, left, right } × Q) is the transition relation.

Example

A simple example of a tree walking automaton is a TWA that performs depth-first search (DFS) on the input tree. The automaton ${\displaystyle A}$ has 3 states, ${\displaystyle Q=\{q_{0},q_{\mathit {left}},q_{\mathit {right}}\}}$. ${\displaystyle A}$ begins in the root in state ${\displaystyle q_{0}}$ and descends to the left subtree. Then it processes the tree recursively. Whenever ${\displaystyle A}$ enters a node ${\displaystyle v}$ in state ${\displaystyle q_{\mathit {left}}}$, it means that the left subtree of ${\displaystyle v}$ has just been processed, so it proceeds to the right subtree of ${\displaystyle v}$. If ${\displaystyle A}$ enters a node ${\displaystyle v}$ in state ${\displaystyle q_{\mathit {right}}}$, it means that the whole subtree with root ${\displaystyle v}$ has been processed and ${\displaystyle A}$ walks to the parent of ${\displaystyle v}$ and changes its state to ${\displaystyle q_{\mathit {left}}}$ or ${\displaystyle q_{\mathit {right}}}$, depending on whether ${\displaystyle v}$ is a left or right child.

Properties

Unlike branching automata, tree walking automata are difficult to analyze and even simple properties are nontrivial to prove. The following list summarizes some known facts related to TWA: