Tree-walking automaton


A tree-walking automaton is a type of finite automaton that deals with tree structures rather than strings. The concept was originally proposed by Aho and Ullman.
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 is a finite state device that walks over an input 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 tree-walking automaton over an alphabet Σ is a tuple
where Q is a finite set of states, its subsets,, and are the sets of initial, accepting and rejecting states, respectively, and is the transition relation.

Example

A simple example of a tree-walking automaton is a TWA that performs depth-first search on the input tree. The automaton has three states,. begins in the root in state and descends to the left subtree. Then it processes the tree recursively. Whenever enters a node in state, it means that the left subtree of has just been processed, so it proceeds to the right subtree of. If enters a node in state, it means that the whole subtree with root has been processed and walks to the parent of and changes its state to or, depending on whether is a left or right child.

Properties

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