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 Aho & Ullman (1971).

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 $A$ has 3 states, $Q = \{ q_{0}, q_{\mathit{left}}, q_{\mathit{right}} \}$. $A$ begins in the root in state $q_{0}$ and descends to the left subtree. Then it processes the tree recursively. Whenever $A$ enters a node $v$ in state $q_{\mathit{left}}$, it means that the left subtree of $v$ has just been processed, so it proceeds to the right subtree of $v$. If $A$ enters a node $v$ in state $q_{\mathit{right}}$, it means that the whole subtree with root $v$ has been processed and $A$ walks to the parent of $v$ and changes its state to $q_{\mathit{left}}$ or $q_{\mathit{right}}$, depending on whether $v$ 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:

• As shown by Bojanczyk & Colcombet (2006), deterministic TWA are strictly weaker than nondeterministic ones ($\mathit{DTWA} \subsetneq \mathit{TWA}$)
• deterministic TWA are closed under complementation (but it is not known whether the same holds for nondeterministic ones)
• the set of languages recognized by TWA is strictly contained in regular tree languages ($\mathit{TWA} \subsetneq \mathit{REG}$), i.e. there exist regular languages which are not recognized by any tree walking automaton (Bojanczyk & Colcombet 2008).

See also

• Pebble automata, an extension of tree walking automata

References

• Aho, A; Ullman, J (1971). "Translations on a context free grammar" (PDF). Information and Control 19 (5): 439–475. doi:10.1016/S0019-9958(71)90706-6.
• Bojanczyk, Mikołaj; Colcombet, Thomas (2006). "Tree-walking automata cannot be determinized" (PDF). Theoretical Computer Science 350 (2-3): 164–173. doi:10.1016/j.tcs.2005.10.031.
• Bojanczyk, Mikołaj; Colcombet, Thomas (2008). "Tree-Walking Automata Do Not Recognize All Regular Languages" (PDF). SIAM J. Comput. 38 (2): 658–701. doi:10.1137/050645427.

External links

• Mikołaj Bojanczyk: Tree-walking automata. A brief survey.