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A tree-walking automaton (TWA) 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 (TWA) 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 (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 subsets I, F, and R are the sets 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 three 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: even simple properties are nontrivial to prove. The following list summarizes some known facts related to TWA:

• As shown by Bojańczyk and Colcombet, deterministic TWA are strictly weaker than nondeterministic ones (${\displaystyle {\mathit {DTWA}}\subsetneq {\mathit {TWA}}}$)
• 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 (${\displaystyle {\mathit {TWA}}\subsetneq {\mathit {REG}}}$), i.e. there exist regular languages that are not recognized by any tree-walking automaton, see Bojańczyk and Colcombet.

See also

• Pebble automata, an extension of tree-walking automata

References

1. Aho, A; Ullman, J (1971). "Translations on a context free grammar". Information and Control. 19 (5): 439–475. doi:10.1016/S0019-9958(71)90706-6.
2. Bojańczyk, 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.
3. Bojańczyk, Mikołaj (2008). Martín-Vide, Carlos; Otto, Friedrich; Fernau, Henning (eds.). "Tree-Walking Automata" (PDF). Language and Automata Theory and Applications. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 1–2. doi:10.1007/978-3-540-88282-4_1. ISBN 978-3-540-88282-4.
4. Bojańczyk, 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 Bojańczyk: Tree-walking automata. A brief survey.

• Trees (data structures)
• Automata (computation)