Tree walking automaton

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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[1].

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[edit]

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[edit]

A simple example of a tree walking automaton is a TWA that performs depth-first search (DFS) on the input tree. The automaton has 3 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[edit]

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 Bojańczyk and Colcombet[2] deterministic TWA are strictly weaker than nondeterministic ones ()
  • 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 (), i.e. there exist regular languages which are not recognized by any tree walking automaton, see Bojańczyk and Colcombet.[3]

See also[edit]

References[edit]

  1. ^ 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. 
  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; Colcombet, Thomas (2008). "Tree-Walking Automata Do Not Recognize All Regular Languages" (PDF). SIAM J. Comput. 38 (2): 658–701. doi:10.1137/050645427. 

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