- Source: 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 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
A
{\displaystyle A}
has three states,
Q
=
{
q
0
,
q
l
e
f
t
,
q
r
i
g
h
t
}
{\displaystyle Q=\{q_{0},q_{\mathit {left}},q_{\mathit {right}}\}}
.
A
{\displaystyle A}
begins in the root in state
q
0
{\displaystyle q_{0}}
and descends to the left subtree. Then it processes the tree recursively. Whenever
A
{\displaystyle A}
enters a node
v
{\displaystyle v}
in state
q
l
e
f
t
{\displaystyle q_{\mathit {left}}}
, it means that the left subtree of
v
{\displaystyle v}
has just been processed, so it proceeds to the right subtree of
v
{\displaystyle v}
. If
A
{\displaystyle A}
enters a node
v
{\displaystyle v}
in state
q
r
i
g
h
t
{\displaystyle q_{\mathit {right}}}
, it means that the whole subtree with root
v
{\displaystyle v}
has been processed and
A
{\displaystyle A}
walks to the parent of
v
{\displaystyle v}
and changes its state to
q
l
e
f
t
{\displaystyle q_{\mathit {left}}}
or
q
r
i
g
h
t
{\displaystyle q_{\mathit {right}}}
, depending on whether
v
{\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 (
D
T
W
A
⊊
T
W
A
{\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 (
T
W
A
⊊
R
E
G
{\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
External links
Mikołaj Bojańczyk: Tree-walking automata. A brief survey.
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