Tree (automata Theory)

In automata theory, a tree is a particular way of representing a tree structure as sequences of natural numbers.

For example, each node of the tree is a word over set of natural numbers ($\mathbb$), which helps this definition to be used in automata theory.

A ''tree'' is a set ''T'' ⊆ $\mathbb$^* such that if ''t''.''c'' ∈ ''T'', with ''t'' ∈ $\mathbb$^* and ''c'' ∈ $\mathbb$, then ''t'' ∈ ''T'' and ''t''.''c''₁ ∈ ''T'' for all 0 ≤ ''c''₁ < ''c''. The elements of ''T'' are known as ''nodes'', and the empty word ε is the (single) ''root'' of ''T''. For every ''t'' ∈ ''T'', the element ''t''.''c'' ∈ ''T'' is a ''successor'' of ''t'' in ''direction'' ''c''. The number of successors of ''t'' is called its ''degree'' or ''arity'', and represented as ''d''(''t''). A node is a ''leaf'' if it has no successors. If every node of a tree has finitely many successors, then it is called a ''finitely'', otherwise an ''infinitely branching'' tree. A ''path'' π is a subset of ''T'' such that ε ∈ π and for every ''t'' ∈ ''T'', either ''t'' is a leaf or there exists a unique ''c'' ∈ $\mathbb$ such that ''t''.''c'' ∈ π. A path may be a finite or infinite set. If all paths of a tree are finite then the tree is called finite, otherwise infinite. A tree is called ''fully infinite'' if all its paths are infinite. Given an alphabet Σ, a ''Σ-labeled tree'' is a pair (''T'',''V''), where ''T'' is a tree and ''V'': ''T'' → Σ maps each node of ''T'' to a symbol in Σ. A labeled tree formally defines a commonly used term tree structure. A set of labeled trees is called a ''tree language''.

A tree is called ''ordered'' if there is an order among the successors of each of its nodes. The above definition of tree naturally suggests an order among the successors, which can be used to make the tree ranked.

In the case of ranked alphabets, an extra function ''Ar'': Σ → $\mathbb$ is defined. This function associates a fixed arity to each symbol of the alphabet. In this case, each ''t'' ∈ ''T'' has to satisfy ''Ar''(''V''(''t'')) = ''d''(''t''). The trees that satisfy this property are called ''ranked'' trees. The trees that do not (necessarily) satisfy that property are called ''unranked''.

For example, the above definition is used in the definition of an infinite tree automaton.

Example

Let ''T'' = ^* and Σ = . We define a labeling function ''V'' as follows: the labeling for the root node is ''V''(ε) = ''a'' and, for every other node ''t'' ∈ ^*, the labellings for its successor nodes are ''V''(''t''.0) = ''a'' and ''V''(''t''.1) = ''b''. It is clear from the picture that ''T'' forms a (fully) infinite binary tree.

References

* {{cite book, first1=Hubert, last1=Comon, first2=Max, last2=Dauchet, first3=Rémi, last3=Gilleron, first4=Florent, last4=Jacquemard, first5=Denis, last5=Lugiez, first6=Christof, last6=Löding, first7=Sophie, last7=Tison, first8=Marc, last8=Tommasi, title=Tree Automata Techniques and Applications, date=November 2008, chapter=Preliminaries , url=https://gforge.inria.fr/frs/download.php/10994/tata.pdf, accessdate=11 February 2014, ref={{harvid, Comon et al., 2008

Trees (data structures)
Automata (computation)
Formal languages
Theoretical computer science