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The actual values are only computed when needed. For example, one could create a function that creates an infinite list (often called a stream) of Fibonacci numbers. The calculation of the n-th Fibonacci number would be merely the extraction of that element from the infinite list, forcing the evaluation of only the first n members of the list.
It can be seen as an extension of top-down finite-tree automata to infinite trees or as an extension of infinite-word automata to infinite trees. A finite automaton which runs on an infinite tree was first used by Michael Rabin [1] for proving decidability of S2S, the monadic second-order theory with two successors. It has been further observed ...
A bottom-up finite tree automaton over F is defined as a tuple (Q, F, Q f, Δ), where Q is a set of states, F is a ranked alphabet (i.e., an alphabet whose symbols have an associated arity), Q f ⊆ Q is a set of final states, and Δ is a set of transition rules of the form f(q 1 (x 1),...,q n (x n)) → q(f(x 1,...,x n)), for an n-ary f ∈ F ...
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 ...
Tree (data structure), a data structure simulating a single-rooted, directed hierarchy (due to the requirement of computer-implementability, only rational trees rather than arbitrary infinite trees are admitted) Tree (graph theory), a connected undirected graph without simple cycles; Tree (set theory), a generalization of a well-ordered set ...
In computer science, tree traversal (also known as tree search and walking the tree) is a form of graph traversal and refers to the process of visiting (e.g. retrieving, updating, or deleting) each node in a tree data structure, exactly once. Such traversals are classified by the order in which the nodes are visited.
Infinite input: An automaton that accepts infinite words . Such automata are called ω-automata. Tree input: The input may be a tree of symbols instead of sequence of symbols. In this case after reading each symbol, the automaton reads all the successor symbols in the input tree.
In fact, for any infinite cardinal κ, every κ-Suslin tree is a κ-Aronszajn tree (the converse does not hold). The Suslin conjecture was originally stated as a question about certain total orderings but it is equivalent to the statement: Every tree of height ω 1 has an antichain of cardinality ω 1 or a branch of length ω 1.