Tree automaton

Tree automaton

A tree automaton is a type of state machine. Tree automata deal with tree structures, rather than the strings of more conventional state machines.

The following article deals with branching tree automata, which correspond to regular languages of trees. For a different notion of tree automaton, see tree walking automaton.

As with classical automata, finite tree automata (FTA) can be either a deterministic automaton or not. According to how the automaton processes the input tree, finite tree automata can be of two types: (a) bottom up, (b) top down. This is an important issue, as although non-deterministic (ND) top-down and ND bottom-up tree automata are equivalent in expressive power, deterministic top-down automata are strictly less powerful than their deterministic bottom-up counterparts, because tree properties specified by deterministic top-down tree automata can only depend on path properties. (Deterministic bottom-up tree automata are as powerful as ND tree automata.)

Contents

Definitions

A ranked alphabet is a pair of ordinary alphabet \mathcal{F} and a function Arity: \mathcal{F}\rightarrow\mathbb{N}. Each letter has its arity so it can be used to build terms. Nullary elements (of zero arity) are also called constants. Terms built with unary symbols and constants can be considered as strings. Higher arity leads to trees.

A bottom-up finite tree automaton over F is defined by: (Q,F,Qf,Δ)

Here Q is a set of unary letters (states), F is a ranked alphabet, Q_{f} \subseteq Q is a set of final states, and Δ is a set of transition rules, that is, rewrite rules from nodes whose childs' roots are states, to nodes whose roots are states. Thus the state of a node is deduced from the states of its children.

There is no initial state as such, but the transition rules for constant symbols (leaves) can be considered as initial states. The tree is accepted if the state labeled at the root is an accepting state.

A top-down finite tree automaton over F is defined by: (Q,F,I,Δ)

There are two differences with bottom-up tree automata : first, I \subseteq Q, the set of its initial states, replaces QF ; second, its transition rules are the converse, that is, rewrite rules from nodes whose roots are states to nodes whose child's roots are states. The tree is accepted if every branch can be gone through this way.

The rewrite rules cause symbols from Q to 'travel' along branches of the tree.

One can easily guess that non-deterministic top-down tree automata are equivalent to non-deterministic bottom-up ones ; the transition rules are simply reversed, and the final states become the initial states.

Why then are deterministic top-down FTA less powerful than their bottom-up counterparts? Because a deterministic TA is by definition one where no two transition rules have the same left-hand side. For tree automata, transition rules are rewrite rules ; and for top-down ones, the left-hand side will be parent nodes. Consequently a deterministic top-down tree automaton will only be able to test for tree properties that are true in all branches, because the choice of the state to write into each child branch is determined at the parent node, without knowing the child branches contents.

Properties

Determinism

As said before, a deterministic tree automaton is one where no two transition rules have the same left-hand side. This definition matches the intuitive idea that for an automaton to be deterministic, one and only one transition must be possible for a given node.

Recognizability

For a bottom-up automaton, a ground term t (that is, a tree) is accepted if there exists a reduction that starts from t and ends with q(t), where q is a final state. For a top-down automaton, a ground term t is accepted if there exists a reduction that starts from q(t) and ends with t, where q(t) is an initial state.

The tree language L(A) recognized by a tree automaton A is the set of all ground terms accepted by A. A set of ground terms is recognizable if there exists a tree automaton that recognizes it.

One important property is that linear (that is, arity-preserving) homomorphisms preserve recognizability.

Completeness and Reduction

A non-deterministic finite tree automaton is complete if there is at least one transition rule available for every possible symbol-states combination. A state q is accessible if there exists a ground term t such that there exists a reduction from t to q(t). An FTA is reduced if all its states are accessible.

Pumping Lemma

Let L be a recognizable set of ground terms. Then, there exists a constant k > 0 satisfying: for every ground term t in L such that Height(t) > k, there exists a context C \in C(F), a non trivial context C' \in C(F) and a ground term u such that t = C[C'[u]] and, for all n \geq 0 C[C'^n[u]] \in L.

Closure

The class of recognizable tree languages is closed under union, under complementation, and under intersection.


Myhill-Nerode Theorem

A congruence on tree languages is a relation such that

u_1 \equiv v_1 \wedge\ldots  \wedge u_n\equiv v_n \Rightarrow f(u_1,\ldots,u_n) \equiv f(v_1,\ldots,v_n)

It is of finite index if its number of equivalence-classes is finite.

For a given tree-language L, define u \equiv_L v if for all contexts C \in C(F), C[u] \in L\Leftrightarrow C[v] \in L.

The Myhill-Nerode Theorem for tree automaton states that the following three statements are equivalent:

  1. L is a recognizable tree language
  2. L is the union of some equivalence classes of congruence of finite index
  3. the relation \equiv_L is a congruence of finite index

External links

All the information in this page was taken from Chapter 1 of http://tata.gforge.inria.fr

Implementations

(OCaml) Grappa - Ranked and Unranked Tree Automata Libraries (http://www.grappa.univ-lille3.fr/~filiot/tata/)

(OCaml) Timbuk - Tools for Reachability Analysis and Tree Automata Calculations (http://www.irisa.fr/celtique/genet/timbuk/)

(Java) LETHAL - Library for working with finite tree and hedge automata (http://lethal.sf.net/)

(Isabelle [OCaml, SML, Haskell]) - Machine-Checked Tree Automata Library (http://afp.sourceforge.net/entries/Tree-Automata.shtml)

(C++) VATA: A Library for Efficient Manipulation of Non-Deterministic Tree Automata - (http://www.fit.vutbr.cz/research/groups/verifit/tools/libvata/)


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Tree walking automaton — A tree walking automaton (TWA) is a type of finite automaton that deals with tree structures rather than strings.The following article deals with tree walking automata. For a different notion of tree automaton, closely related to regular tree… …   Wikipedia

  • Tree-adjoining grammar — (TAG) is a grammar formalism defined by Aravind Joshi. Tree adjoining grammars are somewhat similar to context free grammars, but the elementary unit of rewriting is the tree rather than the symbol. Whereas context free grammars have rules for… …   Wikipedia

  • Automaton — This article is about a self operating machine. For other uses of Automaton, see Automaton (disambiguation) or Automata (disambiguation). An automaton (plural: automata or automatons ) is a self operating machine. The word is sometimes used to… …   Wikipedia

  • Pebble automaton — A pebble automaton is an extension of tree walking automata which allows the automaton to use a finite amount of pebbles , used for marking tree node. The result is a model stronger than ordinary tree walking automata, but still strictly weaker… …   Wikipedia

  • Deterministic pushdown automaton — In automata theory, a pushdown automaton is a finite automaton with an additional stack of symbols; its transitions can take the top symbol on the stack and depend on its value, and they can add new top symbols to the stack. A deterministic… …   Wikipedia

  • Alternating finite automaton — In automata theory, an alternating finite automaton (AFA) is a nondeterministic finite automaton whose transitions are divided into existential and universal transitions. For example, let A be an alternating automaton.* For an existential… …   Wikipedia

  • Nested stack automaton — In automata theory, a nested stack automaton is a finite automaton that can make use of a stack containing data which can be additional stacks.[1] A nested stack automaton may read its stack, in addition to pushing or popping it. A nested stack… …   Wikipedia

  • Deterministic automaton — is a concept of automata theory in which the outcome of a transition from one state to another given a certain input can be predicted for every occurrence. A common deterministic automaton is a deterministic finite state machine (sometimes… …   Wikipedia

  • Büchi automaton — A Büchi automaton is the extension of a finite state automaton to infinite inputs. It accepts an infinite input sequence iff there exists a run of the automaton (in case of a deterministic automaton, there is exactly one possible run) which… …   Wikipedia

  • Embedded pushdown automaton — An embedded pushdown automaton or EPDA is a computational model that parse languages in the tree adjoining grammar (TAG). It is similar to the context free grammar parsing pushdown automaton, except that instead of using a stack (data structure)… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”