Automat of trees

Introduction

A automat of tree is a type of machine in states. The automats of trees treat trees, rather than the Character strings of the more conventional automats.

As for the traditional automats, the automats of finished trees (FTA for finite tree automata in English) can be deterministic or not. According to the way in which the automats " déplacent" on the tree that they treat, the automats of trees can be of two types (A) ascending (b) downward. The distinction is important, because if the automats ascending (ND) not-determinists and descendants are equivalent, the downward deterministic automats are strictly less powerful than their ascending deterministic counterparts, because the properties of trees specified by the downward deterministic automats can depend only on the properties of ways.

Definitions

An automat of ascending tree finished on $F$ is defined by: $(Q,\; F,\; Q\_\; \{F\},\; \backslash \; Delta)$

Here $Q$ is a whole of unary states, $F$ is an ordered alphabet, $Q\_f\; \backslash \; subseteq\; Q$ is a whole of final states, and $\backslash \; Delta$ is a whole of rules of transition, i.e. of rewriting rules which transform the nodes of which the roots of wire are states in nodes of which the roots of which states. Consequently the state of a node is deduced from the states of his/her children.

There is no initial state as such, but the rules of transition for the constant symbols can be regarded as initial states. The tree is accepted if the state of the root is an accepting state.

An automat of finished tree going down on $F$ is defined by: $(Q,\; F,\; I,\; \backslash \; Delta)$

There are two differences with the automats of ascending trees: initially, $I\; \backslash \; subseteq\; Q$, the whole of its initial states, replaces $Q\_F$; then, its rules of transition are the reverse, i.e. rewriting rules which transform the nodes of which the roots are states in nodes of which the roots of wire are states. The tree is accepted if all the branches are completely crossed until the end.

One can easily guess intuitively that the nondeterministic automats of downward trees are equivalent to their ascending counterparts; the rules of transition are simply reversed, and the final states become the initial states.

Why the downward FTA deterministic are then less powerful than their ascending counterparts? Because a deterministic MT cannot by definition never have two rules of transition of which the left part is identical. For the automats of trees, the rules of transition are rewriting rules; and for those which are downward, the left side will correspond to the nodes ancestors. Consequently a deterministic finite-state machine going down will be able to test only properties which are true in all the branches without exception, because the choice of the state to be written in each branch girl is determined with the node relative, without knowing the contents of the branches girls in question.

Properties

Determinism

As he is written higher, an automat of trees is deterministic if he does not have any pair of rules of transition having the same left side. This definition corresponds to the intuitive idea that so that an automat is deterministic, a transition and only one must be possible for a given node.

Reconnaissability

For an ascending automat, a basic term $t$ (i.e. a tree) is accepted if there exists a reduction which starts from T and leads to Q (T), where Q is a final state. For an automat going down, a basic term $t$ is accepted if there exists a reduction which starts from Q (T) and leads to T, where Q (T) is an initial state.

The language of trees $L\; (A)$ recognized by an automat of trees $A$ is the whole of all the basic terms accepted by $A$. A whole of basic terms is recognizable if there exists an automat which recognizes it.

An important property is that linear homomorphisms (i.e., which preserve arite) preserve the reconnaissability .

Complétude and reduction

An automat of trees finished not determinist is complete if there is at least a rule of transition available for each possible combination symbol-state. A state $q$ is accessible if there exists a basic term $t$ such as there exists a reduction of $t$ with $q\; (T)$. A FTA is reduced if all its states are accessible.

Lemma of the star

That is to say $L$ a recognizable whole of basic terms. Then, there exists a constant $k\; >\; 0$ such as: for each basic term $t$ in $L$ such as $Height\; (T)\; >\; k$, there exists a context $C\; \backslash \; in\; C\; (F)$, a noncommonplace context $C\text{'}\; \backslash \; in\; C\; (F)$ and a basic term $u$ such as $t\; =\; C]$ and for all $n\; \backslash \; geq\; 0$ $C]\; \backslash \; in\; L$.

Closing

The class of the languages of recognizable trees is closed for the union, the complementation and the intersection.

Theorem of Myhill-Nerode

The three following assertions are equivalent: (I) L is a recognizable language of trees (II) L is the union of classes of equivalence of a congruence with finished index (III) the relation $\backslash \; equiv\_L$ is a congruence with finished index

Definitions necessary for this theorem: A congruence on languages of trees is a relation such as: $u\_i\; \backslash \; equiv\; v\_i\; 1\; \backslash \; Leq\; I\; \backslash \; Leq\; N\; \backslash \; Rightarrow\; F\; (u\_1,\; \backslash \; ldots,\; u\_n)\; \backslash \; equiv\; F\; (v\_1,\; \backslash \; ldots,\; v\_n)$ It is with finished index if its number of classes of equivalence is finished. For a language of trees $L$ given, $u\; \backslash \; equiv\_L\; v$ so for any context $C\; \backslash \; in\; C\; (F)$, $C\; \backslash \; in\; L$ if $C\; \backslash \; in\; L$.

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