Ciclo solar

Formal languages

Alphabet

One calls alphabet any unit $\ finished Sigma$ , nonempty. The elements of $\ Sigma$ are called letters .

Word

One calls word any continuation of elements of $\ Sigma ^ {\ NR}$ with finished support, one poses $\ epsilon$ the empty continuation said the word empties . The whole of the words on $\ Sigma$ is noted $\ Sigma^ {*}$ .

The fundamental operation on the words is the concatenation , noted $\ cdot$ , it is defined as follows:

That is to say two words $u = (a_ {1},…, a_ {N})$ and $v = (b_ {1},…, b_ {N})$ .

One has then,

$u \ cdot \ epsilon = \ epsilon \ cdot U =u$
$u \ cdot v = (a_ {1},…, a_ {N}, b_ {1},…, b_ {N})$

The concatenation is associative : $u \ cdot {} (v \ cdot W) = (U \ cdot v) \ cdot w$

Consequently:

$(\ Sigma^ {*}, \ cdot)$ is a monoid , i.e. $\ cdot$ is associative and has for neutral element $\ epsilon \ in \ Sigma^ {*}$ .

Definitions

Finite-state machine

One calls Finite-state machine the quintuplet

A=< \ Sigma, Q, \ delta, I, F>

, where:

$\ Sigma$ is an alphabet,
$Q$ is a number of stable conditions,
$I$ is part of $Q$ called together initial states,
$F$ is part of $Q$ called together final states,
$\ delta$ is part of $Q \ times \ Sigma \ times Q$ called together of the transitions. It is a function of transition which with a state of the system and an element of the alphabet associates the passage in another state.

Chemin

A way is a succession of consecutive arrows. A way is noted:

$(q_0, a_1, q_1) \ cdots (q_ {k-1}, a_k, q_k)$ , with $q_i \ in Q$ , $a_i \ in \ Sigma$ , $(q_ {i-1}, a_i, q_i) \ in \ delta$ .

One calls trace or label the continuation of letters recognized $a_1 \ cdots a_k$

It is said that a way is made a success of when $q_0 \ in I$ and $q_k \ in F$

A word is recognized when it is the label of a successful way.

Accessibility

A state,

q \ in Q

, is known as:

accessible if and only if there exists a way on the basis of a state initial and going until $q$ ;
coaccessible if and only if there exists a way on the basis of the state $q$ and going until a final state.

An automat is known as:

accessible if and only if all its states are accessible;
coaccessible if and only if all its states are coaccessibles;
pruned if it is accessible and coaccessible.

Determinism

An automat is deterministic if and only if it has only one initial state and for each state, there exists with more the one outgoing arrow for each letter, i.e. if

\ forall Q \ in Q, \ forall has \ in \ Sigma,  |\ delta (Q, a)| \ Leq 1

See too

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