Markov algorithm: Difference between revisions

Revision as of 06:46, 2 November 2017

Revision as of 00:04, 22 November 2017

Line 16:

The process of applying the normal algorithm to an arbitrary word <math>V</math> in the alphabet of this algorithm is a discrete sequence of elementary steps, consisting of the following. Let’s assume that <math>V'</math> is the word obtained in the previous step of the algorithm (or the original word <math>V</math>, if the current step is the first). If among formulas of substitution there is no left-hand side of which would be included in the <math>V'</math>, then the work of the algorithm is considered completed, and the result of this work is considered to be the word <math>V'</math>. Otherwise, including the substitution of the left side of which is included in the <math>V'</math>, the very first part is selected. If the substitution formula looks like <math>L\to\cdot D</math>, then out of all of possible representations of the word <math>V'</math> that looks like <math>RLS</math> it is chosen one <math>R</math>, which is the shortest. Then the work of the algorithm is considered completed with the result <math>RDS</math>. However, if this substitution formula looks like <math>L\to D</math>, then out of all of possible representations of the word <math>V'</math> in the form of <math>RLS</math> it is chosen one, in which <math>R</math> – the shortest, after which the word <math>RDS</math> is considered to be the result of the current step, subject to further processing in the next step.

For example, during the process of applying the algorithm to the scheme diagram above the word <math>|*||</math> consistently emerging the words <math>|b*|</math>, <math>ba|*|</math>, <math>a|*|</math>, <math>a|b*</math>, <math>aba|*</math>, <math>baa|*</math>, <math>aa|*</math>, <math>aa|c</math>, <math>aac</math>, <math>ac|</math> иand <math>c||</math>, after which the algorithm stops with the result <math>||</math>. Other examples, see below.

Any normal algorithm is equivalent to some [[Turing machine]], and vice versa{{snd}}any [[Turing machine]] is equivalent to some normal algorithm. A version of the [[Church-Turing thesis]] formulated in relation to the normal algorithm is called the "principle of normalization."