Algorithm

An algorithm is an understandable and accurate instruction to an executor to complete a sequence of actions aimed at achieving a goal.
Algorithm (lat. Algorithmi - from the Arabic name of the mathematician Al-Khwarizmi [1]) is a finite set of precisely defined rules for solving an arbitrary class of problems or a set of instructions describing the order of actions of an executor to solve a certain problem.

Schematic illustration of the operation of a Turing machine.

 The basic idea behind the Turing machine is very simple.  A Turing machine is an abstract machine (automatic) that works with a tape of individual cells in which symbols are recorded.  The machine also has a head for writing and reading characters from cells, which can move along the tape.  At each step, the machine reads the character from the cell that the head points to, and, based on the character read and the internal state, takes the next step.  In this case, the machine can change its state, write another character in a cell or move the head one cell to the right or left. [8]

 Based on the study of these machines, Turing's thesis (the main hypothesis of algorithms) was advanced:

  Some algorithm for finding the values ​​of a function defined in a certain alphabet exists if and only if the function is calculated by Turing, that is, when it can be calculated on a Turing machine. 

 This thesis is an axiom, a postulate, and cannot be proved by mathematical methods, since the algorithm is not an exact mathematical concept.

The first attempts to clarify the concept of an algorithm and its research were carried out in the first half of the 20th century by Alan Turing, Emil Post, Jacques Erbran, Kurt Godel, A. A. Markov, Alonzo Church.  Several definitions of the concept of an algorithm were developed, but it was subsequently revealed that they all define the same concept