Computability Problem - Philosophy of Science

Computability problem

After discussing the concept of information, we should move on to the next question. Namely, in simple terms, what, in fact, can be done with this information. From the point of view of "information science", information handling is the execution of the program. A & quot; programming & quot; - this is the writing of rules, following which you can convert information to the required form.

From a technical point of view, programming is the planning of the behavior of some autonomous system. At the engineering level, such an autonomous system can be a computer. Traditionally, it is assumed that a computer (a real calculator) is a Turing Machine implementation, and the calculation itself is the process of data processing in the ideal case.

The problem of solvability, in this extreme version, was investigated by A. Turing ( Entscheidungsproblem ). This problem was formulated as a problem of the existence of a universal mechanical procedure that would solve all mathematical problems. In his article "On computable numbers, with an application to the entscheidungsproblem", published in November 1936, he described the model of the "machine" without affecting any practical aspects of implementation. The universality of this procedure was understood as the "opportunity to simulate the behavior of any other machine of the same type". It was assumed that in a solvable theory, for every problem that can only be formulated in its vocabulary, there is an answer, and this answer can be obtained in a purely mechanical manner, following certain fixed prescriptions.

During the development of computer technology, a lot of real computers and algorithms were constructed that solve many important problems. This circumstance compels us to take seriously the well-known philosophical problem of the correlation of the theoretical level of scientific research and the practical application of the results obtained. For computer sciences, in this sense, it is interesting to compare the concepts of the algorithm, which give A. Turing and D. Whip.

According to A. Turing, for any algorithm there is a Turing machine that can execute it. That is, the Turing machine (or its equivalent) defines the notion of algorithmic (executable = recursive = = mechanical) procedure [13, p. 357].

D. Whip, being the developer of several programming languages, determines the algorithm by enumerating its properties, and not theoretically. The last, among others, indicated by D. Knuth property is the efficiency of the algorithm. The algorithm is usually considered effective if all its operators are simple enough that they can be executed accurately for a finite period of time with the help of a pencil and paper & quot; [11, p. 24]. Such a definition, on the one hand, imposes a number of limitations, on the other hand, guarantees the possibility of its implementation, since it allows one to formalize a particular process in the form of a task.

This contrast stands for two different kinds of activities, intertwined in computer science, namely, mathematics and engineering. In different areas of computer science, these kinds of activities are presented in different ways. We can say that their proportion forms a whole spectrum from the theory of algorithms (with the predominance of discrete mathematics in it) to system programming (with the predominance of the engineering component of constructing complex systems), between which are "mixed" domains, such as cryptography or compiler theory.

A more detailed examination of the relationship between mathematics and engineering would require a detailed examination of each area. The clear distinction between theory, its application and engineering activity, characteristic of the physical sciences, although traced in some areas of computer science, is not so natural for it.

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