Why study computational theory

Computational theory

Algorithm theory, Computability theory, mathematical theory that deals with the formal concept of computability and algorithms.

The task of finding an adequate definition for "calculable" is considered to have been solved on the basis of Church's thesis, since all of the previously proposed concepts of calculability have turned out to be mutually equivalent (Turing machine). The formalization of the concept of calculation has turned out to be necessary after the prevailing idea around 1900 that all of mathematics can be algorithmized in a certain way. This idea was nullified by Gödel's incompleteness theorem in 1931, which is located on the border between mathematical logic and computational theory. This sentence made it clear that there are non-calculable functions or non-decidable problems.

Examples of this are the undecidability of predicate logic (undecidable theory), the holding problem, the domino problem, the Post's correspondence problem, the tenth Hilbert's problem and the word problem for semi-Thue systems.

The advanced computational theory is concerned with, inter alia. with relative predictability and Turing reducibility, which can be defined using so-called oracle-Turing machines. In this way, hierarchies, such as the arithmetic hierarchy, can be considered within the non-decidable sets, and problems that are mutually Turing-reducible can be combined into equivalence classes, so-called Turing degrees. In the theory of the degrees of unsolvability, their association-theoretical structures are examined in particular.

Particular attention is also paid to the structure of the recursively enumerable sets.

A particularly successful proof technique for the construction of sets which on the one hand should receive desired properties and on the other hand should avoid other properties is the priority method.