# Recursion theory is one of the important branches of mathematical logic, which studies feasible computational methods to solve problems

2022-05-26 By Recursion theory (Chinese pinyin: Diguilun;Recursion theory (recursion theory), one of the important branches of mathematical logic, the study of problem solving feasible computational methods and computational complexity of a subject.The calculation method to solve a certain kind of problem is also called algorithm.Algorithms are an old mathematical concept.The analytic geometry created by R. Descartes in the 16th century is a typical algorithm for solving geometric problems with algebra.But there are some problems in mathematics that have long been unsolved by algorithms.It is suspected that there is no such algorithm.To prove this, the algorithm must be defined precisely.In the 1930s, K. Godel proposed a precise definition of the algorithm, S.C.Kling defined the recursive function from this.At the same time, A.M. Turing described algorithms with a Turing machine (a theoretical computer) and proved that Turing’s computable functions were equivalent to recursive functions.The Turing machine led to the general acceptance of Church’s thesis about algorithms: that a recursive function is an exact mathematical description of a computable function.A recursive function is a computable function defined by mathematical logic on a set of natural numbers.If the characteristic function of an n-element set of natural numbers is a recursive function, the set is called a recursive set, and the range of a recursive function is called a recursive enumerable set.A recursive set is a determinable set of algorithms.Recursive sets are recursively enumerable, but there are recursively enumerable sets that are not recursively enumerable.The research of recursion theory enables people to transform some long-unsolved problems into non-recursive recursive enumerable sets, which strictly proves that there is no algorithm to determine these problems.These problems are called undecidable.Recursion theory further studies the degree of complexity between undecidable, that is, non-recursive, recursively enumerable sets.In 1944 E.L. Post proposed the concept of unsolvability.The construction method of relative computability is also given.This led people to compare unsolvability and study the algebraic structure of unsolvability.In this aspect, there are many powerful research methods such as infinite damage priority method and infinite damage priority method, and there are many interesting research results.For computable recursive sets, we can also study the complexity of computation, considering the time and space of computation on Turing machines, we can get the length of computation time and the amount of space occupied by these two complexities.The study of computational complexity has great influence on the development of computer science.