Matiyasevichtételének

Matiyasevichtételének, also known as Matiyasevich's theorem, is a fundamental result in mathematical logic and computer science. It is named after Yuri Matiyasevich, a Russian mathematician who proved it in 1970. The theorem establishes a profound connection between the concept of Diophantine equations and the limits of what can be algorithmically computed.

Specifically, Matiyasevichtételének states that a set of natural numbers is recursively enumerable if and only if it is Diophantine. A set is recursively enumerable if there exists an algorithm that can list all its members. A set is Diophantine if it can be represented as the set of all positive integer solutions to a system of polynomial equations with integer coefficients.

This theorem is a direct consequence of and a generalization of earlier work by Martin Davis, Hilary Putnam, and Julia Robinson, often referred to as the MRDP (Matiyasevich, Davis, Putnam, Robinson) hypothesis. Their work showed that Hilbert's tenth problem, which asked for a general algorithm to determine if a Diophantine equation has integer solutions, was unsolvable. Matiyasevichtételének provided the definitive proof of this unsolvability.

The significance of Matiyasevichtételének lies in its implications for the theory of computation and the foundations of mathematics. It demonstrates that there are mathematical problems that are fundamentally undecidable, meaning no algorithm can solve them for all possible inputs. This has had a lasting impact on fields such as computability theory, logic, and artificial intelligence, highlighting inherent limitations in algorithmic problem-solving.