# A report on Mathematics, Computability theory and Computer science

Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

- Computability theoryMathematics is essential in many fields, including natural sciences, engineering, medicine, finance, computer science and social sciences.

- MathematicsMany problems in mathematics have been shown to be undecidable after these initial examples were established.

- Computability theoryOther first-level areas emerged during the 20th century (for example category theory; homological algebra, and computer science) or had not previously been considered as mathematics, such as Mathematical logic and foundations (including model theory, computability theory, set theory, proof theory, and algebraic logic).

- MathematicsComputer science research also often intersects other disciplines, such as cognitive science, linguistics, mathematics, physics, biology, Earth science, statistics, philosophy, and logic.

- Computer scienceIn an effort to answer the first question, computability theory examines which computational problems are solvable on various theoretical models of computation.

- Computer science2 related topics with Alpha

## Algorithm

0 linksIn mathematics and computer science, an algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation.

Gurevich: '… Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine … according to Savage [1987], an algorithm is a computational process defined by a Turing machine'."Turing machines can define computational processes that do not terminate. The informal definitions of algorithms generally require that the algorithm always terminates. This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general case—due to a major theorem of computability theory known as the halting problem.

## Mathematical logic

0 linksMathematical logic is the study of formal logic within mathematics.

Major subareas include model theory, proof theory, set theory, and recursion theory.

Computer scientists often focus on concrete programming languages and feasible computability, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability.