# A report onComputer science and Computability theory

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 theory

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

- Computer science

## Mathematics

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

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

Other 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).

## Mathematical logic

Study of formal logic within mathematics.

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.

## Algorithm

In 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.

## Formal language

In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.

Therefore, formal language theory is a major application area of computability theory and complexity theory.

## Model of computation

In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input.