Computability theory

Computability theory is a branch of mathematical logic and theoretical computer science that is concerned with the study of the extent to which problems can be solved using a computational model. It encompasses a wide range of topics, including algorithms, computational complexity, and decidability.

History

The origins of computability theory can be traced back to the work of David Hilbert and his list of problems presented at the International Congress of Mathematicians in 1900. One of these problems, known as the Entscheidungsproblem, asked whether there was a general algorithm that could determine the truth or falsity of any mathematical statement. This question led to the development of the concept of algorithmic computability.

Concepts

Algorithms

An algorithm is a finite sequence of well-defined instructions for solving a problem or performing a task. In computability theory, the concept of an algorithm is formalized using models of computation such as Turing machines and lambda calculus.

Computational Complexity

Computational complexity is a subfield of computability theory that classifies computational problems according to their inherent difficulty. This is often measured in terms of the amount of resources required to solve a problem, such as time or space.

Decidability

Decidability is a property of a problem if there exists an algorithm that can determine the answer to the problem. A problem is said to be undecidable if no such algorithm exists.

Models of Computation

Computability theory uses various models of computation to formalize the concept of an algorithm. These include:

• Turing machines: A theoretical machine that manipulates symbols on a strip of tape according to a table of rules.
• Lambda calculus: A formal system in mathematical logic for expressing computation based on function abstraction and application.
• Recursive functions: Functions that are defined in terms of themselves, often used in the study of computability.

See Also

• Church–Turing thesis
• Halting problem
• Gödel's incompleteness theorems