Computability
Introduction
Computability is a central concept in the field of computer science and mathematics, specifically in the areas of computational theory and mathematical logic. It is concerned with the question of what can be computed by a machine, and how efficiently such computations can be performed.
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic that originated in the 1930s with the work of Kurt Gödel, Alonzo Church, Alan Turing, and others. This field is concerned with the formalization of the intuitive notion of a computable function, and it investigates the properties of computable functions and the computational complexity of their solutions.
Turing Machines
A Turing machine is a theoretical computing machine invented by Alan Turing, which serves as an idealized model of mechanical computation. Turing machines are used in theoretical computer science to understand the nature of computation and to prove results about computability.
Church-Turing Thesis
The Church-Turing thesis is a hypothesis about the nature of computable functions. It states that a function is computable if and only if it can be computed by a Turing machine. This thesis, although unprovable, has been widely accepted by the mathematical community.
Decidability
In computability theory, a problem is said to be decidable if there exists an algorithm that can determine the answer to the problem in a finite amount of time. Conversely, a problem is undecidable if no such algorithm exists. The concept of decidability is closely related to the concept of computability, as it essentially asks whether a problem can be solved by a computer.
Halting Problem
The halting problem is a famous example of an undecidable problem. Proposed by Alan Turing in 1936, the halting problem asks whether, given a description of a program and an input, it is possible to decide algorithmically whether the program will halt on that input.
Computational Complexity
Computational complexity theory is a branch of the theory of computation that deals with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes to solve a problem).
P vs NP Problem
The P vs NP problem is one of the most fundamental unsolved problems in computer science. It asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.