Computational Theory
Computational Theory, often known as the theory of computation, is a branch of computer science and mathematics concerned with how problems can be solved using algorithms and whether problems are solvable at all. This field explores the nature of computation, its limits, and its applications, providing a theoretical foundation for understanding the capabilities and limitations of computers.
History
The roots of Computational Theory can be traced back to the early 20th century with contributions from several key figures:
- Alan Turing, who in 1936 introduced the concept of the Turing Machine, a theoretical device that can simulate any computer algorithm. His work laid the foundation for the modern theory of computation.
- Alonzo Church, who developed the Lambda Calculus, which is another foundational model of computation.
- Kurt Gödel, whose incompleteness theorems provided insights into the limits of formal systems, influencing the development of computational theory.
Key Concepts
- Automata Theory: Studies abstract computing machines or "automata" like Finite Automata, Pushdown Automata, and Turing Machines, to understand which problems can be solved by these models.
- Computability Theory: Investigates what can be computed by any machine, focusing on the concept of an algorithm. Key topics include decidability, reducibility, and the Halt Problem.
- Complexity Theory: Examines the resources needed to solve computational problems, particularly focusing on time and space complexity, leading to classifications like P vs NP, NP-completeness, and NP-hardness.
Applications
While Computational Theory is deeply theoretical, its applications are vast: