Computability Theory
A branch of mathematical logic that studies which problems can be solved by algorithmic means, focusing on the fundamental limits of computation and the classification of computational problems.
Computability Theory
Computability theory, also known as recursion theory, explores the fundamental boundaries of mechanical computation and algorithmic problem-solving. Emerging from the foundations of Mathematical Logic, it addresses the central question: "What can be computed?"
Historical Foundations
Origins
- Developed in the 1930s through work of:
- Alan Turing (Turing Machines)
- Alonzo Church (Lambda Calculus)
- Kurt Gödel (Recursive Functions)
- Emerged from the Hilbert's Program for mathematics foundations
Key Historical Problems
Core Concepts
Computability Models
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- Universal Turing machines
- Multi-tape variants
- Non-deterministic Turing Machines
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- Church's formalism
- Type Theory
- Functional Programming
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- Random Access Machines
- Counter machines
Fundamental Results
The Church-Turing Thesis
- Church-Turing Thesis
- Equivalence of computation models
- Computational Universality
Undecidability
Classification of Problems
Decidability Hierarchy
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- Primitive recursive functions
- Regular Languages
- Context-Free Languages