# computability theory

> study of computable functions and Turing degrees

**Wikidata**: [Q818930](https://www.wikidata.org/wiki/Q818930)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Computability_theory)  
**Source**: https://4ort.xyz/entity/computability-theory

## Summary
Computability theory is a subfield of theoretical computer science and mathematics that focuses on the study of computable functions and Turing degrees. It explores what problems can be solved by algorithms and the inherent limitations of computation. This field is foundational for understanding the theoretical capabilities of computers and computational processes.

## Key Facts
*   **Definition:** Computability theory is the study of computable functions and Turing degrees.
*   **Parent Fields:** It is a subfield of theoretical computer science and mathematics.
*   **Related Field:** It is related to computational complexity theory, which classifies problems by difficulty.
*   **Aliases:** Also known as recursion theory, computabilidad, teoria de la computabilidad, théorie de la récursion, Theorie der Berechenbarkeit, Berechnungstheorie, Rekursionstheorie, 計算可能性, 計算可能理論, and вычислимость.
*   **Subclass Of:** It is a subclass of mathematical logic, theoretical computer science, and theory of computation.
*   **Related Individuals:** Notable figures related to the field include Joost Engelfriet, Klaus W. Wagner, Barry Burd, Martin Davis, Boris Trakhtenbrot, Lothar Budach, William Gasarch, Anil Nerode, Mario de Jesús Pérez Jiménez, Yuri Matiyasevich, Yiannis N. Moschovakis, and Antonín Kučera.
*   **Identifiers:** It has various identifiers including lex_id: beregnelighed, psh_id: 6607, foldoc_id: computability+theory, kbpedia_id: ComputabilityTheory, babelnet_id: 00775789n, freebase_id: /m/014c37, quora_topic: Computability-Theory-computer-science, nl_cr_aut_id: ph678150, proofwiki_id: Definition:Computability_Theory, zhihu_topic_id: 19734535, sciencedirect_topic_id: mathematics/recursion-theory, mathematics/computability-theory, jstor_topic_id_(archived): recursion-theory, national_library_of_israel_j9u_id: 987007545779505171, microsoft_academic_id_(discontinued): 111142201, and mathematics_subject_classification_id: 03Dxx.
*   **Wikipedia Presence:** It has a Wikipedia title "Computability theory" and is available in multiple languages.
*   **Main Category:** Its main category is Category:Computability theory.

## FAQs
### Q: What is the primary focus of computability theory?
A: Computability theory primarily focuses on the study of computable functions and Turing degrees. It investigates which problems can be solved by algorithms and the fundamental limits of what can be computed.

### Q: What fields is computability theory considered a part of?
A: Computability theory is considered a subfield of theoretical computer science and mathematics. It is also classified as a subclass of mathematical logic and the broader theory of computation.

### Q: Who are some notable individuals associated with computability theory?
A: Several mathematicians and computer scientists are associated with computability theory, including Martin Davis, Boris Trakhtenbrot, Anil Nerode, and Yuri Matiyasevich, among others who have contributed to the field.

## Why It Matters
Computability theory is a cornerstone of theoretical computer science and mathematics because it establishes the fundamental boundaries of what can be computed. By rigorously defining "computable functions" and "Turing degrees," it provides the theoretical framework necessary to understand the capabilities and limitations of any algorithm or computational process. This foundational understanding is critical for the entire field of computer science, as it informs the design of programming languages, the development of algorithms, and the very concept of problem-solving with computers. It helps to distinguish between problems that are solvable by mechanical means and those that are inherently unsolvable, thereby guiding research and development in areas like artificial intelligence and cryptography. Its insights are also crucial for computational complexity theory, as one must first determine if a problem is computable before classifying its difficulty. Ultimately, computability theory provides the essential theoretical bedrock upon which all practical computation is built.

## Notable For
*   **Defining Computability:** It is notable for formally defining the concept of "computable functions," establishing what can and cannot be solved by mechanical procedures.
*   **Turing Degrees:** It introduces and extensively studies "Turing degrees," which classify the relative computational difficulty of undecidable problems.
*   **Foundational Role:** It serves as a foundational subfield for both theoretical computer science and mathematical logic, providing the theoretical underpinnings for understanding algorithms and computation.
*   **Limits of Computation:** It is distinguished by its exploration of the inherent theoretical limits of computation, identifying problems that are undecidable.

## Body

### Definition and Scope
Computability theory is a branch of theoretical computer science and mathematics. Its core focus is the study of computable functions and Turing degrees. The field investigates the theoretical capabilities and limitations of algorithms and computation.

### Classification and Relationships
Computability theory is categorized as:
*   **Part of / Parent:**
    *   theoretical computer science (subfield of computer science and mathematics)
    *   mathematics
*   **Subclass of:**
    *   mathematical logic
    *   theoretical computer science
    *   theory of computation
*   **Related to:**
    *   computational complexity theory (a theory that classifies problems by difficulty)
    *   theory of mathematical algorithms

### Related Individuals
Several notable individuals have contributed to or are associated with computability theory:
*   Joost Engelfriet (Ph.D. Universiteit Antwerpen 1974; university teacher, mathematician, computer scientist)
*   Klaus W. Wagner (Ph.D. Friedrich-Schiller-Universität Jena 1976; computer scientist, research fellow, university teacher; born 1947)
*   Barry Burd (American mathematics professor; computer scientist, mathematician, university teacher; born 1951)
*   Martin Davis (American mathematician; mathematician, university teacher, computer scientist; 1928–2023; born 1928-03-08)
*   Boris Trakhtenbrot (Russian-Israeli mathematician; mathematician, computer scientist, university teacher; born 1921-02-20)
*   Lothar Budach (German mathematician; mathematician, university teacher, computer scientist; 1935-2007; born 1935-11-14)
*   William Gasarch (American computer scientist; computer scientist, mathematician, university teacher; born 1959)
*   Anil Nerode (American mathematician; mathematician, computer scientist, university teacher; born 1932-06-04)
*   Mario de Jesús Pérez Jiménez (Spanish mathematician, computer scientist; mathematician, computer scientist, university teacher; born 1948-11-13)
*   Yuri Matiyasevich (Soviet and Russian mathematician; mathematician, computer scientist, university teacher; born 1947-03-02)
*   Yiannis N. Moschovakis (Greek-American logician; mathematician, logician, university teacher; born 1938-01-18)
*   Antonín Kučera (mathematician, computer scientist; born 1946-07-01)

### Identifiers and Aliases
Computability theory is recognized across various knowledge bases and platforms:
*   **Lexeme ID:** beregnelighed
*   **PSH ID:** 6607
*   **FOLDOC ID:** computability+theory
*   **KBPedia ID:** ComputabilityTheory
*   **BabelNet ID:** 00775789n
*   **Freebase ID:** /m/014c37
*   **Quora Topic:** Computability-Theory-computer-science
*   **National Library of Czech Republic Authority ID (NL CR AUT ID):** ph678150 (teorie vyčíslitelnosti)
*   **ProofWiki ID:** Definition:Computability_Theory
*   **Zhihu Topic ID:** 19734535 (可计算性理论)
*   **ScienceDirect Topic IDs:** mathematics/recursion-theory, mathematics/computability-theory
*   **JSTOR Topic ID (archived):** recursion-theory
*   **National Library of Israel J9U ID:** 987007545779505171
*   **Microsoft Academic ID (discontinued):** 111142201
*   **Mathematics Subject Classification ID:** 03Dxx
*   **Wikipedia:** The entity has a Wikipedia page titled "Computability theory" and is available in at least 10 languages (ar, as, ast, az, bg, bn, ca, cs, cv, da).
*   **Wikimedia Project:** It is listed on Wikipedia:Vital articles/Level/4 as of 2022-10-31.
*   **Commons Category:** Computer science, with a specific category for Computability theory.

```json
{
  "@context": "https://schema.org",
  "@type": "Thing",
  "name": "computability theory",
  "description": "study of computable functions and Turing degrees",
  "sameAs": [
    "https://www.wikidata.org/wiki/Q6607",
    "https://en.wikipedia.org/wiki/Computability_theory"
  ],
  "additionalType": [
    "https://schema.org/Discipline",
    "https://schema.org/Theory"
  ]
}

## References

1. BabelNet
2. Quora
3. National Library of Israel Names and Subjects Authority File
4. KBpedia
5. [OpenAlex](https://docs.openalex.org/download-snapshot/snapshot-data-format)