# Computability theory

recursion theorycomputablecomputabilitytheory of computabilitycomputability theory in computer sciencerecursion theorists“strong”computability theoristcomputational classeffective
Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.wikipedia
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### Mathematical logic

formal logicsymbolic logiclogic
Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.

### Computable function

computablerecursive functionstotal computable function
Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. Nowadays these are often considered as a single hypothesis, the Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function.
Computable functions are the basic objects of study in computability theory.

### Theory of computation

computational theoristcomputational theorycomputation theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.
The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".

### Computational complexity theory

computational complexitycomplexity theorycomplexity
Although there is considerable overlap in terms of knowledge and methods, mathematical recursion theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages.
Closely related fields in theoretical computer science are analysis of algorithms and computability theory.

### Church–Turing thesis

Church-Turing thesisChurch's thesisTuring's Thesis
Nowadays these are often considered as a single hypothesis, the Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function.
In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a hypothesis about the nature of computable functions.

### Emil Leon Post

Emil PostPostE.L. Post
Recursion theory originated in the 1930s, with work of Kurt Gödel, Alonzo Church, Rózsa Péter, Alan Turing, Stephen Kleene, and Emil Post.
He is best known for his work in the field that eventually became known as computability theory.

### Rózsa Péter

Péter, RózsaPéterR. Politzer
Recursion theory originated in the 1930s, with work of Kurt Gödel, Alonzo Church, Rózsa Péter, Alan Turing, Stephen Kleene, and Emil Post.
She is best known as the "founding mother of recursion theory".

### Stephen Cole Kleene

Stephen KleeneKleeneStephen C. Kleene
Recursion theory originated in the 1930s, with work of Kurt Gödel, Alonzo Church, Rózsa Péter, Alan Turing, Stephen Kleene, and Emil Post.
One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science.

### Effective descriptive set theory

effective
In these areas, recursion theory overlaps with proof theory and effective descriptive set theory.
Thus effective descriptive set theory combines descriptive set theory with recursion theory.

### Julia Robinson

Julia Hall Bowman RobinsonJulia BowmanRobinson
In 1970, Yuri Matiyasevich proved (using results of Julia Robinson) Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution; this problem asked whether there is an effective procedure to decide whether a Diophantine equation over the integers has a solution in the integers.
Julia Hall Bowman Robinson (December 8, 1919 – July 30, 1985) was an American mathematician noted for her contributions to the fields of computability theory and computational complexity theory–most notably in decision problems.

### List of undecidable problems

Undecidable
The list of undecidable problems gives additional examples of problems with no computable solution.
In computability theory, an undecidable problem is a type of computational problem that requires a yes/no answer, but where there cannot possibly be any computer program that always gives the correct answer; that is, any possible program would sometimes give the wrong answer or run forever without giving any answer.

### Algorithm

algorithmsalgorithm designcomputer algorithm
Nowadays these are often considered as a single hypothesis, the Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function.
This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general case—due to of a major theorem of Computability Theory known as the Halting Problem.

### Recursive set

recursivedecidabledecidable set
With a definition of effective calculation came the first proofs that there are problems in mathematics that cannot be effectively decided.
In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set or not.

### Kurt Gödel

GödelGödel, KurtGodel, Kurt
Recursion theory originated in the 1930s, with work of Kurt Gödel, Alonzo Church, Rózsa Péter, Alan Turing, Stephen Kleene, and Emil Post.

### Halting problem

always terminatesavoid the halting problemdetect non-terminating computations
The halting problem, which is the set of (descriptions of) Turing machines that halt on input 0, is a well-known example of a noncomputable set.
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.

### Computer science

computer scientistcomputer sciencescomputer scientists
Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.
In an effort to answer the first question, computability theory examines which computational problems are solvable on various theoretical models of computation.

### Model of computation

models of computationmodelmodels of computing
The use of Turing machines here is not necessary; there are many other models of computation that have the same computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator.
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input.

### Formal language

formal language theoryformal languageslanguage
Although there is considerable overlap in terms of knowledge and methods, mathematical recursion theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages.
Therefore, formal language theory is a major application area of computability theory and complexity theory.

### Turing reduction

Turing reducibleTuring reducibilityCook reduction
Informally, a set of natural numbers A is Turing reducible to a set B if there is an oracle machine that correctly tells whether numbers are in A when run with B as the oracle set (in this case, the set A is also said to be (relatively) computable from B and recursive in B).
In computability theory, a Turing reduction (also known as a Cook reduction) from a problem A to a problem B, is a reduction which solves A, assuming the solution to B is already known (Rogers 1967, Soare 1987).

### Oracle machine

oracleoracle Turing machineoracles
Recursion theory in mathematical logic has traditionally focused on relative computability, a generalization of Turing computability defined using oracle Turing machines, introduced by Turing (1939).
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems.

### Μ-recursive function

partial recursive functionrecursive function theorygeneral recursive function
The use of Turing machines here is not necessary; there are many other models of computation that have the same computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator.
In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis).

### Computation in the limit

Limit lemmalimit-computablelimit-recursive
Many degrees with special properties were constructed: hyperimmune-free degrees where every function computable relative to that degree is majorized by a (unrelativized) computable function; high degrees relative to which one can compute a function f which dominates every computable function g in the sense that there is a constant c depending on g such that g(x) < f(x) for all x > c; random degrees containing algorithmically random sets; 1-generic degrees of 1-generic sets; and the degrees below the halting problem of limit-recursive sets.
In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions.

### Turing jump

Turing Jump notation
Given a set A, the Turing jump of A is a set of natural numbers encoding a solution to the halting problem for oracle Turing machines running with oracle A.
In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem

### Post's theorem

Post's theorem establishes a close relationship between the Turing jump operation and the arithmetical hierarchy, which is a classification of certain subsets of the natural numbers based on their definability in arithmetic.
In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.

### Hilbert's tenth problem

Hilbert's 10th problem10th10th Hilbert problem
In 1970, Yuri Matiyasevich proved (using results of Julia Robinson) Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution; this problem asked whether there is an effective procedure to decide whether a Diophantine equation over the integers has a solution in the integers.
It was the development of computability theory (also known as recursion theory) that provided a precise explication of the intuitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous.