Computable analysis
In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functio
Kleene's recursion theorem
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephe
Busy beaver
In theoretical computer science, the busy beaver game aims at finding a terminating program of a given size that produces the most output possible. Since an endlessly looping program producing infinit
List of undecidable problems
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
ELEMENTARY
In computational complexity theory, the complexity class ELEMENTARY of elementary recursive functions is the union of the classes The name was coined by László Kalmár, in the context of recursive func
Primitive recursive set function
In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They
Reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwa
Low basis theorem
The low basis theorem is one of several basis theorems in computability theory, each of which showing that, given an infinite subtree of the binary tree , it is possible to find an infinite path throu
Lambda calculus
Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.
Automatic group
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell if a given
Computability logic
Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory
Bounded quantifier
In the study of formal theories in mathematical logic, bounded quantifiers (a.k.a. restricted quantifiers) are often included in a formal language in addition to the standard quantifiers "∀" and "∃".
Analytical hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of seco
Low (computability)
In computability theory, a Turing degree [X] is low if the Turing jump [X′] is 0′. A set is low if it has low degree. Since every set is computable from its jump, any low set is computable in 0′, but
Richardson's theorem
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, and exponential and sine functions. It was proved in 1
Enumeration reducibility
In computability theory and computational complexity theory, enumeration reducibility is a method of reduction that determines if there is some effective procedure for determining enumerability betwee
Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is
Krivine machine
In theoretical computer science, the Krivine machine is an abstract machine (sometimes called virtual machine). As an abstract machine, it shares features with Turing machines and the SECD machine. Th
McCarthy Formalism
In computer science and recursion theory the McCarthy Formalism (1963) of computer scientist John McCarthy clarifies the notion of recursive functions by use of the IF-THEN-ELSE construction common to
Myhill isomorphism theorem
In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notion of computability on a set.
Smn theorem
In computability theory the S mn theorem, (also called the translation lemma, parameter theorem, and the parameterization theorem) is a basic result about programming languages (and, more generally, G
Reverse Mathematics: Proofs from the Inside Out
Reverse Mathematics: Proofs from the Inside Out is a book by John Stillwell on reverse mathematics, the process of examining proofs in mathematics to determine which axioms are required by the proof.
Index set (computability)
In computability theory, index sets describe classes of computable functions; specifically, they give all indices of functions in a certain class, according to a fixed Gödel numbering of partial compu
Undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a corre
Algorithm characterizations
Algorithm characterizations are attempts to formalize the word algorithm. Algorithm does not have a generally accepted formal definition. Researchers are actively working on this problem. This article
Post's theorem
In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.
Kőnig's lemma
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to h
Computably inseparable
In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated" with a computable set. These sets arise in the
Arithmetical hierarchy
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on
Model of computation
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 co
Entscheidungsproblem
In mathematics and computer science, the Entscheidungsproblem (pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm], German for 'decision problem') is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. T
Circuit satisfiability problem
In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an as
Maximal set
In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that for every further recursively enumerable s
Turing jump
In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem X a successively harder decision problem X′ with the prop
Kolmogorov complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a
Computation in the limit
In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in the limit, limit recursive and recursively ap
Church–Turing thesis
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
Computable set
In computability theory, a set of natural numbers is called computable, recursive, or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possib
Hyperarithmetical theory
In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such
Admissible numbering
In computability theory, admissible numberings are enumerations (numberings) of the set of partial computable functions that can be converted to and from the standard numbering. These numberings are a
UTM theorem
In computability theory, the UTM theorem, or universal Turing machine theorem, is a basic result about Gödel numberings of the set of computable functions. It affirms the existence of a computable uni
Fast-growing hierarchy
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing f
K-trivial set
In mathematics, a set of natural numbers is called a K-trivial set if its initial segments viewed as binary strings are easy to describe: the prefix-free Kolmogorov complexity is as low as possible, c
Forcing (computability)
Forcing in computability theory is a modification of Paul Cohen's original set-theoretic technique of forcing to deal with computability concerns. Conceptually the two techniques are quite similar: in
Computable ordinal
In mathematics, specifically computability and set theory, an ordinal is said to be computable or recursive if there is a computable well-ordering of a subset of the natural numbers having the order t
Martin measure
In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers, named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an
Computability
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer s
Recursion (computer science)
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems
Ackermann function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive.
Incompressibility method
In mathematics, the incompressibility method is a proof method like the probabilistic method, the counting method or the pigeonhole principle. To prove that an object in a certain class (on average) s
Arithmetical set
In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the
Primitive recursive functional
In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist of a collection of functions in all pure finite t
Decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem
Normal form (abstract rewriting)
In abstract rewriting, an object is in normal form if it cannot be rewritten any further, i.e. it is irreducible. Depending on the rewriting system, an object may rewrite to several normal forms or no
Computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is com
Creative and productive sets
In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic. They are a standard topic in mathematical logic
Π01 class
In computability theory, a Π01 class is a subset of 2ω of a certain form. These classes are of interest as technical tools within recursion theory and effective descriptive set theory. They are also u
Lempel–Ziv complexity
The Lempel–Ziv complexity is a measure that was first presented in the article On the Complexity of Finite Sequences (IEEE Trans. On IT-22,1 1976), by two Israeli computer scientists, Abraham Lempel a
General recursive function
In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "com
Alpha recursion theory
In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals . An admissible set is closed under functions, where denotes a rank of Godel's constru
Grzegorczyk hierarchy
The Grzegorczyk hierarchy (/ɡrɛˈɡɔːrtʃək/, Polish pronunciation: [ɡʐɛˈɡɔrt͡ʂɨk]), named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory. Every f
Course-of-values recursion
In computability theory, course-of-values recursion is a technique for defining number-theoretic functions by recursion. In a definition of a function f by course-of-values recursion, the value of f(n
Craig's theorem
In mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not relat
Chain rule for Kolmogorov complexity
The chain rule for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states: That is, the combined randomness of two sequences X and Y is the sum of the randomness
Kleene's T predicate
In computability theory, the T predicate, first studied by mathematician Stephen Cole Kleene, is a particular set of triples of natural numbers that is used to represent computable functions within fo
Enumerator (computer science)
An enumerator is a Turing machine with an attached printer. The Turing machine can use that printer as an output device to print strings. Every time the Turing machine wants to add a string to the lis
Primitive recursive function
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number
Μ operator
In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the primitive recursi
History of the Church–Turing thesis
The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, fu
Slow-growing hierarchy
In computability theory, computational complexity theory and proof theory, the slow-growing hierarchy is an ordinal-indexed family of slowly increasing functions gα: N → N (where N is the set of natur
Gödel numbering for sequences
In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely p
Friedberg numbering
In computability theory, a Friedberg numbering is a numbering (enumeration) of the set of all uniformly recursively enumerable sets that has no repetitions: each recursively enumerable set appears exa
Double recursion
In recursive function theory, double recursion is an extension of primitive recursion which allows the definition of non-primitive recursive functions like the Ackermann function. Raphael M. Robinson
Recursive language
In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possi
LOOP (programming language)
LOOP is a simple register language that precisely captures the primitive recursive functions.The language is derived from the counter-machine model. Like the counter machines the LOOP language compris
Numbering (computability theory)
In computability theory a numbering is the assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some formal language. A numbering can be used to t
Oracle machine
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, wh
Computably enumerable set
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable
Simple set
In computability theory, a subset of the natural numbers is called simple if it is computably enumerable (c.e.) and co-infinite (i.e. its complement is infinite), but every infinite subset of its comp
Slicing the Truth
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles is a book on reverse mathematics in combinatorics, the study of the axioms needed to pro
Basis theorem (computability)
In computability theory, there are a number of basis theorems. These theorems show that particular kinds of sets always must have some members that are, in terms of Turing degree, not too complicated.
Church–Turing–Deutsch principle
In computer science and quantum physics, the Church–Turing–Deutsch principle (CTD principle) is a stronger, physical form of the Church–Turing thesis formulated by David Deutsch in 1985. The principle
Effective Polish space
In mathematical logic, an effective Polish space is a complete separable metric space that has a . Such spaces are studied in effective descriptive set theory and in constructive analysis. In particul
Description number
Description numbers are numbers that arise in the theory of Turing machines. They are very similar to Gödel numbers, and are also occasionally called "Gödel numbers" in the literature. Given some univ
PA degree
In the mathematical field of computability theory, a PA degree is a Turing degree that computes a complete extension of Peano arithmetic (Jockusch 1987). These degrees are closely related to fixed-poi
Trakhtenbrot's theorem
In logic, finite model theory, and computability theory, Trakhtenbrot's theorem (due to Boris Trakhtenbrot) states that the problem of validity in first-order logic on the class of all finite models i
Computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable funct
High (computability)
In computability theory, a Turing degree [X] is high if it is computable in 0′, and the Turing jump [X′] is 0′′, which is the greatest possible degree in terms of Turing reducibility for the jump of a
Computable number
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective
Hardy hierarchy
In computability theory, computational complexity theory and proof theory, the Hardy hierarchy, named after G. H. Hardy, is a hierarchy of sets of numerical functions generated from an ordinal-indexed
Halting problem
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 ru
Post correspondence problem
The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem and the Entscheidungsproblem it is often use
Effective method
In logic, mathematics and computer science, especially metalogic and computability theory, an effective method or effective procedure is a procedure for solving a problem by any intuitively 'effective
Tarski–Kuratowski algorithm
In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithm which produces an upper bound for the complexity of a given formula in the arithmetical
Computation
Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm). Mechanical or electronic devices (or, historically, people) that perform com
Turing degree
In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set
Automatic semigroup
In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "cano
Complete numbering
In computability theory complete numberings are generalizations of Gödel numbering first introduced by A.I. Mal'tsev in 1963. They are studied because several important results like the Kleene's recur