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Topics referred to by the same term
Recursion theorem can refer to: The recursion theorem in set theory Kleene's recursion theorem, also called the fixed point theorem, in computability
Recursion_theorem
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
Kleene's_recursion_theorem
Process of repeating items in a self-similar way
Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines
Recursion
Mathematical theorem
In mathematics, the transfinite recursion theorem says a function can be defined using a recursion over a well-ordered set; for example, N {\displaystyle
Transfinite_recursion_theorem
Theorem in computability theory
Q_{e}(x)=\varphi _{a}(x)} when e ∉ P {\displaystyle e\notin P} . By Kleene's recursion theorem, there exists e {\displaystyle e} such that φ e = Q e {\displaystyle
Rice's_theorem
Condition for a mathematical function to map some value to itself
computability theory, by applying Kleene's recursion theorem. These results are not equivalent theorems; the Knaster–Tarski theorem is a much stronger result than
Fixed-point_theorem
Topics referred to by the same term
first incompleteness theorem Tarski's undefinability theorem Halting problem Kleene's recursion theorem Lawvere's fixed-point theorem (categorical generalization
Diagonal_argument
American mathematician (1909–1994)
algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to
Stephen_Cole_Kleene
Mathematical proposition equivalent to the axiom of choice
directly using transfinite recursion, still assuming the axiom of choice. For that, see for example Transfinite recursion theorem § Example: a basis construction
Zorn's_lemma
Self-replicating program
Turing-complete programming language, as a direct consequence of Kleene's recursion theorem. For amusement, programmers sometimes attempt to develop the shortest
Quine_(computing)
Tool for analyzing divide-and-conquer algorithms
p(input x of size n): if n < some constant k: Solve x directly without recursion else: Create a subproblems of x, each having size n/b Call procedure p
Master theorem (analysis of algorithms)
Master_theorem_(analysis_of_algorithms)
Mathematical concept
More formally, we can state the Transfinite Recursion Theorem as follows: Transfinite Recursion Theorem (version 1). Given a class function G: V → V
Transfinite_induction
Well-quasi-ordering of finite trees
arithmetical transfinite recursion). In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also
Kruskal's_tree_theorem
Principle of interchangeability of data and code
creating a malformed program. In computational theory, Kleene's second recursion theorem provides a form of code-is-data, by proving that a program can have
Code_as_data
Kanamori–McAloon theorem (mathematical logic) Kirby–Paris theorem (proof theory) Kleene's recursion theorem (recursion theory) König's theorem (set theory
List_of_theorems
Limitative results in mathematical logic
results about undecidable sets in recursion theory. Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of computability
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Set theory concept
also the English-language presentation of von Neumann's "general recursion theorem" by Bernays 1991, pp. 100–109. Moore 2013. See page 279 for the assertion
Von_Neumann_universe
Generalization of Rice's theorem
{\displaystyle p} can get access to its own source code by Kleene's recursion theorem). If this eventually returns true, then this first task continues
Rice–Shapiro_theorem
Arithmetic operation
literature. Taken literally, the above definition is an application of the recursion theorem on the partially ordered set N 2 {\displaystyle \mathbb {N} ^{2}}
Addition
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
On transforming a program by substituting constants for free variables
(+ x y)) 3 g42)), where g42 is a "fresh" symbol. Currying Kleene's recursion theorem Partial evaluation Kleene, S. C. (1936). "General recursive functions
Smn_theorem
Class of mathematical orderings
Initial segments are also used in the statement of the transfinite recursion theorem. Properties of initial segments include: A well-ordered set is never
Well-order
Problem in computer science
for electrical engineers and technical specialists. Discusses recursion, partial-recursion with reference to Turing Machines, halting problem. Has a Turing
Halting_problem
Every polynomial has a real or complex root
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Characterization of hyperarithmetic sets
In effective descriptive set theory, the Suslin–Kleene theorem characterizes the hyperarithmetic subsets of N {\displaystyle \mathbb {N} } . Informally
Suslin–Kleene_theorem
calculus Church–Rosser theorem Calculus of constructions Combinatory logic Post correspondence problem Kleene's recursion theorem Recursively enumerable
List of mathematical logic topics
List_of_mathematical_logic_topics
Use of functions that call themselves
recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves
Recursion_(computer_science)
Study of computable functions and Turing degrees
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated
Computability_theory
Mathematical result or axiom on order relations
required to satisfy the above recursive condition, then the transfinite recursion theorem ensures this defines the function f {\displaystyle f} uniquely (in
Hausdorff_maximal_principle
Abstract machine used to study decision problems
Robert I. (1987). "Fundamentals of Recursively Enumerable Sets and the Recursion Theorem". Recursively Enumerable Sets and Degrees. Perspectives in Mathematical
Oracle_machine
Defining elements of a set in terms of other elements in the set
starting from n = 0 and proceeding onwards with n = 1, 2, 3 etc. The recursion theorem states that such a definition indeed defines a function that is unique
Recursive_definition
Turing machine that halts for any input
index of such a machine. Build a Turing machine M, using Kleene's recursion theorem, that on input 0 first simulates the machine with index e running
Decider_(Turing_machine)
Theorem that every set can be well-ordered
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict
Well-ordering_theorem
Statement in mathematical logic
developed in 1934. The diagonal lemma is closely related to Kleene's recursion theorem in computability theory, and their respective proofs are similar.
Diagonal_lemma
Fundamental theorem in mathematical logic
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability
Gödel's_completeness_theorem
Branch of mathematical logic
results in WKL, etc. Over RCA0, Π1 1 transfinite recursion, ∆0 2 determinacy, and the ∆1 1 Ramsey theorem are all equivalent to each other. Over RCA0, Σ1
Reverse_mathematics
Statement in mathematical combinatorics
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours)
Ramsey's_theorem
In mathematics, a statement that has been proven
mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses
Theorem
sequences, and structures. recursion theorem 1. Master theorem (analysis of algorithms) 2. Kleene's recursion theorem recursive definition A definition
Glossary_of_logic
Subfield of mathematics
Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method
Mathematical_logic
Impossible task in computing
impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only if it
Entscheidungsproblem
Theorem in set theory
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there
Schröder–Bernstein_theorem
Extension of recursion theory to admissible ordinals beyond the natural numbers
In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals α {\displaystyle \alpha } . An admissible
Alpha_recursion_theory
Type of binary relation
and recursion on S gives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The
Well-founded_relation
Two functions defined from each other
In mathematics and computer science, mutual recursion is a form of recursion where two or more mathematical or computational objects, such as functions
Mutual_recursion
⟹ G x = b {\displaystyle x\notin C\implies Gx=b} . By the Second Recursion Theorem, there is a term X which is equal to f applied to the Church numeral
Scott–Curry_theorem
Concept in mathematical logic
type theory (ITT), a discipline within mathematical logic, induction-recursion is a feature for simultaneously declaring a type and function on that
Induction-recursion
Programming paradigm based on applying and composing functions
depth of recursion. This could make recursion prohibitively expensive to use instead of imperative loops. However, a special form of recursion known as
Functional_programming
Square matrices satisfy their characteristic equation
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix
Cayley–Hamilton_theorem
Relation between deterministic and nondeterministic space complexity
In computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, gives a relationship between deterministic and non-deterministic
Savitch's_theorem
nowhere is recursion mentioned. The proof of the equivalence of machine-computability and recursion must wait for Kleene 1943 and 1952: "The theorem that all
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Axiomatic set theories based on the principles of mathematical constructivism
{\displaystyle g(Sn)=f(g(n))} . This iteration- or recursion principle is akin to the transfinite recursion theorem, except it is restricted to set functions and
Constructive_set_theory
Theorem in mathematical logic
compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important
Compactness_theorem
theorem Goodstein's theorem Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's
List_of_mathematical_proofs
studied because several important results like the Kleene's recursion theorem and Rice's theorem, which were originally proven for the Gödel-numbered set
Complete_numbering
Polynomial ideals are finitely generated
fundamental theorems on polynomials, the Nullstellensatz (zero-locus theorem) and the syzygy theorem (theorem on relations). These three theorems were the
Hilbert's_basis_theorem
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Mathematical transform that expresses a function of time as a function of frequency
This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n + 2 and n allowing to compute, e.g., the three-dimensional
Fourier_transform
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
Mathematical-logic system based on functions
calculus may be used to model arithmetic, Booleans, data structures, and recursion, as illustrated in the following sub-sections i, ii, iii, and § iv. There
Lambda_calculus
Function in mathematical logic
Kurt Gödel developed the concept for the proof of his incompleteness theorems. A Gödel numbering can be interpreted as an encoding in which a number
Gödel_numbering
Function computable with bounded loops
mathematics before, but the construction of primitive recursion is traced back to Richard Dedekind's theorem 126 of his Was sind und was sollen die Zahlen? (1888)
Primitive_recursive_function
Generalization of "n-th" to infinite cases
theorems but also to define functions on ordinals. This is known as transfinite recursion. Formally, a function F is defined by transfinite recursion
Ordinal_number
The low basis theorem is one of several basis theorems in computability theory, each of which show that, given an infinite subtree of the binary tree 2
Low_basis_theorem
Theorem used in quantum mechanics for angular momentum calculations
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in
Wigner–Eckart_theorem
Smallest fixed point of a function from a poset
converge with the least fixed point. Unfortunately, whereas Kleene's recursion theorem shows that the least fixed point is effectively computable, the optimal
Least_fixed_point
Area of mathematical logic
It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems) that a theory has a model if and only if it
Model_theory
Proof method in mathematical logic
induction. Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical
Structural_induction
Theorem in quantum information theory
In quantum information and computation, the Solovay–Kitaev theorem says that if a set of single-qubit quantum gates generates a dense subgroup of SU(2)
Solovay–Kitaev_theorem
Fixed-point theorem
mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed-point theorem for partially ordered sets
Bourbaki–Witt_theorem
Functional programming language
can be inferred. In core type theory, induction and recursion principles are used to prove theorems about inductive types. In Agda, dependently typed pattern
Agda_(programming_language)
Mathematical construction
include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization
Ultraproduct
Every set is smaller than its power set
question marks, boxes, or other symbols. In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle
Cantor's_theorem
In mathematical logic, Craig's theorem (also known as Craig's trick) states that any recursively enumerable set of well-formed formulas of a first-order
Craig's_theorem
Yes-or-no question that cannot ever be solved by a computer
between these two is that if a decision problem is undecidable (in the recursion theoretical sense) then there is no consistent, effective formal system
Undecidable_problem
American theoretical physicist
mathematical insights in physics, such as his 1981 proof of the positive energy theorem in general relativity, and his interpretation of the Jones invariants of
Edward_Witten
Theoretical framework of management cybernetics
viable system. Society itself can be seen as a system of recursion. In this case, recursion refers to systems that are nested within other systems. (Axioms
Viable_system_model
Theorem that arithmetical truth cannot be defined in arithmetic
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Method of comparing problems by transforming one into another in computability theory
Odifreddi, 1989. Classical Recursion Theory, North-Holland. ISBN 0-444-87295-7 P. Odifreddi, 1999. Classical Recursion Theory, Volume II, Elsevier.
Reduction (computability theory)
Reduction_(computability_theory)
No-go theorem pertaining the triviality of space-time and internal symmetries
In theoretical physics, the Coleman–Mandula theorem is a no-go theorem stating that spacetime and internal symmetries can only combine in a trivial way
Coleman–Mandula_theorem
Formula on random variables
Propagation of error Markov chain central limit theorem Panjer recursion Inverse-variance weighting Donsker's theorem Paired difference test Klenke, Achim (2013)
Bienaymé's_identity
Type of algorithm in computer science
function must terminate. It is supported by theorem provers Agda and Rocq. Both corecursion and recursion can be thought of as operating on trees, which
Corecursion
Topics referred to by the same term
statements" in mathematical logic. Recursive set, a "decidable set" in recursion theory Decision problem List of undecidable problems Decision (disambiguation)
Decidability
One of several equivalent definitions of a computable function
the rules of primitive recursion as those do not provide a mechanism for "infinite loops" (undefined values). A normal form theorem due to Kleene says that
General_recursive_function
Theorem for proving more complex theorems
also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however
Lemma_(mathematics)
Measure of algorithmic complexity
impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program P computing a
Kolmogorov_complexity
Algorithms which recursively solve subproblems
they use tail recursion, they can be converted into simple loops. Under this broad definition, however, every algorithm that uses recursion or loops could
Divide-and-conquer_algorithm
Thesis on the nature of computability
machine, or λ-function, or carefully invoke recursion axioms, or at best, cleverly invoke various theorems of computability theory. But because the computability
Church–Turing_thesis
System of mathematical set theory
(1–3): 107–234. doi:10.1016/S0168-0072(00)00031-2. P. Odifreddi, Classical Recursion Theory (1989) p.421. North-Holland, 0-444-87295-7 Devlin, Keith J. (1984)
Kripke–Platek_set_theory
Product of numbers from 1 to n
theorem can again be invoked to prove that the numbers of bits in the corresponding products decrease by a constant factor at each level of recursion
Factorial
Mathematical logic concept
properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large
Absoluteness_(logic)
Theorem about fixed points of multiple variables
computability theory, Bekić's theorem or Bekić's lemma is a theorem about fixed-points which allows splitting a mutual recursion into recursions on one variable at
Bekić's_theorem
Mathematical logic concept
arithmetic and that its consistency is therefore less controversial. Gentzen's theorem is concerned with first-order arithmetic: the theory of the natural numbers
Gentzen's_consistency_proof
functions for inversion. Theorem: Any function constructible via the clauses of primitive recursion using the standard primitive recursion schema is constructible
Gödel's_β_function
Computer science and linguistics concept relating to non-terminal production
types of grammars in the Chomsky hierarchy can be recursive and it is recursion that allows the production of infinite sets of words. A non-recursive
Recursive_grammar
Overview of and topical guide to logic
undecidable problems Post correspondence problem Post's theorem Primitive recursive function Recursion (computer science) Recursive language Recursive set
Outline_of_logic
Element mapped to itself by a mathematical function
extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory
Fixed_point_(mathematics)
n theorem, let s:(ωω)2 → ωω be continuous such that for all ϵ, x, t, and w, U(s(ϵ,x),t,w) ↔ (∃y,z)(y ≺ x ∧ U(ϵ,y,z) ∧ U(z,t,w)). By the recursion theorem
Moschovakis_coding_lemma
Undecidability of equality of real numbers
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ln 2
Richardson's_theorem
Branch of mathematical logic
like RT2 2 (Ramsey's theorem for pairs). Research in reverse mathematics often incorporates methods and techniques from recursion theory as well as proof
Proof_theory
RECURSION THEOREM
RECURSION THEOREM
Girl/Female
Arabic, Muslim
Pleasure Trip; Excursion Spot
Girl/Female
Arabic, Hindu, Indian, Kannada, Marathi, Muslim, Sindhi
Pleasure Trip; Excursion Spot
Girl/Female
Muslim/Islamic
Excursion spot
Girl/Female
Muslim
Pleasure trip, Excursion spot
RECURSION THEOREM
RECURSION THEOREM
Boy/Male
Norse
Thunder Finn.
Boy/Male
Spanish Greek
Purple.
Boy/Male
Muslim
One who praises Allah
Boy/Male
Muslim
Intelligent, Brilliant
Girl/Female
Arabic, Assamese, Greek, Hebrew, Indian, Kannada, Muslim
Virgin; Beauty; Fire; Noble
Boy/Male
Hindu, Indian
Best Among the Gods
Boy/Male
Hindu, Indian
Good Times
Boy/Male
Bengali, Hindu, Indian, Kannada, Malayalam, Marathi, Mythological, Tamil
Fifth Incarnation of Lord Vishnu
Girl/Female
Tamil
Literature
Boy/Male
Bengali, Gujarati, Hindu, Indian, Malayalam, Marathi, Oriya, Tamil, Telugu
One with Few Desires; Money
RECURSION THEOREM
RECURSION THEOREM
RECURSION THEOREM
RECURSION THEOREM
RECURSION THEOREM
n.
Attack; occurrence.
n.
The act of ceding back; restoration; repeated cession; as, the recession of conquered territory to its former sovereign.
v. t.
Causing revulsion; revulsive.
n.
A meeting; a clash; a collision.
n.
Same as Occursion.
n.
The principle of repulsion; the quality or capacity of repelling; repulsion.
n.
A pleasure excursion; a trip.
n.
A flowing; also, a hostile incursion.
n.
Reversion.
n.
The act of beating or striking back.
n.
The act of recurring; return.
n.
A running into; hence, an entering into a territory with hostile intention; a temporary invasion; a predatory or harassing inroad; a raid.
n.
An excursion for plundering.
n.
A riding out; an excursion.
n.
The power, either inherent or due to some physical action, by which bodies, or the particles of bodies, are made to recede from each other, or to resist each other's nearer approach; as, molecular repulsion; electrical repulsion.
a.
Causing, or tending to, revulsion.
n.
An excursion.
n.
An excursion.
n.
The office of a decurion.