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Defining elements of a set in terms of other elements in the set
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements
Recursive_definition
Process of repeating items in a self-similar way
answer A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a
Recursion
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Use of functions that call themselves
flexible way. The definition of a recursive function is typically divided into two parts: one or more base case(s) and one or more recursive case(s). This
Recursion_(computer_science)
Limited form of tree data structure
and the right child. That is, it is a k-ary tree where k = 2. A recursive definition using set theory is that a binary tree is a triple (L, S, R), where
Binary_tree
One of several equivalent definitions of a computable function
Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that
General_recursive_function
Statement that attaches a meaning to a term
noted that some definitions are "legal" or "coercive" – their object is to create or alter rights, duties, or crimes. A recursive definition, sometimes also
Definition
Data type that refers to itself in its definition
programming, a recursive data type is a data type whose definition contains values of the same type. It is also known as a recursively defined, inductively
Recursive_data_type
Subroutine call performed as final action of a procedure
target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end
Tail_call
Higher-order function Y for which Y f = f (Y f)
functional programming languages, and provide a means to allow for recursive definitions. Applied to a non-constant function of one variable that treats
Fixed-point_combinator
Curve used in computer graphics and related fields
these properties. Bézier curves can be defined for any degree n. A recursive definition for the Bézier curve of degree n expresses it as a point-to-point
Bézier_curve
Tree data structure with a variable and unbounded number of branches per node
"multidigraph". In a correspondence to the types of entities used in the recursive definition, each node of an apq is assigned a type (1), (2a), (2b), (2c) or
Rose_tree
Rational number sequence
numbers that is more efficient than the one given by their original recursive definition: ( m + 3 m ) B m = { m + 3 3 − ∑ j = 1 m 6 ( m + 3 m − 6 j ) B m
Bernoulli_number
Mathematical set of all subsets of a set
_{k=0}^{n}{\binom {n}{k}}} If S is a finite set, then a recursive definition of P(S) proceeds as follows: If S = {}, then P(S) = { {} }. Otherwise
Power_set
Mathematical function
\dots ,X_{n}).\end{aligned}}} This is equivalent with the (doubly) recursive definition: e 1 ( x 1 ) = X 1 e 0 ( X 1 , … , X n ) = 1 e k ( X 1 , … , X n
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Mathematical logic concept
a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable
Computably_enumerable_set
Arithmetic operation
}}n>0\end{cases}}} The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions
Tetration
Set theory concept
into the definition of the rank of a set gives a self-contained recursive definition: The rank of a set is the smallest ordinal number strictly greater
Von_Neumann_universe
Type system used in computer programming and mathematics
programming practical recursive functions are needed. A central property of the lambda calculus is that recursive definitions are not directly available
Hindley–Milner_type_system
Square root of the determinant of a skew-symmetric square matrix
Pfaffian of a skew-symmetric 2n × 2n matrix A with n > 0 can be computed recursively as pf ( A ) = ∑ j = 1 j ≠ i 2 n ( − 1 ) i + j + 1 + θ ( i − j ) a i
Pfaffian
Two functions defined from each other
the children. This mutually recursive definition can be converted to a singly recursive definition by inlining the definition of a forest: t: v [t[1],
Mutual_recursion
Pair of mathematical objects
entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example, the
Ordered_pair
Recursive mathematical formula
the appropriate value at 0; any value would be consistent with the recursive definition. A smooth extension to the reals would satisfy f ( 0 ) = f ′ ( 1
Exponential_factorial
Mathematical function that can be computed by a program
definitions. The class of computable functions can be defined in many equivalent models of computation, including Turing machines General recursive functions
Computable_function
Theorem in computability theory
defined via recursive definitions. The statement of the theorems refers to an admissible numbering φ {\displaystyle \varphi } of the partial recursive functions
Kleene's_recursion_theorem
Adaptive filter algorithm for digital signal processing
Recursive least squares (RLS) is an adaptive filter algorithm that recursively finds the coefficients that minimize a weighted linear least squares cost
Recursive least squares filter
Recursive_least_squares_filter
Quickly growing function
examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function
Ackermann_function
Acronym whose expansion includes a copy of itself
A recursive acronym is an acronym that refers to itself, and appears most frequently in computer programming. The term was first used in print in 1979
Recursive_acronym
Fundamental theorem in mathematical logic
interpret its own construction, so that this construction is non-recursive (as recursive definitions would be unambiguous). Also, if T {\displaystyle T} is at
Gödel's_completeness_theorem
Function with unusual fractal properties
also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree. One way to define the question-mark
Minkowski's question-mark function
Minkowski's_question-mark_function
Axiomatic logical system
+ y) (4) and (5) are the recursive definition of addition. x·0 = 0 x·Sy = (x·y) + x (6) and (7) are the recursive definition of multiplication. The axioms
Robinson_arithmetic
Number used for counting
first natural number and (2) gives a recursive definition for each subsequent number in terms of previous definitions, as illustrated below. a + 1 = a +
Natural_number
Generalization of "n-th" to infinite cases
Informally, one may define an ordinal recursively as a downward closed set of ordinals. Such a recursive definition is usually justified with the transitive
Ordinal_number
Real number that can be computed within arbitrary precision
notion of computability available at the time. Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal
Computable_number
Quality of zero being an even number
formalized into a recursive definition of the set of even natural numbers: 0 is even. (n + 1) is even if and only if n is not even. This definition has the conceptual
Parity_of_zero
Axioms for the natural numbers
{\begin{aligned}u(0)&=0_{X},\\u(S)&=S_{X}(u).\end{aligned}}} This is precisely the recursive definition of 0X and SX. When the Peano axioms were first proposed, Bertrand
Peano_axioms
Condition for a mathematical function to map some value to itself
view), the development of the theory is quite different. The same definition of recursive function can be given, in computability theory, by applying Kleene's
Fixed-point_theorem
Technique for defining number-theoretic functions by recursion
seen to be a primitive recursive function (assuming an appropriate Gödel numbering is used). In order to convert a definition by course-of-values recursion
Course-of-values_recursion
Sequence of differential equation solutions
somewhat different definitions of the so-called associated Laguerre polynomials.) One can also define the Laguerre polynomials recursively, defining the first
Laguerre_polynomials
Array of numbers
determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1 × 1
Matrix_(mathematics)
partition of space; and with fractal models that give an infinitely recursive definition of the shape. However, these distinctions are often blurred: for
Geometric_modeling
Logical connective
mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover"). Moreover, in the case of a recursive definition
If_and_only_if
Math theory of strings of symbols
especially proof theory. A generative grammar can be seen as a recursive definition in string theory. The most basic operation on strings is concatenation:
Concatenation_theory
Fractal constructible with L-systems
a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most
Dragon_curve
Topics referred to by the same term
Intensional definition Elementary definition Recursive definition Field of definition A continuous function A well-defined function High-definition television
Definition_(disambiguation)
Israeli cryptographer (born 1952)
(archived 2016-11-11) Shamir, Adi (October 1976). The fixedpoints of recursive definitions. Weizmann Institute of Science. OCLC 884951223. Adi Shamir at the
Adi_Shamir
Self-referential description of meaning
Self-reference Self-refuting idea Recursive definition Revision theory Tautology Vicious circle principle "Circular Definition". Glossary of Linguistic Terms
Circular_definition
Divide-and-conquer algorithm to compute a Hadamard transform
size n / 2 {\displaystyle n/2} . This implementation follows the recursive definition of the 2 m × 2 m {\displaystyle 2^{m}\times 2^{m}} Hadamard matrix
Fast_Walsh–Hadamard_transform
Branch of logic
propositional logic is defined recursively by these definitions: Definition 1: Atomic propositional variables are formulas. Definition 2: If c n m {\displaystyle
Propositional_logic
Family of formal knowledge representation
concepts and roles. This is done by using a recursive definition similar to the syntax. The following definitions follow the treatment in Baader et al. A
Description_logic
Equation for two-body bound states
A graphical representation of the Bethe–Salpeter equation, showing its recursive definition
Bethe–Salpeter_equation
3-volume treatise on mathematics, 1910–1913
theory specifies the rules of syntax (rules of grammar) usually as a recursive definition that starts with "0" and specifies how to build acceptable strings
Principia_Mathematica
Subfield of mathematics
uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical
Mathematical_logic
Trait of a player in game theory
game-theoretic one. Hofstadter provided this definition: "Superrational thinkers, by recursive definition, include in their calculations the fact that
Superrationality
Object in category theory
data for u, given in the form of a recursive definition: ⊢ u (z) = q y ∈E N ⊢ u (s y) = f (u (y)) The above definition is the universal property of NNOs
Natural_numbers_object
Digital electronic circuit
6-LUT, hence an entire ALM. An open-source Verilog generator for the recursive priority-encoder is available online. A behavioral description of priority
Priority_encoder
Mathematical-logic system based on functions
means a recursive function definition cannot be written with let. The letrec construction would allow writing recursive function definitions, where the
Lambda_calculus
Recursive function for formal verification case testing
the 91 function only handle the tail-recursive version. This is an equivalent mutually tail-recursive definition: M m t ( n ) = M m t ′ ( n , 0 ) {\displaystyle
McCarthy_91_function
Proof method in mathematical logic
proposition to hold for all x.) A structurally recursive function uses the same idea to define a recursive function: "base cases" handle each minimal structure
Structural_induction
In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix
Topological_recursion
Finite-state machine
composition. Clearly, this process may be recursively continued, giving the following recursive definition of δ ^ : Q × Σ ⋆ → Q {\displaystyle {\widehat
Deterministic finite automaton
Deterministic_finite_automaton
Study of programming languages via mathematical objects
recursively as: int factorial(int n) { if (n == 0) return 1; else return n * factorial(n - 1); } To provide a meaning for this recursive definition,
Denotational_semantics
Binary tree heap data structure
(approximately 1.44 log2 n). Skew heaps may be described with the following recursive definition:[citation needed][clarification needed] A heap with only one element
Skew_heap
Set with algorithmic membership test
enumerable Decidability (logic) Recursively enumerable language Recursive language Recursion That is, under the Set-theoretic definition of natural numbers, the
Computable_set
Specification of a mathematical group by generators and relations
then call a subset U of FS recursive (respectively recursively enumerable) if f(U) is recursive (respectively recursively enumerable). If S is indexed
Presentation_of_a_group
Mathematical proposition equivalent to the axiom of choice
defined using transfinite recursion. It is exactly like the usual recursive definition of a sequence but runs over ordinals. A key difference is that an
Zorn's_lemma
High-temperature expansion in statistical mechanics
identifying correlated clusters. The simplest definitions follow after one identifies the clusters recursively. At the lowest level, one finds the class of
Cluster_expansion
Necessary condition for optimality associated with dynamic programming
x_{1}=T(x_{0},a_{0})} . Therefore, the problem can be rewritten as a recursive definition of the value function: V ( x 0 ) = max a 0 { F ( x 0 , a 0 ) + β
Bellman_equation
Finite sets whose elements are all hereditarily finite sets
and all of its elements are finite sets, recursively all the way down to the empty set. A recursive definition of well-founded hereditarily finite sets
Hereditarily_finite_set
Topics referred to by the same term
artery, arising from the radial artery immediately below the elbow Recursive definition Recurrent neural network, a special artificial neural network Recurrence
Recurrence
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
Variational Principles. Courier Dover Publications. ISBN 0-486-65840-6. A recursive definition requires a first case, which may be taken as δ = 1 for p = 0, or
Kronecker_delta
Algebra associated to any vector space
\bigwedge }^{\!n}(V)\to {\textstyle \bigwedge ^{\!n-k}}(V)} by the recursive definition ι α ∧ β = ι β ∘ ι α . {\displaystyle \iota _{\alpha \wedge \beta
Exterior_algebra
highly connected to systemically important banks, and so on. The recursive definition is equivalent to performing eigendecomposition of a matrix of connectivity
Too_connected_to_fail
Number used in combinatorial game theory
On_p". arXiv:1108.0962v3 [math.RA]. Laubie, François (1999). "A recursive definition of $p$-ary addition without carry". Journal de théorie des nombres
Nimber
Recursive integer sequence
natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were
Catalan_number
Study of computable functions and Turing degrees
definition in terms of μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for
Computability_theory
Generalization of ordinary algorithms that compute more than Turing machines
hypercomputation are obtained by super-recursive algorithms. The Church–Turing thesis in recursion theory relies on a particular definition of the term algorithm. Based
Super-recursive_algorithm
System to identify resources on a network
methods, such as recursive, non-recursive, and iterative. A resolution process may use a combination of these methods. In a non-recursive query, a DNS resolver
Domain_Name_System
Computation model defining an abstract machine
A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine can equivalently be defined as
Turing_machine
complete it is highly tuned for matching text, and even a simple (recursive) definition of the factorial takes almost ten lines. Part of this is because
CRM114_(program)
Thesis on the nature of computability
1933, Kurt Gödel, with Jacques Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily
Church–Turing_thesis
Parallel sorting algorithm
lower order are mostly used for the two pre-sorters; therefore, a recursive definition of a bitonic sorting network from bitonic sorters can be described
Bitonic_sorter
Generalization of the real numbers
0; in other words, { | } is a form of the surreal number 0. The recursive definition of surreal numbers is completed by defining comparison: Given numeric
Surreal_number
Fractal curve resembling a blancmange pudding
above Fourier series for T w ( x ) . {\displaystyle T_{w}(x).} The recursive definition allows the monoid of self-symmetries of the curve to be given. This
Blancmange_curve
Formal language
There are three equivalent definitions of a recursively enumerable language: A recursively enumerable language is a recursively enumerable subset in the
Recursively enumerable language
Recursively_enumerable_language
Computer performance metric
_{1}=H_{2}+MR_{2}\cdot \mathrm {AMP} _{2}} In this manner, this recursive definition can be extended throughout all layers of the memory hierarchy. John
Average_memory_access_time
Instructions a computer can execute
is_a_creature(puff). is_the_mother_of(norberta, puff). Rule (2) is a recursive (inductive) definition. It can be understood declaratively, without the need to understand
Computer_program
Set of all things that may be the input of a mathematical function
of f is X. In modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that X
Domain_of_a_function
Properties of 2D or 3D digital images that correspond to classic topological properties
digital surfaces. The digital manifold was studied in the 1990s. A recursive definition of the digital k-manifold was proposed intuitively by Chen and Zhang
Digital_topology
Set of markup declarations for SGML-family markup language
referenced in its defined content (this also prevents circular or recursive definitions of internal entities). This document is parsed as if it was: <!DOCTYPE
Document_type_definition
Object which stores memory addresses in a computer program
if there is none }; This pointer-recursive definition is essentially the same as the reference-recursive definition from the language Haskell: data LinkedList
Pointer (computer programming)
Pointer_(computer_programming)
Raster graphics file format
Huffman coding and color indexing transform. This format uses a recursive definition: all of the control images, such as the local entropy code selection
WebP
Sequence of operations for a task
The Tower of Hanoi is a puzzle commonly solved using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex)
Algorithm
Computer hardware or software server
may recursively query name servers higher up in the hierarchy. This is known as a recursive query or recursive lookup. A server providing recursive queries
Name_server
Problem in computer science
of an effectively calculable function can be formalized by the general recursive functions or equivalently by the lambda-definable functions. He proves
Halting_problem
which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a (non-empty) binary tree is
Glossary_of_computer_science
Statement that is taken to be true
context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms Σ {\displaystyle \Sigma } of the
Axiom
decision problems that can be solved in time bounded by an elementary recursive function. Equivalently, these are the problems that can be solved in time
ELEMENTARY
Formal language in mathematics and computer science
There are two equivalent major definitions for the concept of a recursive language: A recursive language is a recursive subset of the set of all possible
Recursive_language
Type of error correcting code
synthesized bit-channels used for information symbols. Although the recursive definition of a polar code is explicit, computing the reliabilities of all synthesized
Polar_code_(coding_theory)
RECURSIVE DEFINITION
RECURSIVE DEFINITION
Girl/Female
Hindu, Indian
Number; Definition
Biblical
savior; deliverer, The Greek form of the name Joshua or Jeshua, a contraction of Jehoshua, that is, help of Jehovah or saviour. Latin: Jesus, Iesus, Iesu, Josue. Greek: Ieous from Hebrew Yeshua. Also means safety, victory and who's help is Jehovah or it may be from the verb "Yasha", "to save," and = Jehovah Savior, or simply Savior; a late form of Hebrew "yehosua", the Jesus means of which is "YHWH is salvation" or "YHWH saves/has saved." Online definition of "savior." Latin term drove out Old English "hæland" which means "healer" as the preferred descriptive term for Jesus.
Girl/Female
Indian
Definition
RECURSIVE DEFINITION
RECURSIVE DEFINITION
Biblical
A whole, Congregation
Male
Italian
Italian form of Latin Cyprianus, CIPRIANO means "from Cyprus."
Girl/Female
Hindu, Indian, Marathi, Sanskrit
With Beautiful Hair
Boy/Male
Hindu, Indian
Close
Boy/Male
Hindu
The one who brought Ganga to earth, With glorious chariot
Girl/Female
American, British, English, French, Greek, Jamaican
Lover; City Name; French Capital
Girl/Female
Arabic, Muslim
Beloved
Girl/Female
Hindu
Boy/Male
Hindu
Obtainment
Surname or Lastname
Americanized form of French Petitjean.English
Americanized form of French Petitjean.English : variant spelling of Pettyjohn.
RECURSIVE DEFINITION
RECURSIVE DEFINITION
RECURSIVE DEFINITION
RECURSIVE DEFINITION
RECURSIVE DEFINITION
n.
A character used in cursive writing.
a.
Not amiable; morose; ill-natured; repulsive.
a.
Prone to make excursions; wandering; roving; exploring; as, an excursive fancy.
a.
Affording retirement from society.
n.
The act of recurring; return.
v. t.
Causing revulsion; revulsive.
a.
Repulsive; driving back.
n.
A revulsive medicine.
a.
Running down; decurrent.
adv.
In a decursive manner.
a.
Preceding; introductory; precursory.
a.
Flowing; easy; cursive; as, a running hand.
a.
Repulsive by itself; as, the idiorepulsive power of heat.
a.
Serving, or able, to repulse; repellent; as, a repulsive force.
a.
Manifesting distaste or dislike; repulsive.
a.
Causing, or tending to, revulsion.
n.
That which causes revulsion; specifically (Med.), a revulsive remedy or agent.
a.
Cold; forbidding; offensive; as, repulsive manners.
a.
Going back; receding.
a.
Making an incursion; invasive; aggressive; hostile.