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Technique for defining number-theoretic functions by recursion
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
Course-of-values_recursion
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
Two functions defined from each other
mutual recursion is a form of recursion where two or more mathematical or computational objects, such as functions or datatypes, are defined in terms of each
Mutual_recursion
Subroutine call performed as final action of a procedure
Computer programming portal Look up tail recursion in Wiktionary, the free dictionary. Course-of-values recursion Recursion (computer science) Primitive recursive
Tail_call
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
2019 novel by Blake Crouch
Recursion is a thriller science fiction novel by American writer Blake Crouch, first published in the United States in June 2019 by the Crown Publishing
Recursion_(Crouch_novel)
Function computable with bounded loops
natural for reading or writing. Course-of-values recursion defines primitive recursive functions. Some forms of mutual recursion also define primitive recursive
Primitive_recursive_function
Function in mathematical logic
One may use Gödel numbering to show how functions defined by course-of-values recursion are in fact primitive recursive functions. Once a Gödel numbering
Gödel_numbering
Hierarchy of complexity classes for formulas defining sets
{\displaystyle f(0)\cdot f(1)\cdot \ldots \cdot f(n-1)=0} ; with course-of-values recursion each of these can be defined by a single primitive recursive function
Arithmetical_hierarchy
Higher-order function Y for which Y f = f (Y f)
the use of fixed-point combinators is sometimes called anonymous recursion. Anonymous function Fixed-point iteration Lambda calculus#Recursion and fixed
Fixed-point_combinator
Defining elements of a set in terms of other elements in the set
because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function n
Recursive_definition
Programming language family
functions, recursion, the self-hosting compiler, and the read–eval–print loop. The name LISP derives from "List Processor". Linked lists are one of Lisp's
Lisp_(programming_language)
Subfield of mathematics
logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known
Mathematical_logic
semantics of modal logic, suggesting that objects in possible worlds have counterparts in other possible worlds. course of values recursion A principle
Glossary_of_logic
usually called "rows". recursion Occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging
Glossary_of_computer_science
Study of computable functions and Turing degrees
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
Computability_theory
Logic with discrete truth values
values. The term finite-valued logic encompasses both finitely many-valued logic and bivalent logic. Fuzzy logics, which allow for degrees of values between
Finite-valued_logic
Form of mathematical proof
sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs
Mathematical_induction
Dialect of Lisp
tail recursion" refers to the property of all Scheme implementations, that they perform tail-call optimization so as to support an indefinite number of active
Scheme_(programming_language)
Product of numbers from 1 to n
numbers of bits in the corresponding products decrease by a constant factor at each level of recursion, so the total time for these steps at all levels of recursion
Factorial
Sequence of operations for a task
Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem of Elementary Number Theory
Algorithm
Polynomial sequence
ready derivation of it. This binomial type identity, for α = β = 1/2, has already been encountered in the above section on #Recursion relations.) Since
Hermite_polynomials
Problem optimization method
are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems. In the optimization literature this
Dynamic_programming
Mathematical-logic system
this value were to contain itself by value, it would have to be of infinite size, which is impossible. Other notations, which support recursion natively
Lambda_calculus
Programming language
start) / 2; cilk_spawn recursion(a, start, midpoint); recursion(a, midpoint, end); cilk_sync; } } void loop(int *a, int n) { recursion(a, 0, n); } The reasons
Cilk
Mathematical transform that expresses a function of time as a function of frequency
there is a simple recursion relating the cases n + 2 and n allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from
Fourier_transform
Mathematical model for sequential decision making under uncertainty
point V ∗ {\displaystyle V^{*}} which satisfies the value iteration (Bellman equation) recursion V ∗ ( s ) = max a E [ R a ( s , s ′ ) + γ V ∗ ( s ′ )
Markov_decision_process
Fractal named after mathematician Benoit Mandelbrot
(described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see
Mandelbrot_set
Study of circular causal processes
Cybernetics is the transdisciplinary study of circular causal processes such as feedback and recursion, where the outcomes of actions return as inputs for subsequent
Cybernetics
Type of subroutine call
It also is not capable of expressing recursion. The addition of new registers for this sort of duty was a key design goal of EDSAC 2. This example demonstrates
Wheeler_Jump
Functional programming language
-> a -- Using recursion (with the "ifthenelse" expression) factorial n = if n < 2 then 1 else n * factorial (n - 1) -- Using recursion (with pattern matching)
Haskell
associated with the work of Alonzo Church and Alan Turing. The debate and discovery of the meaning of "computation" and "recursion" has been long and contentious
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Functions of an angle
these series from elementary recursion relations derived from the differential equation. Being defined as fractions of entire functions, the other trigonometric
Trigonometric_functions
Algorithm for finding zeros of functions
Hensel's lemma, which uses the recursion from Newton's method on the p-adic numbers. Because of the more stable behavior of addition and multiplication in
Newton's_method
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
some large integer m. Use that as an approximation for the value of Hm+x. Then use the recursion relation Hn = Hn−1 + 1/n backwards m times, to unwind it
Harmonic_number
Representation theory
elliptic modular function j. Peterson gave a recursion formula for the multiplicities mult(β) of the roots β of a symmetrizable (generalized) Kac–Moody algebra
Weyl_character_formula
Numbers arranged in a triangle
Pelillo, Marcello (2012), "Efficient computation of Ihara coefficients using the Bell polynomial recursion", Linear Algebra and Its Applications, 436 (5):
Triangular_array
Algorithm in computer graphics to add color or texture
stack space is severely constrained (e.g. Microcontrollers). Moving the recursion into a data structure (either a stack or a queue) prevents a stack overflow
Flood_fill
Computer programming textbook by Matthias Felleisen and colleagues
book assumes less domain knowledge than that of SICP. The paper distinguishes between structural recursion, where the related data definition happens to
How_to_Design_Programs
Coefficients in angular momentum eigenstates of quantum systems
the Hilbert space inner product. From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found.
Clebsch–Gordan_coefficients
Thesis on the nature of computability
functions: the smallest class of functions (with arbitrarily many arguments) that is closed under composition, recursion, and minimization, and includes
Church–Turing_thesis
Problem in combinatorial optimization
can break the recursion and convert it into a tree. Then we can cut some leaves and use parallel computing to expedite the running of this method. To
Knapsack_problem
Association of one output to each input
the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the
Function_(mathematics)
Automated methods for the creation of mazes
A disadvantage of the first approach is a large depth of recursion – in the worst case, the routine may need to recur on every cell of the area being
Maze_generation_algorithm
Programming language standard
and VALUES for returning multiple values: (defun magic-eight-ball () "Return an outlook prediction, with the probability as a secondary value" (values "Outlook
Common_Lisp
Mathematical function characterizing set membership
function. (See paragraph below about the use of the inverse in classical recursion theory.) Given a probability space ( Ω , F , P ) {\displaystyle \textstyle
Indicator_function
General-purpose programming language
Early FORTRAN compilers supported no recursion in subroutines. Early computer architectures supported no concept of a stack, and when they did directly
Fortran
Function definition that is not bound to an identifier
frequently employed to tailor the behavior of a generically defined function, such as a loop or recursion pattern. Anonymous functions are a convenient
Anonymous_function
Infinitely detailed mathematical structure
17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable
Fractal
American theoretical physicist
Britto on a method calculating scattering amplitudes known as the BCFW recursion relations. Witten has been honored with numerous awards including a MacArthur
Edward_Witten
Algebraic manipulation of "true" and "false"
algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually
Boolean_algebra
Software library for interpreting regular expressions
evaluated as being part of the pattern) nor the (?{}) construct, although the latter can be emulated using (?Cn). Recursion control verbs added in the
Perl Compatible Regular Expressions
Perl_Compatible_Regular_Expressions
Paradox in set theory
has to do with the " 'transformation of the generalization of an equality into an equality of courses-of-values. For Frege a function is something incomplete
Russell's_paradox
On solvability of Diophantine equations
known as recursion theory) that provided a precise explication of the intuitive notion of algorithmic computability, thus making the notion of recursive
Hilbert's_tenth_problem
Symbol representing a mathematical object
object is the value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may
Variable_(mathematics)
Set of rules defining correctly structured Prolog programs
to infinite recursion, as in: predicate1(X) :- predicate2(X,X). predicate2(X,Y) :- predicate1(X), X \= Y. Given this ordering, any query of the form ?-
Prolog_syntax_and_semantics
Variable used in a concurrent system
The value of the semaphore S represents the number of units of available resource units when non-negative. In some implementations, negative values indicate
Semaphore_(programming)
Branch of logic
one of two possible truth values, the truth value of true or the truth value of false. The principle of bivalence and the law of excluded middle are upheld
Propositional_logic
S2CID 115779583. Kadvany, John (2008-02-08). "Positional Value and Linguistic Recursion". Journal of Indian Philosophy. 35 (5–6): 487–520. CiteSeerX 10.1
History_of_mathematics
3-volume treatise on mathematics, 1910–1913
First of all, "function" means "propositional function", something taking values true or false. Second, functions are not determined by their values: it
Principia_Mathematica
Mathematical puzzle game
representation of the move number, m. For example, all the details for move m=216 of an 8-disk Tower of Hanoi can be computed without any iteration or recursion, and
Tower_of_Hanoi
Fractal sets in complex dynamics of mathematics
the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and
Julia_set
Logical deductive system
"infinite recursion," represented by ∅ {\displaystyle \emptyset } . Of course, the same construction can be carried out with any other set instead of N {\displaystyle
Scott_information_system
Formalism to describe programming languages
number) and end-of-line, or a personal part followed by a name part (this rule illustrates the use of recursion in BNFs, covering the case of people who use
Backus–Naur_form
Algorithm for computing greatest common divisors
\end{aligned}}} where m−1 = n−2 = 1 and m−2 = n−1 = 0 are the initial values of the recursion. The convergent mk/nk is the best rational number approximation
Euclidean_algorithm
Mathematical functions
and N {\displaystyle N} by recursion. Gauss' M {\displaystyle M} and N {\displaystyle N} satisfy the following system of differential equations: M (
Lemniscate_elliptic_functions
Particular class of sets which can be described entirely in terms of simpler sets
z_{n}\in X{\Bigr \}}.} L {\displaystyle L} is defined by transfinite recursion as follows: L 0 := ∅ . {\textstyle L_{0}:=\varnothing .} L α + 1 := Def
Constructible_universe
Social deduction game
moderator Derivation is by recursion on decreasing numbers of players in the following round, see: Yao, E. (2008). "A Theoretical Study of Mafia Games". p. 7
Mafia_(party_game)
Axiomatic set theories based on the principles of mathematical constructivism
in this sense. It further allows definitions of class functions by transfinite recursion. The study of the various principles that grant set definitions
Constructive_set_theory
mathematical puzzles and mathematical games. Recursion theory see computability theory Representation theory a subfield of abstract algebra; it studies algebraic
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Technique invented by Paul Cohen for proving consistency and independence results
concepts of P {\displaystyle \mathbb {P} } -names, interpretations, and x ˇ {\displaystyle {\check {x}}} may be defined by transfinite recursion. With ∅
Forcing_(mathematics)
Programming language
demonstrates the usage of the self-reference verb $: to recursively calculate fibonacci numbers: 1:`($:@-&2+$:@<:)@.(>&2) This recursion can also be accomplished
J_(programming_language)
Filter for nonlinear state estimation
The update equations are identical to those of discrete-time extended Kalman filter. The above recursion is a first-order extended Kalman filter (EKF)
Extended_Kalman_filter
Mathematical logic concept
course in proof theory (textbook draft ed.), archived from the original on 2011-06-07 Khlodovskii, I. N. (1959), "A new proof of the consistency of arithmetic"
Gentzen's_consistency_proof
Set theory concept
recursion as follows: Let V0 be the empty set: V 0 := ∅ . {\displaystyle V_{0}:=\varnothing .} For any ordinal number β, let Vβ+1 be the power set of
Von_Neumann_universe
Every polynomial has a real or complex root
definition, the multiplicity of the root. The proof that this statement results from the previous ones is done by recursion on n: when a root r 1 {\displaystyle
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
All-encompassing set or class
be defined by structural recursion as follows: Let S0X be X itself. Let S1X be the union of X and PX. Let S2X be the union of S1X and P(S1X). In general
Universe_(mathematics)
Programming language
The following code calculates the Fibonacci sequence of a number n inputted. It uses tail recursion and pattern matching. let fib n = let rec fib_aux m
OCaml
List of statements that appear to contradict themselves
precordium when the radial pulse is not felt. Hidders, J. "Expressive Power of Recursion and Aggregates in XQuery" (PDF). Retrieved 23 May 2012.: Chapter 1, Introduction
List_of_paradoxes
General-purpose programming language
documentation. Retrieved 30 April 2026. van Rossum, Guido (22 April 2009). "Tail Recursion Elimination". Neopythonic.blogspot.be. Archived from the original on 19
Python_(programming_language)
Axiom in Russell's ramified theory of types
version of the Peano axioms in 4 axioms including mathematical induction, some definitions that "have the character of axioms, and certain recursion axioms
Axiom_of_reducibility
Programming language
updater of the doublet. Of course, updater is a doublet and can be used to change the updater component of a doublet. Variables can hold values of any type
POP-2
Machine learning algorithm
a solution compared to Shafer-Shenoy. Computed recursively Multiple recursions of the Shafer-Shenoy algorithm results in Hugin algorithm Found by the
Junction_tree_algorithm
Algorithms for calculating square roots
should satisfy the recursion X m = X m − 1 − Y m , {\displaystyle X_{m}=X_{m-1}-Y_{m},} where X m {\displaystyle X_{m}} is the sum of all the terms after
Square_root_algorithms
In logic, a statement which is always true
contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables. The double turnstile notation ⊨
Tautology_(logic)
Theory of truth in the philosophy of language
Concept of Truth in Formal Languages" (1935), attempted to formulate a new theory of truth in order to resolve the liar paradox. In the course of this he
Semantic_theory_of_truth
Péter was one of the founders of recursion theory, a branch of mathematical logic, of computer science, and of the theory of computation that originated in
List of inventions and discoveries by women
List_of_inventions_and_discoveries_by_women
Publications. pp. 45–67. ISBN 978-1853594892. Ang, Roslynn (2017). "Recursions of Colonial Desire for Differences: The Doubly Erased And/or Hyper-Visible
Colonisation_of_Hokkaido
Relationship between language and human evolution
of modern human imagination. This hypothesis proposes that there were two phases that led to modern recursive language. The phenomenon of recursion occurs
Origin_of_language
Programming language that uses first order logic
called tail call optimization for deterministic predicates exhibiting tail recursion or, more generally, tail calls: A clause's stack frame is discarded before
Prolog
Measure of algorithmic complexity
as S ( n ) {\displaystyle S(n)} ). By modifying the function at lower values of n {\displaystyle n} we get an upper bound on B B {\displaystyle BB} ,
Kolmogorov_complexity
Case study
development and that of other similar systems of grammar is more sensitive than vocabulary or more basic grammar, such as word order or recursion, requiring a
Linguistic development of Genie
Linguistic_development_of_Genie
Institute Professor at MIT, developed Levinson recursion Daniel Levitin – neuroscientist, music producer, author of This Is Your Brain on Music Soung Chang Liew
List of Massachusetts Institute of Technology alumni
List_of_Massachusetts_Institute_of_Technology_alumni
French polymath (1623–1662)
triangle can also be represented: He defined the numbers in the triangle by recursion: Call the number in the (m + 1)th row and (n + 1)th column tmn. Then tmn = tm−1
Blaise_Pascal
Counting from "0" instead of "1" first
is thus often used as the base case for many kinds of numerical recursion. Proofs and other sorts of mathematical reasoning in computer science often begin
Zero-based_numbering
Solutions of Legendre's differential equation
{\displaystyle x=\pm 1} . The Legendre functions of the second kind can also be defined recursively via Bonnet's recursion formula Q n ( x ) = { 1 2 log 1 + x
Legendre_function
Computation model defining an abstract machine
two students Stephen Kleene and J. B. Rosser by use of Church's lambda-calculus and Gödel's recursion theory (1934). Church's paper (published 15 April
Turing_machine
Numerical technique
sampling is a generalization of one-dimensional adaptive quadratures to multi-dimensional integrals. On each recursion step the integral and the error
Monte_Carlo_integration
Several equations of degree 1 to be solved simultaneously
{1}{2}}y-z=0\end{cases}}} is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such
System_of_linear_equations
List of concepts in artificial intelligence
allocation, data types, recursion, functions as arguments, generators, and cooperative multitasking. IPL invented the concept of list processing, albeit
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION
COURSE OF-VALUES-RECURSION