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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
Functions whose execution you can pause
Using coroutines for state machines or concurrency is similar to using mutual recursion with tail calls, as in both cases the control changes to a different
Coroutine
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)
Subroutine call performed as final action of a procedure
tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end recursion) is particularly useful, and is often easy to optimize
Tail_call
Declaration of an identifier in computer programming
languages that require declaration before use; it is necessary for mutual recursion in such languages, as it is impossible to define such functions (or
Forward_declaration
Mathematical term in group theory
the child trees of ∅. The elements b, c, and d are defined through a mutual recursion. To understand the effect of the latter operations, consider the rightmost
Grigorchuk_group
In programming languages, the object or class the currently running code belongs to
method in a base class, or in cases of mutual recursion. The fragile base class problem has been blamed on open recursion, with the suggestion that invoking
This_(computer_programming)
Fast approximate median algorithm
previous step:. Note that pivot calls select; this is an instance of mutual recursion. function pivot(list, left, right) // for 5 or less elements just get
Median_of_medians
Data type that refers to itself in its definition
types can also be defined by mutual recursion. The most important basic example of this is a tree, which can be defined mutually recursively in terms of a
Recursive_data_type
Function computable with bounded loops
reading or writing. Course-of-values recursion defines primitive recursive functions. Some forms of mutual recursion also define primitive recursive functions
Primitive_recursive_function
Programming language
declarations to define pointer types, or when record declarations led to mutual recursion, or when an identifier may or may not have been used in an enumeration
Pascal_(programming_language)
Higher-order function Y for which Y f = f (Y f)
y(\lambda z.xxyz))\ (\lambda xy.y(\lambda z.xxyz))} The analog for mutual recursion is a polyvariadic fix-point combinator, which may be denoted Y*. In
Fixed-point_combinator
Software development tool
Multiple passes obviate the need for forward declarations, allowing mutual recursion to be implemented elegantly. The prime examples of languages requiring
Multi-pass_compiler
Form of logic that allows quantification over predicates
second-order logic. More expressive fragments are defined for any k > 0 by mutual recursion: Σ k + 1 1 {\displaystyle \Sigma _{k+1}^{1}} has the form ∃ R 0 … ∃
Second-order_logic
GNU profiling tool
(estimated as 30% to 260%) for higher-order or object-oriented programs. Mutual recursion and non-trivial cycles are not resolvable by the gprof approach (context-insensitive
Gprof
bringing Rintaro's time with Amadeus to an end. 12 "Mother Goose of Mutual Recursion: Recursive Mother Goose" Transliteration: "Sōgosaiki no Mazā Gūsu"
List of Steins;Gate 0 episodes
List_of_Steins;Gate_0_episodes
Part of a computer program where a given name binding is valid
them, and requires forward declaration in some cases, notably for mutual recursion. In other languages, such as Python, a name's scope begins at the start
Scope_(computer_programming)
Link analysis algorithm for webpages
Authority and hub values are defined in terms of one another in a mutual recursion. An authority value is computed as the sum of the scaled hub values
HITS_algorithm
One-way software control-flow statement
through single tail recursion (tail recursion calling the same function). Further, tail call optimization allows mutual recursion of unbounded depth,
Goto
Globalization meta-process
allow mutual recursion, which is, in a sense, more lifted than is supported in lambda calculus. Lambda calculus does not support mutual recursion and only
Lambda_lifting
Top-down parser utilizing recursion
left recursion. Any context-free grammar can be transformed into an equivalent grammar that has no left recursion, but removal of left recursion does
Recursive_descent_parser
Set-theoretic function
( α ) {\displaystyle \Psi _{\pi }^{\xi }(\alpha )} are defined in mutual recursion in the following way: M0 = K ∩ L i m {\displaystyle K\cap {\mathsf
Ordinal_collapsing_function
Recursion without calling a function by name
In computer science, anonymous recursion is recursion which does not explicitly call a function by name. This can be done either explicitly, by using a
Anonymous_recursion
Concept in computer science
(\lambda x.y))} This approach is then generalized to support mutual recursion. A mutually recursive let expression may be composed by rearranging the expression
Let_expression
Formal model in concurrency theory
they are equivalent in the traces model. UFP can also be extended to mutual recursions (by using vectors of processes) and other models of CSP (e.g., in
Communicating sequential processes
Communicating_sequential_processes
Theorem about fixed points of multiple variables
lemma is a theorem about fixed-points which allows splitting a mutual recursion into recursions on one variable at a time. It was created by Austrian Hans
Bekić's_theorem
Type of grammar for describing formal languages
parsing, left recursion must be eliminated. Direct recursion, be that left or right, is important in context-free grammars, because there recursion is the only
Parsing_expression_grammar
Parsing technique
Frost, Hafiz, and Callaghan, which do accommodate ambiguity and left recursion in polynomial time and which generate polynomial-sized representations
Top-down_parsing
Mathematical constructs and creation rules
be self-referential, but usually only in a way that permits structural recursion. The standard example is encoding the natural numbers using Peano's encoding
Inductive_type
Programming paradigm based on objects
self, to refer to the current object. In languages that support open recursion, a method in an object can call other methods in the same object, including
Object-oriented_programming
Type of parser
But it fails to accept 'aaaa' {\displaystyle {\texttt {'aaaa'}}} . Left recursion happens when a grammar production refers to itself as its left-most element
Packrat_parser
Integration of programming as part of running program
and recursion solutions, but timing had been a major issue. While the general form of a temporal recursion, being any asynchronous function recursion through
Live_coding
Synchronization mechanism for enforcing limits on access to a resource
In computer science, a lock or mutex (from mutual exclusion) is a synchronization primitive that prevents state from being modified or accessed by multiple
Lock_(computer_science)
Problematic interdependence of software modules
errors, compile errors). Circular dependencies can also result in infinite recursions or other unexpected failures. Circular dependencies may also cause memory
Circular_dependency
Computer science and recursion theory
In computer science and recursion theory the McCarthy formalism (1963) of computer scientist John McCarthy clarifies the notion of recursive functions
McCarthy_Formalism
Software programming optimization technique
alias when a call to the actual function is required (to avoid endless recursion), as illustrated below: function construct-memoized-functor (F is a function
Memoization
Variable used in a concurrent system
OS can release the mutex and signal waiting tasks of this condition. Recursion deadlock: a task is allowed to lock a reentrant mutex multiple times as
Semaphore_(programming)
Dialect of Lisp
iteration construct, do, but it is more idiomatic in Scheme to use tail recursion to express iteration. Standard-conforming Scheme implementations are required
Scheme_(programming_language)
Fractal composed of triangles
"the part that prevents leaks in motors". Apollonian gasket, a set of mutually tangent circles with the same combinatorial structure as the Sierpiński
Sierpiński_triangle
Sentence, idea or formula that refers to itself
varying degrees of expressiveness. Additionally, self-reference is seen in recursion (related to the mathematical recurrence relation) in functional programming
Self-reference
Mathematical puzzle game
proof with mathematical induction and is often used as an example of recursion when teaching programming. As in many mathematical puzzles, finding a
Tower_of_Hanoi
Mathematical ways to group elements of a set
and B6 = 203 (sequence A000110 in the OEIS). Bell numbers satisfy the recursion B n + 1 = ∑ k = 0 n ( n k ) B k {\displaystyle B_{n+1}=\sum _{k=0}^{n}{n
Partition_of_a_set
Early programming language for lists
problem solving. such as lists, dynamic memory allocation, data types, recursion, functions as arguments, generators, and cooperative multitasking. IPL
Information Processing Language
Information_Processing_Language
General-purpose programming language
commonly provide tail call optimization to allow for extensive use of recursion without stack overflow problems. Limitations in JVM bytecode complicate
Scala_(programming_language)
Mathematical function
this simplifies the equations. McMurchie and Davidson (1978) introduced recursion relations, which greatly reduces the amount of calculations. Pople and
Gaussian_orbital
Type of error correcting code
the physical channel into virtual outer channels. When the number of recursions becomes large, the virtual channels tend to either have high reliability
Polar_code_(coding_theory)
Machine learning algorithm
derived subset in a recursive manner called recursive partitioning. The recursion is completed when the subset at a node has all the same values of the
Decision_tree_learning
Procedural, imperative computer programming language
computation, scientific computing, and system programming. It supports recursion, structured programming, linked data structure handling, fixed-point,
PL/I
Properties linking logical conjunction and disjunction
problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic Algebraic
Conjunction/disjunction duality
Conjunction/disjunction_duality
Methodological basis for 3D CAD/CAM solid modeling and image rendering
eliminated. Recursion Given a deep composition tree, recursion can be expensive in combination with allocating and freeing up memory. Recursion can be simulated
Ray_casting
Inherent flaw in computer instructions
does not have the expected behavior, such as infinite looping, infinite recursion, incorrect comparison in a conditional such as using the wrong comparison
Software_bug
Types of numerical variables in mathematics
compchemeng.2005.11.008. Odifreddi, Piergiorgio (February 18, 1992). Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers. North Holland
Continuous or discrete variable
Continuous_or_discrete_variable
Abstract machine used in a formal logic and theoretical computer science
function) Successor function Identity function Composition function Primitive recursion (induction) μ operator (unbounded search operator) The authors show that
Counter_machine
2023. ISBN 978-4-08-883628-7. Vol. 24 (ch. 209–217): Fearsome Womb: Recursion. ISBN 978-4-08-883670-6. and 呪胎戴天-再帰- (in Japanese). October 4, 2023.
List of Jujutsu Kaisen characters
List_of_Jujutsu_Kaisen_characters
Computational model used in machine learning
particle swarm optimization are other learning algorithms. Convergent recursion is a learning algorithm for cerebellar model articulation controller (CMAC)
Neural network (machine learning)
Neural_network_(machine_learning)
Algorithm that estimates unknowns from a series of measurements over time
above system is known as the inverse Wiener-Hopf factor. The backward recursion is the adjoint of the above forward system. The result of the backward
Kalman_filter
Languages of the Yupik natives of Alaska
can be varied to communicate different meanings, principally through recursion. The only exception lies with case suffixes on nouns and person suffixes
Yupik_languages
Decentralized hacktivist collective
on March 6, 2012. Retrieved March 13, 2012. "Kretsinger, Sony hacker Recursion, jailed for year". BBC News. April 19, 2013. Archived from the original
Anonymous_(hacker_group)
primitive recursion A form of recursion where a function is defined in terms of itself, using simpler cases, with a base case to stop the recursion. primitive
Glossary_of_logic
Interdisciplinary study of systems
Multidimensional systems Open and closed systems in social science Pattern language Recursion (computer science) Reductionism Redundancy (engineering) Reversal theory
Systems_theory
{I} (X_{1};\cdots ;X_{n-1})\mid X_{n}{\big )}.} The first step in the recursion yields Shannon's definition I ( X 1 ; X 2 ) = H ( X 1 ) − H ( X 1 ∣
Information theory and measure theory
Information_theory_and_measure_theory
Programming language with English-like syntax
These other features included: Free-form code User-defined functions Recursion Locale-based processing Support for extended character sets such as Unicode
COBOL
Recursive function for formal verification case testing
function was chosen for being nested-recursive (contrasted with single recursion, such as defining f ( n ) {\displaystyle f(n)} by means of f ( n − 1 )
McCarthy_91_function
more advanced concepts that are occasionally needed: concurrency and recursion. Hierarchy is the most fundamental of all of the Warnier/Orr constructs
Warnier/Orr_diagram
Oldest scriptures of Hinduism
Machine, Frontline Kadvany, John (2007). "Positional Value and Linguistic Recursion". Journal of Indian Philosophy. 35 (5–6): 487–520. CiteSeerX 10.1.1.565
Vedas
Israeli military operation against Syria
men in Daraa". SOHR. 4 November 2025. Retrieved 7 January 2026. "New recursion: Israeli forces infiltrate Khan Arnaba and arrest four members of a family"
Israeli invasion of Syria (2024–present)
Israeli_invasion_of_Syria_(2024–present)
Relationship between language and human evolution
language is recursion, (in this context, the iterative embedding of phrases within phrases). Other scholars—notably Daniel Everett—deny that recursion is universal
Origin_of_language
(mathematical logic) Kirby–Paris theorem (proof theory) Kleene's recursion theorem (recursion theory) König's theorem (set theory, mathematical logic) Lindström's
List_of_theorems
Recursive function
function is often used as a benchmark for languages with optimization for recursion. The original definition by Takeuchi was as follows: def tarai(x: int
Tak_(function)
Infinite sequence of differential equations
given in terms of commuting Hamiltonian vector fields or the Lenard–Magri recursion scheme. In all of these formulations, the distinguishing feature of the
Korteweg–De_Vries_hierarchy
Process in which a first language is being acquired
infinite number of sentences, which is based on a syntactic principle called recursion. Evidence suggests that every individual has three recursive mechanisms
Language_acquisition
Diagram that shows all possible logical relations between a collection of sets
problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic Algebraic
Venn_diagram
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
Process where information about current status is used to influence future status
unintended results Phase margin – Parameter of electronic amplifiers Recursion – Process of repeating items in a self-similar way Resonance – Physical
Feedback
Japanese multinational corporation
into production in Japan the following year for the domestic market. Recursion turbo parallel-twin middleweight, shown at the 2013 Tokyo Motor Show.
Suzuki
Language that uses manual communication and body language to convey meaning
languages. Such fundamental properties include duality of patterning and recursion. Duality of patterning means that languages are composed of smaller, meaningless
Sign_language
Logical incompatibility between two or more propositions
the topic of the definition of tautologous, Nagel and Newman create two mutually exclusive and exhaustive classes K1 and K2, into which fall (the outcome
Contradiction
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
Mathematical problem set on a chessboard
Martin (1997). Backtracking Algorithms in MCPL using Bit Patterns and Recursion (PDF) (Technical report). University of Cambridge Computer Laboratory
Eight_queens_puzzle
Algorithm to smooth data points
normal equation matrix. The calculation is further simplified by using recursion to build orthogonal Gram polynomials. The whole calculation can be coded
Savitzky–Golay_filter
Dutch computer scientist
starting in 1969. After an MSc he wrote a PhD thesis, defended in 1976, on recursion theory in higher types, under the supervision of Dirk van Dalen. Bergstra
Jan_Bergstra
American cartoonist (born 1948)
diagram—an orthographic projection!" His comics experiment with time, space, recursion, and representation. He uses the word "decode" to express the action of
Art_Spiegelman
Branch of linguistics and semiotics relating context to meaning
its effects on the speaker's utility U S {\displaystyle U_{S}} . The recursion updates the participants' beliefs about the state of the world and about
Pragmatics
Logic theorem
proposition "the house is white" and its negation "the house is not white" are mutually exclusive. To express the fact that the law is tenseless and to avoid equivocation
Law_of_noncontradiction
Programming statement for branching control based on a value
the basis of the McCarthy formalism: its usage replaces both primitive recursion and the mu-operator. The earliest Fortran compilers supported the computed
Switch_statement
Mathematical theory
inductive inference with an emphasis on queries". Complexity, logic, and recursion theory, Lecture Notes in Pure and Appl. Math., 187, Dekker, New York,
Solomonoff's theory of inductive inference
Solomonoff's_theory_of_inductive_inference
Methods in artificial intelligence research
languages include: Garbage collection Dynamic typing Higher-order functions Recursion Conditionals Programs were themselves data structures that other programs
Symbolic artificial intelligence
Symbolic_artificial_intelligence
Task of computing complete subgraphs
graph. These partial solutions are used to shortcut the backtracking recursion. The fastest algorithm known today is a refined version of this method
Clique_problem
Set of elements in any of some sets
2009-07-14. "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian
Union_(set_theory)
Set of elements common to all of some sets
2012-05-08. "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian
Intersection_(set_theory)
Flashsort Introsort: begin with quicksort and switch to heapsort when the recursion depth exceeds a certain level Timsort: adaptative algorithm derived from
List_of_algorithms
Established norm or requirement to facilitate consistency
emerges from situations in which all parties realize mutual gains, but only by making mutually consistent decisions. Examples: Private standards are
Technical_standard
Logical connective OR
the definition of logical addition in a form that does not necessitate mutual exclusiveness. Aloni, Maria (2016), "Disjunction", in Zalta, Edward N. (ed
Logical_disjunction
3-volume treatise on mathematics, 1910–1913
"either p is true or q is true," where the alternatives are to be not mutually exclusive, will be represented by "p ∨ q" ". (cf. section B) The first
Principia_Mathematica
Unpredictable phenomenon in complex systems
their interrelation. Unlike machines, living systems at all levels of recursion - be it a sentient body, a tree, a family, an organisation, the education
Emergence
Axiom of set theory
formulation the combinatorial axiom of choice: CAC: Any collection of mutually disjoint nonempty sets has a transversal. Blass, Andreas (1984) [1983]
Axiom_of_choice
About mathematical functions
proposition] holds", and lastly how to cast it into the choice function. Recursion theory and computability: But the unexpected outcome of Hilbert's and
History of the function concept
History_of_the_function_concept
Dynamical system
)[rx_{n}(1-x_{n})]_{s}+(1-\epsilon )[rx_{n}(1-x_{n})]_{s-1}} Even though the recursion is chaotic, a more solid form develops in the evolution. Elongated convective
Coupled_map_lattice
Mathematical optimization algorithm
_{k+1}:=\mathbf {b} -\mathbf {Ax} _{k+1}} for the implicit one by the recursion subject to round-off error accumulation, and is thus recommended for an
Conjugate_gradient_method
Abstraction of linear independence of vectors
Tutte's original definition. There is a further definition in terms of recursion by deletion and contraction. The deletion-contraction identity is F (
Matroid
MUTUAL RECURSION
MUTUAL RECURSION
Boy/Male
Arabic, Muslim, Sindhi
Messenger; Ambassador; Prophet
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
A Saint
Boy/Male
Australian, Chinese, Vietnamese
Mulberry; Bright; Noble; Mutual
Boy/Male
Arabic, Muslim
Absolute; Unlimited
Boy/Male
Arabic, Muslim
Exalted; Supreme
Boy/Male
Hindu, Indian, Punjabi, Sanskrit, Sikh
Bud
Boy/Male
Muslim
Servant of the most high
Boy/Male
Indian
Messenger, Prophet, Ambassador
Boy/Male
Hindu
King of gujarat
Boy/Male
Hindu
Lord Murugan
Boy/Male
Greek Latin
Mutual love.
Boy/Male
Muslim/Islamic
Servant of the Most High
Male
Hindi/Indian
(মà§à¦•à§à¦²) Hindi name MUKUL means "blossom, bud."
Girl/Female
Arabic, Muslim
Harmony; Mutual Affection
Boy/Male
Muslim
Messenger, Prophet, Ambassador
Girl/Female
Gujarati, Hindu, Indian, Tamil
Friendly
Boy/Male
Arabic, Muslim
Servant of the Most High
Boy/Male
Indian, Tamil
Like a Pearl
Boy/Male
Sikh
Friendly, Friendship, Sweet
Boy/Male
Gujarati, Indian, Jain
King of Water
MUTUAL RECURSION
MUTUAL RECURSION
Boy/Male
Hindu, Indian, Marathi
The Moon
Girl/Female
Hebrew
Cedar tree.
Surname or Lastname
English
English : variant of Hoskin.
Girl/Female
Muslim
Boy/Male
Indian
Good deeds, Kind acts
Girl/Female
Hindu, Indian, Tamil
Shine of Gloriness
Boy/Male
Hebrew American Spanish
God is with us; god is among us.
Girl/Female
Hindu, Indian
Indian Goddess Name
Girl/Female
Biblical
Muddy, eggs, fine linen or silk.
Girl/Female
Australian, Irish
Good Health
MUTUAL RECURSION
MUTUAL RECURSION
MUTUAL RECURSION
MUTUAL RECURSION
MUTUAL RECURSION
adv.
In a mutual manner.
a.
Of or pertaining to rites or ritual; as, ritual service or sacrifices; the ritual law.
a.
A prescribed exercise in the systematic handing of a weapon; as, the manual of arms; the manual of the sword; the manual of the piece (cannon, mortar, etc.).
a.
Resembling a wall; perpendicular or steep; as, a mural precipice.
a.
Involving, or accompanied by, mutual slaughter; mutually destructive.
a.
Reciprocally acting or related; reciprocally receiving and giving; reciprocally given and received; reciprocal; interchanged; as, a mutual love, advantage, assistance, aversion, etc.
a.
Of or pertaining to the hand; done or made by the hand; as, manual labor; the king's sign manual.
v. i.
To communicate mutually; to hold mutual communication.
a.
Possessed, experienced, or done by two or more persons or things at the same time; common; joint; as, mutual happiness; a mutual effort.
a.
In action at the time being; now exiting; present; as the actual situation of the country.
n.
Mutial accommodation; mutual giving.
a.
Pertaining to a miter; resembling a miter; as, the mitral valve between the left auricle and left ventricle of the heart.
n.
Hence, the code of ceremonies observed by an organization; as, the ritual of the freemasons.
n.
A prescribed form of performing divine service in a particular church or communion; as, the Jewish ritual.
n.
Ore from which a metal is derived; -- so called by miners.
v. i.
To have reciprocal or mutual relations; to be mutually related.
a.
Existing in act or reality; really acted or acting; in fact; real; -- opposed to potential, possible, virtual, speculative, conceivable, theoretical, or nominal; as, the actual cost of goods; the actual case under discussion.
v. t.
To cover with metal; as, to metal a ship's bottom; to metal a road.
a.
Mutual.
a.
Taking place at a suture; as, a sutural de/iscence.