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One of several equivalent definitions of a computable function
and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to
General_recursive_function
Function computable with bounded loops
Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions
Primitive_recursive_function
Topics referred to by the same term
Recursive function may refer to: Recursive function (programming), a function which references itself General recursive function, a computable partial
Recursive_function
Quickly growing function
recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions
Ackermann_function
Use of functions that call themselves
smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach
Recursion_(computer_science)
Association of one output to each input
the Riemann hypothesis. In computability theory, a general recursive function is a partial function from the integers to the integers whose values can
Function_(mathematics)
Function whose actual domain of definition may be smaller than its apparent domain
function is generally simply called a function. In computability theory, a general recursive function is a partial function from the integers to the integers;
Partial_function
Mathematical function that can be computed by a program
and general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and
Computable_function
Process of repeating items in a self-similar way
and recursive rule, one can generate the set of all natural numbers. Other recursively defined mathematical objects include factorials, functions (e.g
Recursion
Thesis on the nature of computability
Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments) that is closed
Church–Turing_thesis
Subroutine call performed as final action of a procedure
The special case of tail-recursive calls, when a function calls itself, may be more amenable to call elimination than general tail calls. When the language
Tail_call
Concept in computability theory
Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions. Suppose that R(y, x1, ..., xk) is a fixed
Mu_operator
Total order in computer science
There may also be systems for certain general recursive functions, for example a system for the Ackermann function may contain the rule A(a+, b+) → A(a
Path ordering (term rewriting)
Path_ordering_(term_rewriting)
Defining elements of a set in terms of other elements in the set
the general recursive definition will be given below. Let A be a set and let a0 be an element of A. If ρ is a function which assigns to each function f
Recursive_definition
Family of higher-order functions
higher-order function that analyzes a recursive data structure and, through use of a given combining operation, recombines the results of recursively processing
Fold_(higher-order_function)
Process where AI repeatedly improves itself
Recursive self-improvement (RSI) is a process in which an artificial intelligence (AI) system iteratively designs a more capable successor. This may trigger
Recursive_self-improvement
Programming language
simple register language designed to precisely capture the primitive recursive functions. The language is derived from the counter-machine model. Like the
LOOP_(programming_language)
Control flow construct for executing code repeatedly
program terminates, such as web servers. Primitive recursive function General recursive function Repeat loop (disambiguation) LOOP (programming language)
Loop_(statement)
Mathematical logic concept
the function which enumerates any maximal recursively enumerable set dominates every general recursive function. There exists maximal recursively enumerable
Computably_enumerable_set
Formalization of the natural numbers
arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation
Primitive recursive arithmetic
Primitive_recursive_arithmetic
Top-down parser utilizing recursion
computer science, a recursive descent parser is a kind of top-down parser built from a set of mutually recursive procedures (or a non-recursive equivalent) where
Recursive_descent_parser
Mathematical function having a characteristic S-shaped curve or sigmoid curve
Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward loops M29:
Sigmoid_function
Problem-solving procedures with certain characteristics
effective calculability led to a variety of proposed definitions (general recursive functions, Turing machines, λ-calculus) that later were shown to be equivalent
Effective_method
Problem in computer science
effectively calculable function can be formalized by the general recursive functions or equivalently by the lambda-definable functions. He proves that the
Halting_problem
system trees, or threaded comments. They are special cases of more general recursive fixed-point queries, which compute transitive closures. In standard
Hierarchical and recursive queries in SQL
Hierarchical_and_recursive_queries_in_SQL
Process for estimating a probability density function
probabilistic approach for estimating an unknown probability density function (PDF) recursively over time using incoming measurements and a mathematical process
Recursive_Bayesian_estimation
Set with algorithmic membership test
Introduction to Recursive Function Theory. Cambridge University Press. ISBN 0-521-22384-9. Rogers, Hartley, Jr (1967). Theory of Recursive Functions and Effective
Computable_set
Sequence of operations for a task
and a discussion of, his proof. Kleene, Stephen C. (1936). "General Recursive Functions of Natural Numbers". Mathematische Annalen. 112 (5): 727–742
Algorithm
Two functions defined from each other
tail call optimization in general (when the function called is not the same as the original function, as in tail-recursive calls) may be more difficult
Mutual_recursion
Mathematical model describing how an output of a function is computed given an input
models include: Abstract rewriting systems Combinatory logic General recursive functions Lambda calculus Concurrent models include: Actor model Cellular
Model_of_computation
Mathematical-logic system
M; this means a recursive function definition cannot be written with let. The letrec construction would allow writing recursive function definitions, where
Lambda_calculus
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
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
System of arithmetic in proof theory
defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure under
Elementary function arithmetic
Elementary_function_arithmetic
Linear recurrence equation
mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations
P-recursive_equation
Technique for defining number-theoretic functions by recursion
computation of a value of a function requires only the previous value; for example, for a 1-ary primitive recursive function g the value of g(n+1) is computed
Course-of-values_recursion
Arithmetic operation
^{2}} ) is not an elementary recursive function. One can prove by induction that for every elementary recursive function f, there is a constant c such
Tetration
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Types of special mathematical functions
incomplete gamma function since Tricomi". Atti Convegni Lincei. 147: 203–237. MR 1737497. Gautschi, Walter (1999). "A Note on the recursive calculation of
Incomplete_gamma_function
American mathematician (1909–1994)
(1936). "General recursive functions of natural numbers". Mathematische Annalen (112): 727–742. —— (1936a). "λ-definability and recursiveness". Duke Mathematical
Stephen_Cole_Kleene
Ability of a computing system to simulate Turing machines
incompleteness theorem. This work, along with Gödel's work on general recursive functions, established that there are sets of simple instructions, which
Turing_completeness
Topics referred to by the same term
science fiction novel by Blake Crouch Recursive science fiction, science fiction about science fiction Recursive function (disambiguation) This disambiguation
Recursion_(disambiguation)
Sequence of program instructions invokable by other software
defined by mathematical induction and recursive divide and conquer algorithms. Here is an example of a recursive function in C to find Fibonacci numbers: int
Function (computer programming)
Function_(computer_programming)
Software programming optimization technique
recursive calls will be made (7 and 6), and the value for 5! will have been stored from the previous call. In this way, memoization allows a function
Memoization
Attempts to formalize the concept of algorithms
schemes—both in formal mathematics and in routine life—are: (1) the recursive functions calculated by a person with paper and pencil, and (2) the Turing
Algorithm_characterizations
Concept in theoretical computer science
Retrieved 7 July 2022. Green recursively constructs machines for any number of states and provides the recursive function that computes their score (computes
Busy_beaver
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
Type of error-correcting code using convolution
with a transfer function through Z-transform. Transfer functions for the first (non-recursive) encoder are: H 1 ( z ) = 1 + z − 1 + z − 2 , {\displaystyle
Convolutional_code
Formal power series
properties that a sequence be P-recursive and have a holonomic generating function are equivalent. Holonomic functions are closed under the Hadamard product
Generating_function
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Study of computable functions and Turing degrees
μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable
Computability_theory
defines "general recursive" functions and "partial recursive functions" in his paper Recursive Predicates and Quantifiers. The representing function, mu-operator
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
Computer programming function
tail-recursive, so it may build up a lot of frames on the stack when called with a large list. Many languages alternately provide a "reverse map" function
Map_(higher-order_function)
Higher-order function which returns some fixed point of the input function
recursive definitions. Applied to a non-constant function of one variable that treats its argument as a piece of data (such as e.g. the sine function)
Fixed-point_combinator
Type of neural network which utilizes recursion
A recursive neural network is a kind of deep neural network created by applying the same set of weights recursively over a structured input, to produce
Recursive_neural_network
Infinite sequence of numbers satisfying a linear equation
recursive functions; and in the theory of formal languages, where they count strings up to a given length in a regular language. Constant-recursive sequences
Constant-recursive_sequence
Mathematical function characterizing set membership
representing function ϕ(x1,…xn)=0 if R(x1,…xn) and ϕ(x1,…xn)=1 if ¬R(x1,…xn). Kleene offers up the same definition in the context of the primitive recursive functions
Indicator_function
Computer science and recursion theory
of recursive functions by use of the IF-THEN-ELSE construction common to computer science, together with four of the operators of primitive recursive functions:
McCarthy_Formalism
Yes/no problem in computer science
ISBN 978-1-4612-1844-9. Hartley, Rogers Jr (1987). The Theory of Recursive Functions and Effective Computability. MIT Press. ISBN 978-0-262-68052-3. Sipser
Decision_problem
On transforming a program by substituting constants for free variables
recursion theorem Partial evaluation Kleene, S. C. (1936). "General recursive functions of natural numbers". Mathematische Annalen. 112 (1): 727–742
Smn_theorem
Algorithms which recursively solve subproblems
designates an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related
Divide-and-conquer_algorithm
Operation on mathematical functions
In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function
Function_composition
Function, homomorphism, or morphism
more general functions. In category theory, "map" is often used as a synonym for "morphism" or "arrow", which is a structure-respecting function and thus
Map_(mathematics)
Mathematical functions
restricted to the recursive subset{x∈OT|x<D0Dω0} in the same sense as above is not provable under Π11−CA0. The normal form for Buchholz's function can be defined
Buchholz_psi_functions
Hypothetical computing devices
instructions Minsky offered a proof that with such numbering the general recursive functions were indeed possible; he offers proof that μ recursion is indeed
Turing_machine_equivalents
Type of hypothetical AI with wide-ranging abilities
architectures can programmers implement to maximise the probability that their recursively-improving AI would continue to behave in a friendly, rather than destructive
Artificial general intelligence
Artificial_general_intelligence
Computation model defining an abstract machine
text; most of Chapter XIII "Computable functions" is on Turing machine proofs of computability of recursive functions, etc. Knuth, Donald E. (1973). The Art
Turing_machine
Symbol representing a mathematical object
of functions. In printed mathematics, the norm is to set variables and constants in an italic typeface. For example, a general quadratic function is conventionally
Variable_(mathematics)
Type of Gödel numbering in mathematics
mathematics. It is a specific case of the more general idea of Gödel numbering. For example, recursive function theory can be regarded as a formalization of
Gödel_numbering_for_sequences
Mathematician and philosopher (1906–1978)
ideas of computability and recursive functions to the point that he was able to present a lecture on general recursive functions and the concept of truth
Kurt_Gödel
Condition for a mathematical function to map some value to itself
meaning is the same: a recursive function can be described as the least fixed point of a certain functional, mapping functions to functions. The above technique
Fixed-point_theorem
Axiom of set theory
"let F(s) be one of the members of s for all s in X" to define a function F. In general, it is impossible to prove that F exists without the axiom of choice
Axiom_of_choice
Branch of mathematical logic
initials "RCA" stand for "recursive comprehension axiom", where "recursive" means "computable", as in computable function. This name is used because
Reverse_mathematics
Programming function applied recursively to its previous result
function that generates a sequence by repeated application of the function to its previous result. You begin with some value A and apply a function f
Anamorphism
what is called the mu operator (see also mu recursive functions) (p. 213)): Any general recursive function can be computed by a program computer using
Counter-machine_model
Standard system of axiomatic set theory
membership symbol ∈ {\displaystyle \in } Brackets (, ) With this alphabet, the recursive rules for forming well-formed formulae (wff) are as follows: Let x {\displaystyle
Zermelo–Fraenkel_set_theory
computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand
List of pioneers in computer science
List_of_pioneers_in_computer_science
Pattern defining an infinite sequence of numbers
recurrence relation means obtaining a closed-form solution: a non-recursive function of n {\displaystyle n} . The concept of a recurrence relation can
Recurrence_relation
Subset of a function's codomain
the range of a function may refer to either of two closely related concepts: the codomain of the function, or the image of the function. In some cases
Range_of_a_function
Fundamental theorem in mathematical logic
) However, the definition expressed by this formula is not recursive (but is, in general, Δ2). An important consequence of the completeness theorem is
Gödel's_completeness_theorem
Mathematical logic concept
but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation
Gentzen's_consistency_proof
open-source parallel library for sparse matrix computations using the Recursive Sparse Blocks (RSB) matrix format. librsb provides cache efficient multi-threaded
Librsb
Function uniquely mapping two numbers into a single number
{\displaystyle \mathbb {N} } . The Cantor pairing function is a primitive recursive pairing function π : N × N → N {\displaystyle \pi :\mathbb {N} \times
Pairing_function
Data type defined by combining other types
is recursive. Operations on algebraic data types can be defined by using pattern matching to retrieve the arguments. For example, consider a function to
Algebraic_data_type
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
System to identify resources on a network
this function implemented in the name server, user applications gain efficiency in design and operation. The combination of DNS caching and recursive functions
Domain_Name_System
Infinite cardinal number
same cardinality of S {\displaystyle S} in general, but all its elements do.) Beth number Gimel function Regular cardinal Infinity Transfinite number
Aleph_number
Axioms for the natural numbers
Peano axioms. Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as: a + 0 = a , (1) a + S
Peano_axioms
Axiomatic set theories based on the principles of mathematical constructivism
insights about totality of functions. In computability theory, the μ operator enables all partial general recursive functions (or programs, in the sense
Constructive_set_theory
Symbolic description of a mathematical object
(∪), or intersection (∩). (Functions can be understood as unary operations) Brackets ( ) With this alphabet, the recursive rules for forming a well-formed
Expression_(mathematics)
Sequence of words formed by specific rules
the basis for a 1947 proof "that the word problem for semigroups was recursively insoluble", and later devised the canonical system for the creation of
Formal_language
Function returning one of only two values
switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the
Boolean_function
Non-contradiction of a theory
strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable
Consistency
Target set of a mathematical function
mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in
Codomain
Result of repeatedly applying a mathematical function
derivative of an iterated function is given by the recursive formula: δfN(x)δf(y)=f′(fN−1(x))δfN−1(x)δf(y)+δ(fN−1(x)−y) Iterated functions crop up in the series
Iterated_function
Input to a mathematical function
of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x
Argument_of_a_function
added construct to make recursive anonymous functions. The symbol '#0' refers to the entire function. The following function calculates the factorial
Examples of anonymous functions
Examples_of_anonymous_functions
Mathematical model of analog computers
383–423. 2008. D. S. Graça and J. F. Costa. Analog computers and recursive functions over the reals. Journal of Complexity, 19(5):644–664, 2003 O. Bournez
General purpose analog computer
General_purpose_analog_computer
Function with unusual fractal properties
as can be seen by a recursive definition closely related to the Stern–Brocot tree. One way to define the question-mark function involves the correspondence
Minkowski's question-mark function
Minkowski's_question-mark_function
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GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
GENERAL RECURSIVE-FUNCTION
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