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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
One of several equivalent definitions of a computable function
computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural
General_recursive_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)
Subroutine call performed as final action of a procedure
different functions available to call. When dealing with recursive or mutually recursive functions where recursion happens through tail calls, however, the
Tail_call
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
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
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
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
Recursive function for formal verification case testing
The McCarthy 91 function is a recursive function, defined by the computer scientist John McCarthy as a test case for formal verification within computer
McCarthy_91_function
Mathematical function that can be computed by a program
general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for
Computable_function
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
Concept in computability theory
the class of elementary recursive functions ("Kalmár elementary functions") as a subset of the primitive recursive functions — specifically, those that
Elementary_recursive_function
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
elementary recursive function. Equivalently, these are the problems that can be solved in time bounded by an iterated exponential function with a bounded
ELEMENTARY
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
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
mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or
Primitive recursive set function
Primitive_recursive_set_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
Elementary operation on a natural number
{\displaystyle S(2)=3} . The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known
Successor_function
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)
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)
Thesis on the nature of computability
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
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
Two functions defined from each other
single recursive function by inlining the forest function in the tree function, which is commonly done in practice: directly recursive functions that operate
Mutual_recursion
Functions in computability theory
functions used in computability theory. Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function
Grzegorczyk_hierarchy
Association of one output to each input
recursive functions are partial functions from integers to integers that can be defined from constant functions, successor, and projection functions via
Function_(mathematics)
Academic subfield of computer science
μ-recursive functions a computation consists of a mu-recursive function, i.e. its defining sequence, any input value(s) and a sequence of recursive functions
Theory_of_computation
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
Theorem in computability theory
numbering φ {\displaystyle \varphi } of the partial recursive functions, such that the function corresponding to index e {\displaystyle e} is φ e {\displaystyle
Kleene's_recursion_theorem
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)
it is possible to achieve hierarchical queries with user-defined recursive functions. A common table expression, or CTE, (in SQL) is a temporary named
Hierarchical and recursive queries in SQL
Hierarchical_and_recursive_queries_in_SQL
Type of Gödel numbering in mathematics
concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions. It is usually used to build sequential "data
Gödel_numbering_for_sequences
Type of software bug
primitive recursive functions is equivalent to the class of LOOP computable functions. Consider this example in C++-like pseudocode: A primitive recursive function
Stack_overflow
Concept in artificial intelligence
Recursive self-improvement (RSI) is a hypothesized process in which artificial general intelligence (AGI) systems rewrite their own computer code, causing
Recursive_self-improvement
Set with algorithmic membership test
computable if and only if the indicator function 1 S {\displaystyle \mathbb {1} _{S}} is computable. Every recursive language is computable. Every finite
Computable_set
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
Recursive function
science, and in particular functional programming, a hylomorphism is a recursive function, corresponding to the composition of an anamorphism (which first builds
Hylomorphism (computer science)
Hylomorphism_(computer_science)
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
Well-quasi-ordering of finite trees
phenomenally fast as a function of n {\displaystyle n} , far faster than any primitive recursive function or the Ackermann function, for example.[citation
Kruskal's_tree_theorem
Sudan function is an example of a function that is recursive, but not primitive recursive. This is also true of the better-known Ackermann function. In
Sudan_function
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
construct such as a recursive function call, it is no longer capable of full μ-recursion, but only primitive recursion. Ackermann's function is the canonical
Loop_variant
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
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
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
Abstract model of computation
(indirect addressing) can compute all the "partial recursive sequential functions" (the mu recursive functions) (p. 397-398). Cook and Reckhow (1973) say it
Random-access_machine
recursive function theory, double recursion is an extension of primitive recursion which allows the definition of non-primitive recursive functions like
Double_recursion
Abstract machine used in a formal logic and theoretical computer science
address. Counter machines with three counters can compute any partial recursive function of a single variable. Counter machines with two counters are Turing
Counter_machine
Recursion without calling a function by name
functions. This is particularly important for the lambda calculus, which has anonymous unary functions, but is able to compute any recursive function
Anonymous_recursion
Hungarian mathematician
applied recursive function theory to computers. Her final book, published in 1976, was Rekursive Funktionen in der Komputer-Theorie (Recursive Functions in
Rózsa_Péter
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
One-to-one correspondence
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the
Bijection
Limit of a uniformly computable sequence of functions
computable in the limit, limit recursive and recursively approximable are also used. One can think of limit computable functions as those admitting an eventually
Computation_in_the_limit
Concept in computability theory
machine that computes the characteristic function of A when run with oracle B. In this case, we also say A is B-recursive and B-computable. If there is an oracle
Turing_reduction
Concept in computability theory
{\displaystyle T_{1}} predicate is primitive recursive in the sense that there is a primitive recursive function that, given inputs for the predicate, correctly
Kleene's_T_predicate
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
Ability to solve a problem by an effective procedure
studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent
Computability
Order type of the set of all recursive ordinals
non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal
Nonrecursive_ordinal
Problem in finite group theory
uniform, this is a recursive function of two variables. It follows that: h ( w ) = g ( w , a ) {\displaystyle h(w)=g(w,a)} is recursive. By construction:
Word_problem_for_groups
Number of arguments required by a function
science, arity (/ˈærɪti/ ) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank,
Arity
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
Recursive function
In computer science, the Tak function is a recursive function, named after Ikuo Takeuchi [ja]. It is defined as follows: τ ( x , y , z ) = { τ ( τ ( x
Tak_(function)
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
Named function defined within a function
enclosing functions) without passing parameters or using global variables. A nested function typically acts as a helper function or a recursive function. Nested
Nested_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
Computational problem with high complexity
algorithmic solution with time bounded by an elementary recursive function. These functions grow no faster than a fixed-height tower of exponentiation
Nonelementary_problem
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
Mathematical set of all subsets of a set
\left|2^{S}\right|=2^{n}=\sum _{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) = { {} }
Power_set
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
Operation on mathematical functions
multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function. Given f, a n-ary function, and
Function_composition
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
for recursive function theory involving programs of only the simplest arithmetic operations". His "Theorem Ia" asserts that any partial recursive function
Counter-machine_model
Defining elements of a set in terms of other elements in the set
an infinite regress. That recursive definitions are valid – meaning that a recursive definition identifies a unique function – is a theorem of set theory
Recursive_definition
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
2008 textbook
objects; recursive function calls; and more. At the end, the reader is left with an "interpreter" that uses nothing but tail-recursive function calls and
Essentials of Programming Languages
Essentials_of_Programming_Languages
Branch of mathematical logic
consequence of the interpretation one usually obtains the result that any recursive function whose totality can be proven either in I or in C is represented by
Proof_theory
all primitive recursive functions—or, equivalently, the set of all formal languages that can be decided in time bounded by such a function. This includes
PR_(complexity)
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
primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist of a collection of functions in all
Primitive recursive functional
Primitive_recursive_functional
arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The β function was introduced
Gödel's_β_function
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
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
Infinite cardinal number
defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"),
Aleph_number
Hierarchy of complexity classes for formulas defining sets
allow the use of primitive recursive functions, as now the quantifiers may be bounded by any primitive recursive function of the arguments. The Σ 0 0
Arithmetical_hierarchy
needed] and is a special case of a general formula for the exponential function: e x / y = 1 + 2 x 2 y − x + x 2 6 y + x 2 10 y + x 2 14 y + x 2 18 y +
List_of_representations_of_e
Paradox in set theory
the function F(fx) could be its own argument: in that case there would be a proposition F(F(fx)), in which the outer function F and the inner function F
Russell's_paradox
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
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
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
Principia_Mathematica
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)
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
Browser-based graphing calculator
restriction, simultaneous graphing, piecewise function graphing, recursive function graphing, polar function graphing, two types of graphing grids – among
Desmos
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
Axiom of set theory
a choice function. Even if infinitely many sets are collected from the natural numbers, it will always be possible to form a choice function from choosing
Axiom_of_choice
Diagram that shows all possible logical relations between a collection of sets
Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image
Venn_diagram
Limitative results in mathematical logic
axiomatized (also called effectively generated) if its set of theorems is recursively enumerable. This means that there is a computer program that, in principle
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
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
Mathematical set containing no elements
exists precisely one function f {\displaystyle f} from ∅ {\displaystyle \varnothing } to A , {\displaystyle A,} the empty function. As a result, the empty
Empty_set
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
delta function. The Kronecker delta forms the multiplicative identity element of an incidence algebra. The Kronecker delta is an elementary recursive function
Kronecker_delta
Yes-or-no question that cannot ever be solved by a computer
called decidable or effectively solvable if the formalized set of A is a recursive set. Otherwise, A is called undecidable. A problem is called partially
Undecidable_problem
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