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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)
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
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
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
Mathematical-logic system based on functions
M; this means a recursive function definition cannot be written with let. The letrec construction would allow writing recursive function definitions, where
Lambda_calculus
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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)
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
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
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
elementary recursive function. Equivalently, these are the problems that can be solved in time bounded by an iterated exponential function with a bounded
ELEMENTARY
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
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
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
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
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
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
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
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
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
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
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
Logical connective AND
concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction (AND-ing over an empty set of
Logical_conjunction
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)
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
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
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
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
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
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
Higher-order function Y for which Y f = f (Y f)
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
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
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
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
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
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
Concept in artificial intelligence
Recursive self-improvement (RSI) is a process in which early artificial general intelligence (AGI) systems rewrite their own computer code, causing an
Recursive_self-improvement
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
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
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
Method of deriving conclusions
inherent in logical operators found in statements, making the meaning and function of these operators explicit without adding any additional information.
Rule_of_inference
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 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
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
Branch of mathematics that studies sets
0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types. Principles such as the axiom of
Set_theory
Subfield of automated reasoning and mathematical logic
required that each individual proof step can be verified by a primitive recursive function or program, and hence the problem is always decidable. Since the proofs
Automated_theorem_proving
Real number that can be computed within arbitrary precision
available at the time. Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms
Computable_number
Any one of the distinct objects that make up a set in set theory
Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image
Element_of_a_set
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
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
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
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
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)
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
arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The β function was introduced
Gödel's_β_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
Mathematical operation with two operands
arity two. More specifically, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples
Binary_operation
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
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
Subfield of mathematics
numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the
Mathematical_logic
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
Algebraic manipulation of "true" and "false"
complement function, the dual function and the contradual function (complemented dual). These four functions form a group under function composition
Boolean_algebra
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
Total order in computer science
bound given on the order types of recursive path orderings with n function symbols is φ(n,0), using Veblen's function for large countable ordinals. The
Path ordering (term rewriting)
Path_ordering_(term_rewriting)
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
for recursive function theory involving programs of only the simplest arithmetic operations". His "Theorem Ia" asserts that any partial recursive function
Counter-machine_model
recursive function theory, double recursion is an extension of primitive recursion which allows the definition of non-primitive recursive functions like
Double_recursion
Mathematical use of "there exists"
union of sets. A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The ¬ {\displaystyle \lnot
Existential_quantification
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
class of functions coincides with FP. These are functions which are defined, like the primitive recursive functions, by a set of base functions and operators
Implicit computational complexity
Implicit_computational_complexity
Collection of mathematical objects
symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what
Set_(mathematics)
Function in mathematical logic
use Gödel numbering to show how functions defined by course-of-values recursion are in fact primitive recursive functions. Once a Gödel numbering for a
Gödel_numbering
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
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
Collection of sets in mathematics that can be defined based on a property of its members
"classes". In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usual sense, since it is not
Class_(set_theory)
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
RECURSIVE FUNCTION
RECURSIVE FUNCTION
Male
Egyptian
, Functionary of the Interior.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Egyptian
, an Egyptian functionary.
Male
Celtic
, great justiciary, or functionary.
Biblical
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Male
Egyptian
, a great functionary.
Male
Egyptian
, the son of the functionary Heknofre.
Male
Egyptian
, a high Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, an Egyptian functionary.
RECURSIVE FUNCTION
RECURSIVE FUNCTION
Girl/Female
Greek
Unheeded prophetess. Cassandra was a Trojan prophetess, daughter of King Priam. In Homer's 'The...
Girl/Female
Indian
Fire
Boy/Male
Hebrew American
Son of Simon.
Girl/Female
Tamil
Boy/Male
Tamil
Bholenath | போலேநாத
Kind hearted Lord
Boy/Male
Tamil
Generous
Boy/Male
Indian, Sanskrit, Tamil
White; Chivalrous
Male
Finnish
Finnish form of Slavic Crnobog, TÅ ERNOBOG means "black god." In Slavic mythology, this is the name of a god of evil and darkness, the counterpart of Belobog ("white god").
Female
Irish
Pet form of Irish CaitrÃona, CAITIE means "pure."
Surname or Lastname
English
English : variant of Alcott.
RECURSIVE FUNCTION
RECURSIVE FUNCTION
RECURSIVE FUNCTION
RECURSIVE FUNCTION
RECURSIVE FUNCTION
a.
Repulsive by itself; as, the idiorepulsive power of heat.
a.
Causing, or tending to, revulsion.
a.
Repulsive; driving back.
a.
Affording retirement from society.
a.
Running down; decurrent.
a.
Manifesting distaste or dislike; repulsive.
n.
That which causes revulsion; specifically (Med.), a revulsive remedy or agent.
a.
Making an incursion; invasive; aggressive; hostile.
adv.
In a decursive manner.
a.
Prone to make excursions; wandering; roving; exploring; as, an excursive fancy.
n.
The act of recurring; return.
a.
Going back; receding.
a.
Flowing; easy; cursive; as, a running hand.
a.
Cold; forbidding; offensive; as, repulsive manners.
n.
A revulsive medicine.
a.
Serving, or able, to repulse; repellent; as, a repulsive force.
a.
Preceding; introductory; precursory.
n.
A character used in cursive writing.
v. t.
Causing revulsion; revulsive.
a.
Not amiable; morose; ill-natured; repulsive.