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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
recursive functions. However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function.
General_recursive_function
Formalization of the natural numbers
Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician Skolem
Primitive recursive arithmetic
Primitive_recursive_arithmetic
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
Topics referred to by the same term
function, a computable partial function from natural numbers to natural numbers Primitive recursive function, a function which can be computed with loops
Recursive_function
In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets
Primitive recursive set function
Primitive_recursive_set_function
Concept in computability theory
property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions. Suppose that R(y, x1, ..., xk) is
Mu_operator
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
Mathematical function that can be computed by a program
functions. Another example is the Ackermann function, which is recursively defined but not primitive recursive. For definitions of this type to avoid circularity
Computable_function
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
Programming language
simple register language designed to precisely capture the primitive recursive functions. The language is derived from the counter-machine model. Like
LOOP_(programming_language)
In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist
Primitive recursive functional
Primitive_recursive_functional
Subroutine call performed as final action of a procedure
dictionary. Course-of-values recursion Recursion (computer science) Primitive recursive function Inline expansion Leaf subroutine Corecursion Like this: if (ls)
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
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)
any recursively enumerable set of well-formed formulas of a first-order language is recursively axiomatizable, and even primitively recursively axiomatizable
Craig's_theorem
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
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
arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The β function was introduced
Gödel's_β_function
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
Hierarchy of complexity classes for formulas defining sets
that allow the use of primitive recursive functions, as now the quantifiers may be bounded by any primitive recursive function of the arguments. The Σ
Arithmetical_hierarchy
Abstract model of computation
indirection – and thereby compute the sub-class of primitive recursive functions – by using a primitive recursive "operator" called "definition by cases" (defined
Random-access_machine
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
Logical connective AND
{\displaystyle A\land B} : In systems where logical conjunction is not a primitive, it may be defined as A ∧ B = ¬ ( A → ¬ B ) {\displaystyle A\land B=\neg
Logical_conjunction
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
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
Kleene's_T_predicate
Mathematical logic concept
function can be chosen to be injective. The set S is the range of a primitive recursive function or empty. Even if S is infinite, repetition of values may be
Computably_enumerable_set
Computational problem with high complexity
example, O ( 2 2 n ) {\displaystyle O(2^{2^{n}})} ). Not all primitive recursive functions are elementary; for example, tetration grows too rapidly to
Nonelementary_problem
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
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
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
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
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
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
System of arithmetic in proof theory
reverse mathematics (Simpson 2009). Elementary recursive arithmetic (ERA) is a subsystem of primitive recursive arithmetic (PRA) in which recursion is restricted
Elementary function arithmetic
Elementary_function_arithmetic
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
Subset of a function's codomain
a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are
Range_of_a_function
Study of computable functions and Turing degrees
for example the μ-recursive functions obtained from primitive recursion and the μ operator. The terminology for computable functions and sets is not completely
Computability_theory
Branch of mathematical combinatorics
must be extraordinarily large, sometimes even greater than any primitive recursive function; see the Paris–Harrington theorem for an example. Graham's number
Ramsey_theory
Ordinal-indexed family of rapidly increasing functions
hierarchy, every primitive recursive function is dominated by some fα with α < ω. Hence, in the Wainer hierarchy, every primitive recursive function is dominated
Fast-growing_hierarchy
Computational learning model
text presentation, where the string given by the teacher is a primitive recursive function of the current step number, and the learner encodes a language
Language identification in the limit
Language_identification_in_the_limit
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
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
1931 paper by Kurt Gödel
enough to define the primitive recursive functions. (The contemporary terminology for recursive functions and primitive recursive functions had not yet been
On Formally Undecidable Propositions of Principia Mathematica and Related Systems
On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems
Book by Stephen Wolfram
several varieties of substitution and network systems; recursive functions; nested recursive functions; combinators; tag systems; register machines; and reversal-addition
A_New_Kind_of_Science
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
Control flow construct for executing code repeatedly
until a program terminates, such as web servers. Primitive recursive function General recursive function Repeat loop (disambiguation) LOOP (programming
Loop_(statement)
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
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
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)
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
On transforming a program by substituting constants for free variables
{\displaystyle \varphi } of partial computable functions, there is a primitive recursive function s {\displaystyle s} of two arguments with the following property:
Smn_theorem
Simple programming languages
can express all computable functions. For example, it can express the Ackermann function, which (not being primitive recursive) cannot be written in BlooP
BlooP_and_FlooP
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)
Limitative results in mathematical logic
number has a particular property, where that property is given by a primitive recursive relation (Smith 2007, p. 141). As such, the Gödel sentence can be
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
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
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 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
differentiating. Prime number Infinitude of the prime numbers Primitive recursive function Principle of bivalence no propositions are neither true nor false
List_of_mathematical_proofs
primitive recursive functions and μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions
Integer-valued_function
Theory of truth in the philosophy of language
of meaning for natural languages, which involves treating "truth" as a primitive, rather than a defined, concept. (See truth-conditional semantics.) Tarski
Semantic_theory_of_truth
Whether a decision problem has an effective method to derive the answer
Entscheidungsproblem Existential quantification Boris Trakhtenbrot (1953). "On recursive separability". Doklady Akademii Nauk SSSR (in Russian). 88: 935–956. Monk
Decidability_(logic)
Attempts to formalize the concept of algorithms
(1) the recursive functions calculated by a person with paper and pencil, and (2) the Turing machine or its Turing equivalents—the primitive register-machine
Algorithm_characterizations
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
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
Mathematical function characterizing set membership
offers up the same definition in the context of the primitive recursive functions as a function φ of a predicate P takes on values 0 if the predicate
Indicator_function
Set of all true first-order statements about the arithmetic of natural numbers
{R}}} ) of the recursively enumerable Turing degrees, in the signature of partial orders. In particular, there are computable functions S and T such that:
True_arithmetic
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
Automated_theorem_proving
Ordered listing of items in collection
enumerated set must be computable. The set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary language),
Enumeration
Mathematical proof expressed visually
systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of
Proof_without_words
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
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
Subfield of mathematics
logics that allow inductive definitions, like one writes for primitive recursive functions. One can formally define an extension of first-order logic —
Mathematical_logic
Possible axiom for set theory in mathematics
The existence of a primitive recursive class surjection F : O r d → V {\displaystyle F:\mathrm {Ord} \to V} , i.e., a class function from O r d {\displaystyle
Axiom_of_constructibility
Problem in computer science
halting problem is decidable for a lossy Turing machine but non-primitive recursive. A machine with an oracle for the halting problem can determine whether
Halting_problem
Class of formal logics
a special case. It explains the quantifiers in terms of mathematical functions. It was also the first logic capable of dealing with the problem of multiple
Classical_logic
Set of the elements not in a given subset
systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of
Complement_(set_theory)
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
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
Base set of symbols with which a language is formed
systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of
Alphabet_(formal_languages)
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
Ability of a computing system to simulate Turing machines
Kronecker formulated notions of computability, defining primitive recursive functions. These functions can be calculated by rote computation, but they are
Turing_completeness
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
3-volume treatise on mathematics, 1910–1913
the notion of "matrix" or "predicative function" (a "primitive idea", PM 1962:164) and presents four new Primitive propositions as ✱8.1–✱8.13. ✱88. Multiplicative
Principia_Mathematica
recursive function theory, double recursion is an extension of primitive recursion which allows the definition of non-primitive recursive functions like
Double_recursion
Logical operation
generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa). In intuitionistic logic,
Negation
Any one of the distinct objects that make up a set in set theory
systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of
Element_of_a_set
computable function that is not primitive recursive. Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but a distribution
List of mathematical functions
List_of_mathematical_functions
Concept in mathematical logic
gates, or only binary NOR gates. Modern texts on logic typically take as primitive some subset of the connectives: conjunction ( ∧ {\displaystyle \land }
Functional_completeness
Type of logical system
first-order logic with a primitive ordered pair is equivalent to a relation algebra with two ordered pair projection functions. A first-order theory of
First-order_logic
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
Function, homomorphism, or morphism
In mathematics, a map or mapping is a function in its general sense.[vague] These terms may have originated as from the process of making a geographical
Map_(mathematics)
Undecidability of equality of real numbers
composition, and the sin, exp, and abs functions. For some classes of expressions generated by other primitives than in Richardson's theorem, there exist
Richardson's_theorem
Topics referred to by the same term
that succeed one another in chronological order Successor function, a primitive recursive function in mathematics used to define addition Simultaneity succession
Succession
Type of binary relation
graph of the successor function x ↦ x+1. Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion. If we consider
Well-founded_relation
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
Index of articles associated with the same name
readily to languages with more predicates and with term constructions. Each primitive predicate needs to have specified required displacements between values
Stratification_(mathematics)
of 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)
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
Description of non-logical symbols
} which assigns a natural number called arity to every function or relation symbol. A function or relation symbol is called n {\displaystyle n} -ary if
Signature_(logic)
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
Male
Egyptian
, a high Egyptian functionary.
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.
Girl/Female
American, Australian, Chinese, Finnish, French, Latin, Portuguese, Swedish
Ancient; Primitive; Venerable
Male
Egyptian
, the son of the functionary Heknofre.
Male
Celtic
, great justiciary, or functionary.
Girl/Female
Danish, Finnish, French, German, Latin, Swedish
Ancient; Primitive; Venerable
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
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.
Male
Egyptian
, an Egyptian functionary.
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.
Girl/Female
American, Australian, Biblical, British, Chinese, Christian, Danish, English, Finnish, French, German, Gothic, Italian, Latin, Portuguese, Swedish
Ancient; Primitive; Venerable
Biblical
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Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : probably for the most part a topographic name for someone who lived near the trunk or stump of a large tree, Middle English stocke (Old English stocc). In some cases the reference may be to a primitive foot-bridge over a stream consisting of a felled tree trunk. Some early examples without prepositions may point to a nickname for a stout, stocky man or a metonymic occupational name for a keeper of punishment stocks.German : from Middle German stoc ‘tree’, ‘tree stump’, hence a topographic name equivalent to 1, but sometimes also a nickname for an impolite or obstinate person.Jewish (Ashkenazic) : ornamental name from German Stock ‘stick’, ‘pole’.
Boy/Male
Arabic, Hindu, Indian, Muslim, Sindhi
Ancient; Antique; Old; Primitive; Without Any Beginning or End
Girl/Female
German, Latin
Archaic; Ancient; Old; Primitive
Male
Egyptian
, Functionary of the Interior.
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
, a great functionary.
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
Girl/Female
Muslim
Attractive
Girl/Female
Arabic, Muslim
Elated; Exultant; Flushed; Enraptured; Feminine of Nashwan
Girl/Female
Indian
Girl/Female
Hindu
Prasitham
Surname or Lastname
English
English : patronymic from Leather.
Boy/Male
Arabic, Muslim
Solid
Boy/Male
Tamil
Shivendu | ஷிவேஂதà¯Â
Pure Moon
Boy/Male
Arabic
Handsome
Girl/Female
Italian
name Cynthia - one of the names of the mythological mood goddess Artemis referring to her birth...
Boy/Male
Indian, Sanskrit
Flowing out
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
n.
A privative prefix or suffix. See Privative, a., 3.
a.
Serving, or able, to repulse; repellent; as, a repulsive force.
adv.
In a decursive manner.
pl.
of Primitia
n.
The primitive perivisceral cavity.
a.
Original; primary; radical; not derived; as, primitive verb in grammar.
n.
A character used in cursive writing.
a.
Prone to make excursions; wandering; roving; exploring; as, an excursive fancy.
a.
Being of the first production; primitive; original.
a.
Involving a limit; as, a limitive law, one designed to limit existing powers.
a.
Cold; forbidding; offensive; as, repulsive manners.
n.
That which causes revulsion; specifically (Med.), a revulsive remedy or agent.
n.
A revulsive medicine.
n.
A term indicating the absence of any quality which might be naturally or rationally expected; -- called also privative term.
a.
Implying privation or negation; giving a negative force to a word; as, alpha privative; privative particles; -- applied to such prefixes and suffixes as a- (Gr. /), un-, non-, -less.
a.
Pristine; primitive.
pl.
of Primitia
a.
Of or pertaining to the beginning or origin, or to early times; original; primordial; primeval; first; as, primitive innocence; the primitive church.
a.
Of or pertaining to a former time; old-fashioned; characterized by simplicity; as, a primitive style of dress.
a.
Primitive; primary; original.