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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
express arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation
Primitive recursive arithmetic
Primitive_recursive_arithmetic
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
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
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
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
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
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)
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
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
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
any recursively enumerable set of well-formed formulas of a first-order language is recursively axiomatizable, and even primitively recursively axiomatizable
Craig's_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
arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The β function was introduced
Gödel's_β_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
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
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
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
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
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)
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
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
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)
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
more natural style of expressing computation than simply using primitive recursive functions. Since the halting problem cannot be solved in general, there
Walther_recursion
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
Abstract machine used in a formal logic and theoretical computer science
and partial- recursive functions: Zero function (or constant function) Successor function Identity function Composition function Primitive recursion (induction)
Counter_machine
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
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
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
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
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
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)
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
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
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
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
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
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)
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
primitive recursive functions and μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions
Integer-valued_function
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
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
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
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
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
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
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
Type of abstract computing machine
RASP can compute any primitive recursive function (e.g. multiplication) but not all mu recursive functions (e.g. the Ackermann function). Elgot–Robinson investigate
Register_machine
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
function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is effectively uniformly continuous if there exists a primitive recursive function d
Computable_real_function
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
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
Symbol representing a mathematical concept
the interpretation of a function symbol must be defined over the whole domain of discourse. Function symbols are a primitive notion, and are therefore
Function_symbol
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
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
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
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
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
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
Diagram that shows all possible logical relations between a collection of sets
systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of
Venn_diagram
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
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
recursive function theory, double recursion is an extension of primitive recursion which allows the definition of non-primitive recursive functions like
Double_recursion
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
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
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
Being equally consistent
proven that the theories ZFC+A and ZFC+B are equiconsistent. Usually, primitive recursive arithmetic can be adopted as the metatheory in question, but even
Equiconsistency
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
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
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
Logical quantifier
systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of
Uniqueness_quantification
Academic subfield of computer science
equivalent to context-free grammars. Primitive recursive functions are a defined subclass of the recursive functions. Different models of computation have
Theory_of_computation
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 of the Theory of Arithmetic is
Axiom
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
Measure of algorithmic complexity
U:2^{*}\to 2^{*}} be a computable function mapping finite binary strings to binary strings. It is a universal function if, and only if, for any computable
Kolmogorov_complexity
Mathematical logic concept
{\displaystyle t\equiv c\mid x\mid f(t_{1},\dotsc ,t_{n})} , that is, a term is recursively defined to be a constant c (a named object from the domain of discourse)
Atomic_formula
Description of non-logical symbols
every function or relation symbol. A function or relation symbol is called n-ary if its arity is n. Some authors define a nullary (0-ary) function symbol
Signature_(logic)
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
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
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
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
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)
Summary of a mathematical proof
Importantly, all the formulas in the proof can be defined by primitive recursive functions, which themselves can be defined in first-order Peano arithmetic
Proof sketch for Gödel's first incompleteness theorem
Proof_sketch_for_Gödel's_first_incompleteness_theorem
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)
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
Collection of mathematical objects
geometric shapes, variables, functions, or even other sets. Sets cannot be mathematically defined, since they form a primitive notion. Instead, they are
Set_(mathematics)
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
Programming statement for branching control based on a value
formally proves that the case function (the if-then-else function being its simplest form) is a primitive recursive function, where he defines the notion
Switch_statement
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—a
Mathematical_logic
Type of logical argument that applies deductive reasoning
systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of
Syllogism
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)
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
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PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
PRIMITIVE RECURSIVE-FUNCTION
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