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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
mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals
Primitive recursive set function
Primitive_recursive_set_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
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
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
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
Association of one output to each input
a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y
Function_(mathematics)
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
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
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable, or recursive) if there is an algorithm that computes the membership of every
Computable_set
Mathematical logic concept
the 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
Computably_enumerable_set
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
Programming language
the primitive recursive functions. The language is derived from the counter-machine model. Like the Counter machines the LOOP language comprises a set of
LOOP_(programming_language)
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)
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
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
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
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
any recursively enumerable set of well-formed formulas of a first-order language is recursively axiomatizable, and even primitively recursively axiomatizable
Craig's_theorem
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
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 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
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
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
Study of computable functions and Turing degrees
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
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
Concept in theoretical computer science
Heiner Marxen and Jürgen Buntrock described it as "a non-trivial (not primitive recursive) lower bound". This lower bound can be calculated but is too complex
Busy_beaver
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
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
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
Standard system of axiomatic set theory
Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that
Zermelo–Fraenkel_set_theory
Axiomatic set theories based on the principles of mathematical constructivism
existence of any primitive recursive function in x × ω → y {\displaystyle x\times \omega \to y} , and in particular in the uncountable function spaces out of
Constructive_set_theory
Branch of mathematics that studies sets
homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher
Set_theory
primitive recursive functions and μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions
Integer-valued_function
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)
Paradox in set theory
the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function: There is just one point where I have encountered
Russell's_paradox
Two functions defined from each other
mutually recursive functions are primitive recursive, which can be proven by course-of-values recursion, building a single function F that lists the values
Mutual_recursion
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
Abstract machine used in a formal logic and theoretical computer science
demonstrations of how to form the five primitive recursive function "operators" (1-5 below) from the base set (1). But what about full Turing equivalence
Counter_machine
Principle in set theory
adjunction operation is also used as one of the operations of primitive recursive set functions. Tarski and Szmielew showed that Robinson arithmetic ( Q {\displaystyle
Axiom_of_adjunction
Set of the elements not in a given subset
In set theory, the complement of a set A, often denoted by Ac (or A′), is the set of elements not in A. When all elements in the universe, i.e. all elements
Complement_(set_theory)
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
Hierarchy of complexity classes for formulas defining sets
defined by a single primitive recursive function. Just as we can define what it means for a set X to be recursive relative to another set Y by allowing the
Arithmetical_hierarchy
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)
Mathematical set containing no elements
the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories
Empty_set
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 elements common to all of some sets
In set theory, the intersection of two sets A and B, denoted by A∩B, is the set containing all elements of A that also belong to B or equivalently, all
Intersection_(set_theory)
Concept in mathematical logic
a set F of Boolean functions fi : Bni → B is functionally complete if the clone on B generated by the basic functions fi contains all functions f :
Functional_completeness
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
Size of a set in mathematics
those points blue again. By recursively alternating the images of these sets, one obtains a "coloring" of the original set (see illustration). For each
Cardinality
Informal set theories
most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving
Naive_set_theory
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
Collection of sets in mathematics that can be defined based on a property of its members
also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula Φ ( x , y ) {\displaystyle
Class_(set_theory)
Axiom of set theory
Axiom—For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an element of that set. Formally, this
Axiom_of_choice
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
Mathematical functions
well-foundedness of < restricted to the recursive subset {x∈OT|x<D0Dv+10} in the sense of the non-existence of a primitive recursive infinite descending sequence
Buchholz_psi_functions
Fractal named after mathematician Benoit Mandelbrot
recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a fractal curve. The "style" of this recursive detail
Mandelbrot_set
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations
Union_(set_theory)
Synchronization primitive that can be locked multiple times by the same thread
science, the reentrant mutex (also known as a recursive mutex or recursive lock) is a synchronization primitive that may be locked multiple times by the same
Reentrant_mutex
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
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
Types of special mathematical functions
incomplete gamma function since Tricomi". Atti Convegni Lincei. 147: 203–237. MR 1737497. Gautschi, Walter (1999). "A Note on the recursive calculation of
Incomplete_gamma_function
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
Computation model defining an abstract machine
of valid strings of an alphabet. A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine
Turing_machine
Function uniquely mapping two numbers into a single number
{\displaystyle \mathbb {N} } . The Cantor pairing function is a primitive recursive pairing function π : N × N → N {\displaystyle \pi :\mathbb {N} \times
Pairing_function
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
form of the NFU definition facilitates set manipulations while the form of the ZFC definition facilitates recursive definitions, but either theory supports
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Theorem about natural numbers
fact that there is no primitive recursive strictly decreasing infinite sequence of ordinals, so limiting bn to primitive recursive sequences would have
Goodstein's_theorem
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
then A is not recursive. Let f:X→Y be any function. If S is any set then its preimage C:=f−1(S) under f is necessarily an f-saturated set. In particular
Saturated_set
Turing machine that halts for any input
functions are guaranteed to halt. For example, the Ackermann function, which is not primitive recursive, is nevertheless a total computable function,
Decider_(Turing_machine)
is a hierarchy of sets of numerical functions generated from an ordinal-indexed family of functions hα: N → N (where N is the set of natural numbers
Hardy_hierarchy
Formal power series
properties that a sequence be P-recursive and have a holonomic generating function are equivalent. Holonomic functions are closed under the Hadamard product
Generating_function
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
Yes/no problem in computer science
is decidable or effectively solvable if the set of inputs for which the answer is YES is a recursive set. A decision problem is partially decidable, semidecidable
Decision_problem
Arithmetic operation
the nonzero elements of 𝔽q. There are φ(p−1) primitive elements in 𝔽q, where φ is Euler's totient function. In 𝔽q, the freshman's dream identity (x+y)p=xp+yp
Exponentiation
Size of a possibly infinite set
functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets.
Cardinal_number
Mathematical set containing all objects
it the singleton function is provably a set, which leads immediately to paradox in New Foundations. Another example is positive set theory, where the
Universal_set
Infinitely detailed mathematical structure
generating heightmaps for computer graphics Droste effect – Recursive visual effect Feigenbaum function – Concept in dynamical systems Form constant – Recurringly
Fractal
Set whose elements all belong to another set
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A
Subset
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
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
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
About mathematical functions
computing a function. Various models for algorithms appeared, in rapid succession, including Church's lambda calculus (1936), Stephen Kleene's μ-recursive functions(1936)
History of the function concept
History_of_the_function_concept
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
Abstract model of computation
demonstration, or a proof, etc. Moreover, from base sets 1, 2, or 3 we can create any of the primitive recursive functions ( cf Minsky (1967), Boolos-Burgess-Jeffrey
Random-access_machine
Symbol representing a mathematical object
constant. Variables are often used for representing matrices, functions, their arguments, sets and their elements, vectors, spaces, etc. In mathematical logic
Variable_(mathematics)
Kind of transfinite induction
membership, for any set, can be derived using some additional standard axioms. This needs the axiom of infinity for ω as a set, recursive functions on ω, the axiom
Epsilon-induction
Function returning one of only two values
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {−1,1})
Boolean_function
Mathematical technique used in proof theory
characterizations of the classes of provably recursive, hyperarithmetical, or Δ 2 1 {\displaystyle \Delta _{2}^{1}} functions of the theory. The field of ordinal
Ordinal_analysis
Mathematical set that can be enumerated
countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural
Countable_set
Identities and relationships involving sets
points in space, lines, other geometric shapes, variables, functions, or even other sets. A set is often denoted by an upper-case letter and curly braces
Algebra_of_sets
Yes-or-no question that cannot ever be solved by a computer
is 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
Proposition in mathematical logic
specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose
Continuum_hypothesis
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
Set theory concept
back into the definition of the rank of a set gives a self-contained recursive definition: The rank of a set is the smallest ordinal number strictly greater
Von_Neumann_universe
Limited form of tree data structure
k = 2. A recursive definition using set theory is that a binary tree is a triple (L, S, R), where L and R are binary trees or the empty set and S is a
Binary_tree
Subset of a function's codomain
codomain and the image of a function are the same set; such a function is called surjective or onto. For any non-surjective function f : X → Y , {\displaystyle
Range_of_a_function
Equalities for combinations of sets
refers to a set whose elements are sets. An indexed family of sets is a function from some set, called its indexing set, into some family of sets. An indexed
List of set identities and relations
List_of_set_identities_and_relations
Extension of recursion theory to admissible ordinals beyond the natural numbers
) are α {\displaystyle \alpha } -recursively-enumerable. It's of note that α {\displaystyle \alpha } -recursive sets are members of L α + 1 {\displaystyle
Alpha_recursion_theory
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PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
PRIMITIVE RECURSIVE-SET-FUNCTION
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