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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
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
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
Process of repeating items in a self-similar way
references can occur. A process that exhibits recursion is recursive. Video feedback displays recursive images, as does an infinity mirror. In mathematics and
Recursion
Use of functions that call themselves
Open recursion Sierpiński curve McCarthy 91 function μ-recursive functions Primitive recursive functions Tak (function) Logic programming Graham, Ronald;
Recursion_(computer_science)
Mathematician and philosopher (1906–1978)
Mathematical Platonism Original proof of Gödel's completeness theorem Primitive recursive functional Gödel–Löb logic Strange loop Tarski's undefinability theorem
Kurt_Gödel
a more natural style of expressing computation than simply using primitive recursive functions. Since the halting problem cannot be solved in general
Walther_recursion
Two functions defined from each other
common in functional programming and in some problem domains, such as recursive descent parsers, where the datatypes are naturally mutually recursive. The
Mutual_recursion
Concept in computability theory
defined the class of elementary recursive functions ("Kalmár elementary functions") as a subset of the primitive recursive functions — specifically, those
Elementary_recursive_function
Technique for defining number-theoretic functions by recursion
for a 1-ary primitive recursive function g the value of g(n+1) is computed only from g(n) and n. The factorial function n! is recursively defined by the
Course-of-values_recursion
Programming language
is a simple register language designed to precisely capture the primitive recursive functions. The language is derived from the counter-machine model
LOOP_(programming_language)
Concept in mathematical logic
Alfred Tarski's paper "On the Primitive Term of Logistic" proved that { ↔ } {\displaystyle \{\leftrightarrow \}} is functionally complete, but this only works
Functional_completeness
Subroutine call performed as final action of a procedure
called "properly tail recursive". Besides space and execution efficiency, tail-call elimination is important in the functional programming idiom known
Tail_call
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
Computably_enumerable_set
Control flow construct for executing code repeatedly
calculation until a program terminates, such as web servers. Primitive recursive function General recursive function Repeat loop (disambiguation) LOOP (programming
Loop_(statement)
Mathematical model describing how an output of a function is computed given an input
tree model External memory model Functional models include: Abstract rewriting systems Combinatory logic General recursive functions Lambda calculus Concurrent
Model_of_computation
Arithmetical concept
intuitionistic logic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel
Dialectica_interpretation
Mathematical function that can be computed by a program
these is the primitive recursive functions. Another example is the Ackermann function, which is recursively defined but not primitive recursive. For definitions
Computable_function
Integer side lengths of a right triangle
are the sides of this type of primitive Pythagorean triple then the solution to the Pell equation is given by the recursive formula a n = 6 a n − 1 − a
Pythagorean_triple
Association of one output to each input
mathematics, the Riemann hypothesis. In computability theory, a general recursive function is a partial function from the integers to the integers whose
Function_(mathematics)
Branch of mathematical logic
reverse mathematics. The initials "RCA" stand for "recursive comprehension axiom", where "recursive" means "computable", as in computable function. This
Reverse_mathematics
Family of higher-order functions
In functional programming, a fold is a higher-order function that analyzes a recursive data structure and, through use of a given combining operation
Fold_(higher-order_function)
Programming language with Arabic keywords
a functional programming language allowing a programmer to write programs completely in Arabic. Its name means "heart" in Arabic and is a recursive acronym
Qalb_(programming_language)
Study of computable functions and Turing degrees
computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator. The terminology for computable
Computability_theory
In logic, a statement which is always true
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Tautology_(logic)
Branch of mathematical logic
natural class of functions, such as the primitive recursive or polynomial-time computable functions. Functional interpretations have also been used to
Proof_theory
Thesis on the nature of computability
with Jacques Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments)
Church–Turing_thesis
Mathematical-logic system
24 Every recursively defined function can be seen as a fixed point of some suitably defined higher order function (also known as functional) closing over
Lambda_calculus
Functional programming construct
been developed in a number of recursive and non-recursive varieties. More complex patterns can be built from the primitive ones of the previous section
Pattern_matching
3-volume treatise on mathematics, 1910–1913
theory specifies the rules of syntax (rules of grammar) usually as a recursive definition that starts with "0" and specifies how to build acceptable
Principia_Mathematica
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
Gödel_numbering_for_sequences
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
Diagram that shows all possible logical relations between a collection of sets
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Venn_diagram
Audio programming language
Free and open-source software portal FAUST (Functional AUdio STream) is a domain-specific purely functional, text-based visual programming language for
FAUST_(programming_language)
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
Programming language
unit f In addition to being constructed from primitives by functionals, a function may be defined recursively by an equation, the simplest kind being: f
FP_(programming_language)
Text-based ray-tracing program
adaptive, non-recursive, super-sampling method. It is adaptive because not every pixel is super-sampled. Type 2 is an adaptive and recursive super-sampling
POV-Ray
Axiom
U} , as functions with return values. Here they are expressed as primitive recursive predicates. Write T U ( e , x , w , y ) {\displaystyle TU(e,x,w,y)}
Church's thesis (constructive mathematics)
Church's_thesis_(constructive_mathematics)
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
Set with algorithmic membership test
computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every natural
Computable_set
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
Ability of a computing system to simulate Turing machines
Leopold Kronecker formulated notions of computability, defining primitive recursive functions. These functions can be calculated by rote computation
Turing_completeness
Dialect of Lisp
optimization, giving stronger support for functional programming and associated techniques such as recursive algorithms. It was also one of the first programming
Scheme_(programming_language)
Relationship in which one statement follows from another
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Logical_consequence
Academic subfield of computer science
formalism equivalent to context-free grammars. Primitive recursive functions are a defined subclass of the recursive functions. Different models of computation
Theory_of_computation
Branch of logic
branches of the definition of ϕ {\displaystyle \phi } ), also acts as a recursive definition, and therefore specifies the entire language. To expand it
Propositional_logic
Symbol representing a mathematical object
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Variable_(mathematics)
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
for differentiating. Prime number Infinitude of the prime numbers Primitive recursive function Principle of bivalence no propositions are neither true
List_of_mathematical_proofs
Mathematical logic concept
contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite
Gentzen's_consistency_proof
Programming style in which control is passed explicitly
In functional programming, continuation-passing style (CPS) is a style of programming in which control is passed explicitly in the form of a continuation
Continuation-passing_style
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
Characteristic of some logical systems
intended. A set of logical connectives associated with a formal system is functionally complete if it can express all propositional functions. Semantic completeness
Completeness_(logic)
Symbol connecting formulas in logic
logical operators, propositional operators, or, in classical logic, truth-functional connectives. For the rules which allow new well-formed formulas to be
Logical_connective
Mathematical set formed from two given sets
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Cartesian_product
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
coincides with FP. These are functions which are defined, like the primitive recursive functions, by a set of base functions and operators for constructing
Implicit computational complexity
Implicit_computational_complexity
Symbol representing a mathematical concept
symbols are a primitive notion, and are therefore not defined in terms of other, more basic concepts. In typed logic, F is a functional symbol with domain
Function_symbol
Logical principle
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Law_of_excluded_middle
Operation on mathematical functions
involve several other functions as arguments, as in the definition of primitive recursive function. Given f, a n-ary function, and n m-ary functions g1, .
Function_composition
Sequence of words formed by specific rules
the basis for a 1947 proof "that the word problem for semigroups was recursively insoluble", and later devised the canonical system for the creation of
Formal_language
Computation model defining an abstract machine
A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine can equivalently be defined as
Turing_machine
Axiom of set theory
{\displaystyle X} .) Functional analysis The Hahn–Banach theorem in functional analysis, allowing the extension of linear functionals. The theorem that every
Axiom_of_choice
first-order multidimensional arrays containing primitive types, but was extended to handle higher-order and recursive data types in the work on Data Parallel
Flattening_transformation
Structure of a formal language
practical language translation tools. A recursive grammar is a grammar that contains production rules that are recursive. For example, a grammar for a context-free
Formal_grammar
Proposition in mathematical logic
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Continuum_hypothesis
Programming language
Erlang (/ˈɜːrlæŋ/ UR-lang) is a general-purpose, concurrent, functional high-level programming language, and a garbage-collected runtime system. The term
Erlang_(programming_language)
Infinite cardinal number
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Aleph_number
Logic principle
Q} : if P ⟺ Q {\displaystyle P\iff Q} then P = Q {\displaystyle P=Q} Functional extensionality of functions f , g {\displaystyle f,g} : if ∀ x , f x =
Extensionality
Paradox in set theory
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Russell's_paradox
Logical connective OR
abbreviates "it is warm". In classical logic, disjunction is given a truth functional semantics according to which a formula ϕ ∨ ψ {\displaystyle \phi \lor
Logical_disjunction
Turing machine that halts for any input
finite size (like the FOR loop in BASIC), we can express all of the primitive recursive functions (Meyer and Ritchie, 1967). An example of such a machine
Decider_(Turing_machine)
Form of mathematical proof
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Mathematical_induction
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
Mathematical theory of data types
individuals and truth-values, respectively, and defines the set of types recursively as follows: if a {\displaystyle a} and b {\displaystyle b} are types
Type_theory
Way to represent data types in the lambda calculus
definition without regard whether they are recursive or not. This is unlike Church encoding which treats recursive data types specially, representing them
Mogensen–Scott_encoding
Basic notion of sameness in mathematics
are not equal are said to be distinct. Equality is often considered a primitive notion, meaning it is not formally defined, but rather informally said
Equality_(mathematics)
Hierarchy of complexity classes for formulas defining sets
_{0}^{0}} that allow the use of primitive recursive functions, as now the quantifiers may be bounded by any primitive recursive function of the arguments.
Arithmetical_hierarchy
Area of mathematical logic
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Model_theory
Axioms for the natural numbers
{\begin{aligned}u(0)&=0_{X},\\u(S)&=S_{X}(u).\end{aligned}}} This is precisely the recursive definition of 0X and SX. When the Peano axioms were first proposed, Bertrand
Peano_axioms
Collection of mathematical objects
other sets. Sets cannot be mathematically defined, since they form a primitive notion. Instead, they are characterized by basic properties (axioms) that
Set_(mathematics)
of intuitionism, the finitism of Hilbert and Bernays, the constructive recursive mathematics of mathematicians Shanin and Markov, and Bishop's program
Mathematical_object
Logical operation
formulate classical negation in a natural deduction setting is to take as primitive rules of inference negation introduction (from a derivation of P {\displaystyle
Negation
Logic theorem
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Law_of_noncontradiction
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Technique of representing an aggregate data structure
general in the sense that it can be adapted to lists, trees, and other recursively defined data structures. Such modified data structures are usually referred
Zipper_(data_structure)
Measure of algorithmic complexity
found across insect species, which correspond to the circuit that is both functional and requires the minimum Kolmogorov complexity to be generated from self-replicating
Kolmogorov_complexity
Function that preserves distinctness
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Injective_function
Set of all things that may be the input of a mathematical function
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Domain_of_a_function
Fundamental theorem in mathematical logic
interpret its own construction, so that this construction is non-recursive (as recursive definitions would be unambiguous). Also, if T {\displaystyle T}
Gödel's_completeness_theorem
Representation of data of various types in lambda calculus
calculus the only primitive data type are functions, represented by lambda abstraction terms. Types that are usually considered primitive in other notations
Church_encoding
Yes/no problem in computer science
effectively solvable if the set of inputs for which the answer is YES is a recursive set. A decision problem is partially decidable, semidecidable, solvable
Decision_problem
Mathematical set containing no elements
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Empty_set
Mathematical logic concept
P {\displaystyle \neg Q\to \neg P} "; or as the statement of a truth-functional tautology or theorem of propositional logic. The principle was stated
Contraposition
Function, homomorphism, or morphism
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Map_(mathematics)
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
Theorem for proving more complex theorems
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Lemma_(mathematics)
One-to-one correspondence
is a bijection. A bijection f with domain X (indicated by f: X → Y in functional notation) also defines a converse relation starting in Y and going to
Bijection
General-purpose programming language
programming language that supports both object-oriented programming and functional programming. Designed to be concise, many of Scala's design decisions
Scala_(programming_language)
Set of elements in any of some sets
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Union_(set_theory)
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PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
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