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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
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
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
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
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)
Mathematical-logic system based on functions
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
Mathematical logician and philosopher
Mathematical Platonism Original proof of Gödel's completeness theorem Primitive recursive functional Gödel–Löb logic Strange loop Tarski's undefinability theorem
Kurt_Gödel
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
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
a more natural style of expressing computation than simply using primitive recursive functions. Since the halting problem cannot be solved in general
Walther_recursion
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
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
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)
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 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
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
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
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
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
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
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
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)
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
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
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
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 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 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)
for differentiating. Prime number Infinitude of the prime numbers Primitive recursive function Principle of bivalence no propositions are neither true
List_of_mathematical_proofs
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
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)
Axiomatic set theories based on the principles of mathematical constructivism
partial general recursive functions (or programs, in the sense that they are Turing computable), including ones e.g. non-primitive recursive but P A {\displaystyle
Constructive_set_theory
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
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
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)
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)
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
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
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
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
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
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
Programming language family
design in a paper in Communications of the ACM on April 1, 1960, entitled "Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part
Lisp_(programming_language)
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
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
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)
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)
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
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
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
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
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
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)
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
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)
Programming language
means 'recursive'. *) match integers with | [] -> 0 (* Yield 0 if integers is the empty list []. *) | first :: rest -> first + sum rest;; (* Recursive call
OCaml
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
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
Non-contradiction of a theory
Gödel's theorem applies to the theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic. Moreover, Gödel's
Consistency
Sequence of operations for a task
arXiv:2506.13131 [cs.AI]. Axt, P (1959). "On a Subrecursive Hierarchy and Primitive Recursive Degrees". Transactions of the American Mathematical Society. 92 (1):
Algorithm
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)
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
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
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)
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)
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)
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
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
Pair of mathematical objects
The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example
Ordered_pair
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
Representation of natural numbers and other data 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
Relationship where 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
Product of numbers from 1 to n
valid at n = 1 {\displaystyle n=1} . Therefore, with this convention, a recursive computation of the factorial needs to have only the value for zero as
Factorial
Basic framework of mathematics
reduces the consistency of the Peano axioms to the weaker system of Primitive recursive arithmetic with an additional axiom asserting the existence of a
Foundations_of_mathematics
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
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
Data serialization format
plain lists, y points to the next cell (if any), thus forming a list. The recursive clause of the definition means that both this representation and the S-expression
S-expression
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
Von_Neumann_universe
Programming statement for branching control based on a value
is primitive recursive in φ1, ..., φm+1, Q1, ..., Qm+1. — Stephen Kleene, Kleene provides a proof of this in terms of the Boolean-like recursive functions
Switch_statement
PowerBook – PowerPC – PowerPC G4 – Prefix grammar – Preprocessor – Primitive recursive function – Programming language – Prolog – PSPACE-complete – Pulse-code
Index_of_computing_articles
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
Function in mathematical logic
how functions defined by course-of-values recursion are in fact primitive recursive functions. Once a Gödel numbering for a formal theory is established
Gödel_numbering
Properties linking logical conjunction and disjunction
of each other, and consequently, only one of them needs to be taken as primitive. If φ D {\displaystyle \varphi ^{D}} is used as notation to designate
Conjunction/disjunction duality
Conjunction/disjunction_duality
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
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
Symbolic description of a mathematical object
understood as unary operations) Brackets ( ) With this alphabet, the recursive rules for forming a well-formed expression (WFE) are as follows: Any constant
Expression_(mathematics)
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)
Mathematical table used in logic
propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination
Truth_table
Ability to solve a problem by an effective procedure
widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally
Computability
Attempt to persuade or to determine the truth of a conclusion
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Argument
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
Algebraic manipulation of "true" and "false"
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Boolean_algebra
Subfield of mathematics
fixed-point logics that allow inductive definitions, like one writes for primitive recursive functions. One can formally define an extension of first-order logic
Mathematical_logic
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
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
Symbol representing a property or relation in logic
{\displaystyle a} and b {\displaystyle b} . Predicates are considered a primitive notion of first-order, and higher-order logic and are therefore not defined
Predicate_(logic)
Axiomatic logical system
interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable. The background logic of Q
Robinson_arithmetic
Reasoning for mathematical statements
versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic
Mathematical_proof
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
Girl/Female
German, Latin
Archaic; Ancient; Old; Primitive
Girl/Female
Danish, Finnish, French, German, Latin, Swedish
Ancient; Primitive; Venerable
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
American, Australian, Chinese, Finnish, French, Latin, Portuguese, Swedish
Ancient; Primitive; Venerable
Girl/Female
American, Australian, Biblical, British, Chinese, Christian, Danish, English, Finnish, French, German, Gothic, Italian, Latin, Portuguese, Swedish
Ancient; Primitive; Venerable
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
Girl/Female
Hindu, Indian
Fire
Girl/Female
Assamese, Bengali, Christian, Hindu, Indian, Kannada, Marathi, Tamil, Telugu
Full of Grace; Like a Peacock
Girl/Female
Tamil
Incomplete
Girl/Female
Tamil
Aswathy | அஸà¯à®µà®¾à®¤à¯€
An Angel
Boy/Male
Indian
Handsome
Girl/Female
Hindu
Equal to sankarshana
Girl/Female
Teutonic
Renowned.
Boy/Male
Hindu
Girl/Female
Arabic, Islamic, Muslim, Pakistani, Urdu
Produce Good Thing
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi
One Adorned with Knowledge of the Vedas
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
PRIMITIVE RECURSIVE-FUNCTIONAL
a.
Of or pertaining to the beginning or origin, or to early times; original; primordial; primeval; first; as, primitive innocence; the primitive church.
a.
Pristine; primitive.
n.
A privative prefix or suffix. See Privative, a., 3.
a.
Original; primary; radical; not derived; as, primitive verb in grammar.
pl.
of Primitia
a.
Involving a limit; as, a limitive law, one designed to limit existing powers.
a.
Prone to make excursions; wandering; roving; exploring; as, an excursive fancy.
pl.
of Primitia
n.
The primitive perivisceral cavity.
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.
adv.
In a decursive manner.
n.
A term indicating the absence of any quality which might be naturally or rationally expected; -- called also privative term.
a.
Of or pertaining to a former time; old-fashioned; characterized by simplicity; as, a primitive style of dress.
a.
Serving, or able, to repulse; repellent; as, a repulsive force.
n.
That which causes revulsion; specifically (Med.), a revulsive remedy or agent.
a.
Cold; forbidding; offensive; as, repulsive manners.
a.
Being of the first production; primitive; original.
n.
A revulsive medicine.
a.
Primitive; primary; original.
n.
A character used in cursive writing.