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Concept in computability theory
elementary was originally introduced by László Kalmár in the context of computability theory. He defined the class of elementary recursive functions ("Kalmár
Elementary_recursive_function
{\displaystyle {\mathsf {ELEMENTARY}}} consists of the decision problems that can be solved in time bounded by an elementary recursive function. Equivalently, these
ELEMENTARY
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
System of arithmetic in proof theory
and defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure
Elementary function arithmetic
Elementary_function_arithmetic
Type of mathematical function
the elementary functions and, recursively, the integrals of the Liouvillian functions. In late-nineteenth-century analysis, elementary functions were
Elementary_function
Formalization of the natural numbers
arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation
Primitive recursive arithmetic
Primitive_recursive_arithmetic
Arithmetic operation
\mathbb {N} ^{2}} ) is not an elementary recursive function. One can prove by induction that for every elementary recursive function f, there is a constant c
Tetration
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
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)
Computational problem with high complexity
has no algorithmic solution with time bounded by an elementary recursive function. These functions grow no faster than a fixed-height tower of exponentiation
Nonelementary_problem
Mathematical function that can be computed by a program
general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for
Computable_function
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
delta function. The Kronecker delta forms the multiplicative identity element of an incidence algebra. The Kronecker delta is an elementary recursive function
Kronecker_delta
Mathematical logic concept
a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable
Computably_enumerable_set
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
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
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
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
Defining elements of a set in terms of other elements in the set
Münchausen trilemma. That recursive definitions are valid – meaning that a recursive definition identifies a unique function – is a theorem of set theory
Recursive_definition
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
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 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
3-volume treatise on mathematics, 1910–1913
72. If φp and ψp are elementary propositional functions which take elementary propositions as arguments, φp ∨ ψp is an elementary proposition. Pp Together
Principia_Mathematica
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
Study of computable functions and Turing degrees
μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable
Computability_theory
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
Function definition that is not bound to an identifier
anonymous function (function literal, lambda function, or block) is a function definition that is not bound to an identifier. Anonymous functions are often
Anonymous_function
Decidable first-order theory of the natural numbers with addition
Presburger-definable set. Robinson arithmetic Skolem arithmetic Elementary recursive function There is no number which, when added by 1 {\displaystyle 1}
Presburger_arithmetic
Programming language
simple register language designed to precisely capture the primitive recursive functions. The language is derived from the counter-machine model. Like the
LOOP_(programming_language)
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
Problem in computer science
Elementary Number Theory", which proposes that the intuitive notion of an effectively calculable function can be formalized by the general recursive functions
Halting_problem
Hungarian mathematician (1905–1976)
alternative form of primitive recursive arithmetic, known as elementary recursive arithmetic, based on primitive functions that differ from the usual kind
László_Kalmár
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
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
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
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
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
Mathematical function, inverse of an exponential function
summands n is large enough. In elementary calculus, the series is said to converge to the function ln(z), and the function is the limit of the series. It
Logarithm
Type of computing function
exponential CPU time: Recursive_Fibonacci (n) { if (n == 0) return 0 if (n == 1) return 1 return Recursive_Fibonacci (n-1) + Recursive_Fibonacci (n-2) } While
Memory-bound_function
Algorithm for integer multiplication
with fewer than n digits. Therefore, those products can be computed by recursive calls of the Karatsuba algorithm. The recursion can be applied until the
Karatsuba_algorithm
definition of a primitive recursive function to the intermediate schema is done inductively, where an elementary pairing function w {\displaystyle w} is
Gödel's_β_function
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
Area of mathematical logic
the complex exponential function. The most general semantic framework in which stability is studied are abstract elementary classes, which are defined
Model_theory
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
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
Sequence of differential equation solutions
Laguerre polynomials.) One can also define the Laguerre polynomials recursively, defining the first two polynomials as L0(x)=1 L1(x)=1−x and then using
Laguerre_polynomials
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
Concept in model theory
in M. If N is an elementary substructure of M, then M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N
Elementary_equivalence
Algebraic manipulation of "true" and "false"
mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth
Boolean_algebra
Analytic function in mathematics
}}.} This can be used recursively to extend the Dirichlet series definition to all complex numbers. The Riemann zeta function also appears in a form
Riemann_zeta_function
Multivalued function in mathematics
terms of elementary (Liouvillian) functions, the first published proof did not appear until 2008. There are countably many branches of the W function, denoted
Lambert_W_function
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
Symbol representing a mathematical concept
systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though
Function_symbol
Order type of the set of all recursive ordinals
non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal
Nonrecursive_ordinal
Symbolic description of a mathematical object
(∪), or intersection (∩). (Functions can be understood as unary operations) Brackets ( ) With this alphabet, the recursive rules for forming a well-formed
Expression_(mathematics)
Diagram that shows all possible logical relations between a collection of sets
by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability
Venn_diagram
Elementary functions and their finitely iterated integrals
Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined
Liouvillian_function
Mathematical use of "there exists"
union of sets. A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The ¬ {\displaystyle \lnot
Existential_quantification
Branch of mathematical logic
games. Weaker systems than recursive comprehension can be defined. The weak system RCA* 0 consists of elementary function arithmetic EFA (the basic axioms
Reverse_mathematics
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
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
Mathematical theory of data types
:{\mathsf {type}}} . A term in logic is recursively defined as a constant symbol, variable, or a function application, where a term is applied to another
Type_theory
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
Ordinals in mathematics and set theory
Computable ordinals (or recursive ordinals) are certain countable ordinals: loosely speaking those represented by a computable function. There are several
Large_countable_ordinal
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
Symbol representing a mathematical object
Independent Variables" Edwards 1892, pp. 2-3, Articles 6-7, "Functions" Edwards, Joseph (1892). An Elementary Treatise on the Differential Calculus (2nd ed.). London:
Variable_(mathematics)
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
Axiom of set theory
states that a choice function exists for any countable family of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis
Axiom_of_choice
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
Programming paradigm restricted to provably terminating programs
recursion. For example, quicksort is not trivially shown to be substructural recursive, but it only recurs to a maximum depth of the length of the vector (worst-case
Total_functional_programming
Proof in set theory
interesting classes of functions decidable, by Rice's theorem, i.e. the set of counting numbers for the subcountable sets may not be recursive and can thus fail
Cantor's_diagonal_argument
Sequence of operations for a task
of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result). Kleene, Stephen C. (1943). "Recursive Predicates
Algorithm
Representation of data of various types in lambda calculus
the identity function, λ x . x {\displaystyle \lambda x.x} . The variable name r {\displaystyle r} is chosen as a mnemonic for "recursive result." This
Church_encoding
One-to-one correspondence
must not be confused with one-to-one function, which means injective but not necessarily surjective. The elementary operation of counting establishes a
Bijection
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
Non-contradiction of a theory
strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable
Consistency
Proof that only uses basic techniques
existence of iterated exponential functions that cannot be proven in this theory. Diamond, Harold G. (1982), "Elementary methods in the study of the distribution
Elementary_proof
In mathematics, a statement that has been proven
proposition is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should
Theorem
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
Function whose domain is the positive integers
Euler's Totient Function Huard, Ou, Spearman, and Williams. Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions Dineva, Rosica
Arithmetic_function
Subfield of mathematics
numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the
Mathematical_logic
Theorem about natural numbers
Furthermore, with the relatively elementary technique of the Grzegorczyk hierarchy, it can be shown that every primitive recursive strictly decreasing infinite
Goodstein's_theorem
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
Algorithmic runtime requirements for common math procedures
in Borwein & Borwein. The elementary functions are constructed by composing arithmetic operations, the exponential function ( exp {\displaystyle \exp
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Limitative results in mathematical logic
axiomatized (also called effectively generated) if its set of theorems is recursively enumerable. This means that there is a computer program that, in principle
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
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
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
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
Type of logical system
of symbols from that signature. The set of axioms is often finite or recursively enumerable, in which case the theory is called effective. Some authors
First-order_logic
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
Set whose elements all belong to another set
Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image
Subset
Form of mathematical proof
natural number. The successor function s of every natural number yields a natural number (s(x) = x + 1). The successor function is injective. 0 is not in
Mathematical_induction
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
Result of repeatedly applying a mathematical function
derivative of an iterated function is given by the recursive formula: δfN(x)δf(y)=f′(fN−1(x))δfN−1(x)δf(y)+δ(fN−1(x)−y) Iterated functions crop up in the series
Iterated_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
Pattern defining an infinite sequence of numbers
recurrence relation means obtaining a closed-form solution: a non-recursive function of n {\displaystyle n} . The concept of a recurrence relation can
Recurrence_relation
Axiomatic set theories based on the principles of mathematical constructivism
axioms postulating (total) function existence lead to the requirement for halting recursive functions. From their function graph in individual interpretations
Constructive_set_theory
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
Mathematical function
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Mathematical set containing no elements
exists precisely one function f {\displaystyle f} from ∅ {\displaystyle \varnothing } to A , {\displaystyle A,} the empty function. As a result, the empty
Empty_set
Undecidability of equality of real numbers
sine function entirely. Constant problem – Problem of deciding whether an expression equals zero Elementary function – Type of mathematical function Tarski's
Richardson's_theorem
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ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, the son of the functionary Heknofre.
Biblical
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Male
Celtic
, great justiciary, or functionary.
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
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.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
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.
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.
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.
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
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