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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
Type of mathematical function
elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are
Elementary_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
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
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
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
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
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
Functions in computability theory
functions used in computability theory. Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function
Grzegorczyk_hierarchy
Mathematical function that can be computed by a program
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
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)
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
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
Mathematical-logic system based on functions
M; this means a recursive function definition cannot be written with let. The letrec construction would allow writing recursive function definitions, where
Lambda_calculus
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
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
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
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
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
Branch of mathematical logic
VI.5.4 Weaker systems than recursive comprehension can be defined. The weak system RCA* 0 consists of elementary function arithmetic EFA (the basic axioms
Reverse_mathematics
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
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Set with algorithmic membership test
computable if and only if the indicator function 1 S {\displaystyle \mathbb {1} _{S}} is computable. Every recursive language is computable. Every finite
Computable_set
Defining elements of a set in terms of other elements in the set
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements
Recursive_definition
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
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
function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying
List_of_types_of_functions
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
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
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 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
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
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
Mathematical function characterizing set membership
offers up the same definition in the context of the primitive recursive functions as a function φ of a predicate P takes on values 0 if the predicate is true
Indicator_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
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
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
Alternate way to define a function in APL
function is 0≤⍵, 1 if ⍵ is 0 or 1 and 0 otherwise. The recursive step is highly multiply recursive. For example, pn 200 would result in the function being
Direct_function
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)
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
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
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
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
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
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
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
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)
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
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
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
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)
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
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
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)
any recursively enumerable set of well-formed formulas of a first-order language is recursively axiomatizable, and even primitively recursively axiomatizable
Craig's_theorem
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
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
Sequence of differential equation solutions
Laguerre polynomials.) One can also define the Laguerre polynomials recursively, defining the first two polynomials as L 0 ( x ) = 1 {\displaystyle L_{0}(x)=1}
Laguerre_polynomials
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
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
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
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
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
Basic framework of mathematics
sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic on the (infinite)
Foundations_of_mathematics
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
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
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
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
Theorem that arithmetical truth cannot be defined in arithmetic
but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Subset of a function's codomain
a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are
Range_of_a_function
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
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
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
Result of repeatedly applying a mathematical function
xg(i)\}\right)^{b-a+1}\{a,1\}} The functional derivative of an iterated function is given by the recursive formula: δ f N ( x ) δ f ( y ) = f ′ ( f N − 1 ( x ) ) δ f
Iterated_function
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
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
In logic, a statement which is always true
be deduced from the truth or falsity of each variable. A valuation is a function that assigns each propositional variable to either T (for truth) or F (for
Tautology_(logic)
Size of a possibly infinite set
strictly more real numbers than natural numbers. The cardinality function is a cardinal function that takes in a set A {\displaystyle A} and returns its cardinal
Cardinal_number
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
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
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
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
Numerical calculations carrying along derivatives
executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos
Automatic_differentiation
Programming mechanism
of code and data together with the treatment of functions lend themselves extremely well for a recursive definition of a variadic compositional operator
Function composition (computer science)
Function_composition_(computer_science)
Assignment of meaning to the symbols of a formal language
determined recursively. Now it is easier to see what makes a formula logically valid. Take the formula F: (Φ ∨ ¬Φ). If our interpretation function makes Φ
Interpretation_(logic)
Branch of mathematical logic
consequence of the interpretation one usually obtains the result that any recursive function whose totality can be proven either in I or in C is represented by
Proof_theory
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
Size of a set in mathematics
points that are definitively colored blue, map these points onto the next recursive image (i.e. by applying f {\displaystyle f} then g {\displaystyle
Cardinality
Problem optimization method
a recursive relationship called the Bellman equation. For i = 2, ..., n, Vi−1 at any state y is calculated from Vi by maximizing a simple function (usually
Dynamic_programming
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
Measure of algorithmic complexity
function GenerateString2() return "4c1j5b2p0cv4w1x8rx2y39umgw5q85s7" whereas the first string is output by the (much shorter) pseudo-code: function GenerateString1()
Kolmogorov_complexity
Mathematical proposition equivalent to the axiom of choice
is defined using transfinite recursion. It is exactly like the usual recursive definition of a sequence but runs over ordinals. A key difference is that
Zorn's_lemma
Logical principle
significance of the principle of excluded middle in mathematics, especially in function theory [reprinted with commentary, p. 334, van Heijenoort] Andrei Nikolaevich
Law_of_excluded_middle
Mathematical logic concept
but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation
Gentzen's_consistency_proof
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
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
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.
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.
Male
Egyptian
, the son of the functionary Heknofre.
Male
Celtic
, great justiciary, or functionary.
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.
Male
Egyptian
, a great functionary.
Male
Egyptian
, a high Egyptian functionary.
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.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, Functionary of the Interior.
Biblical
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Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
Boy/Male
Hindu, Indian
Best Among the Truthful
Boy/Male
Czech
Great glory.
Boy/Male
Arabic
Perspicacity; Also a Horseman or Knight
Girl/Female
Hindu, Indian
God Name; Heaven
Boy/Male
Hindu, Indian, Malayalam, Marathi
Oneself; Alone
Girl/Female
Hindu, Indian
Cute; Victory Person
Girl/Female
Hindu, Indian, Marathi
Lord of Intelligence
Boy/Male
Armenian
Name of a saint.
Boy/Male
Hindu, Indian, Traditional
Fresh Butter
Surname or Lastname
English
English : habitational name from a place in Norfolk, so called from the Old Norse personal name Fili or Fila (of uncertain origin) + Old Norse býr ‘farm’, ‘settlement’.
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
ELEMENTARY RECURSIVE-FUNCTION
a.
Pertaining to one of the four elements, air, water, earth, fire.
a.
Cold; forbidding; offensive; as, repulsive manners.
a.
Prone to make excursions; wandering; roving; exploring; as, an excursive fancy.
a.
Having only one principle or constituent part; consisting of a single element; simple; uncompounded; as, an elementary substance.
n.
Unorganized material; elementary matter.
a.
Pertaining to aliment or food, or to the function of nutrition; nutritious; alimental; as, alimentary substances.
adv.
In a decursive manner.
a.
Serving, or able, to repulse; repellent; as, a repulsive force.
n.
That which causes revulsion; specifically (Med.), a revulsive remedy or agent.
adv.
According to elements; literally; as, the words, "Take, eat; this is my body," elementally understood.
n.
A character used in cursive writing.
n.
A revulsive medicine.
a.
Repulsive; driving back.
v. t.
Causing revulsion; revulsive.
a.
Pertaining to, or treating of, the elements, rudiments, or first principles of anything; initial; rudimental; introductory; as, an elementary treatise.
a.
Elementary; rudimental.
a.
Elementary.
a.
Elementary.
a.
Pertaining to rudiments or first principles; rudimentary; elementary.
a.
Pertaining to the elements, first principles, and primary ingredients, or to the four supposed elements of the material world; as, elemental air.