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In computability theory, the assignment of natural numbers to a set of objects
In computability theory a numbering is an assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some
Numbering (computability theory)
Numbering_(computability_theory)
Study of computable functions and Turing degrees
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated
Computability_theory
Ability to solve a problem by an effective procedure
Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical
Computability
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every
Computable_set
Real number that can be computed within arbitrary precision
Stoltenberg-Hansen, V.; Tucker, J.V. (1999). "Computable Rings and Fields". In Griffor, E.R. (ed.). Handbook of Computability Theory. Elsevier. pp. 363–448. ISBN 978-0-08-053304-9
Computable_number
Academic subfield of computer science
Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1 S. Barry Cooper (2004). Computability Theory. Chapman and Hall/CRC. ISBN 1-58488-237-9
Theory_of_computation
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
Function in mathematical logic
and completeness properties of formal systems. In computability theory, the term "Gödel numbering" is used in settings more general than the one described
Gödel_numbering
Classes of partial recursive functions
In computability theory, index sets describe classes of computable functions; specifically, they give all indices of functions in a certain class, according
Index_set_(computability)
Branch of pure mathematics
Number theory is a branch of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers
Number_theory
Mathematical logic concept
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable
Computably_enumerable_set
Inherent difficulty of computational problems
analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computational complexity theory is that the former is
Computational complexity theory
Computational_complexity_theory
Method of comparing problems by transforming one into another in computability theory
In computability theory, many reducibility relations (also called reductions, reducibilities, and notions of reducibility) are studied. They are motivated
Reduction (computability theory)
Reduction_(computability_theory)
In computability theory two sets A , B {\displaystyle A,B} of natural numbers are computably isomorphic or recursively isomorphic if there exists a total
Computable_isomorphism
Branch of model theory that deals with computation
Computable model theory is a branch of model theory that deals with questions of computability as they apply to model-theoretic structures. Computable
Computable_model_theory
Concept in computability theory
In computability theory, admissible numberings are enumerations (numberings) of the set of partial computable functions that can be converted to and from
Admissible_numbering
Yes-or-no question that cannot ever be solved by a computer
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct
Undecidable_problem
Yes/no problem in computer science
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a
Decision_problem
Thesis on the nature of computability
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers
Church–Turing_thesis
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 rules for assigning mathematical values to database items
whose table definitions require a database design. In computability theory, the simplest numbering scheme is the assignment of natural numbers to a set
Numbering_scheme
Set of all true first-order statements about the arithmetic of natural numbers
295 see theories associated with a structure Shore 2011, p. 184 Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002), Computability and logic
True_arithmetic
Whether a decision problem has an effective method to derive the answer
many-one reduction in computability theory. A property of a theory or logical system weaker than decidability is semidecidability. A theory is semidecidable
Decidability_(logic)
Ordered listing of items in collection
in this theory, the existence of a surjection from I onto S need not imply the existence of an injection from S into I. In computability theory one often
Enumeration
Framework for studying interactive computational tasks through logic
Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed
Computability_logic
Problem in computer science
In computability theory, the halting problem is the decision problem of, given an arbitrary computer program and an input, determining whether said program
Halting_problem
Computation model defining an abstract machine
machines has yielded many insights into computer science, computability theory, and complexity theory. In his 1948 essay, "Intelligent Machinery", Turing wrote
Turing_machine
In computability theory, a Friedberg numbering is a computable numbering (enumeration) of the set of all computably enumerable sets that has no repetitions:
Friedberg_numbering
Theorem in computability theory
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions
Kleene's_recursion_theorem
Study of algorithms for performing number theoretic computations
number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory
Computational_number_theory
Theorems whose content is effectively computable
contradiction. The logic involved is closer to proof theory than to that of computability theory and computable functions. It is rather loosely conjectured that
Effective results in number theory
Effective_results_in_number_theory
Subfield of mathematics
Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic
Mathematical_logic
functions on Nn. Gödel numbering, defined on well-formed formulae of some formal language, is a natural-valued function. Computability theory is essentially based
Integer-valued_function
American mathematician
set theory and philosophy of set theory (particularly the idea of the set-theoretic multiverse), in computability theory, and in group theory. After
Joel_David_Hamkins
Ordinal-indexed family of rapidly increasing functions
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or
Fast-growing_hierarchy
Mathematical theory
uncomputable. In fact, he showed that computability and completeness are mutually exclusive: any complete theory must be uncomputable. The proof of this
Solomonoff's theory of inductive inference
Solomonoff's_theory_of_inductive_inference
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with
List of computability and complexity topics
List_of_computability_and_complexity_topics
British computer scientist
scientist and expert on computability theory, also known as recursion theory. Computability theory is about what can and cannot be computed by people and machines
John_V._Tucker
Type of computational problem
In computational complexity theory and computability theory, a counting problem is a type of computational problem that is obtained by strengthening a
Counting_problem_(complexity)
Affirms the existence of a computable universal function
In computability theory, the UTM theorem, or universal Turing machine theorem, is a basic result about Gödel numberings of the set of computable functions
UTM_theorem
Study of mathematical analysis seen through computability theory
mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the
Computable_analysis
Logics for computability are formulations of logic that capture some aspect of computability as a basic notion. This usually involves a mix of special
Logics_for_computability
Countable ordinal that is the order type of a computable well-ordering of natural numbers
specifically computability and set theory, a computable (or recursive) ordinal is an ordinal number that can be represented as a computable well-ordering
Computable_ordinal
Measure of algorithmic complexity
14words". It is also possible to show the non-computability of K by reduction from the non-computability of the halting problem H, since K and H are Turing-equivalent
Kolmogorov_complexity
Measure of unsolvability
unsolvability of the set. The concept of Turing degree is fundamental in computability theory, where sets of natural numbers are often regarded as decision problems
Turing_degree
In computability theory complete numberings are generalizations of Gödel numbering first introduced by A.I. Mal'tsev in 1963. They are studied because
Complete_numbering
Generalization of Turing computability
In computability theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order
Hyperarithmetical_theory
Numbers expressible as integrals of algebraic functions
In mathematics, specifically number theory, a period or algebraic period is a complex number that can be expressed as an integral of an algebraic function
Period_(number_theory)
On transforming a program by substituting constants for free variables
In computability theory the S m n theorem, written also as "smn-theorem" or "s-m-n theorem" (also called the translation lemma, parameter theorem, and
Smn_theorem
In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notion
Myhill_isomorphism_theorem
Theorem in computability theory
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the
Rice's_theorem
Basic framework of mathematics
mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, parts of computer
Foundations_of_mathematics
Concept in theoretical computer science
become larger than any computable function. This has implications in computability theory, the halting problem, and complexity theory. The concept of a busy
Busy_beaver
In the mathematical field of computability theory, a PA degree is a Turing degree that computes a complete extension of Peano arithmetic (Jockusch 1987)
PA_degree
discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Sequence of words formed by specific rules
expensive). Therefore, formal language theory is a major application area of computability theory and complexity theory. Formal languages may be classified
Formal_language
Abstract machine used to study decision problems
In complexity theory and computability theory, an oracle machine is an abstract machine that can query a black box called an oracle, which is able to
Oracle_machine
Area of mathematical logic
in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different
Model_theory
Listing all imaginary quadratic fields with a given class number
compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed)
Class_number_problem
American mathematician (1926–2015)
1926 – July 17, 2015) was an American mathematician who worked in computability theory, and was a professor in the Mathematics Department of the Massachusetts
Hartley_Rogers_Jr.
Mathematical result on infinite trees
The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory. This
Kőnig's_lemma
Computer hardware technology that uses quantum mechanics
can be simulated by a Turing machine. Quantum computers provide no computability power over classical computers. Thus, quantum computers cannot solve
Quantum_computing
Concept in computability theory
In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property
Mu_operator
Exploring properties of the integers with complex analysis
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.
Analytic_number_theory
Generalization of "n-th" to infinite cases
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite
Ordinal_number
In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithm that produces an upper bound for the complexity
Tarski–Kuratowski_algorithm
Concept in computability theory
In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated"
Computably_inseparable
Subfield of computer science and mathematics
described in a finite number of English words". Rogers, Hartley Jr. (1967). Theory of Recursive Functions and Effective Computability. McGraw-Hill. Page
Theoretical_computer_science
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
Zermelo–Fraenkel_set_theory
Category of mathematical proof
Turing's computing machine model (see Post–Turing machine for details). John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages
Proof_of_impossibility
Limit of a uniformly computable sequence of functions
computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in
Computation_in_the_limit
Theory of truth in the philosophy of language
A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences. The semantic conception
Semantic_theory_of_truth
Large number coined by Ronald Graham
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger
Graham's_number
Problem-solving procedures with certain characteristics
In metalogic, mathematical logic, and computability theory, an effective method or effective procedure is a finite-time, deterministic procedure for solving
Effective_method
Study of computation
perform those computations. In an effort to answer the first question, computability theory examines which computational problems are solvable on various theoretical
Computer_science
Theorem that arithmetical truth cannot be defined in arithmetic
language of arithmetic is assigned a distinct number. This procedure is known variously as Gödel numbering, coding and, more generally, as arithmetization
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Axioms for the natural numbers
Kaye 1991, Section 11.3. Kaye 1991, pp. 70ff.. Davis, Martin (1974). Computability. Notes by Barry Jacobs. Courant Institute of Mathematical Sciences,
Peano_axioms
about simplicial sets and cubical sets. Synthetic computability theory develops computability theory in constructive mathematics by postulating, among
Synthetic_mathematics
Basic notion of sameness in mathematics
naive set theory, it was shown that the parallel postulate cannot be proved, the existence of mathematical objects that cannot be computed or explicitly
Equality_(mathematics)
Size of a possibly infinite set
mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets. A natural number can be used for two purposes: to
Cardinal_number
Axiomatic logical system
Andrzej; Robinson, Raphael M. (1953). Undecidable theories. North Holland. Tourlakis, George (2022). Computability. Cham, Switzerland: Springer. ISBN 978-3-030-83202-5
Robinson_arithmetic
Type of infinite structure
In mathematical logic, and more specifically in model theory, an infinite structure ( M , < , … ) {\displaystyle (M,<,\dots )} that is totally ordered
O-minimal_theory
Transformation of one computational problem to another
In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently
Reduction_(complexity)
Infinite cardinal number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They
Aleph_number
Axiomatic set theories based on the principles of mathematical constructivism
In computability theory, the μ operator enables all partial general recursive functions (or programs, in the sense that they are Turing computable), including
Constructive_set_theory
3-volume treatise on mathematics, 1910–1913
photographically reprinted with the same page numbering; corrections were still made. The total number of pages (excluding the endpapers) in the first
Principia_Mathematica
Statement in mathematical logic
of computable functions was not yet developed in 1934. The diagonal lemma is closely related to Kleene's recursion theorem in computability theory, and
Diagonal_lemma
Computational problems no algorithm can solve
In computability theory, an undecidable problem is a decision problem for which an effective method (algorithm) to derive the correct answer does not exist
List_of_undecidable_problems
Non-contradiction of a theory
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T {\displaystyle T} is consistent if there is no
Consistency
Formal language
are not recursive include: Post correspondence problem Mortality (computability theory) Entscheidungsproblem Recursively enumerable languages (REL) are
Recursively enumerable language
Recursively_enumerable_language
Complexity class used to classify decision problems
More unsolved problems in computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify
NP_(complexity)
Branch of mathematical logic
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating
Proof_theory
Biennial conference series on computational number theory
number theory. They are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number
Algorithmic Number Theory Symposium
Algorithmic_Number_Theory_Symposium
science, including algorithms, data structures, computability, computational complexity, automata theory and formal languages: CCC - Computational Complexity
List of computer science conferences
List_of_computer_science_conferences
Integration by parts version of Abel's method for summation by parts
in analytic number theory and the study of special functions to compute series. Wikibooks has a book on the topic of: Analytic Number Theory/Useful summation
Abel's_summation_formula
One of several equivalent definitions of a computable function
recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines
General_recursive_function
Informal set theories
Naive set theory is any of several set theories used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined
Naive_set_theory
Collection of sets in mathematics that can be defined based on a property of its members
In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined
Class_(set_theory)
Computer system simulating intelligence
Zadeh, the founder of the fuzzy set theory, who differentiated machine intelligence into hard and soft computing techniques, which are used in artificial
Computational_intelligence
Functions in computability theory
logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory. Every function in the Grzegorczyk hierarchy is a primitive recursive
Grzegorczyk_hierarchy
NUMBERING COMPUTABILITY-THEORY
NUMBERING COMPUTABILITY-THEORY
Boy/Male
Biblical
Right hand; numbering; preparing.
Girl/Female
Biblical
Numbering, showing, increase of tribute.
Surname or Lastname
English, Scottish, French, and German
English, Scottish, French, and German : from Middle English, Old French, Middle High German olifant ‘elephant’ (medieval Latin olifantus, from classical Latin elephantus, Greek elephas, genitive elephantos). The circumstances in which this word was applied as a surname are not clear. It may have been a nickname for a large, lumbering individual, or a metonymic occupational name for a worker in ivory, or a habitational name from a house distinguished by the sign of an elephant.
Surname or Lastname
English and Scottish
English and Scottish : topographic name for someone who lived by a patch of wet ground overgrown with brushwood, northern Middle English kerr (Old Norse kjarr). A legend grew up that the Kerrs were left-handed, on theory that the name is derived from Gaelic cearr ‘wrong-handed’, ‘left-handed’.Irish : see Carr.This surname has also absorbed examples of German Kehr.
Biblical
scribe, numbering
Biblical
or Timnath-serah, image of the sun; numbering of the rest
Surname or Lastname
English
English : according to Reaney this is a nickname from an unattested Old English word cybbe meaning ‘clumsy’ or ‘thickset’. Reaney’s speculation is apparently based on taking the Middle English word kibble ‘cudgel’ as a diminutive of an unattested Old English word. Corresponding personal names have been postulated for the place names Kibworth (‘enclosure of a man called Cybba’) and Kibblesworth (‘enclosure of a man called Cybbel’); so, in theory, the surname could be a reflex of these Old English personal names.North German : nickname for a cantankerous person, from Middle Low German, Middle High German kiven ‘to quarrel’.
Surname or Lastname
English
English : perhaps an occupational name for a maker of bottles or cups, from Old French gourde ‘water vessel’, ‘flask’, but possibly of the same derivation as 2.French : from Old French gourd ‘heavy’, ‘dull’, ‘sluggish’, hence a nickname for a slow lumbering person.
Boy/Male
Biblical
Right hand; numbering; preparing.
Surname or Lastname
English
English : from a short form of the personal names Giles, Julian, or William. In theory the name would have a soft initial when derived from the first two of these, and a hard one when from William or from the other possibilities discussed in 2–4 below. However, there has been much confusion over the centuries.Northern English : topographic name for someone who lived by a ravine or deep glen, Middle English gil(l), Old Norse gil ‘ravine’.Scottish and Irish : reduced Anglicized form of Gaelic Mac Gille (Scottish), Mac Giolla (Irish), patronymics from an occupational name for a servant or a short form of the various personal names formed by attaching this element to the name of a saint. See McGill. The Old Norse personal name Gilli is probably of this origin, and may lie behind some examples of the name in northern England.Scottish and Irish : reduced Anglicized form of Gaelic Mac An Ghoill (see Gall 1).Norwegian : habitational name from any of three farmsteads in western Norway named Gil, from Old Norse gil ‘ravine’.Dutch : cognate of Giles.Jewish (Israeli) : ornamental name from Hebrew gil ‘joy’.German : from a vernacular short form of the medieval personal name Aegidius (see Gilger).Indian (Panjab) : Sikh name, probably from Panjabi gil ‘moisture’, also meaning ‘prosperity’. There is a Jat tribe that bears this name; the Ramgarhia Sikhs also have a clan called Gill.
Girl/Female
Biblical
Numbering, showing, increase of tribute.
Girl/Female
Biblical
Image of the sun, numbering of the rest.
Biblical
Mispereth, numbering; showing; increase of tribute
Biblical
right hand; numbering; preparing
Surname or Lastname
English (mainly Gloucestershire), Dutch, and German (also Türk)
English (mainly Gloucestershire), Dutch, and German (also Türk) : from Middle English, Old French turc, Middle High and Low German Turc ‘Turk’, from Turkish türk. In theory this could be an ethnic name but, both in England and northwest Europe, it is generally a nickname for a person with black hair and a swarthy complexion or a cruel, rowdy, or unruly person. The Dutch and German surname also represents a house name, derived from the use of a picture of a Turk as a house sign. It is also found as a nickname for someone who had taken part in the wars against the Turks.English : from a medieval personal name, a back-formation from Turkel, misanalyzed as containing the Old French diminutive suffix -el.Scottish : reduced Anglicized form of Gaelic Mac Tuirc, a patronymic from the byname Torc ‘boar’.Jewish (Ashkenazic) : ethnic name denoting someone from Turkey or anywhere in the Ottoman Empire, or a nickname for someone thought to resemble a Turk.Americanized form of the Greek ethnic name Tourkos ‘Turk’. See also Turco.
Girl/Female
Biblical
Image of the sun, numbering of the rest.
Surname or Lastname
English, Scottish, and Irish (of Norman origin)
English, Scottish, and Irish (of Norman origin) : of disputed origin. It may be from a Celtic personal name derived from the element cam ‘bent’, ‘crooked’ (compare Cameron and Campbell). This was relatively frequent in Norfolk, Lincolnshire, and Yorkshire in the 12th and 13th centuries, perhaps as a result of Breton immigration. According to another theory it is a habitational name from Comines near Lille, but there is no evidence for this (no early forms with de have been found). In southern Ireland this Anglo-Norman name has been confused with 2.Irish : Anglicized form of Gaelic Mac CuimÃn (or Ó CuimÃn) ‘son (or ‘descendant’) of CuimÃn’, a personal name formed from a diminutive of cam ‘crooked’.Americanized form of French Canadian Vien, Viens, based on the misconception that these derive from French venire ‘to come’.
Surname or Lastname
English
English : unexplained. It may be a variant of a medieval name, Preville, a habitational name from a Norman place named with the elements pré ‘meadow’ + ville ‘settlement’. However, this theory is not supported by evidence of early forms.
Boy/Male
Biblical
Scribe, numbering'.
NUMBERING COMPUTABILITY-THEORY
NUMBERING COMPUTABILITY-THEORY
NUMBERING COMPUTABILITY-THEORY
NUMBERING COMPUTABILITY-THEORY
NUMBERING COMPUTABILITY-THEORY
NUMBERING COMPUTABILITY-THEORY
NUMBERING COMPUTABILITY-THEORY