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Theorem in computability theory
Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were
Kleene's_recursion_theorem
Topics referred to by the same term
Recursion theorem can refer to: The recursion theorem in set theory Kleene's recursion theorem, also called the fixed point theorem, in computability
Recursion_theorem
American mathematician (1909–1994)
hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions
Stephen_Cole_Kleene
On transforming a program by substituting constants for free variables
3 g42)), where g42 is a "fresh" symbol. Currying Kleene's recursion theorem Partial evaluation Kleene, S. C. (1936). "General recursive functions of natural
Smn_theorem
Limitative results in mathematical logic
results about undecidable sets in recursion theory. Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of computability
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Condition for a mathematical function to map some value to itself
computability theory, by applying Kleene's recursion theorem. These results are not equivalent theorems; the Knaster–Tarski theorem is a much stronger result
Fixed-point_theorem
Use of functions that call themselves
recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves
Recursion_(computer_science)
Kanamori–McAloon theorem (mathematical logic) Kirby–Paris theorem (proof theory) Kleene's recursion theorem (recursion theory) König's theorem (set theory
List_of_theorems
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
Theorem in computability theory
Q_{e}(x)=\varphi _{a}(x)} when e ∉ P {\displaystyle e\notin P} . By Kleene's recursion theorem, there exists e {\displaystyle e} such that φ e = Q e {\displaystyle
Rice's_theorem
calculus Church–Rosser theorem Calculus of constructions Combinatory logic Post correspondence problem Kleene's recursion theorem Recursively enumerable
List of mathematical logic topics
List_of_mathematical_logic_topics
Characterization of hyperarithmetic sets
In effective descriptive set theory, the Suslin–Kleene theorem characterizes the hyperarithmetic subsets of N {\displaystyle \mathbb {N} } . Informally
Suslin–Kleene_theorem
Self-replicating program
Turing-complete programming language, as a direct consequence of Kleene's recursion theorem. For amusement, programmers sometimes attempt to develop the shortest
Quine_(computing)
Topics referred to by the same term
first incompleteness theorem Tarski's undefinability theorem Halting problem Kleene's recursion theorem Lawvere's fixed-point theorem (categorical generalization
Diagonal_argument
Problem in computer science
1965, p. 115 Lucas 2021. Kleene 1952, p. 382. Rosser, "Informal Exposition of Proofs of Gödel's Theorem and Church's Theorem", reprinted in Davis 1965
Halting_problem
Generalization of Rice's theorem
p {\displaystyle p} can get access to its own source code by Kleene's recursion theorem). If this eventually returns true, then this first task continues
Rice–Shapiro_theorem
Impossible task in computing
impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of First-order logic, a statement is universally valid if and only if it
Entscheidungsproblem
Thesis on the nature of computability
machine, or λ-function, or carefully invoke recursion axioms, or at best, cleverly invoke various theorems of computability theory. But because the computability
Church–Turing_thesis
Subfield of mathematics
Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method
Mathematical_logic
Turing machine that halts for any input
the index of such a machine. Build a Turing machine M, using Kleene's recursion theorem, that on input 0 first simulates the machine with index e running
Decider_(Turing_machine)
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Smallest fixed point of a function from a poset
not converge with the least fixed point. Unfortunately, whereas Kleene's recursion theorem shows that the least fixed point is effectively computable, the
Least_fixed_point
One of several equivalent definitions of a computable function
of primitive recursion as those do not provide a mechanism for "infinite loops" (undefined values). A normal form theorem due to Kleene says that for
General_recursive_function
Statement in mathematical logic
yet developed in 1934. The diagonal lemma is closely related to Kleene's recursion theorem in computability theory, and their respective proofs are similar
Diagonal_lemma
Theorem about fixed points of multiple variables
computability theory, Bekić's theorem or Bekić's lemma is a theorem about fixed-points which allows splitting a mutual recursion into recursions on one variable at
Bekić's_theorem
Logical principle
(see Nouveaux Essais, IV,2)" (ibid p 421) The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica
Law_of_excluded_middle
studied because several important results like the Kleene's recursion theorem and Rice's theorem, which were originally proven for the Gödel-numbered
Complete_numbering
Function computable with bounded loops
mathematics before, but the construction of primitive recursion is traced back to Richard Dedekind's theorem 126 of his Was sind und was sollen die Zahlen? (1888)
Primitive_recursive_function
sequences, and structures. recursion theorem 1. Master theorem (analysis of algorithms) 2. Kleene's recursion theorem recursive definition A definition
Glossary_of_logic
Programming paradigm based on applying and composing functions
Darlington developed the functional language NPL. NPL was based on Kleene Recursion Equations and was first introduced in their work on program transformation
Functional_programming
Mathematical-logic system
calculus may be used to model arithmetic, Booleans, data structures, and recursion, as illustrated in the following sub-sections i, ii, iii, and § iv. There
Lambda_calculus
Fixed-point theorem
mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed-point theorem for partially ordered sets
Bourbaki–Witt_theorem
Sequence of operations for a task
Reprinted in The Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An
Algorithm
Order type of the set of all recursive ordinals
Mathematics, 19 (2): 213–262, doi:10.1016/0001-8708(76)90187-0 P. G. Hinman, Recursion-Theoretic Hierarchies (1978), pp.419--420. Perspectives in Mathematical
Nonrecursive_ordinal
Non-contradiction of a theory
and this formula is said to be (formally) provable or be a (formal) theorem" cf Kleene 1952, p. 83. Carnielli, Walter; Coniglio, Marcelo Esteban (2016).
Consistency
Generalization of Turing computability
Embedding Theorem for the automorphism group of the α-enumeration degrees (p. 27), Ph.D. thesis, University of Leeds, 2019. C. T. Chong, L. Yu, Recursion Theory:
Hyperarithmetical_theory
Principle of interchangeability of data and code
of creating a malformed program. In computational theory, Kleene's second recursion theorem provides a form of code-is-data, by proving that a program
Code_as_data
Mathematical logic concept
that Goodstein's theorem cannot be proven in Peano arithmetic. Their proof was based on Gentzen's theorem. Gentzen (1936). See Kleene (2009) harvtxt error:
Gentzen's_consistency_proof
Study of mathematics itself
finitary methods are used to study various axiomatized mathematical theorems (Kleene 1952, p. 55). Other prominent figures in the field include Bertrand
Metamathematics
nowhere is recursion mentioned. The proof of the equivalence of machine-computability and recursion must wait for Kleene 1943 and 1952: "The theorem that all
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Computation model defining an abstract machine
Church and his two students Stephen Kleene and J. B. Rosser by use of Church's lambda-calculus and Gödel's recursion theory (1934). Church's paper (published
Turing_machine
Foundational controversy in twentieth-century mathematics
axiom. Rather, his recursion steps through integers assigned to variable k (cf his (2) on page 602). His skeleton-proof of Theorem V, however, "use(s)
Brouwer–Hilbert_controversy
Mathematical function that can be computed by a program
and projection functions, and is closed under composition, primitive recursion, and the μ operator. Equivalently, computable functions can be formalized
Computable_function
Basic framework of mathematics
generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical
Foundations_of_mathematics
Hierarchy of complexity classes for formulas defining sets
arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain
Arithmetical_hierarchy
Generalization of "n-th" to infinite cases
theorems but also to define functions on ordinals. This is known as transfinite recursion. Formally, a function F is defined by transfinite recursion
Ordinal_number
System of formal deduction in logic
other logics as well. It is defined as a deductive system that generates theorems from axioms and inference rules, especially if the only postulated inference
Hilbert_system
{\displaystyle (f)\leq _{e}} graph ( g ) . {\displaystyle (g).} Kleene's recursion theorem introduces the notion of relative partial recursiveness, which
Enumeration_reducibility
Mathematical model for deduction or proof systems
formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the
Formal_system
Size of a possibly infinite set
cannot happen with proper subsets of finite sets. However, a fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to
Cardinal_number
Complexity class used to classify decision problems
only known strict inclusions come from the time hierarchy theorem and the space hierarchy theorem, and respectively they are N P ⊊ N E X P T I M E {\displaystyle
NP_(complexity)
Paradox in set theory
already realized that his theory would lead to a contradiction (to Cantor's theorem), as he told Hilbert and Richard Dedekind by letter. Hilbert also formulated
Russell's_paradox
Ordinals in mathematics and set theory
Accessed 2022-12-01. Barwise (1976), theorem 7.2. Simpson, Stephen G. (1978-01-01). "Short Course on Admissible Recursion Theory". Studies in Logic and the
Large_countable_ordinal
Type of logical system
to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization
First-order_logic
Particular class of sets which can be described entirely in terms of simpler sets
Barwise (1975), p. 60, comment following proof of theorem 5.9. P. Odifreddi, Classical Recursion Theory, pp.427. Studies in Logic and the Foundations
Constructible_universe
System including an indeterminate value
or false, but in many cases we don't know which. Similarly, Stephen Cole Kleene used a third value to represent predicates that are "undecidable by [any]
Three-valued_logic
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
Logical quantifier
edu. Retrieved 2019-12-14. This is a consequence of the compactness theorem. Kleene, Stephen (1952). Introduction to Metamathematics. Ishi Press International
Uniqueness_quantification
Sequence of words formed by specific rules
The last sentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions
Formal_language
Proof by Alan Turing
to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture
Turing's_proof
Branch of mathematics
effective descriptive set theory combines descriptive set theory with recursion theory. An effective Polish space is a complete separable metric space
Effective descriptive set theory
Effective_descriptive_set_theory
Computer science and logic conference
fragment of intuitionistic linear logic" Dexter Kozen, "A completeness theorem for Kleene algebras and the algebra of regular events" Thomas Henzinger, Xavier
Symposium on Logic in Computer Science
Symposium_on_Logic_in_Computer_Science
Whether a decision problem has an effective method to derive the answer
are not adequately represented by the set of theorems alone. (For example, Kleene's logic has no theorems at all.) In such cases, alternative definitions
Decidability_(logic)
ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal is an admissible ordinal. By a theorem of Sacks, the countable admissible
Admissible_ordinal
Formal language
only if L {\displaystyle L} is also recursive. Computably enumerable set Recursion Sipser, Michael (1997). Introduction to the Theory of Computation (1st ed
Recursively enumerable language
Recursively_enumerable_language
functions for inversion. Theorem: Any function constructible via the clauses of primitive recursion using the standard primitive recursion schema is constructible
Gödel's_β_function
Measure of unsolvability
on some Tn such that machines <i that halt on X do so <n−i steps (by recursion, this is uniformly computable from 0′). X is noncomputable since otherwise
Turing_degree
3-volume treatise on mathematics, 1910–1913
set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume
Principia_Mathematica
Kind of proof calculus
language L {\displaystyle {\mathcal {L}}} is usually defined (here: by recursion) as follows: Each propositional variable is a formula. " ⊥ {\displaystyle
Natural_deduction
Italian mathematician
types and proved a completeness theorem for type checking using a model that was created based on the idea of recursion theory. Longo's research in the
Giuseppe_Longo
Mathematical logic concept
of the Löwenheim–Skolem theorem; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity
Skolem's_paradox
1931 paper by Kurt Gödel
to prove the incompleteness theorems. The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact
On Formally Undecidable Propositions of Principia Mathematica and Related Systems
On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems
Attempts to formalize the concept of algorithms
"machine computable" then it is "hand-calculable by partial recursion". Kleene's Theorem XXIX : "Theorem XXIX: "Every computable partial function φ is partial
Algorithm_characterizations
In logic, a statement which is always true
is complete if every tautology is a theorem (derivable from axioms). An axiomatic system is sound if every theorem is a tautology. The problem of constructing
Tautology_(logic)
Theory of truth in the philosophy of language
notably Tarski's undefinability theorem using the same formal technique Kurt Gödel used in his incompleteness theorems. Roughly, this states that a truth-predicate
Semantic_theory_of_truth
Enderton, Herbert B. (2010), Computability Theory: An Introduction to Recursion Theory, Academic Press, ISBN 978-0-12-384958-8. Joseph, Deborah; Young
Creative_and_productive_sets
Computer science and recursion theory
In computer science and recursion theory the McCarthy formalism (1963) of computer scientist John McCarthy clarifies the notion of recursive functions
McCarthy_Formalism
Mathematical technique used in proof theory
elimination). ACA0, arithmetical comprehension. ATR0, arithmetical transfinite recursion. Martin-Löf type theory with arbitrarily many finite level universes.
Ordinal_analysis
Mathematical set of all subsets of a set
existential quantifier is the left adjoint. Cantor's theorem Family of sets Field of sets Combination Kleene star The notation 2S, meaning the set of all functions
Power_set
Functions in computability theory
characteristic function of the predicate T {\displaystyle T} from the Kleene normal form theorem are definable in a way such that they lie at level E 0 {\displaystyle
Grzegorczyk_hierarchy
Finite or infinite ordered list of elements
using recursion. This is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one
Sequence
Symbolic description of a mathematical object
arithmetical operations, the logarithm and the exponential (Richardson's theorem). The earliest written mathematics likely began with tally marks, where
Expression_(mathematics)
Syntactically correct logical formula
mathematical software such as model checkers, automated theorem provers, interactive theorem provers) tend to retain of the notion of formula only the
Well-formed_formula
Countable ordinal that is the order type of a computable well-ordering of natural numbers
Computability, MIT Press, ISBN 0-07-053522-1 Sacks, Gerald (1990), Higher Recursion Theory, Perspectives in mathematical logic, Springer-Verlag, ISBN 0-387-19305-7
Computable_ordinal
Relationship between programs and proofs
advocated by total functional programming, is to eliminate unrestricted recursion (and forgo Turing completeness, although still retaining high computational
Curry–Howard_correspondence
Relationship in which one statement follows from another
introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935). Syntactic consequence does not depend on any interpretation
Logical_consequence
Mathematical logic hierarchy
Church–Kleene ordinal in the definition of the lightface hierarchy. Projective hierarchy Wadge hierarchy Veblen hierarchy P. G. Hinman, *Recursion-Theoretic
Borel_hierarchy
Axiom
"function" and "computable" of the theory at hand. A common context is recursion theory as established since the 1930's. Adopting C T {\displaystyle {\mathrm
Church's thesis (constructive mathematics)
Church's_thesis_(constructive_mathematics)
Abstract machine used in a formal logic and theoretical computer science
function) Successor function Identity function Composition function Primitive recursion (induction) μ operator (unbounded search operator) The authors show that
Counter_machine
Size of a set in mathematics
are proven to be uncountable by so-called diagonal arguments. Cantor's theorem generalizes these arguments to show there is an infinite hierarchy of infinities
Cardinality
Method of deriving conclusions
inferential steps and often use various rules of inference to establish the theorem they intend to demonstrate. Rules of inference are definitory rules—rules
Rule_of_inference
Axiom in Russell's ramified theory of types
that "have the character of axioms, and certain recursion axioms that result from a general recursion schema" plus some formation rules that "govern the
Axiom_of_reducibility
Concept in logic
to Axiomatic Set Theory (1982) by Gaisi Takeuti and Wilson M. Zaring. Theorem—if a = b {\displaystyle a=b} , then, for any well-formed formula ϕ {\displaystyle
Substitution_(logic)
Academic subfield of computer science
theory is closely related to the branch of mathematical logic called recursion theory, which removes the restriction of studying only models of computation
Theory_of_computation
Leopold Löwenheim publishes a proof of the (downward) Löwenheim-Skolem theorem, implicitly using the axiom of choice. 1918 - C. I. Lewis writes A Survey
Timeline of mathematical logic
Timeline_of_mathematical_logic
School of thought in philosophy of mathematics
incompleteness theorems are 'proved with logic just like any other theorems'. However, that argument appears not to acknowledge the distinction between theorems of
Logicism
Informal set theories
they do exclude some paradoxes, like Russell's paradox. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at
Naive_set_theory
Basic notion of sameness in mathematics
scientists like John Alan Robinson in their work on resolution and automated theorem proving. The substitution property can produce false statements when applied
Equality_(mathematics)
Axiomatic set theories based on the principles of mathematical constructivism
{\displaystyle g(Sn)=f(g(n))} . This iteration- or recursion principle is akin to the transfinite recursion theorem, except it is restricted to set functions and
Constructive_set_theory
Mathematical theory of data types
inductive types. Two methods of generating inductive types are induction-recursion and induction-induction. A method that only uses lambda terms is Scott
Type_theory
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KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
KLEENES RECURSION-THEOREM
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