Kleene’s recursion theorem
E607898
Kleene’s recursion theorem is a fundamental result in computability theory that guarantees the existence of self-referential programs, allowing a program to effectively obtain and use its own description.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kleene fixed-point theorem | 1 |
| Kleene’s recursion theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6594144 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Kleene’s recursion theorem Context triple: [Stephen Kleene, notableConcept, Kleene’s recursion theorem]
-
A.
Kleene’s normal form theorem
Kleene’s normal form theorem is a fundamental result in computability theory that characterizes all partial recursive (effectively computable) functions using a universal primitive recursive function and the μ-operator.
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B.
Rice's theorem
Rice's theorem is a fundamental result in computability theory stating that any non-trivial semantic property of the language recognized by a Turing machine is undecidable.
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C.
Löb's theorem
Löb's theorem is a fundamental result in mathematical logic that characterizes when a sufficiently strong formal system can prove statements about its own provability, closely refining the insights of Gödel’s incompleteness theorems.
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D.
Kleene numbering
Kleene numbering is a method in computability theory for effectively assigning natural numbers to partial recursive functions, refining Gödel numbering to study algorithmic properties of functions.
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E.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Kleene’s recursion theorem Target entity description: Kleene’s recursion theorem is a fundamental result in computability theory that guarantees the existence of self-referential programs, allowing a program to effectively obtain and use its own description.
-
A.
Kleene’s normal form theorem
Kleene’s normal form theorem is a fundamental result in computability theory that characterizes all partial recursive (effectively computable) functions using a universal primitive recursive function and the μ-operator.
-
B.
Rice's theorem
Rice's theorem is a fundamental result in computability theory stating that any non-trivial semantic property of the language recognized by a Turing machine is undecidable.
-
C.
Löb's theorem
Löb's theorem is a fundamental result in mathematical logic that characterizes when a sufficiently strong formal system can prove statements about its own provability, closely refining the insights of Gödel’s incompleteness theorems.
-
D.
Kleene numbering
Kleene numbering is a method in computability theory for effectively assigning natural numbers to partial recursive functions, refining Gödel numbering to study algorithmic properties of functions.
-
E.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | theorem in computability theory ⓘ |
| appearsInWorkOf | Stephen Cole Kleene NERFINISHED ⓘ |
| appliesTo |
Gödel numbers of programs
ⓘ
Turing machine indices ⓘ partial computable functions ⓘ |
| field |
computability theory
ⓘ
mathematical logic ⓘ theoretical computer science ⓘ |
| formalizes | self-referential definitions of computable functions ⓘ |
| guarantees |
existence of fixed points for computable operators on indices
ⓘ
existence of self-referential programs ⓘ |
| hasConsequence |
existence of fixed points for effective transformations of programs
ⓘ
no general method can avoid self-reference in certain constructions ⓘ |
| hasFormulation | for every total computable function f on program indices there exists e such that φ_e = φ_{f(e)} ⓘ |
| hasVariant |
parameterized recursion theorem
NERFINISHED
ⓘ
second recursion theorem NERFINISHED ⓘ |
| historicalContext | developed in mid-20th century ⓘ |
| implies |
a program can obtain its own description
ⓘ
a program can use its own description in its computation ⓘ |
| influenced |
formal study of computer viruses
ⓘ
theory of self-reproducing automata ⓘ |
| involvesConcept |
Gödel numbering
ⓘ
effective enumeration of partial computable functions ⓘ fixed point of a computable operator ⓘ |
| isAnalogousTo |
fixed-point theorems in domain theory
ⓘ
fixed-point theorems in logic such as the diagonal lemma ⓘ |
| isPrerequisiteFor |
formal constructions of quines in programming language theory
ⓘ
understanding Rice’s theorem ⓘ |
| isTaughtIn |
advanced logic courses
ⓘ
graduate courses on computability theory ⓘ |
| isToolFor |
analyzing semantics of programming languages
ⓘ
studying properties of computable functions ⓘ |
| namedAfter | Stephen Cole Kleene NERFINISHED ⓘ |
| relatedTo |
Kleene’s fixed-point theorem
NERFINISHED
ⓘ
Rice’s theorem NERFINISHED ⓘ quines ⓘ s-m-n theorem NERFINISHED ⓘ self-reference in computation ⓘ self-reproducing programs ⓘ |
| strengthens | basic diagonalization techniques ⓘ |
| supports | construction of diagonal arguments in computability ⓘ |
| usedIn |
construction of quines
ⓘ
formalization of viruses and self-replicating code ⓘ proofs involving program self-inspection ⓘ proofs of undecidability results ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Kleene’s recursion theorem Description of subject: Kleene’s recursion theorem is a fundamental result in computability theory that guarantees the existence of self-referential programs, allowing a program to effectively obtain and use its own description.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kleene fixed-point theorem