Rice's theorem
E588867
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Rice's theorem canonical | 2 |
| Rice–Shapiro theorem | 1 |
| Rice’s theorem | 1 |
| undecidability theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6370985 — 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: Rice's theorem Context triple: [Halting problem, relatedTo, Rice's theorem]
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A.
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.
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B.
Church–Turing thesis
The Church–Turing thesis is a foundational principle in computability theory stating that any function that can be effectively computed by an algorithm can be computed by a Turing machine (or equivalently by other formal models of computation).
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C.
Halting problem
The halting problem is a fundamental decision problem in computability theory that asks whether a given program will eventually stop running or continue to run forever, and is famously proven to be undecidable.
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D.
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two fundamental results in mathematical logic showing that any sufficiently powerful, consistent formal system cannot prove all true statements about arithmetic, and cannot prove its own consistency.
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E.
Cook–Levin theorem
The Cook–Levin theorem is a foundational result in computational complexity theory that established the Boolean satisfiability problem (SAT) as the first NP-complete problem, launching the theory of NP-completeness.
- 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: Rice's theorem Target entity description: 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.
-
A.
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.
-
B.
Church–Turing thesis
The Church–Turing thesis is a foundational principle in computability theory stating that any function that can be effectively computed by an algorithm can be computed by a Turing machine (or equivalently by other formal models of computation).
-
C.
Halting problem
The halting problem is a fundamental decision problem in computability theory that asks whether a given program will eventually stop running or continue to run forever, and is famously proven to be undecidable.
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D.
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two fundamental results in mathematical logic showing that any sufficiently powerful, consistent formal system cannot prove all true statements about arithmetic, and cannot prove its own consistency.
-
E.
Cook–Levin theorem
The Cook–Levin theorem is a foundational result in computational complexity theory that established the Boolean satisfiability problem (SAT) as the first NP-complete problem, launching the theory of NP-completeness.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | theorem in computability theory ⓘ |
| appliesTo |
Turing machines
ⓘ
linked to:
Turing machine
indices of partial computable functions in acceptable numberings ⓘ partial recursive functions ⓘ recursively enumerable sets ⓘ |
| assumes |
effective enumeration of partial computable functions
ⓘ
standard model of Turing computability ⓘ |
| characterizes | undecidability of non-trivial semantic properties of computable functions ⓘ |
| concerns |
languages recognized by Turing machines
ⓘ
semantic properties of partial computable functions ⓘ undecidability ⓘ |
| countryOfOrigin |
United States
ⓘ
linked to:
United States of America
|
| defines | non-trivial property as one that holds for at least one and not all computable functions ⓘ |
| excludes | trivial properties that hold for all or no computable functions ⓘ |
| field |
computability theory
ⓘ
mathematical logic ⓘ theoretical computer science ⓘ |
| hasConsequence |
many questions about program behavior are algorithmically unsolvable
ⓘ
no algorithm can decide any non-trivial semantic property of Turing-recognizable languages ⓘ |
| implies |
undecidability of many program analysis problems
ⓘ
undecidability of non-trivial properties of program output sets ⓘ undecidability of program equivalence ⓘ |
| importance | fundamental result in computability theory ⓘ |
| involvesConcept |
decidable set
ⓘ
many-one reduction ⓘ recursively enumerable sets of indices ⓘ semantic property ⓘ trivial property ⓘ |
| mainStatement | Every non-trivial semantic property of the language recognized by a Turing machine is undecidable ⓘ |
| namedAfter | Henry Gordon Rice ⓘ |
| proofTechnique |
diagonalization
ⓘ
reduction from the halting problem ⓘ |
| relatedTo |
Church–Turing thesis
ⓘ
Halting problem ⓘ Post's theorem ⓘ Recursion theorem ⓘ
linked to:
Kleene’s recursion theorem
Rice–Shapiro theorem ⓘ
linked to:
Rice's theorem
|
| scope | semantic properties rather than syntactic properties ⓘ |
| status | proven ⓘ |
| taughtIn |
graduate logic and recursion theory courses
ⓘ
undergraduate computability courses ⓘ |
| typeOf |
meta-theorem about computable functions
ⓘ
undecidability theorem ⓘ
linked to:
Rice's theorem
|
| typicalFormulation | For any non-trivial property of partial computable functions, the set of indices of functions with that property is undecidable ⓘ |
| usedIn |
computability theory textbooks
ⓘ
program verification theory ⓘ proofs of undecidability in programming language theory ⓘ static analysis impossibility 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: Rice's theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Rice's theorem
linked to: Rice's theorem
linked to: Rice's theorem