PCP theorem
E74456
The PCP theorem is a fundamental result in computational complexity theory stating that every problem in NP has probabilistically checkable proofs that can be verified by examining only a constant number of bits, with major implications for the hardness of approximation.
All labels observed (3)
| Label | Occurrences |
|---|---|
| PCP theorem canonical | 6 |
| NP in terms of probabilistically checkable proofs | 1 |
| Probabilistically Checkable Proofs theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T594763 — 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: PCP theorem Context triple: [Interactive Proofs and the Hardness of Approximating Cliques, relatedTo, PCP theorem]
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A.
Interactive Proofs and the Hardness of Approximating Cliques
"Interactive Proofs and the Hardness of Approximating Cliques" is a seminal theoretical computer science paper that introduced powerful interactive proof techniques to show that finding near-maximum cliques in graphs is computationally intractable to approximate within strong bounds.
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B.
The Knowledge Complexity of Interactive Proof Systems
"The Knowledge Complexity of Interactive Proof Systems" is a seminal theoretical computer science paper that introduced the notion of zero-knowledge proofs, fundamentally shaping modern cryptography and complexity theory.
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C.
Randomness and Computation
"Randomness and Computation" is Shafi Goldwasser's influential doctoral thesis that helped lay the foundations of modern complexity theory and cryptography by rigorously exploring the role of randomness in efficient computation.
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D.
Introduction to the Theory of Computation
Introduction to the Theory of Computation is a widely used textbook in theoretical computer science that covers formal languages, automata, computability, and complexity theory.
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E.
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).
- 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: PCP theorem Target entity description: The PCP theorem is a fundamental result in computational complexity theory stating that every problem in NP has probabilistically checkable proofs that can be verified by examining only a constant number of bits, with major implications for the hardness of approximation.
-
A.
Interactive Proofs and the Hardness of Approximating Cliques
"Interactive Proofs and the Hardness of Approximating Cliques" is a seminal theoretical computer science paper that introduced powerful interactive proof techniques to show that finding near-maximum cliques in graphs is computationally intractable to approximate within strong bounds.
-
B.
The Knowledge Complexity of Interactive Proof Systems
"The Knowledge Complexity of Interactive Proof Systems" is a seminal theoretical computer science paper that introduced the notion of zero-knowledge proofs, fundamentally shaping modern cryptography and complexity theory.
-
C.
Randomness and Computation
"Randomness and Computation" is Shafi Goldwasser's influential doctoral thesis that helped lay the foundations of modern complexity theory and cryptography by rigorously exploring the role of randomness in efficient computation.
-
D.
Introduction to the Theory of Computation
Introduction to the Theory of Computation is a widely used textbook in theoretical computer science that covers formal languages, automata, computability, and complexity theory.
-
E.
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).
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf | theorem in computational complexity theory ⓘ |
| characterizes |
NP in terms of probabilistically checkable proofs
ⓘ
linked to:
PCP theorem
|
| field |
computational complexity theory
ⓘ
theoretical computer science ⓘ |
| fullName |
Probabilistically Checkable Proofs theorem
ⓘ
linked to:
PCP theorem
|
| hasConsequence |
development of modern hardness-of-approximation theory
ⓘ
gap amplification techniques in reductions ⓘ tight inapproximability results for many classical optimization problems ⓘ |
| hasParameter |
completeness
ⓘ
query complexity ⓘ randomness complexity ⓘ soundness error ⓘ |
| historicalDevelopment | evolved from work on interactive proofs and multi-prover interactive proofs ⓘ |
| implies |
many NP-optimization problems are hard to approximate within certain constant factors
ⓘ
strong inapproximability bounds for Max-3SAT ⓘ strong inapproximability bounds for Max-Clique ⓘ strong inapproximability bounds for Set Cover ⓘ strong inapproximability bounds for Vertex Cover ⓘ strong inapproximability bounds for various constraint satisfaction problems ⓘ there exist fixed constants for which approximating some NP-hard problems within those factors is NP-hard ⓘ |
| importance |
central to understanding limits of efficient approximation algorithms
ⓘ
one of the most influential results in theoretical computer science of the 1990s ⓘ |
| proofType |
adaptive probabilistically checkable proofs
ⓘ
non-adaptive probabilistically checkable proofs ⓘ |
| relatedConcept |
gap-preserving reductions
ⓘ
locally decodable codes ⓘ locally testable codes ⓘ |
| relatedTheorem |
IP equals PSPACE
ⓘ
MIP equals NEXP ⓘ |
| relatesToClass |
NP
ⓘ
PCP ⓘ |
| relatesToConcept |
NP
ⓘ
NP-completeness ⓘ approximation algorithms ⓘ error-correcting codes ⓘ gap-introducing reductions ⓘ hardness of approximation ⓘ inapproximability results ⓘ interactive proofs ⓘ probabilistically checkable proofs ⓘ |
| statesThat |
NP equals PCP(log n, 1)
ⓘ
every language in NP has a probabilistically checkable proof verifiable by reading only a constant number of bits of the proof ⓘ |
| typicalFormulation | NP equals PCP(O(log n), O(1)) ⓘ |
| usesResource |
constant query complexity
ⓘ
logarithmic randomness ⓘ randomness ⓘ |
| verificationModel | randomized polynomial-time verifier ⓘ |
| verificationProperty |
verifier reads only a constant number of bits of the proof
ⓘ
verifier uses logarithmically many random bits in the input size ⓘ |
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: PCP theorem Description of subject: The PCP theorem is a fundamental result in computational complexity theory stating that every problem in NP has probabilistically checkable proofs that can be verified by examining only a constant number of bits, with major implications for the hardness of approximation.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: PCP theorem
linked to: PCP theorem
subject linked to:
“Inapproximability results for SAT and other problems”
subject linked to:
“Inapproximability results for SAT and other problems”
subject linked to:
MIP equals NEXP