Håstad’s switching lemma
E120579
Håstad’s switching lemma is a fundamental result in computational complexity theory that provides powerful bounds on the simplification of Boolean formulas under random restrictions, with major applications in circuit lower bounds.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Håstad switching lemma | 1 |
| Håstad’s multi-switching lemma | 1 |
| Håstad’s switching lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1043641 — 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: Håstad’s switching lemma Context triple: [Johan Håstad, knownFor, Håstad’s switching lemma]
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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.
-
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.
Blum complexity measures
Blum complexity measures are a formal framework in computational complexity theory that rigorously define and compare the resource usage (such as time or space) of algorithms via axiomatic conditions.
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D.
PCP theorem
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.
-
E.
Blum axioms
Blum axioms are a set of formal conditions introduced by Manuel Blum that rigorously define what constitutes a valid complexity measure in computational complexity theory.
- 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: Håstad’s switching lemma Target entity description: Håstad’s switching lemma is a fundamental result in computational complexity theory that provides powerful bounds on the simplification of Boolean formulas under random restrictions, with major applications in circuit lower bounds.
-
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.
Blum complexity measures
Blum complexity measures are a formal framework in computational complexity theory that rigorously define and compare the resource usage (such as time or space) of algorithms via axiomatic conditions.
-
D.
PCP theorem
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.
-
E.
Blum axioms
Blum axioms are a set of formal conditions introduced by Manuel Blum that rigorously define what constitutes a valid complexity measure in computational complexity theory.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in computational complexity theory ⓘ |
| appearsIn | Johan Håstad’s work on almost optimal lower bounds for small-depth circuits ⓘ |
| appliesTo |
CNF formulas
ⓘ
DNF formulas ⓘ small-depth Boolean circuits ⓘ |
| assumption | random restriction chosen independently for each variable ⓘ |
| characterizes | simplification of Boolean formulas under random restrictions ⓘ |
| field |
circuit complexity
ⓘ
computational complexity theory ⓘ theoretical computer science ⓘ |
| implies | that random restrictions often yield shallow decision trees ⓘ |
| importance |
central tool in modern circuit lower bound proofs
ⓘ
foundational result in the theory of AC0 circuits ⓘ |
| influenced | subsequent switching lemmas and refinements ⓘ |
| mainConcept |
Boolean formulas
ⓘ
circuit lower bounds ⓘ decision trees ⓘ random restrictions ⓘ |
| namedAfter | Johan Håstad ⓘ |
| provides | probabilistic bounds on decision tree depth after restriction ⓘ |
| relatedTo |
Ajtai’s lower bounds for constant-depth circuits
ⓘ
Fourier analysis of Boolean functions ⓘ Furst–Saxe–Sipser lower bounds ⓘ random restriction method ⓘ |
| relates | width of CNF or DNF to resulting decision tree depth ⓘ |
| shows |
probability that restricted formula has large decision tree depth is exponentially small in depth bound
ⓘ
random restrictions simplify small-width CNF and DNF formulas ⓘ |
| strengthenedBy |
Håstad’s switching lemma
self-linksurface differs
ⓘ
surface form:
Håstad’s multi-switching lemma
multi-switching lemmas ⓘ |
| toolFor |
“Almost optimal lower bounds for small depth circuits”
ⓘ
surface form:
Håstad’s optimal lower bounds for small-depth circuits
Razborov–Smolensky style arguments for AC0[⊕] ⓘ |
| typicalConclusion | restricted formula has small-depth decision tree with high probability ⓘ |
| typicalDomain | formulas of bounded width ⓘ |
| usedFor |
AC0 circuit lower bounds
ⓘ
correlation bounds between AC0 circuits and parity ⓘ hardness amplification arguments in circuit complexity ⓘ hierarchy theorems for small-depth circuits ⓘ proving lower bounds for constant-depth circuits ⓘ pseudorandomness constructions for AC0 ⓘ showing limitations of small-depth circuits computing majority ⓘ showing limitations of small-depth circuits computing parity ⓘ |
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: Håstad’s switching lemma Description of subject: Håstad’s switching lemma is a fundamental result in computational complexity theory that provides powerful bounds on the simplification of Boolean formulas under random restrictions, with major applications in circuit lower bounds.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Håstad’s multi-switching lemma
subject surface form:
Almost optimal lower bounds for small depth circuits
this entity surface form:
Håstad switching lemma