1
Boolean Analysis — Core Definitions
▶
1.1
Overview
1.2
The Boolean hypercube and functions
1.3
Uniform measure and expectation
1.4
Walsh–Fourier characters
1.5
Fourier coefficients
1.6
Walsh expansion
1.7
Orthonormality and Parseval’s identity
1.8
Coordinate influence
1.9
Noise operator
1.10
Notable Boolean functions
1.11
Sensitivity
1.12
Properties for Arrow’s theorem
1.13
Additional declarations
2
Arrow’s Impossibility Theorem
▶
2.1
Overview
2.2
Social choice definitions
2.3
Voter ordering model
2.4
Key correlation lemmas
2.5
Fourier formula for the cycle correlation
2.6
Acyclicity implies degree-1 structure
2.7
Degree-1 implies dictatorship
2.8
Arrow’s Impossibility Theorem
2.9
Additional declarations
3
Håstad’s Switching Lemma
▶
3.1
Overview
3.2
The formalised statement
3.3
Faithfulness critique
3.4
Dependency graph
3.5
Corollary: no small CNF approximations
4
Error-Correcting Codes — Core Definitions
▶
4.1
Overview
4.2
Basic objects: codewords and codes
4.3
Distance, weight, and rate
4.4
Basic lemma: distance vs. block length
4.5
Additional declarations
5
Singleton Bound
▶
5.1
Overview
5.2
Main result
6
Hamming Bound
▶
6.1
Overview
6.2
Volume formula
6.3
Disjointness of decoding balls
6.4
Hamming bound
7
Entropy and Asymptotic Bounds
▶
7.1
Overview
7.2
Asymptotic upper bound on ball size
7.3
Entropy algebra lemmas
7.4
Analytic helpers
7.5
Stirling-based binomial lower bound
7.6
Positivity of \(q\)-ary entropy
7.7
Additional declarations
8
Linear Codes and the Uniformity Lemma
▶
8.1
Overview
8.2
Distance equals minimum weight in linear codes
8.3
Uniform distributions and random matrices
8.4
Uniformity lemma
9
Gilbert–Varshamov Bound
▶
9.1
Overview
9.2
Probabilistic bound on low-weight outputs
9.3
Union bound over bad matrices
9.4
Gilbert–Varshamov bound
10
List Decoding
▶
10.1
Overview
10.2
List-decodable codes
10.3
Counting lemmas
10.4
List-decoding capacity
11
Johnson Bound
▶
11.1
Overview
11.2
Johnson bound
11.3
Additional declarations
12
CSS Codes (Not Yet Formalized)
▶
12.1
Overview
12.2
Symplectic vector space
12.3
CSS construction
12.4
CSS code parameters
13
Quantum Hamming Bound
▶
13.1
Overview
13.2
Pauli strings
13.3
\(n\)-qubit Hilbert space
13.4
Knill–Laflamme conditions
13.5
Error sphere and sphere-packing
13.6
Quantum Hamming bound
13.7
Additional declarations
14
Quantum Singleton Bound
▶
14.1
Overview
14.2
Symplectic vector space
14.3
Support, weight, and isotropic subspaces
14.4
Code parameters and erasure correctability
14.5
Key lemma: two disjoint correctable sets
14.6
Quantum Singleton bound
14.7
Additional declarations
15
Boolean Analysis — Gate Merge
▶
15.1
Overview
15.2
Declarations
16
Boolean Analysis — Bonami
▶
16.1
Overview
16.2
Declarations
17
Boolean Analysis — One Bit
▶
17.1
Overview
17.2
Declarations
18
Boolean Analysis — Simple
▶
18.1
Overview
18.2
Declarations
18.3
Additional declarations
19
Boolean Analysis — LMN
▶
19.1
Overview
19.2
Declarations
20
Boolean Analysis — Bernoulli Cost
▶
20.1
Overview
20.2
Declarations
21
Boolean Analysis — Circuit Compression
▶
21.1
Overview
21.2
Declarations
22
Boolean Analysis — Circuit Helpers
▶
22.1
Overview
22.2
Declarations
23
Boolean Analysis — Circuit Layer Reduction
▶
23.1
Overview
23.2
Declarations
24
Boolean Analysis — Circuit Reindex
▶
24.1
Overview
24.2
Declarations
25
Boolean Analysis — Circuit Tree Manip
▶
25.1
Overview
25.2
Declarations
26
Boolean Analysis — Compression Step
▶
26.1
Overview
26.2
Declarations
27
Boolean Analysis — Depth3 Switching
▶
27.1
Overview
27.2
Declarations
28
Boolean Analysis — Gate Switching
▶
28.1
Overview
28.2
Declarations
29
Boolean Analysis — Iterative Reduction
▶
29.1
Overview
29.2
Declarations
30
Boolean Analysis — Normal Form Conversion
▶
30.1
Overview
30.2
Declarations
31
Boolean Analysis — Recursive Reduction
▶
31.1
Overview
31.2
Declarations
32
Boolean Analysis — Restriction Compose
▶
32.1
Overview
32.2
Declarations
33
Boolean Analysis — Restriction Monotonicity
▶
33.1
Overview
33.2
Declarations
34
Boolean Analysis — Switching Bernoulli
▶
34.1
Overview
34.2
Declarations
35
Boolean Analysis — Switching
▶
35.1
Overview
35.2
Declarations
35.3
Additional declarations
36
Boolean Analysis — Bernoulli Restriction
▶
36.1
Overview
36.2
Declarations
37
Boolean Analysis — Canonical D Tree
▶
37.1
Overview
37.2
Declarations
38
Boolean Analysis — Circuit
▶
38.1
Overview
38.2
Declarations
38.3
Base literal, formula, and decision-tree definitions
39
Boolean Analysis — Encoding
▶
39.1
Overview
39.2
Declarations
40
Boolean Analysis — Encoding Properties
▶
40.1
Overview
40.2
Declarations
40.3
Additional declarations
41
Boolean Analysis — Restriction
▶
41.1
Overview
41.2
Declarations
42
Boolean Analysis — Round Trip
▶
42.1
Overview
42.2
Declarations
43
Communication Complexity — Det Basic
▶
43.1
Overview
43.2
Declarations
44
Communication Complexity — Det Complexity
▶
44.1
Overview
44.2
Declarations
45
Communication Complexity — Det Composition
▶
45.1
Overview
45.2
Declarations
46
Communication Complexity — Det Rectangle
▶
46.1
Overview
46.2
Declarations
47
Communication Complexity — Finite Message
▶
47.1
Overview
47.2
Declarations
48
Communication Complexity — Func Disjointness
▶
48.1
Overview
48.2
Declarations
49
Communication Complexity — Helper
▶
49.1
Overview
49.2
Declarations
50
Communication Complexity — One Way
▶
50.1
Overview
50.2
Declarations
51
Communication Complexity — Rectangle
▶
51.1
Overview
51.2
Declarations
52
Communication Complexity — Subprotocol
▶
52.1
Overview
52.2
Declarations
53
Communication Complexity — Transcript
▶
53.1
Overview
53.2
Declarations
54
Communication Complexity — Trees
▶
54.1
Overview
54.2
Declarations
55
Communication Complexity — Upper Bounds
▶
55.1
Overview
55.2
Declarations
56
Communication Complexity — Coin Tape
▶
56.1
Overview
56.2
Declarations
57
Communication Complexity — Comparison
▶
57.1
Overview
57.2
Declarations
58
Communication Complexity — Finite Probability Space
▶
58.1
Overview
58.2
Declarations
59
Communication Complexity — Func Hash
▶
59.1
Overview
59.2
Declarations
60
Communication Complexity — Minimax
▶
60.1
Overview
60.2
Declarations
61
Communication Complexity — One Way Minimax
▶
61.1
Overview
61.2
Declarations
62
Communication Complexity — Private Coin Approximation
▶
62.1
Overview
62.2
Declarations
63
Communication Complexity — Private Coin Basic
▶
63.1
Overview
63.2
Declarations
64
Communication Complexity — Private Coin Complexity
▶
64.1
Overview
64.2
Declarations
65
Communication Complexity — Private Coin Composition
▶
65.1
Overview
65.2
Declarations
66
Communication Complexity — Private Coin Finite Message
▶
66.1
Overview
66.2
Declarations
67
Communication Complexity — Public Coin Approximation
▶
67.1
Overview
67.2
Declarations
68
Communication Complexity — Public Coin Basic
▶
68.1
Overview
68.2
Declarations
69
Communication Complexity — Public Coin Complexity
▶
69.1
Overview
69.2
Declarations
70
Communication Complexity — Public Coin Composition
▶
70.1
Overview
70.2
Declarations
71
Communication Complexity — Public Coin Finite Message
▶
71.1
Overview
71.2
Declarations
72
Communication Complexity — Public Coin One Way
▶
72.1
Overview
72.2
Declarations
73
Cryptography — Schnorr Protocol
▶
73.1
Overview
73.2
Declarations
74
Boolean Analysis — Bool BLR
▶
74.1
Overview
74.2
Declarations
74.3
Additional declarations
75
Boolean Analysis — Bool Fourier
▶
75.1
Overview
75.2
Declarations
75.3
Additional declarations
76
Boolean Analysis — Low Degree
▶
76.1
Overview
76.2
Declarations
77
Boolean Analysis — Zk BLR
▶
77.1
Overview
77.2
Declarations
78
Boolean Analysis — Zk Fourier
▶
78.1
Overview
78.2
Declarations
79
Communication Complexity — Balanced Simulation
▶
79.1
Overview
79.2
Declarations
80
Communication Complexity — Bit String
▶
80.1
Overview
80.2
Declarations
81
Communication Complexity — Func Equality
▶
81.1
Overview
81.2
Declarations
82
Communication Complexity — Hamming
▶
82.1
Overview
82.2
Declarations
83
Communication Complexity — Rank
▶
83.1
Overview
83.2
Declarations
84
Communication Complexity — Newman Theorem
▶
84.1
Overview
84.2
Declarations
85
Communication Complexity — Derandomization
▶
85.1
Overview
85.2
Declarations
86
Communication Complexity — Discrepancy
▶
86.1
Overview
86.2
Declarations
87
Communication Complexity — Entropy
▶
87.1
Overview
87.2
Declarations
88
Communication Complexity — Func Disjointness Lower Bound
▶
88.1
Overview
88.2
Declarations
89
Communication Complexity — Func Inner Product
▶
89.1
Overview
89.2
Declarations
90
Communication Complexity — KL Divergence
▶
90.1
Overview
90.2
Declarations
91
Communication Complexity — Pinsker
▶
91.1
Overview
91.2
Declarations
92
Communication Complexity — TV Distance
▶
92.1
Overview
92.2
Declarations
93
Complexity — NAESAT To Coloring
▶
93.1
Overview
93.2
Declarations
94
Complexity — SAT To3 SAT
▶
94.1
Overview
94.2
Declarations
95
Complexity — Three SAT To Clique
▶
95.1
Overview
95.2
Declarations
96
Complexity — Three SAT To Coloring
▶
96.1
Overview
96.2
Declarations
97
Graph Theory — Basic
▶
97.1
Overview
97.2
Declarations
98
Graph Theory — Reach
▶
98.1
Overview
98.2
Declarations
99
Learning Theory — Halving
▶
99.1
Overview
99.2
Declarations
100
Learning Theory — Hedge
▶
100.1
Overview
100.2
Declarations
101
Learning Theory — Convex Prediction
▶
101.1
Overview
101.2
Declarations
102
Learning Theory — Episode
▶
102.1
Overview
102.2
Declarations
103
Learning Theory — Hoeffding
▶
103.1
Overview
103.2
Declarations
104
Learning Theory — Bernstein
▶
104.1
Overview
104.2
Declarations
105
Learning Theory — Concentration Bound
▶
105.1
Overview
105.2
Declarations
106
Learning Theory — Johnson–Lindenstrauss Main
▶
106.1
Overview
106.2
Declarations
107
Learning Theory — Rademacher
▶
107.1
Overview
107.2
Declarations
108
Learning Theory — CCE
▶
108.1
Overview
108.2
Declarations
109
Learning Theory — Convex Minimax Core
▶
109.1
Overview
109.2
Declarations
110
Learning Theory — Convex Minimax No Regret
▶
110.1
Overview
110.2
Declarations
111
Learning Theory — Finite Minimax
▶
111.1
Overview
111.2
Declarations
112
Learning Theory — Weighted Majority
▶
112.1
Overview
112.2
Declarations
113
Error-Correcting Codes — MRRW Bound
▶
113.1
Overview
113.2
Declarations
114
Kikuchi LDC — Background Facts
▶
114.1
Overview
114.2
Declarations
115
Kikuchi LDC — Decomposition
▶
115.1
Overview
115.2
Declarations
116
Kikuchi LDC — Defs
▶
116.1
Overview
116.2
Declarations
117
Kikuchi LDC — High Value
▶
117.1
Overview
117.2
Declarations
118
Kikuchi LDC — Main Theorem
▶
118.1
Overview
118.2
Declarations
119
Kikuchi LDC — Three XOR
▶
119.1
Overview
119.2
Declarations
120
Kikuchi LDC — Two XOR
▶
120.1
Overview
120.2
Declarations
121
Boolean Analysis — General Hypercontractivity
▶
121.1
Overview
121.2
Declarations
122
Boolean Analysis — KKL
▶
122.1
Overview
122.2
Declarations
123
Boolean Analysis — Decision Tree Fourier
▶
123.1
Overview
123.2
Declarations
124
Boolean Analysis — Restriction Card Tail
▶
124.1
Overview
124.2
Declarations
125
Boolean Analysis — Restriction Fourier
▶
125.1
Overview
125.2
Declarations
125.3
Additional declarations
126
Boolean Analysis — A Cp Gates
▶
126.1
Overview
126.2
Declarations
127
Boolean Analysis — Circuit Degree
▶
127.1
Overview
127.2
Declarations
128
Boolean Analysis — Circuit Size
▶
128.1
Overview
128.2
Declarations
129
Boolean Analysis — Feed Forward Circuit
▶
129.1
Overview
129.2
Declarations
130
Boolean Analysis — Low Degree Obstruction
▶
130.1
Overview
130.2
Declarations
131
Boolean Analysis — Smolensky Algebra
▶
131.1
Overview
131.2
Declarations
132
Complexity — Subset Sum To Partition
▶
132.1
Overview
132.2
Declarations
133
Cryptography — RSA
▶
133.1
Overview
133.2
Declarations
134
Graph Theory — Karger Min Cut
▶
134.1
Overview
134.2
Declarations
135
Graph Theory — Karger Min Cut Trace
▶
135.1
Overview
135.2
Declarations
136
Graph Theory — Exchange
▶
136.1
Overview
136.2
Declarations
137
Graph Theory — Optimality
▶
137.1
Overview
137.2
Declarations
138
Graph Theory — Union Find
▶
138.1
Overview
138.2
Declarations
139
Learning Theory — Minimax
▶
139.1
Overview
139.2
Declarations
140
Learning Theory — Convex Minimax Separation
▶
140.1
Overview
140.2
Declarations
141
Connectivity
▶
141.1
Connectivity
142
Cycle Space Bond Space
▶
142.1
Cycle Space Bond Space
143
Directed Graphs
▶
143.1
Directed Graphs
144
Edge Colourings
▶
144.1
Edge Colourings
145
Euler Hamilton
▶
145.1
Euler Hamilton
146
Graphs And Subgraphs
▶
146.1
Graphs And Subgraphs
147
Independent Sets Cliques
▶
147.1
Independent Sets Cliques
148
Matchings
▶
148.1
Matchings
149
Networks
▶
149.1
Networks
150
Planar Graphs
▶
150.1
Planar Graphs
151
Trees
▶
151.1
Trees
152
Vertex Colourings
▶
152.1
Vertex Colourings
Dependency graph
TCSLib
1
Boolean Analysis — Core Definitions
1.1
Overview
1.2
The Boolean hypercube and functions
1.3
Uniform measure and expectation
1.4
Walsh–Fourier characters
1.5
Fourier coefficients
1.6
Walsh expansion
1.7
Orthonormality and Parseval’s identity
1.8
Coordinate influence
1.9
Noise operator
1.10
Notable Boolean functions
1.11
Sensitivity
1.12
Properties for Arrow’s theorem
1.13
Additional declarations
2
Arrow’s Impossibility Theorem
2.1
Overview
2.2
Social choice definitions
2.3
Voter ordering model
2.4
Key correlation lemmas
2.5
Fourier formula for the cycle correlation
2.6
Acyclicity implies degree-1 structure
2.7
Degree-1 implies dictatorship
2.8
Arrow’s Impossibility Theorem
2.9
Additional declarations
3
Håstad’s Switching Lemma
3.1
Overview
3.2
The formalised statement
3.3
Faithfulness critique
3.4
Dependency graph
3.5
Corollary: no small CNF approximations
4
Error-Correcting Codes — Core Definitions
4.1
Overview
4.2
Basic objects: codewords and codes
4.3
Distance, weight, and rate
4.4
Basic lemma: distance vs. block length
4.5
Additional declarations
5
Singleton Bound
5.1
Overview
5.2
Main result
6
Hamming Bound
6.1
Overview
6.2
Volume formula
6.3
Disjointness of decoding balls
6.4
Hamming bound
7
Entropy and Asymptotic Bounds
7.1
Overview
7.2
Asymptotic upper bound on ball size
7.3
Entropy algebra lemmas
7.4
Analytic helpers
7.5
Stirling-based binomial lower bound
7.6
Positivity of \(q\)-ary entropy
7.7
Additional declarations
8
Linear Codes and the Uniformity Lemma
8.1
Overview
8.2
Distance equals minimum weight in linear codes
8.3
Uniform distributions and random matrices
8.4
Uniformity lemma
9
Gilbert–Varshamov Bound
9.1
Overview
9.2
Probabilistic bound on low-weight outputs
9.3
Union bound over bad matrices
9.4
Gilbert–Varshamov bound
10
List Decoding
10.1
Overview
10.2
List-decodable codes
10.3
Counting lemmas
10.4
List-decoding capacity
11
Johnson Bound
11.1
Overview
11.2
Johnson bound
11.3
Additional declarations
12
CSS Codes (Not Yet Formalized)
12.1
Overview
12.2
Symplectic vector space
12.3
CSS construction
12.4
CSS code parameters
13
Quantum Hamming Bound
13.1
Overview
13.2
Pauli strings
13.3
\(n\)-qubit Hilbert space
13.4
Knill–Laflamme conditions
13.5
Error sphere and sphere-packing
13.6
Quantum Hamming bound
13.7
Additional declarations
14
Quantum Singleton Bound
14.1
Overview
14.2
Symplectic vector space
14.3
Support, weight, and isotropic subspaces
14.4
Code parameters and erasure correctability
14.5
Key lemma: two disjoint correctable sets
14.6
Quantum Singleton bound
14.7
Additional declarations
15
Boolean Analysis — Gate Merge
15.1
Overview
15.2
Declarations
16
Boolean Analysis — Bonami
16.1
Overview
16.2
Declarations
17
Boolean Analysis — One Bit
17.1
Overview
17.2
Declarations
18
Boolean Analysis — Simple
18.1
Overview
18.2
Declarations
18.3
Additional declarations
19
Boolean Analysis — LMN
19.1
Overview
19.2
Declarations
20
Boolean Analysis — Bernoulli Cost
20.1
Overview
20.2
Declarations
21
Boolean Analysis — Circuit Compression
21.1
Overview
21.2
Declarations
22
Boolean Analysis — Circuit Helpers
22.1
Overview
22.2
Declarations
23
Boolean Analysis — Circuit Layer Reduction
23.1
Overview
23.2
Declarations
24
Boolean Analysis — Circuit Reindex
24.1
Overview
24.2
Declarations
25
Boolean Analysis — Circuit Tree Manip
25.1
Overview
25.2
Declarations
26
Boolean Analysis — Compression Step
26.1
Overview
26.2
Declarations
27
Boolean Analysis — Depth3 Switching
27.1
Overview
27.2
Declarations
28
Boolean Analysis — Gate Switching
28.1
Overview
28.2
Declarations
29
Boolean Analysis — Iterative Reduction
29.1
Overview
29.2
Declarations
30
Boolean Analysis — Normal Form Conversion
30.1
Overview
30.2
Declarations
31
Boolean Analysis — Recursive Reduction
31.1
Overview
31.2
Declarations
32
Boolean Analysis — Restriction Compose
32.1
Overview
32.2
Declarations
33
Boolean Analysis — Restriction Monotonicity
33.1
Overview
33.2
Declarations
34
Boolean Analysis — Switching Bernoulli
34.1
Overview
34.2
Declarations
35
Boolean Analysis — Switching
35.1
Overview
35.2
Declarations
35.3
Additional declarations
36
Boolean Analysis — Bernoulli Restriction
36.1
Overview
36.2
Declarations
37
Boolean Analysis — Canonical D Tree
37.1
Overview
37.2
Declarations
38
Boolean Analysis — Circuit
38.1
Overview
38.2
Declarations
38.3
Base literal, formula, and decision-tree definitions
39
Boolean Analysis — Encoding
39.1
Overview
39.2
Declarations
40
Boolean Analysis — Encoding Properties
40.1
Overview
40.2
Declarations
40.3
Additional declarations
41
Boolean Analysis — Restriction
41.1
Overview
41.2
Declarations
42
Boolean Analysis — Round Trip
42.1
Overview
42.2
Declarations
43
Communication Complexity — Det Basic
43.1
Overview
43.2
Declarations
44
Communication Complexity — Det Complexity
44.1
Overview
44.2
Declarations
45
Communication Complexity — Det Composition
45.1
Overview
45.2
Declarations
46
Communication Complexity — Det Rectangle
46.1
Overview
46.2
Declarations
47
Communication Complexity — Finite Message
47.1
Overview
47.2
Declarations
48
Communication Complexity — Func Disjointness
48.1
Overview
48.2
Declarations
49
Communication Complexity — Helper
49.1
Overview
49.2
Declarations
50
Communication Complexity — One Way
50.1
Overview
50.2
Declarations
51
Communication Complexity — Rectangle
51.1
Overview
51.2
Declarations
52
Communication Complexity — Subprotocol
52.1
Overview
52.2
Declarations
53
Communication Complexity — Transcript
53.1
Overview
53.2
Declarations
54
Communication Complexity — Trees
54.1
Overview
54.2
Declarations
55
Communication Complexity — Upper Bounds
55.1
Overview
55.2
Declarations
56
Communication Complexity — Coin Tape
56.1
Overview
56.2
Declarations
57
Communication Complexity — Comparison
57.1
Overview
57.2
Declarations
58
Communication Complexity — Finite Probability Space
58.1
Overview
58.2
Declarations
59
Communication Complexity — Func Hash
59.1
Overview
59.2
Declarations
60
Communication Complexity — Minimax
60.1
Overview
60.2
Declarations
61
Communication Complexity — One Way Minimax
61.1
Overview
61.2
Declarations
62
Communication Complexity — Private Coin Approximation
62.1
Overview
62.2
Declarations
63
Communication Complexity — Private Coin Basic
63.1
Overview
63.2
Declarations
64
Communication Complexity — Private Coin Complexity
64.1
Overview
64.2
Declarations
65
Communication Complexity — Private Coin Composition
65.1
Overview
65.2
Declarations
66
Communication Complexity — Private Coin Finite Message
66.1
Overview
66.2
Declarations
67
Communication Complexity — Public Coin Approximation
67.1
Overview
67.2
Declarations
68
Communication Complexity — Public Coin Basic
68.1
Overview
68.2
Declarations
69
Communication Complexity — Public Coin Complexity
69.1
Overview
69.2
Declarations
70
Communication Complexity — Public Coin Composition
70.1
Overview
70.2
Declarations
71
Communication Complexity — Public Coin Finite Message
71.1
Overview
71.2
Declarations
72
Communication Complexity — Public Coin One Way
72.1
Overview
72.2
Declarations
73
Cryptography — Schnorr Protocol
73.1
Overview
73.2
Declarations
74
Boolean Analysis — Bool BLR
74.1
Overview
74.2
Declarations
74.3
Additional declarations
75
Boolean Analysis — Bool Fourier
75.1
Overview
75.2
Declarations
75.3
Additional declarations
76
Boolean Analysis — Low Degree
76.1
Overview
76.2
Declarations
77
Boolean Analysis — Zk BLR
77.1
Overview
77.2
Declarations
78
Boolean Analysis — Zk Fourier
78.1
Overview
78.2
Declarations
79
Communication Complexity — Balanced Simulation
79.1
Overview
79.2
Declarations
80
Communication Complexity — Bit String
80.1
Overview
80.2
Declarations
81
Communication Complexity — Func Equality
81.1
Overview
81.2
Declarations
82
Communication Complexity — Hamming
82.1
Overview
82.2
Declarations
83
Communication Complexity — Rank
83.1
Overview
83.2
Declarations
84
Communication Complexity — Newman Theorem
84.1
Overview
84.2
Declarations
85
Communication Complexity — Derandomization
85.1
Overview
85.2
Declarations
86
Communication Complexity — Discrepancy
86.1
Overview
86.2
Declarations
87
Communication Complexity — Entropy
87.1
Overview
87.2
Declarations
88
Communication Complexity — Func Disjointness Lower Bound
88.1
Overview
88.2
Declarations
89
Communication Complexity — Func Inner Product
89.1
Overview
89.2
Declarations
90
Communication Complexity — KL Divergence
90.1
Overview
90.2
Declarations
91
Communication Complexity — Pinsker
91.1
Overview
91.2
Declarations
92
Communication Complexity — TV Distance
92.1
Overview
92.2
Declarations
93
Complexity — NAESAT To Coloring
93.1
Overview
93.2
Declarations
94
Complexity — SAT To3 SAT
94.1
Overview
94.2
Declarations
95
Complexity — Three SAT To Clique
95.1
Overview
95.2
Declarations
96
Complexity — Three SAT To Coloring
96.1
Overview
96.2
Declarations
97
Graph Theory — Basic
97.1
Overview
97.2
Declarations
98
Graph Theory — Reach
98.1
Overview
98.2
Declarations
99
Learning Theory — Halving
99.1
Overview
99.2
Declarations
100
Learning Theory — Hedge
100.1
Overview
100.2
Declarations
101
Learning Theory — Convex Prediction
101.1
Overview
101.2
Declarations
102
Learning Theory — Episode
102.1
Overview
102.2
Declarations
103
Learning Theory — Hoeffding
103.1
Overview
103.2
Declarations
104
Learning Theory — Bernstein
104.1
Overview
104.2
Declarations
105
Learning Theory — Concentration Bound
105.1
Overview
105.2
Declarations
106
Learning Theory — Johnson–Lindenstrauss Main
106.1
Overview
106.2
Declarations
107
Learning Theory — Rademacher
107.1
Overview
107.2
Declarations
108
Learning Theory — CCE
108.1
Overview
108.2
Declarations
109
Learning Theory — Convex Minimax Core
109.1
Overview
109.2
Declarations
110
Learning Theory — Convex Minimax No Regret
110.1
Overview
110.2
Declarations
111
Learning Theory — Finite Minimax
111.1
Overview
111.2
Declarations
112
Learning Theory — Weighted Majority
112.1
Overview
112.2
Declarations
113
Error-Correcting Codes — MRRW Bound
113.1
Overview
113.2
Declarations
114
Kikuchi LDC — Background Facts
114.1
Overview
114.2
Declarations
115
Kikuchi LDC — Decomposition
115.1
Overview
115.2
Declarations
116
Kikuchi LDC — Defs
116.1
Overview
116.2
Declarations
117
Kikuchi LDC — High Value
117.1
Overview
117.2
Declarations
118
Kikuchi LDC — Main Theorem
118.1
Overview
118.2
Declarations
119
Kikuchi LDC — Three XOR
119.1
Overview
119.2
Declarations
120
Kikuchi LDC — Two XOR
120.1
Overview
120.2
Declarations
121
Boolean Analysis — General Hypercontractivity
121.1
Overview
121.2
Declarations
122
Boolean Analysis — KKL
122.1
Overview
122.2
Declarations
123
Boolean Analysis — Decision Tree Fourier
123.1
Overview
123.2
Declarations
124
Boolean Analysis — Restriction Card Tail
124.1
Overview
124.2
Declarations
125
Boolean Analysis — Restriction Fourier
125.1
Overview
125.2
Declarations
125.3
Additional declarations
126
Boolean Analysis — A Cp Gates
126.1
Overview
126.2
Declarations
127
Boolean Analysis — Circuit Degree
127.1
Overview
127.2
Declarations
128
Boolean Analysis — Circuit Size
128.1
Overview
128.2
Declarations
129
Boolean Analysis — Feed Forward Circuit
129.1
Overview
129.2
Declarations
130
Boolean Analysis — Low Degree Obstruction
130.1
Overview
130.2
Declarations
131
Boolean Analysis — Smolensky Algebra
131.1
Overview
131.2
Declarations
132
Complexity — Subset Sum To Partition
132.1
Overview
132.2
Declarations
133
Cryptography — RSA
133.1
Overview
133.2
Declarations
134
Graph Theory — Karger Min Cut
134.1
Overview
134.2
Declarations
135
Graph Theory — Karger Min Cut Trace
135.1
Overview
135.2
Declarations
136
Graph Theory — Exchange
136.1
Overview
136.2
Declarations
137
Graph Theory — Optimality
137.1
Overview
137.2
Declarations
138
Graph Theory — Union Find
138.1
Overview
138.2
Declarations
139
Learning Theory — Minimax
139.1
Overview
139.2
Declarations
140
Learning Theory — Convex Minimax Separation
140.1
Overview
140.2
Declarations
141
Connectivity
141.1
Connectivity
142
Cycle Space Bond Space
142.1
Cycle Space Bond Space
143
Directed Graphs
143.1
Directed Graphs
144
Edge Colourings
144.1
Edge Colourings
145
Euler Hamilton
145.1
Euler Hamilton
146
Graphs And Subgraphs
146.1
Graphs And Subgraphs
147
Independent Sets Cliques
147.1
Independent Sets Cliques
148
Matchings
148.1
Matchings
149
Networks
149.1
Networks
150
Planar Graphs
150.1
Planar Graphs
151
Trees
151.1
Trees
152
Vertex Colourings
152.1
Vertex Colourings