StrongSensitivityConjecture.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 3 contain sorry, 0 declared as axioms

Strong Sensitivity Conjecture (`bs(f) ≤ s(f)^2`)

This file formalizes the strong sensitivity conjecture, asserting:

For every Boolean function f : {0,1}^n → {0,1}, bs(f) ≤ s(f)^2, where bs(f) denotes block sensitivity and s(f) denotes sensitivity.

Huang's theorem proves a quartic upper bound, bs(f) ≤ s(f)^4, thereby resolving the most widely known form of the sensitivity conjecture.

We now ask whether a stronger upper bound holds. Interestingly, the original paper of Nisan and Szegedy, where the sensitivity conjecture first appeared, already speculated that a quadratic upper bound might be the correct relation. On the lower bound side, Rubinstein (https://link.springer.com/article/10.1007/BF01200762) constructed Boolean functions exhibiting the first quadratic separation. The best currently known gap, due to Ambainis and Sun (https://arxiv.org/abs/1108.3494), is bs(f) ≥ (2/3)⋅s(f)^2.

References:

Induced Subgraphs of Hypercubes and a Proof of the Sensitivity Conjecture by Hao Huang (see Section 3, Concluding Remarks)
Variations on the Sensitivity Conjecture by Pooya Hatami, Raghav Kulkarni, and Denis Pankratov (see Question 3.1)
On the Degree of Boolean Functions as Real Polynomials by Noam Nisan, and Mario Szegedy (see Section 4, Open Problems)

Imports (1)

FormalConjecturesUtil

def StrongSensitivityConjecture.flip Flip operator, flip x B returns input x with bits in block B inverted line 62 source
Source documentation
Flip operator, flip x B returns input x with bits in block B inverted.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Finset
Proof uses: ite, Membership.mem, Bool.not
4 instances, coercions and auxiliary constants left out.
```
def flip (x : Fin n → Bool) (B : Finset (Fin n)) : Fin n → Bool :=
…
```
def StrongSensitivityConjecture.sensitivityAt Local sensitivity s(f,x), number of indices where flipping one bit changes the value of f line 67 source
Source documentation
Local sensitivity s(f,x), number of indices where flipping one bit changes the value of f.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool
Proof uses: Finset.card, Finset.filter, Ne, StrongSensitivityConjecture.flip, Singleton.singleton, Finset, Eq, Finset.univ
4 instances, coercions and auxiliary constants left out.
```
def sensitivityAt (f : (Fin n → Bool) → Bool) (x : Fin n → Bool) : ℕ :=
…
```
def StrongSensitivityConjecture.sensitivity Global sensitivity s(f), maximum sensitivity of f over all inputs line 72 source
Source documentation
Global sensitivity s(f), maximum sensitivity of f over all inputs.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool
Proof uses: Finset.sup, Finset.univ, StrongSensitivityConjecture.sensitivityAt
7 instances, coercions and auxiliary constants left out.
```
def sensitivity (f : (Fin n → Bool) → Bool) : ℕ :=
…
```
def StrongSensitivityConjecture.IsValidBlockConfig Check validity of block collection (disjoint and sensitive), A collection of blocks cB is valid for f at x if the blocks are disjoint and flipping any block changes f(x) line 78 source
Source documentation
Check validity of block collection (disjoint and sensitive), A collection of blocks cB is valid for f at x if the blocks are disjoint and flipping any block changes f(x).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool, Finset
Proof uses: And, Set.PairwiseDisjoint, id, Membership.mem, Ne, StrongSensitivityConjecture.flip
5 instances, coercions and auxiliary constants left out.
```
def IsValidBlockConfig (f : (Fin n → Bool) → Bool) (x : Fin n → Bool)
    (cB : Finset (Finset (Fin n))) : Prop :=
…
```
def StrongSensitivityConjecture.blockSensitivityAt Local block sensitivity bs(f,x), maximum size of a collection of sensitive, disjoint blocks for f at x line 85 source
Source documentation
Local block sensitivity bs(f,x), maximum size of a collection of sensitive, disjoint blocks for f at x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool
Proof uses: Finset.sup, Finset, Finset.filter, StrongSensitivityConjecture.IsValidBlockConfig, Classical.propDecidable, Finset.univ, Finset.card
5 instances, coercions and auxiliary constants left out.
```
noncomputable def blockSensitivityAt (f : (Fin n → Bool) → Bool) (x : Fin n → Bool) : ℕ :=
…
```
def StrongSensitivityConjecture.blockSensitivity Global block sensitivity of f, maximum block sensitivity of f over all inputs line 91 source
Source documentation
Global block sensitivity of f, maximum block sensitivity of f over all inputs.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool
Proof uses: Finset.sup, Finset.univ, StrongSensitivityConjecture.blockSensitivityAt
7 instances, coercions and auxiliary constants left out.
```
noncomputable def blockSensitivity (f : (Fin n → Bool) → Bool) : ℕ :=
…
```
theorem StrongSensitivityConjecture.strong_sensitivity_conjecture Strong Sensitivity Conjecture, for every Boolean function f : {0,1}^n → {0,1}, bs(f) ≤ s(f)^2 sorry line 103 source
Source documentation
Strong Sensitivity Conjecture, for every Boolean function f : {0,1}^n → {0,1}, bs(f) ≤ s(f)^2.
We call this the strong sensitivity conjecture because the original sensitivity conjecture only asked for a polynomial bound in terms of s(f). Huang's celebrated result (often called the sensitivity theorem) gives a quartic bound, bs(f) ≤ s(f)^4, thereby settling the original conjecture.

depends on sorry what this means
Statement uses: Nat, Fin, Bool, LE.le, StrongSensitivityConjecture.blockSensitivity, HPow.hPow, StrongSensitivityConjecture.sensitivity, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem strong_sensitivity_conjecture {n : ℕ} (f : (Fin n → Bool) → Bool) :
    blockSensitivity f ≤ sensitivity f ^ 2 := by
…
```
Mentions: StrongSensitivityConjecture.blockSensitivity, StrongSensitivityConjecture.sensitivity
def StrongSensitivityConjecture.nisanExample Simple test example line 118 source
Source documentation
Simple test example. For multiples of four with $n \ge 12$, this Boolean function has block sensitivity strictly greater than its sensitivity. Source: Nisan1989.
nisanExample(x) = 1 iff the Hamming weight of $x$ is either $n/2$ or $n/2 + 1$. The function is symmetric, so its value only depends on the Hamming weight of the input.

L4DEL explanation
Simple test example. For multiples of four with $n \ge 12$, this Boolean function has block sensitivity strictly greater than its sensitivity. Source: Nisan1989. nisanExample(x) = 1 iff the Hamming weight of $x$ is either $n/2$ or $n/2 + 1$. The function is symmetric, so its value only depends on the Hamming weight of the input.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Bool
Proof uses: Finset.card, Finset.filter, Eq, Bool.true, Finset.univ, Decidable.decide, Membership.mem, Rat, Finset, Insert.insert, HDiv.hDiv, OfNat.ofNat, Singleton.singleton, HAdd.hAdd
15 instances, coercions and auxiliary constants left out.
```
def nisanExample (n : ℕ) (x : Fin n → Bool) : Bool :=
…
```
lemma StrongSensitivityConjecture.nisanExample_sensitivity Assuming n is a positive multiple of 4, the sensitivity of nisanExample is $n/2 + 2$ sorry line 126 source
Source documentation
Assuming n is a positive multiple of 4, the sensitivity of nisanExample is $n/2 + 2$. It is achieved by any $x$ with Hamming weight $n/2 + 2$: flipping any of its $n/2 + 2$ one-bits moves the input into the accepting layer of weight $n/2 + 1$.

depends on sorry what this means
Statement uses: Nat, Dvd.dvd, OfNat.ofNat, LT.lt, Eq, StrongSensitivityConjecture.sensitivity, StrongSensitivityConjecture.nisanExample, HAdd.hAdd, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma nisanExample_sensitivity (n : ℕ) (hn : 4 ∣ n) (hn_pos : 0 < n) :
    sensitivity (nisanExample n) = n / 2 + 2 := by
…
```
Mentions: StrongSensitivityConjecture.sensitivity, StrongSensitivityConjecture.nisanExample
lemma StrongSensitivityConjecture.nisanExample_blockSensitivity Assuming n is a multiple of 4 and $n \ge 8$, the block sensitivity of nisanExample is $3n/4$, achieved by any $x$ with Hamming weight $n/2$ sorry line 135 source
Source documentation
Assuming n is a multiple of 4 and $n \ge 8$, the block sensitivity of nisanExample is $3n/4$, achieved by any $x$ with Hamming weight $n/2$. An optimal block configuration uses all $n/2$ one-bits as singleton blocks and forms $n/4$ disjoint size-2 blocks from the zero-bits.

depends on sorry what this means
Statement uses: Nat, Dvd.dvd, OfNat.ofNat, LE.le, Eq, StrongSensitivityConjecture.blockSensitivity, StrongSensitivityConjecture.nisanExample, HDiv.hDiv, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma nisanExample_blockSensitivity (n : ℕ) (hn : 4 ∣ n) (hn_large : 8 ≤ n) :
    blockSensitivity (nisanExample n) = 3 * n / 4 := by
…
```
Mentions: StrongSensitivityConjecture.blockSensitivity, StrongSensitivityConjecture.nisanExample

FormalConjectures/Paper/StrongSensitivityConjecture.lean

Strong Sensitivity Conjecture (bs(f) ≤ s(f)^2)

`FormalConjectures/Paper/StrongSensitivityConjecture.lean`

Strong Sensitivity Conjecture (`bs(f) ≤ s(f)^2`)