1. Negation-Limited Formulas
Siyao Guo Ilan Komargodski
New York University Weizman Institute of Science

2. Boolean Circuits and Formulas
Circuit: directed acyclic graph. Gates labeled by 𝐀𝐍𝐃, 𝐎𝐑 and 𝐍𝐎𝐓 operations. Fan-out 2.
Formula: a circuit with fan-out 1.
Size 𝑠 = # of gates.
output wire ∧ ∨
input wires
𝑥 1
𝑥 2
𝑥 1

3. Monotone vs. Non Monotone Computation
Monotone computation: no 𝐍𝐎𝐓 gates.
“The effect of negation gates on circuit size remains to a large extent a mystery”, Stasys Jukna
General
Monotone
5𝑛 [ILMR01]
2 Ω(𝑛 log 𝑛) 1/3 [RH00]
Circuit size lower bound
𝑛 3−𝑜(1) [Tal14]
2 Ω(𝑛/log(𝑛)) [GP14]
Formula size lower bound

4. Bound the # of Negations
Circuits [Markov’58,Fischer’75,BNT98]:
size: 𝑠, negations: 𝑡 
size: 2𝑠+𝑂(𝑛⋅log 𝑛), negations: ⌈log 𝑛+1 ⌉
Formulas [Nechiporuk’62,Morizumi’09]:
size: 𝑠⋅ 𝑛 6.4 , negations: 𝑛 2
# of negations above is tight.
𝑛 is the input size of the functions.
The size is not known to be tight.

5. Bound the # of Negations
[Markov’58,Fischer’75]
log 𝑛
# of negations in a circuit
[Nechiporuk’62,Morizumi’09]
𝑛 2
# of negations in a formula
𝑛 is the input size of the functions.

6. Bound the # of Negations
[Markov’58,Fischer’75]
????
log 𝑛
# of negations
in a circuit
[Nechiporuk’62,Morizumi’09]
????
𝑛 2
# of negations in a formula
𝑛 is the input size of the functions.

7. Extending Monotone Results
[AM05,Ros15] (size lower bounds),
[BCOST15] (learning),
[GMOR15] (crypto)
[Markov’58,Fischer’75] ??
log 𝑛
# of negations in a circuit
[Nechiporuk’62,Morizumi’09]
????
IDEA: Decompose the 𝒕-negation circuit into 𝟐 𝒕 monotone parts and apply the known results on them (in a smart way).
𝑛 2
# of negations in a formula
What about negation-limited formulas?
𝑛 is the input size of the functions.

8. What about Negation-Limited Formulas?
[AM05,Ros15] (size lower bounds),
[BCOST15] (learning),
[GMOR15] (crypto)
[Markov’58,Fischer’75] ??
log 𝑛
# of negations in a circuit
Trivial
[Nechiporuk’62,Morizumi’09]
????
𝑛 2
# of negations in a formula
𝑛 is the input size of the functions.

9. What about Negation-Limited Formulas?
[AM05,Ros15] (size lower bounds),
[BCOST15] (learning),
[GMOR15] (crypto)
[Markov’58,Fischer’75] ??
log 𝑛
# of negations in a circuit
This Work
[Nechiporuk’62,Morizumi’09] ??
IDEA: Decompose the 𝒕-negation formula into 𝒕 monotone parts and apply the known results on them (in a smart way).
𝑛 2
# of negations in a formula
𝑛 is the input size of the functions.

10. Our Decomposition Theorem
Every formula 𝐹 of size 𝑠 and 𝑡 negations can be rewritten as a formula of size 2𝑠 of the form
𝐹≡𝐻 𝐺 1 𝑥 ,…, 𝐺 𝑇 𝑥 , where
𝑇=𝑂(𝑡)
𝐻 is a read-once formula
every 𝐺 𝑖 is a monotone formula. 𝐻 𝐹
𝑂(𝑡) inputs
𝐺 1
𝐺 2
𝐺 𝑇
Pushing NOTs to the top
Monotone

11. Application 1 – Formulas to Circuits
size: 𝑠
negations: 𝑡
Formula
size: 2 𝑠
negations: 2 𝑡
Known
Circuit
size: 𝑠
negations: 𝑡
Formula
size: 𝑠
negations: 𝑡
Trivial
Is this the best we can do?

12. Application 1 – Formulas to Circuits
Theorem 1: Formula, size: 𝑠, negations: 𝑡, depth: 𝑑 circuit, size: 2𝑠+𝑂 𝑡⋅log 𝑡 , negations: log 𝑡+𝑂 1 , depth 𝑑+𝑂 log 𝑡 .
Result for negation-limited circuits 
non-trivial result for negation-limited formulas
Circuit
size:
2𝑠+𝑂(𝑡⋅log 𝑡)
negations:
log 𝑡+𝑂(1)
Formula
size: 𝑠
negations: 𝑡
This Work

13. Average-Case Lower Bound Extension
Theorem [Ros15]: For any 𝜀>0, there exists an explicit function f which is ½ + o(1) hard for NC1 circuits with −𝜀 ⋅log 𝑛 negation gates under uniform distribution.
Corollary: For any 𝜀>0, there exists an explicit function f which is ½ + o(1) hard for polynomial size formulas with 𝑛 1 2 −𝜀 negation gates under uniform distribution.

14. Proof of Theorem 1
Theorem 1: Formula, size: 𝑠, negations: 𝑡, depth: 𝑑 circuit, size: 2𝑠+𝑂 𝑡⋅log 𝑡 , negations: log 𝑡+𝑂 1 , depth 𝑑+𝑂 log 𝑡 .
Decomposition Theorem
Apply [BNT98] theorem 𝐻′ 𝐻 𝐹
𝐺 1
𝐺 2
𝐺 𝑇
𝐺 1
𝐺 2
𝐺 𝑇
𝐻 is a formula with O(𝑡) inputs
View 𝐻 as a circuit with O(𝑡) inputs
𝐻′ is a circuit with log 𝑡+𝑂(1)
negations

15. Application 2 – Shrinkage of Formulas
Definition: 𝐹 |𝜌 is 𝐹 where each input is fixed w.p 1−𝑝 to a uniformly random bit.
What happens to 𝐿 𝐹 |𝜌 ; the size of 𝐹 |𝜌 ??
Applications to PRGs, lower bounds, Fourier results, #SAT algorithms, etc.
Definition: Γ is the largest constant s.t. ∀ formula F
E 𝜌 𝐿 𝐹 |𝜌 =𝑂 𝑝 Γ ⋅𝐿 𝐹 +1
Trivial: Γ≥1

16. Application 2 – Shrinkage of Formulas
Γ=2 - shrinkage exponent of formulas [Tal14].
Open: Γ 0 - shrinkage exponent of monotone formulas (conjecture =3.27).
Γ 𝑡 - shrinkage exponent of formulas with 𝑡 negations.
Γ 0 = ? ? 3.27
Conj. [PZ93]
Γ=2 [Subb’61,...,Tal’14]
Γ 𝑡 =? 𝑡
# of negations

17. Application 2 – Shrinkage of Formulas
Definition: Γ 𝑡 is the largest constant s.t. ∀ formula F that contains 𝑡 negations
E 𝜌 𝐿 𝐹 |𝜌 =𝑂 𝑝 Γ 𝑡 ⋅𝐿 𝐹 +1
Theorem 2: Let 𝐹 be a formula with 𝑡>0 negations
E 𝜌 𝐿 𝐹 |𝜌 =𝑂 𝑝 Γ 0 ⋅𝐿 𝐹 +𝑡
Corollary: If 𝑡=Θ 1 , then Γ 𝑡 = Γ 0

18. Decomposition Theorem
Proof of Theorem 2
Theorem 2: Let 𝐹 be a formula with 𝑡>0 negations
E 𝜌← 𝑅 𝑝 𝐿 𝐹 |𝜌 =𝑂 𝑝 Γ 0 ⋅𝐿 𝐹 +𝑡
Restrict each 𝐺 𝑖 with 𝜌
Decomposition Theorem 𝐻 𝐻 𝐹
𝐺 1
𝐺 2
𝐺 𝑇
𝐺 1
𝐺 2
𝐺 𝑇
Each 𝐺 𝑖 is monotone
𝐺 𝑖 shrinks at rate Γ 0
There are 𝑂(𝑡) 𝐺 𝑖 ’s

19. Proof of Decomposition
Statement:
Pushing NOTs to the top
𝑟𝑒𝑎𝑑 𝑜𝑛𝑐𝑒 𝐻 𝐹
𝑚𝑜𝑛 𝐺 1
𝑚𝑜𝑛 𝐺 2
𝑚𝑜𝑛 𝐺 𝑇 …
size: s, negations: t
size: S <= 2s, T <= max{5t-2,1}
Proof: Induction on t.
Base case: t=0.
- Induction hypothesis: holds for < t (t>=1). 𝐹 𝐹

20. Induction Step (NOT Root)
Statement:
Pushing NOTs to the top
𝑟𝑒𝑎𝑑 𝑜𝑛𝑐𝑒 𝐻 𝐹
𝑚𝑜𝑛 𝐺 1
𝑚𝑜𝑛 𝐺 2
𝑚𝑜𝑛 𝐺 𝑇 …
size: s, negations: t
size: S <= 2s, T <= max{5t-2,1}
Proof:
Decompose F’ by IH
𝑟𝑜 H′ 𝐹′
𝑚𝐺 1
𝑚𝐺 𝑇′ …
s’<=s, t’= t-1
size: S <= 2s, T = T’ <= max{5(t-1)-2,1}

21. Induction Step (AND/OR Root)
Statement:
Pushing NOTs to the top
𝑟𝑒𝑎𝑑 𝑜𝑛𝑐𝑒 𝐻 𝐹
𝑚𝑜𝑛 𝐺 1
𝑚𝑜𝑛 𝐺 2
𝑚𝑜𝑛 𝐺 𝑇 …
size: s, negations: t
size: S <= 2s, T <= max{5t-2,1}
∧ / ∨
Proof:
Decompose F1, F2 by IH
∧ / ∨
𝑟𝑜 H1
𝑟𝑜 H2 𝐹1 𝐹2
Require: t1,t2<t
𝑚𝐺 1
𝑚𝐺 𝑇1
𝑚𝐺′ 1
𝑚𝐺′ 𝑇2 … …
s1+s2=s, t1+t2 = t
S1+S2 <= 2s, T1+T2 <= 5t-4

22. Induction Step (Overall)
Statement:
Pushing NOTs to the top
𝑟𝑒𝑎𝑑 𝑜𝑛𝑐𝑒 𝐻 𝐹
𝑚𝑜𝑛 𝐺 1
𝑚𝑜𝑛 𝐺 2
𝑚𝑜𝑛 𝐺 𝑇 …
size: s, negations: t
size: S <= 2s, T <= max{5t-2,1}
F’: t’ =t, each subformula of F’ has < t negations
Proof: ∨
𝐹′′
Decompose F’
(previous slides)
𝐹′′
F’’ is monotone ∧
𝑟𝑜 H′ …
𝑟𝑜 H′
𝐹′′
𝐹′′ 𝐹′
𝑚𝐺 1
𝑚𝐺 𝑇′
𝑚𝐺 1
𝑚𝐺 𝑇′ … … 1
s’+s’’=s, t’= t
T’ <= 5t-4
s’’+ s’’ + 2s’<= 2s, T = T’+2 <= 5t-2

23. Summary
Efficient decomposition theorem for negation-limited formulas
Extending results to negation limited formulas from:
Negation limited circuits (size lower bounds, learning, crypto, etc). Generic, exp. improvement.
Monotone formulas (shrinkage)
This Work
[Nechiporuk’62,Morizumi’09]
Thanks! ??
𝑛 2
# of negations

24. Circuits to Formulas 𝒕-negation formula 𝒕-negation circuit 𝑛 1 2 −𝜀
𝑛 1 2 −𝜀
1 2 −𝜀 ⋅log 𝑛
Average-case lower bound for NC 1
Ω(𝑛)
Ω(log 𝑛)
PRF
𝑛 𝑂(𝑡⋅ 𝑛 /𝜀)
𝑛 𝑂( 2 𝑡 ⋅ 𝑛 /𝜀)
Learning b