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

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

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

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.

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.

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.

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.

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.

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.

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

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?

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

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 1 2 −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.

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

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

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

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

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

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

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}

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

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

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

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