Size Problem as Oracle Shuichi Hirahara The University of Tokyo - - PowerPoint PPT Presentation
Limits of Minimum Circuit Size Problem as Oracle Shuichi Hirahara The University of Tokyo Osamu Watanabe Tokyo Institute of Technology CCC 2016/05/30 Minimum Circuit Size Problem (MCSP) Input Output Truth table 0,1
Shuichi Hirahara（The University of Tokyo） Osamu Watanabe（Tokyo Institute of Technology）
CCC 2016/05/30
𝒚𝟐 𝒚𝟑 𝒚𝟐 xor 𝒚𝟑 1 1 1 1 1 1
Decide if ∃ circuit of size ≤ 𝑡 whose truth table is 𝑈.
[Kabanets & Cai (2000)]
[ABKvMR06]
[Kabanets & Cai (2000)]
Few evidences
≤𝑈
P or ≤𝑈 BPP
[ABKvMR06]
[Kabanets & Cai (2000)]
Few evidences
≤𝑈
P or ≤𝑈 BPP
Our Main Contributions
Provide strong evidences that “current reduction techniques” cannot establish NP-hardness of MCSP (under ≤𝑈
𝑞 nor ≤𝑛 BPP).
General reductions Restricted reductions
[Allender & Das (2014)]
𝑞 .
[Murry & Williams (2015)] Hardness under general reductions ≤𝑈
BPP
≤𝑛
𝑞
Difficulty of proving hardness under restricted reductions
[Allender, Buhrman, Koucký, van Melkebeek and Ronneburger (2006)]
Integer Factorization
Discrete log
[Allender & Das (2014)]
Integer Factorization
Discrete log
[Murray & Williams (CCC 2015)]
𝑞 MCSP ⟹ ZPP ≠ EXP.
An extension of [Kabanets & Cai (2000)]
General reductions Restricted reductions
[Allender & Das (2014)]
𝑞 .
[Murry & Williams (2015)] Hardness under general reductions ≤𝑈
BPP
≤𝑛
𝑞
Difficulty of proving hardness under restricted reductions Our Results
techniques” (for ≤𝑈
𝑞 and ≤𝑛 BPP)
𝑞 and ≤𝑈 𝑞
≤𝑛
BPP
≤𝑈
𝑞
≤𝑢𝑢
𝑞
𝐵
[Allender & Das (2014)]
The reduction can be generalized to a reduction to MCSP𝐵 for all oracle 𝐵.
Def (Minimum 𝑩-Oracle Circuit Size Problem; 𝐍𝐃𝐓𝐐𝑩) Input: Truth table 𝑈 ∈ 0, 1 2𝑜 and size parameter 𝑡 ∈ ℕ Output: Does there exists an 𝐵-oracle circuit of size ≤ 𝑡 whose truth table is 𝑈?
𝑦1 𝑦2 𝑦3 𝑦4 𝐵 𝑦1, … , 𝑦4
In addition to
𝐵-oracle gates can be used. gates,
Def (Oracle-independent Reductions) A reduction to MCSP is oracle-independent if the reduction can be generalized to a reduction to MCSP𝐵 for any oracle 𝐵. 𝑀 reduces to MCSP via an oracle-independent P-Turing reduction 𝑀 ∈
𝐵
PMCSP𝐵 .
def
For example:
𝑀 ∈ PMCSP𝐵 for any oracle 𝐵. Idea: The reduction relies on common properties of MCSP𝐵 for all 𝐵 (instead of a non-relativizing property of MCSP)
𝐵
Instead, we will show [Ko (1991)] There exists an oracle 𝐵 such that NP𝐵 ⊈ PMCSP𝐵, 𝐵.
A specific oracle 𝐵 All oracles 𝐵
(MCSP is not NP-hard relative to 𝐵)
Do not allow direct access to 𝐵
𝐵
𝐵
𝐵
[Kabanets & Cai (2000)] [Allender, Grochow & Moore (2015)]
[Allender & Das (2014)]
Important Observation
[Hastad, Impagliazzo, Levin & Luby (1999)]
[Allender & Das (2014)]
Uniform distribution 𝑉2𝑜 Pseudorandom distribution 𝐻 𝑉𝑛
(by a counting argument). ← small circuit complexity ← high circuit complexity Important Observation
Uniform distribution 𝑉2𝑜 Pseudorandom distribution 𝐻 𝑉𝑛
(by a counting argument). ← small circuit complexity ← high circuit complexity Important Observation
Remains true even if 𝐵-oracle gates can be used. A similar counting arguments can be applied.
Uniform distribution 𝑉2𝑜 Pseudorandom distribution 𝐻 𝑉𝑛
(by a counting argument). ← small circuit complexity ← high circuit complexity Important Observation
Remains true even if 𝐵-oracle gates can be used. A similar counting arguments can be applied.
Corollary [Allender & Das (2014)]
𝐵
Adding 𝐵-oracle gates does not increase the circuit complexity.
We know very few lower bounds for general circuits.
Pseudorandom distribution 𝐻 𝑉𝑛 Uniform distribution 𝑉2𝑜
This is “weakness” of current reduction techniques.
Theorem 1. (Limit of Oracle-independent P-Reductions)
If 𝑀 reduces to MCSP via an oracle-independent polynomial- time Turing reduction, then 𝑀 ∈ P.
If 𝑀 ∈ PMCSP𝐵 for any oracle 𝐵, then 𝑀 ∈ P.
𝐵
PMCSP𝐵 = P.
(In other words)
Theorem 2. (Limit of Oracle-independent BPP-Reductions)
If 𝑀 reduces to MCSP via an oracle-independent one-query BPP-Turing reduction (with negligible error probability), then 𝑀 ∈ AM ∩ coAM. In short,
𝐵
BPPMCSP𝐵[1] ⊆ AM ∩ coAM.
(unless polynomial hierarchy collapses).
∀𝐵, ∃𝑁, 𝑁MCSP𝐵 𝑦 = 𝑀 𝑦 ⟹ 𝑀 ∈ P.
Theorem 1. (Limit of P-Reductions)
𝐵
PMCSP𝐵 = P. The theorem says:
Lemma.
∃𝑁, ∀𝐵, 𝑁MCSP𝐵 𝑦 = 𝑀 𝑦 ⟹ 𝑀 ∈ P However, it is sufficient to prove: (Proof of Lemma ⟹ Theorem) A simple diagonalization argument.
(omitted)
∀𝐵, ∃𝑁, 𝑁MCSP𝐵 𝑦 = 𝑀 𝑦 ⟹ 𝑀 ∈ P.
Theorem 1. (Limit of P-Reductions)
𝐵
PMCSP𝐵 = P. The theorem says:
Lemma.
∃𝑁, ∀𝐵, 𝑁MCSP𝐵 𝑦 = 𝑀 𝑦 ⟹ 𝑀 ∈ P However, it is sufficient to prove: (Proof of Lemma ⟹ Theorem) A simple diagonalization argument.
(omitted)
Lemma.
∃𝑁, ∀𝐵, ∀𝑦, 𝑁MCSP𝐵 𝑦 = 𝑀 𝑦 ⟹ 𝑀 ∈ P Choose an oracle 𝐵 so that 𝑁MCSP𝐵 𝑦 can be easily simulated
1, … , 𝑈 𝑛 be truth tables queried by 𝑁.
𝑗
Circuit complexity relative to 𝐵 It takes 2𝑃 log 𝑜 = 𝑜𝑃 1 time if we regard “circuit size” as description length.
Lemma.
∃𝑁, ∀𝐵, ∀𝑦, 𝑁MCSP𝐵 𝑦 = 𝑀 𝑦 ⟹ 𝑀 ∈ P Choose an oracle 𝐵 so that 𝑁MCSP𝐵 𝑦 can be easily simulated
(𝑗, 𝑘)
𝑈𝑗𝑘
is equal to 𝑈𝑗 (for each 𝑗).
Theorem 1.
𝐵
PMCSP𝐵 = P.
Theorem 2. (Limit of BPP-Reductions)
𝐵
BPPMCSP𝐵[1] ⊆ AM ∩ coAM.
BPP MCSP𝐵.
𝑅(𝑦, −) 𝑠 𝑅 𝑦, 𝑠 < 2𝑡+1 instances
Suppose Pr 𝑅 𝑦, 𝑠 ∈ MCSP𝐵 ≈ 1.
< 2𝑡+1 instances
YES
Conversely, suppose that 𝑅(𝑦, −) concentrates on .
For some 𝑙 ≤ 𝑡 − 𝑃 log 𝑜 ,
1, 𝑈2, … 𝑈2𝑙 (Claim: Pr 𝑅 𝑦, 𝑠 ∈ MCSP𝐵 ≈ 1.)
Conversely, suppose that 𝑅(𝑦, −) concentrates on .
For some 𝑙 ≤ 𝑡 − 𝑃 log 𝑜 ,
1, 𝑈2, … 𝑈2𝑙 (Claim: Pr 𝑅 𝑦, 𝑠 ∈ MCSP𝐵 ≈ 1.)
Conversely, suppose that 𝑅(𝑦, −) concentrates on .
For some 𝑙 ≤ 𝑡 − 𝑃 log 𝑜 ,
1, 𝑈2, … 𝑈2𝑙
(Claim: Pr 𝑅 𝑦, 𝑠 ∈ MCSP𝐵 ≈ 1.)
It suffices to check if such a concentration occurs in AM ∩ coAM.
It suffices to check if such a concentration occurs in AM ∩ coAM.
(or lower and upper bound protocols [Goldwasser & Sipser ’86] [Fortnow ‘87])
It suffices to check if such a concentration occurs in AM ∩ coAM.
When the query distribution is concentrated When the query distribution is not concentrated
Pr 𝑅 𝑦, 𝑠 is heavy is large.
(Here, 𝑧 is heavy ⇔ the inverse of 𝑧 is large)
Pr 𝑅 𝑦, 𝑠 is heavy is small.
𝑞 MCSP ⟹ ZPP ≠ EXP.
[Murray & Williams (2015)]
𝑞 MCSP ⟹ ZPP ≠ EXP.
Proof: Based on firm links between circuit complexity and Levin’s Kolmogorov complexity. Exntension to the case of nonadaptive reductions
[Allender, Buhrman, Koucký, van Melkebeek and Ronneburger (2006)] [Allender, Koucky, Ronneburger & Roy (2011)]
𝑞 GapMCSP ⟹ ZPP ≠ EXP.
[Murray & Williams (2015)]
𝑞 MCSP ⟹ ZPP ≠ EXP.
Exntension to the case of Turing reductions A promise problem of approximating log of circuit complexity within constant factor