The Complexity of Complexity

Transcription

1 The Complexity of Complexity New Insights on the (Non)hardness of Circuit Minimization and Related Problems Eric Allender Rutgers University Joint work with Shuichi Hirara (U. Tokyo) RodFest, January 5, 2017 Eric Allender: The Complexity of Complexity 1

2 The Complexity of Complexity New Insights on the (Non)hardness of Circuit Minimization and Related Problems Eric Allender Rutgers University Joint work with Shuichi Hirara (U. Tokyo) RodFest, January 5, 2017 Eric Allender: The Complexity of Complexity 2

3 NP-Intermediate Problems NPC P Eric Allender: The Complexity of Complexity 3

4 NP-Intermediate Problems Ladner (1975) showed that intermediate problems exist unless P=NP. NPC P But these problems are unnatural. Eric Allender: The Complexity of Complexity 4

5 NP-Intermediate Problems Some prominent candidates were studied early on NPC P Eric Allender: The Complexity of Complexity 5

6 NP-Intermediate Problems Some prominent candidates were studied early on NPC P Graph Genus Linear Programming but subsequently moved. Eric Allender: The Complexity of Complexity 6

7 NP-Intermediate Problems MCSP is still here: NPC P Eric Allender: The Complexity of Complexity 7

8 The Context! The Minimum Circuit Size Problem (MCSP) = {(f,i) : f is the truth-table of a function that has a circuit of size i}.! That is, MCSP is the overgraph of the circuit size complexity measure SIZE(f). Eric Allender: The Complexity of Complexity 8

9 The Context! The Minimum Circuit Size Problem (MCSP) = {(f,i) : f is the truth-table of a function that has a circuit of size i}.! In NP, but not known (or widely believed) to be NP-complete. [Kabanets, Cai], [Murray, Williams], [A, Holden, Kabanets], [Hirahara, Watanabe]! Lots of reasons to believe it s not in P. Eric Allender: The Complexity of Complexity 9

10 More Context! MCSP is more like a family of problems, than a single problem.! For instance size could mean # of wires or # of gates, or # of bits to describe the circuit, etc.! None of these is known to be reducible to any other but all can stand in for MCSP.! One more such variant: O KT = {(x,i) : KT(x) i} Eric Allender: The Complexity of Complexity 10

11 Complexity Classes EXPSPACE NEX PSPACE P NP P BPP P/poly Eric Allender: The Complexity of Complexity 11

12 More Complexity Classes P/poly BPP RP ZPP P DET L Uniform AC 0 AC 0 Eric Allender: The Complexity of Complexity 12

13 Yet More Context! Factoring and Discrete Log are in ZPP MCSP.! Graph Isomorphism is in RP MCSP.! Every promise problem in SZK is in (Promise) BPP MCSP.! But is MCSP NP-hard? Statistical Zero Knowledge SZK is contained in NP/poly conp/poly Eric Allender: The Complexity of Complexity 13

14 Yet More Context! Factoring and Discrete Log are in ZPP MCSP.! Graph Isomorphism is in RP MCSP.! Every promise problem in SZK is in (Promise) BPP MCSP.! But is MCSP NP-hard? All these hardness results use probabilistic reductions. Eric Allender: The Complexity of Complexity 14

15 Consequences of Hardness! Eric Allender: The Complexity of Complexity 15

16 Local Reductions Output Given i, can compute i th bit of output in time t(n) Input Eric Allender: The Complexity of Complexity 16

17 Local Reductions! Eric Allender: The Complexity of Complexity 17

18 First Theorem! O KT is hard for DET under non-uniform NC 0 reductions.! Corollary: O KT is not in AC 0 [p] for any prime p.! Circuit lower bounds imply O KT is hard for DET under uniform AC 0 Turing reductions.! O KT hard for PARITY under these same uniform reductions implies (similar) circuit lower bounds. Eric Allender: The Complexity of Complexity 18

19 First Theorem! O KT is hard for DET under non-uniform NC 0 reductions.! Corollary: O KT is not in AC 0 [p] for any prime p.! Independently, [Oliveira & Santhanam] have shown that DET reduces to MCSP via nonuniform TC 0 reductions. (This still leaves open the question of whether MCSP is in AC 0 [p].) Eric Allender: The Complexity of Complexity 19

20 First Theorem! O KT is hard for DET under non-uniform NC 0 reductions.! Corollary: O K and O C are hard for DET under non-uniform NC 0 reductions.! Open question: Is O K or O C hard for P or hard for Σ 0 1 under non-uniform AC0 or NC 0 m reductions?! Non-uniformity is key. No uniform m reduction can reduce anything nontrivial to O K or O C. Eric Allender: The Complexity of Complexity 20

21 First Theorem! O KT is hard for DET under non-uniform NC 0 reductions.! Corollary: O K and O C are hard for DET under non-uniform NC 0 reductions.! Open question: Is O K or O C hard for P or hard for Σ 0 1 under non-uniform AC0 or NC 0 m reductions?! O K and O C are hard for Σ 0 1 under (non-uniform) P/poly-Turing reductions [ABKMR]. Eric Allender: The Complexity of Complexity 21

22 Three Bizarre Inclusions! NEXP is contained in NP O K (for every U). The decidable sets that are in P RK tt for every U are in PSPACE. [AFG11]! PSPACE is contained in P O K (for every U).! BPP is contained in P tt OK (for every U). The decidable sets that are in P OK tt for every U are in PSPACE. [AFG11] [CDELM] The sets that are in P tt OK for every U are decidable. Eric Allender: The Complexity of Complexity 22

23 Three Bizarre Inclusions! NEXP is contained in NP O K (for every U). The sets that are in NP O K for every U are in EXPSPACE.! PSPACE is contained in P O K (for every U).! BPP is contained in P tt OK (for every U). The sets that are in P tt OK for every U are in PSPACE.! Thus there are reasons to care about efficient reductions to O K and O C. Eric Allender: The Complexity of Complexity 23

24 Context for the 2 nd Theorem! Recall: Every promise problem in SZK is in (Promise) BPP MCSP.! SZK is usually defined in terms of promise problems. Eric Allender: The Complexity of Complexity 24

25 Promise Problems Yes No Ordinary decision problems. Eric Allender: The Complexity of Complexity 25

26 Promise Problems Yes No A solution Ordinary decision problems. Yes Don t Care No Promise Problems. Eric Allender: The Complexity of Complexity 26

27 Promise Problems! Eric Allender: The Complexity of Complexity 27

28 The 2 nd Theorem! If cryptographically-secure one-way functions exist, then there is a function ε(n)=o(1) such that Gap ε MCSP is not in P/poly, and Gap ε MCSP is not NP-hard under P/poly- Turing reductions. Eric Allender: The Complexity of Complexity 28

29 The 2 nd Theorem! If cryptographically-secure one-way functions exist, then there is a function ε(n)=o(1) such that Gap ε O KT is not in P/poly, and Gap ε O KT is not NP-hard under P/poly-Turing reductions. Eric Allender: The Complexity of Complexity 29

30 Three Bizarre Inclusions, Reprise! NEXP is contained in NP GapO K, which is contained in EXPSPACE.! PSPACE is contained in P GapO K.! BPP is contained in P tt GapOK, which is contained in PSPACE.! I m on record as conjecturing BPP = P tt OK! Might it be easier to show BPP = P tt GapOK?! In particular, is there an easier way than [CDELM] to show that P tt GapOK is decidable? Eric Allender: The Complexity of Complexity 30

31 1 st theorem (O KT hard for DET)! [Toran] showed that DET AC 0 -reduces to the isomorphism problem for rigid graphs.! [A,Grochow,Moore] showed that rigid graph isomorphism is in RP O KT. We ll modify that proof. Eric Allender: The Complexity of Complexity 31

32 1 st Theorem: Proof! On input (G 0,G 1 ) Randomly pick a bit string w=w 1 w 2 w t. Pick random permutations π 1 π t. Let z= wπ 1 (G w1 )π 2 (G w2 ) π t (G wt )! If G 0 and G 1 are not isomorphic, then z allows us to reconstruct w and π 1 π t, so that z has (non-time-bounded) K-complexity around t+ts (where s = log n!), whp. Hence KT(z) > t+ts.! Otherwise, KT(z) is around n 2 +ts. Eric Allender: The Complexity of Complexity 32

33 1 st Theorem: Proof! On input (G 0,G 1 ) Randomly pick a bit string w=w 1 w 2 w t. Pick random permutations π 1 π t. Let z= wπ 1 (G w1 )π 2 (G w2 ) π t (G wt )! The mapping (G 0,G 1,π 1,π 2, π t ) z is computable in AC 0 if the permutations are presented as a list of the form (i,π(i)).! But a random bit string is not going to encode a sequence of permutations. Eric Allender: The Complexity of Complexity 33

34 1 st Theorem: Proof! On input (G 0,G 1 ) Randomly pick a bit string w=w 1 w 2 w t. Pick random permutations π 1 π t. Let z= wπ 1 (G w1 )π 2 (G w2 ) π t (G wt )! Solution: There is a logspace-computable function f that takes (nt) O(1) random bits and with high probability outputs a (nearly)-uniformly random sequence π 1,π 2, π t. Eric Allender: The Complexity of Complexity 34

35 1 st Theorem: Proof! On input (G 0,G 1 ) Randomly pick a bit string w=w 1 w 2 w t. Pick random permutations π 1 π t. Let z= wπ 1 (G w1 )π 2 (G w2 ) π t (G wt )! Thus there is an AC 0 function g such that g(g 0,G 1,f(r)) is a probabilistic reduction from Rigid Graph Isomorphism to O KT. Hardwiring in a good choice of f(r) yields a non-uniform AC 0 reduction. Eric Allender: The Complexity of Complexity 35

36 1 st Theorem: Proof! On input (G 0,G 1 ) Randomly pick a bit string w=w 1 w 2 w t. Pick random permutations π 1 π t. Let z= wπ 1 (G w1 )π 2 (G w2 ) π t (G wt )! Combining this with [Toran], O KT is hard for DET under AC 0 reductions.! Now, by [Agrawal,A,Rudich], O KT is also hard for DET under NC 0 reductions. Eric Allender: The Complexity of Complexity 36

37 Congratulations, Rod!!Thanks! Eric Allender: The Complexity of Complexity 37