A survey on derandomizing BPP and AM

Danny Gutfreund, Hebrew U.Ronen Shaltiel, Weizmann

Inst.Amnon Ta-Shma, Tel-Aviv U.

Arthur-Merlin Games [BM] Interactive games in which the all-

powerful prover Merlin attempts to prove some statement to a probabilistic poly-time verifier.

Merlin Arthur“xL”

toss coinsmessage

message

I accept

Arthur-Merlin Games [BM] Completeness: If the statement is

true then Arthur accepts. Soundness: If the statement is

false then Pr[Arthur accepts]<½.

Merlin Arthur“xL”

toss coinsmessage

message

I accept

Arthur-Merlin Games [BM] Completeness: If the statement is

true then Arthur accepts. Soundness: If the statement is

false then Pr[Arthur accepts]<½.

The class AM: All languages L which have an Arthur-Merlin protocol.

Contains many interesting problems not known to be in NP.

Example: Co-isomorphism of Graphs. L={G1,G2: the labeled graphs G1,G2 are

not isomorphic}. L in coNP and is not known to be in NP.

Merlin Arthur(G1,G2 ) L

Randonly chooses:

b {1,2} random permutation of

Gb

“The graph Gc was permuted”

Decides which of the two graphs

was permuted.

Verifies that c=b.

The big question:

Does AM=NP?

In other words: Can every Arthur-Merlin protocol be replaced with one in which Arthur is deterministic?

Note that such a protocol is an NP proof.

Derandomization: a brief overview A paradigm that attempts to transform:

Probabilistic algorithms => deterministic algorithms. (P BPP EXP NEXP).

Probabilistic protocols => deterministic protocols. (NP AM EXP NEXP).

We don’t know how to separate BPP and NEXP.

Can derandomize BPP and AM under natural complexity theoretic assumptions.

Hardness versus Randomness Initiated by [BM,Yao,Shamir].

Assumption: hard functions exist.

Conclusion: Derandomization.

A lot of works: [BM82,Y82,HILL,NW88,BFNW93, I95,IW97,IW98,KvM99,STV99,ISW99,MV99, ISW00,SU01,U02,TV02]

A quick surveyAssumption: There exists a function in

DTIME(2O(n)) which is hard for “small” circuits.

ClassBPPAM

A hard function for:

Deterministic circuits

Nondeterministic circuits

High-endBPP=PAM=NP

Low-endBPPSUBEXPAM NSUBEXP

Hardness versus Randomness

Assumption: hard functions exist.

Conclusion: Derandomization.

Hardness versus Randomness

Assumption: hard functions exist.

Exists pseudo-random generator

Conclusion: Derandomization.

Pseudo-random generators A pseudo-random generator (PRG) is an algorithm

that stretches a short string of truly random bits into a long string of pseudo-random bits.

pseudo-random bits

PRG seed

Pseudo-random bits are indistinguishable from truly random bits for feasible algorithms.

For derandomizing AM: Feasible algorithms = nondeterministic circuits.

??????????????

Pseudo-random generators for nondeterministic circuits Nondeterministic circuits can identify pseudo-

random strings. Given a long string, guess a short seed and check

that PRG(seed)=long string. Can distinguish between random strings and

pseudo-random strings. Assuming the circuit can run the PRG!! The Nisan-Wigderson setup: The circuit cannot run

the PRG!! For example: The PRG runs in time n5 and fools

(nondeterministic) circuits of size n3. Sufficient for derandomization!!

Hardness versus Randomness

Assumption: hard functions exist.

Exists pseudo-random generator

Conclusion: Derandomization.

PRG’s for nondeterministic circuits derandomize AM We can model the AM protocol as a

nondeterministic circuit which gets the random coins as input.

Merlin Arthur“xL”

random message

message

I accept

Hardwire input

PRG’s for nondeterministic circuits derandomize AM We can model the AM protocol as a

nondeterministic circuit which gets the random coins as input.

Merlin Arthur“xL”

random input

Nondeterministic guess

I accept

inputNondeterministic guessHardwire input

PRG’s for nondeterministic circuits derandomize AM We can model the AM protocol as a

nondeterministic circuit which gets the random coins as input.

We can use pseudo-random bits instead of truly random bits.

Merlin Arthur“xL”

pseudo-random input

Nondeterministic guess

I accept

Nondeterministic guess inputHardwire input

PRG’s for nondeterministic circuits derandomize AM We have an AM protocol in which Arthur

acts deterministically. (Arthur sends all pseudo-random strings

and Merlin replies on each one.) Deterministic protocol => NP proof.

Merlin Arthur“xL”

pseudo-random input

Nondeterministic guess

I accept

A quick surveyAssumption: There exists a function in

DTIME(2O(n)) which is hard for “small” circuits.

ClassBPPAM

A hard function for:

Deterministic circuits

Nondeterministic circuits

High-endBPP=PAM=NP

Low-endBPPSUBEXPAMNSUBEXP

The Nisan-Wigderson setting We’re given a function f which is:

Hard for small circuits. Computable by uniform machines with “slightly”

larger time. Basic idea:

G(x)=x,f(x) “f(x) looks random to a small circuit that sees x”.

Warning: no composition theorems. Correctness proof of PRG can’t use it’s efficiency.

The PRG runs in time “slightly” larger than the size of the circuit.

The rest At this point I moved to the blackboard and covered: The Nisan-Wigderson generator. (You can find a

presentation (as well as an introduction to derandomization)on my homepage www.wisdom.weizmann.ac.il/~ronens under the title “derandomizing BPP”. This was written in 1998 so the part about hardness amplification is slightly outdated. However, the first chapter is still relevant and contains both the BMY and NW generators.

I also explained that PRGs for nondeterministic circuits (which derandomize AM) can be constructed using methods for constructing PRGs for deterministic circuits. This was pointed out by Klivans and van-Melekbeek and you can get the paper at http://www.cs.wisc.edu/~dieter/Research/r-gni.html.