Karp-Lipton Results for Zero-Error Exponential Time

I Lemma 58 (Fully uniform simulations using easy witness and truth-table concatenation).

Either BPP⊆ZPQP, or ZPEXP⊆i.o.ESUBEXP.

Proof. We use the “easy witness” method of Kabanets [37]. LetM be any probabilistic Turing machine with zero error running in time 2mj

for somej ≥ 1, such that on each random computation path ofM on any inputx, the output is either the correct answer for

M onxor ‘?’, and moreover the probability of outputting ‘?’ is less than 2−2m_{for any input}

x∈ {0,1}m_{. For eachε >0, we define the following attempted deterministic simulationM} ε

forM. On inputxof lengthm,Mεcycles over all circuitsC of size 2m

ε/2

onmj _{inputs. For} each such circuit, it explicitly computes the truth tablett(C) of the circuitC, and runsM

onxwithtt(C) as randomness. If the run accepts, it accepts; if the run rejects, it rejects. If the run outputs ‘?’, it moves on to the next circuitC in lexicographic order of circuit encodings. If all runs output ‘?’, the machine rejects. It should be clear that the simulation

Mεruns in deterministic time≤22

mε

on inputs of length mfor any sufficiently largem, and only accepts inputsx∈L(M).

If for eachε >0, we have that the simulationMεsolvesL(M) correctly on all inputs of lengthmfor infinitely many input lengthsm, we have thatL(M)⊆i.o.ESUBEXP.

Suppose, on the contrary, that there is someε >0 such that the simulationMε fails on at least one inputxof each large enough input lengthm. We show how to use this to decide every language inBPPinZPQP.

LetN be any bounded-error probabilistic machine running in time at mostnk _{for some} constantkand large enoughn. Assume without loss of generality thatN has error ≤1/10 on any input of lengthn. We use M to give a zero-error simulationN0 of N on all inputs of large enough length. Given an inputy of length n, N0 simulates M on each input x

of lengthm(n) def= (dlogne)d _{in turn, for some constant d≥1 to be specified later. If M} outputs ‘?’,N0 outputs ‘?’, otherwise it moves on to the next input in lexicographic order. If runningM gives ‘?’ outputs for every inputxof lengthm(n),N0 outputs ‘?’. Otherwise,

N0 concatenates the random strings used on the computation paths ofM for each input of lengthm(n) into a single stringRn of lengthO(2polylog(n)). It then usesRn as the truth-table of the hard function for the generator in Theorem 13, setting parameters so that at leastn2k pseudorandom bits are produced by the generator. It cycles over all possible seeds of the generator and runsN using each output in turn as the sequence of random choices, accepting if and only if a majority of runs accepts.

Setting dto be a large enough constant depending onj, k, εand the constantc in the statement of Theorem 13, it can be shown that this simulation can be done in quasi-polynomial time, and that it is correct for each inputy of large enough length wheneverN0 does not output ‘?’. The key here is that by the failure of Mε for at least one input of any large enough length, the stringRn is guaranteed to be hard enough that the generator is correct. This is becauseRn contains a subfunction of sufficiently large worst-case circuit complexity. Hence cycling over all seeds of the generator and taking the majority value gives the correct answer forN on inputy. Finally, under our initial assumption that M has exponentially small failure probability, by a union bound the probability thatN0 outputs ‘?’ on any large enough input is small. This concludes the proof thatBPP⊆ZPQP. _J ITheorem 59(High-end fully uniform Karp-Lipton theorem for zero-error exponential time).

IfZPEXP⊆i.o.Circuit[2n/3_{], thenZPEXP⊆}

i.o.ESUBEXP.

Proof. Observe that the proof of Theorem 56 establishes that ifREXP 6⊆ i.o.ESUBEXP, then PSPACE ⊆ BPQP. By Lemma 58, if ZPEXP 6⊆ i.o.ESUBEXP, then BPP ⊆ ZPQP, and hence by upward translation,BPQP=ZPQP. Putting these together, we have that if

ZPEXP6⊆i.o.ESUBEXP, thenPSPACE⊆ZPQP. Now by upward translation, we have that

EXPSPACE=ZPEXP, and hence by Corollary 39, we getZPEXP6⊆i.o.Circuit[2n/3_]. J We have learned from Valentine Kabanets (private communication) that he has independ- ently established Theorem 59 in an unpublished manuscript.

In fact, we can get a non-trivial consequence from the weakest possible non-trivial assumption about the circuit size of Boolean functions computable in zero-error exponential time. This extension of Theorem 59 relies on the following simple lemma.

ILemma 60(Maximally hard functions in exponential space). Let smax:N→Nbe such that

for eachn∈N,smax(n) is the maximum circuit complexity among Boolean functions onn bits. ThenEXPSPACE6⊆i.o.Circuit[smax−1].

Proof (Sketch). The proof is by simple diagonalization. In exponential space, we can systematically list the truth tables of Boolean functions onnbits, and maintain the one with the highest circuit complexity. To compute the circuit complexity of a listed truth table can be done by cycling over all circuits, starting from the smallest one, and checking for each circuit whether it computes the given truth table. Once the truth table of a function with maximum circuit complexity has been computed, we simply look up the corresponding entry

in the truth table for any particular input. _J

Now by using the same proof as for Theorem 59 but applying Lemma 60 instead of Corol- lary 39, we have the following stronger version of Theorem 59. (We note that Theorems 54 and 56 admit similar extensions.)

ITheorem 61(Strong Karp-Lipton Theorem for zero-error probabilistic exponential time). Let

smax:N→Nbe such that for eachn∈N, smax(n)is the maximum circuit complexity among Boolean functions on nbits. IfZPEXP⊆i.o.Circuit[smax−1], thenZPEXP⊆i.o.ESUBEXP.