NP with Small Advice Lance Fortnow Adam R. Klivans∗ Department of Computer Science Department of Computer Science University of Chicago The University of Texas at Austin Chicago, IL 60637 Austin, TX 78712 [email protected] [email protected] Abstract • It is known that if EXP cannot be computed by nondeterministic polynomial-size circuits then it We prove a new equivalence between the non-uniform is possible to obtain similar derandomizations of and uniform complexity of exponential time. We show AM [18, 21, 25]. Shaltiel and Umans [24] were NP that EXP ⊆ NP/log if and only if EXP = P|| . Our equiv- the first to prove that if EXP 6⊂ NP/poly then alence makes use of a recent result due to Shaltiel and AM ⊆ NSUBEXP for infinitely many input lengths. Umans showing EXP in PNP implies EXP in NP/poly. || Collapses Perhaps less well known than the above derandom- 1. Introduction izations are equally important results showing that uni- form complexity classes such as EXP or NEXP collapse Let A and B be uniform complexity classes such that if they are contained in smaller, non-uniform classes: B ⊆ A. If A seems much “larger” than B then it is of- • Babal et al. [2] showed that EXP ⊆ P/poly implies ten the case that we can prove that B is strictly con- that EXP = MA, improving on work due to Meyer tained in A, e.g. let B = P and A = EXP. Is the same [17] who first proved that EXP ⊆ P/poly implies true if we consider a non-uniform analogue of B? That EXP = ΣP. is to say, augment B by giving it access to some ad- 2 vice string b such that b depends only on the length of • Impagliazzo et al. [13] improved the above collapse x; can we still separate A from B/b? If not, can we de- and showed that NEXP ⊆ P/poly if and only if rive interesting consequences on A if it is contained in NEXP = EXP = MA. This result is crucial to Ka- B/b, i.e. can we show that A collapses to some smaller banets and Impagliazzo’s breakthrough paper [15] complexity class? showing that derandomizing BPP implies proving These questions are of central importance in com- circuit lower bounds. putational complexity theory, particularly in the area If we pay particular attention to MA, then the above of derandomization, where both separations of uni- separations and collapses match up nicely– if EXP ⊆ form from non-uniform classes or collapses of uni- P/poly then EXP collapses to MA, and if EXP 6⊂ P/poly form classes have important consequences: then MA can be derandomized (and will be contained in NSUBEXP). Separations The same is not true, however, for AM. Separat- • If EXP 6⊂ P/poly, then Babai et al. [2], building on ing EXP from NP/poly implies that AM is contained in the “Hardness versus Randomness” paradigm [22], non-deterministic, sub-exponential time [24]. Placing P have shown that BPP is contained in subexpo- EXP ⊆ NP/poly, however, implies only that EXP = Σ3 , 1 nential time and that MA is contained in non- the third level of the polynomial-time hierarchy . deterministic subexponential time (both contain- Is it true that EXP ⊆ NP/poly implies that EXP = ments are for infinitely many input lengths). AM? If so, combining this fact with the above deran- ∗ Work done while visiting the Toyota Technological Institute at 1 Actually one can prove that under the assumption that EXP ⊆ P Chicago. NP/poly, EXP ⊆ ZPPΣ2 [7] NP domization of AM [24] would yield a rare uncondi- (it is also known that NEXP 6⊂ P|| [11]). We can con- tional derandomization of AM, namely that AM is con- sider, however, the consequences of NEXP being con- tained in Σ2 − SUBEXP, the subexponential time ana- tained in randomized complexity classes that take ad- P P logue of Σ2 (AM is currently only known to be in Π2 )– vice (such classes have been a focus of research inter- see Gutfreund et al. [12] for a discussion. Shaltiel and est as of late [4, 10]). We observe that the techniques Umans [25] have asked if EXP ⊆ NP/ log implies that of Impagliazzo et al. [13] can be used to prove that EXP = AM, as even this is not known. NEXP ⊆ BPP/ log implies NEXP = BPP, strengthen- ing a result of Trevisan and Vadhan [27]. 1.1. Our Results 1.2. Related Work We give a new collapse for exponential time if it is computed by a nondeterministic, slightly non-uniform The first important collapse of a uniform class con- complexity class. More precisely we show that if EXP ⊆ tained in a non-uniform class is due to Karp and Lip- NP/ log then EXP = PNP, i.e. EXP is computed by a ton [17] who showed that NP ⊆ P/poly implies that || P polynomial-time turing machine with non-adaptive ac- PH = Σ2 and that NP ⊂ P/ log implies P = NP. cess to an NP-oracle. Further, we can also prove the For exponential time, aside from the collapse results converse: mentioned above due to Babai et al. [2] and Impagli- azzo et al. [13], Buhrman and Homer [8] showed that if Theorem 1 The following are equivalent. EXPNP ⊆ EXP/poly then EXPNP = EXP and Buhrman, Fortnow, and Pavan [7] showed a weak relativization of 1. EXP ⊆ PNP || Impagliazzo et al. [13], namely that for any A ∈ EXP, A A A 2. EXP ⊆ NP/ log NEXP ⊆ PA/poly implies NEXP = EXP and if A P A A is complete for Σk then NEXP ⊆ P /poly implies The forward direction of our equivalence makes use NEXPA = EXP = MAA. of a new hardness amplification result due to Shaltiel Buhrman, Chang and Fortnow [6] give an equiva- and Umans. They prove that if EXP 6⊂ NP/poly then lence of a non-uniform collapse to NP and a uniform NP EXP 6⊂ P|| /poly. The contrapositive gives a partial inclusion. collapse of exponential time which we show how to strengthen via a non-standard method of computing Theorem 4 (Buhrman-Chang-Fortnow) The fol- NP lowing are equivalent. advice. As a result we obtain EXP ⊆ P|| implies EXP ⊆ NP/ log, improving on the conclusion EXP ⊆ 1. coNP ⊆ NP/1 AM/ log obtained by Shaltiel and Umans [25]. 2. The polynomial-time hierarchy collapses to Dp The backwards direction requires two collapses. First we prove that if EXP ⊆ NP/ log then EXP = PNP, where Dp is the set of languages that are the difference of and then we use the fact that the ODDMAXBIT func- two NP languages. tion is complete for PNP to show how the above advice Buhrman, Chang and Fortnow also generalize Theo- strings can be computed and verified non-adaptively. rem 4 to show that coNP in NP/k if and only if the We also prove variations of Theorem 1 for other polynomial-time hierarchy collapses to the 2kth level classes. of the Boolean hierarchy where the first level of the Theorem 2 The following are equivalent. Boolean hierarchy is NP and the i + 1st level is the set of differences of sets in NP and the sets in the ith level. NP 1. PSPACE ⊆ P|| This extension only works for finite k but Buhrman, 2. PSPACE ⊆ NP/ log Fortnow and Chang conjecture that it extends to k = O(log n). Theorem 3 The following are equivalent. Conjecture 5 (Buhrman-Chang-Fortnow) The #P NP following are equivalent. 1. P ⊆ P|| 1. coNP ⊆ NP/ log 2. P#P ⊆ NP/ log 2. The polynomial-time hierarchy collapses to PNP Is it possible to prove something similar to Theo- || P rem 1 for NEXP? We show that, in fact, such a state- Since EXP in NP/poly implies EXP ⊆ Σ3 [1, 29], ment is vacuously true for NEXP since one can sepa- Theorem 4 implies EXP ⊆ NP/1 if and only if EXP = rate NEXP from NP/ log outright via diagonalization Dp. Likewise Conjecture 5 implies Theorem 1 so we can view our Theorem 1 as a partial resolution of Conjec- game where on input x, Arthur sends a random chal- ture 5. lenge r to Merlin who responds with y; Arthur then probabilistically verifies y to determine acceptance of 2. Preliminaries x (see the survey by Kabanets [14]). 2.1. Complexity Classes 2.3. Alternation and Games We assume the reader is familiar with complex- We will make use of the characterization of PSPACE k k ity classes P = ∪kDTIME(n ), NP = ∪kNTIME(n ), due to Chandra, Kozen, and Stockmeyer as a game nk nk [9]. Chandra et al. showed that PSPACE is equivalent EXP = ∪kDTIME(2 ), NEXP = ∪kNTIME(2 ), k to the following two person game: on input x, play- PSPACE = ∪kDSPACE(n ) as well as notions of ora- cle turing machines and the polynomial-time hierarchy ers alternate announcing bits for a polynomial num- (see e.g. [3] for further explanations). ber of rounds and a polynomial-time computable judge The non-uniform class NP/ log is the set of languages chooses a winner based on x and the announced bits: L such that there exists a language A in NP and a func- Theorem 7 (Chandra-Kozen-Stockmeyer) A ∗ tion a : N → Σ with |a(n)| = O(log n) such that for language L is in PSPACE if there exists a polynomial- ∗ all x in Σ , x is in L if and only if (x, a(|x|)) is in A. time relation R on 2k + 1 strings where k = nO(1) and NP/poly has the same definition except that we allow players P1 and P2 such that |a(n)| = O(nk) for some k.