Upward Separation for FewP and Related ClassesRajesh P. N. Rao 1Department of Computer ScienceUniversity of RochesterRochester, NY 14627, USAJ�org Rothe 2Fakult�at f�ur Mathematik und InformatikFriedrich-Schiller-Universit�at Jena07743 Jena, GermanyOsamu Watanabe 3Department of Computer ScienceTokyo Institute of TechnologyMeguro-ku, Ookayama, Tokyo 152, JapanFebruary 22, 19941Research supported in part by NSF research grant CDA-8822724.2Research supported in part by a grant from the DAAD and by the NSF under grant NSF-CCR-8957604. Work done in part while visiting the University of Rochester.3Research supported in part by the JSPS and the NSF under grant JSPS-ENGR-207/NSF-INT-9116781. Work done in part while visiting the University of Rochester.

AbstractThis paper studies the range of application of the upward separation tech-nique that has been introduced by Hartmanis to relate certain structural prop-erties of polynomial-time complexity classes to their exponential-time analogsand was �rst applied to NP [Har83]. Later work revealed the limitations ofthe technique and identi�ed classes defying upward separation. In particu-lar, it is known that coNP as well as certain promise classes such as BPP,R, and ZPP do not possess upward separation in all relativized worlds [HIS85;HJ93], and it had been suspected that this was also the case for other promiseclasses such as UP and FewP [All91].In this paper, we refute this conjecture by proving that, in particular, FewPdoes display upward separation, thus providing the �rst upward separation re-sult for a promise class. In fact, this follows from a more general result the proofof which heavily draws on Buhrman, Longpr�e, and Spaan's recently discoveredtally encoding of sparse sets. As consequences of our main result, we obtainupward separations for various known counting classes such as �P, coC=P, SPP,and LWPP. Some applications and open problems are discussed.Keywords: Computational Complexity; Upward Separation.1 IntroductionA main task in complexity theory is to prove collapses or separations between complexityclasses, or, if this doesn't succeed as is often the case, to provide structural consequencesfrom some collapse or separation. The techniques of upward and downward separation dealwith the link of small and large classes: downward separation typically shows that theseparation of large classes is downwards translated to smaller ones (e.g., if some level ofthe polynomial hierarchy di�ers from the succeeding one, then all smaller levels form astrict hierarchy [Sto77; MS72]), whereas upward separation results state that if small (i.e.,polynomial-time) classes di�er on sets of small density such as sparse or tally sets, thentheir exponential-time counterparts are separated. The �rst results of this kind are due toBook who has shown that E 6= NE if and only if there exist tally sets in NP � P [Boo74](see Lemma 2.3), and to Hartmanis et al. who have shown that E 6= NE if and only if thereexist sparse sets in NP�P [Har83; HIS85]. E and NE here refer to Sc>0 DTIME(2cn) andSc>0 NTIME(2cn), respectively. Any class satisfying this latter \if and only if" proposition,as NP does above with respect to sparse sets, is said to display upward separation.1

In contrast to the NP case, several results have been established that reveal the limita-tions of the upward separation technique by showing that certain classes do not robustly(i.e., with respect to all oracles) display upward separation. Hartmanis, Immerman andSewelson have shown that the upward separation technique fails for coNP relative to anoracle [HIS85], and Hemaspaandra and Jha provided relativizations in which the promiseclasses1 BPP, R, and ZPP defy upward separation [HJ93]. They posed the question ofwhether one can prove similar failings regarding upward separation for other promiseclasses, and even the non-promise class PP. Allender constructed an oracle relative towhich Sc>0 DTIME(2c2n) = Sc>0 NTIME(2c2n) and yet NP�P contains extremely sparsesets [All91] (see also [AW90]). In addition, his paper presents some new|even thoughrestricted|upward separation results regarding the (promise) classes UP and FewP: thereexist sets of constant (respectively, logarithmic) density in UP�P (respectively, FewP�P)if and only if the respective exponential-time analogs di�er [All91]. The natural questionarises whether or not, in FewP � P, the existence of log-sparse sets is equivalent to theexistence of sparse sets; Allender conjectured that this equivalence does not robustly hold.In this paper, we refute Allender's conjecture by showing that FewP does robustly displayupward separation. In fact, this follows from a more general result (Theorem 3.2) thatprovides a simple su�cient condition for a class to possess upward separation:2 all the classis required to satisfy is closure under the Few operator (de�ned in Section 3). As a con-sequence, upward separation results are obtained for a variety of known counting classes,including �P, coC=P, SPP, and LWPP. In contrast to the work of Hemaspaandra and Jha[HJ93], who gave the �rst examples of promise classes that fail to robustly display upwardseparation, we show that this behavior is not typical for promise classes in general by pro-viding the �rst examples of promise classes, speci�cally FewP, SPP, and LWPP, that dohave upward separation.Buhrman, Longpr�e, and Spaan's tally encoding of sparse sets, introduced to provethe surprising result that any sparse set conjunctively truth-table reduces to some tallyset [BLS93] (see [Sal93] for an alternative proof and [Sch93] for another application of theirtechnique), is central to the proof of our main result. Buhrman, Longpr�e, and Spaan'scoding of a sparse set improves upon the one used by Hartmanis, Immerman, and Sewelson[HIS85] in order to establish (and to apply to NP) the upward separation technique.1I.e., classes de�ned via nondeterministic polynomial-time Turing machines the acceptance criterion ofwhich is not the logical negation of the rejection criterion|these machines \promise" to satisfy yet exactlyone of the criterions on any input.2Another structural su�cient condition for a di�erent type of upward separation (giving results of theform: \NP�BPP contains sparse sets if and only if NE 6� BPE") is observed in [HJ93]. Unlike our results,those are in fact established via the technique of Hartmanis et al. [Har83; HIS85].2

2 PreliminariesIn general, we adopt the standard notations of Hopcroft and Ullman [HU79]. We use thestandard alphabet � := f0; 1g; unless otherwise stated, all sets considered will be subsets of��. For any string x 2 ��, jxj denotes the length of x. To encode a pair of strings, we use apolynomial-time computable pairing function, h�; �i : ����� �! ��, that has polynomial-time computable inverses; this notion is extended to encode every k-tuple of strings, in thestandard way. For any set A, kAk represents the cardinality of A, A := �� � A denotesthe complement of A, and A=n is the set of all length n strings in A. The census functionof A, denoted by cenA(n), is de�ned as the number of strings in A of length at most n.A is said to be d-sparse (or of density d) if cenA(n) is bounded above by the function d(n);A is simply called sparse if d is a polynomial. A set T is tally if T � 0�. For any class K ofsets, de�ne co � K := fA jA 2 Kg (sometimes, we simply write coK).We shall use the term K-machine to refer to some nondeterministic Turing machine(NTM) whose speci�c mode of acceptance de�nes K. Oracle machines are de�ned in thestandard way [HU79], and AB denotes the class of sets accepted by some A-machine thataccesses an oracle set from B. Sometimes we shall use the common operator notation, inwhich, e.g., NP = 9 �P, and the 8 operator is de�ned to be 8 := co � 9 �co, e.g., coNP = 8 �P.In this paper, we focus on the following counting classes de�ned and intensely studied inthe literature|rather than formally de�ning them, we just refer the reader to the respectivereferences: UP [Val76], FewP [All86], �P [PZ83; GP86], PP, C=P [Sim75; Wag86], SPP[OH90; FFK91], and LWPP [FFK91]. Below we summarize the known relations amongthese classes and state some properties to be applied in proving Corollary 3.3.3Fact 2.1 1. UP � FewP � NP � coC=P � PP.2. FewP � SPP � LWPP � C=P � PP.3. SPP � �P.4. �P, SPP, and LWPP are self-low.The upward separation technique relates certain structural properties of polynomial-timecomplexity classes to their \exponential-time analogs." Adopting the notation of [HJ93],we can precisely formalize such a coupling of classes in a unifying way. As usual, de�ne3Note that the inclusions in Fact 2.1 straightforwardly translate into operator notation, e.g., Few�K � 9�Kfor any class K. The proof of the \self-lowness" of �P, i.e., �P�P = �P, is due to Papadimitriou and Zachos[PZ83]. Using a similar technique, Fenner, Fortnow, and Kurtz have shown this property to hold for SPPand LWPP as well [FFK91]. 3

Rtr(C) := fL j (9C 2 C) [L �tr C]g for any class C and any r and t for which �tr is de-�ned. A �em B if A exponential-time (i.e., Sc>0 DTIME(2cn)) many-one reduces to B,and A �pm; e`d B if A �pm B via a reduction f that is exponentially length-decreasing (i.e.,(9c > 0) (8x) [2cjf(x)j � jxj]).De�nition 2.2 [HJ93] We say that a pair of classes (A; B) is an associated pair ifRem(A) � B and Rpm; e`d(B) � A.Consider any class K that is de�ned via a certain acceptance mode of polynomial-timeNTMs. Then, the associated exponential-time analog , L, is de�ned via the same acceptancemechanism in terms of 2cn-time bounded NTMs|notationally, L thus di�ers from K justin the extension \E" rather than \P" indicating the di�erent time bound. For example,(P; E), (NP; NE), (FewP; FewE),4 (�P; �E), (PP; PE), (C=P; C=E), and (SPP; SPE) areassociated pairs.Given any set L 2 ��, we can pre�x its strings x by a 1 and then interpret as naturalnumbers bin(1x) in binary representation (see [Boo74; Har83]), thus converting L to a tallyset: tally(L) := f0bin(1x) j x 2 Lg. Conversely, any tally set T can be transformed into a setbin(T ) := fx j 0bin(1x) 2 Tg over � that contains the same information in \logarithmicallycompressed" form. Clearly, for any set L, bin(tally(L)) = L. Using the above notations,the key observation Book's results essentially draw upon [Boo74] can be stated as follows:L �em tally(L) and tally(L) �pm; e`d L. For completeness, the straightforward generalizationof Book's results about (NP; NE) to every associated pair containing (P; E) is presented.Lemma 2.3 If P � K and (K; L) is an associated pair, then K � P contains tally sets ifand only if L 6= E.Proof. Since (K; L) and (P; E) are both associated pairs, we have Rem(K) � L,Rpm; e`d(L) � K, Rem(P) � E, and Rpm; e`d(E) � P. Assume L 6= E, and let L � �� be someset in L � E. Then, tally(L) �pm; e`d L implies tally(L) 2 K. Suppose tally(L) 2 P. Then,L �em tally(L) implies L 2 E, a contradiction. Thus, there exists a tally set T = tally(L) inK�P. Conversely, let T be some tally set in K�P. A similar argument as above|now usingthat Rem(K) � L and Rpm; e`d(E) � P|shows that the binary encoding of T , L = bin(T ), isin L � E. 24The promise of the FewP machine to have at most polynomially many accepting paths translates in theFewE case to the promise of having at most 2O(n) accepting paths, which still are few compared with thedouble-exponential total number of paths of an exponential-time NTM [All86].4

3 Upward Separation ResultsIn this section, we provide a structural su�cient condition for upward separation. We showthat any complexity class K that is closed under the Few operator (de�ned below) possessesthis property.De�nition 3.1 Let K be any class of sets. A set L is in Few � K if and only if there exista set A 2 K and polynomials p and q such that for every x 2 ��,1. kfy j jyj = p(jxj) ^ hx; yi 2 Agk � q(jxj), and2. x 2 L () kfy j jyj = p(jxj) ^ hx; yi 2 Agk > 0.The Few operator formalizes, in the sense of FewP, a generalized type of polynomial-time many-one reducibility, as is common use for, e.g., polynomial-time randomized many-one reducibility with bounded error formalized by the BP operator [Sch87; Tod91]. Clearly,Few � P = Few �UP = Few � FewP = FewP.Theorem 3.2 Let (K; L) be an associated pair such that P � K and Few � K = K. Then,K� P contains sparse sets if and only if L 6= E.Proof. The \if" part holds by Lemma 2.3. For proving the \only if," we show the contra-positive: the supposition L = E forces all sparse sets from K into P. To this end, Buhrman,Longpr�e, and Spaan's tally encoding of any sparse set [BLS93] is used to make Lemma 2.3applicable. Below we give a short description of the encoding used (see [BLS93] for somealgebraic background that explains the speci�c choice of the parameters).Suppose L = E, and let S be any sparse set in K of density d for some polynomial d. For�xed n � 0, de�ne r(n) := l 2nlognm and let pn be the smallest prime larger than r(n) � d(n).Consider the �nite �eld GF(pn) with pn elements. As each polynomial over GF(pn) of degree� r(n) can be represented by its r(n)+1 coe�cients, it may be viewed as an (r(n)+1)-digitnumber in base pn. Thus, each string x 2 �=n corresponds to some polynomialqx(a) := xr(n)ar(n) + xr(n)�1ar(n)�1 + � � �+ x1a+ x0;where xr(n) : : :x0 is the representation of x in base pn. To encode the length n strings of S,n � 0, de�ne the nth segment of the tally set BLS(S) := Sn�0 Tn(S) byTn(S) := n0hn;a;qx(a)i a 2 GF(pn) ^ x 2 S=n o :5

Buhrman, Longpr�e, and Spaan introduced this coding to prove the surprising result thatany sparse set S conjunctively truth-table reduces to the tally set BLS(S).Consider the following algorithm for BLS(S): On input 0hn;a;bi, guess all strings x oflength n. For each guessed x, compute r(n) and pn, and verify a 2 GF(pn) and qx(a) = b.If this is not the case, then reject, otherwise simulate the K-machine for S on input x andaccept accordingly. Since there are only a polynomial number of strings in S=n, this showsthat BLS(S) 2 Few � K, and as Few � K = K, we have BLS(S) 2 K. Thus, bin(BLS(S)) isin L, which equals E by our supposition. Hence, tally(bin(BLS(S))) = BLS(S) is in P, andsince S conjunctively truth-table reduces to BLS(S), it follows that S 2 P. 2Corollary 3.3 Let K be any of the classes FewP; �P; SPP; LWPP; NP, or coC=P, and let(K; L) be the respective associated pair. Then, (K; L) displays upward separation, that is,K� P contains sparse sets if and only if L 6= E.Proof. By Theorem 3.2 it su�ces to show that each of the classes K considered is closedunder the Few operator. This is easily observed for FewP, and by Fact 2.1, Few � �P ��P�P = �P, Few � SPP � SPPSPP = SPP, and Few � LWPP � LWPPLWPP = LWPP.For K = NP and K = coC=P, the result follows from the well known or obvious factsthat Few � K � 9 � K, 9 � NP = NP [Sto77; MS72], and 8 � C=P = C=P [Tod91]. Thus,Few �NP � 9 �NP = NP and Few � coC=P � 9 � coC=P = co � 8 � C=P = coC=P. 2Note that, in the above proof, there is nothing special about the mod 2 de�ning �P[PZ83; GP86]|all we need is its self-lowness and that FewP � �P [CH90]. Thus, the resultholds as well for all classes ModpP (de�ned in [CH90; BGH90]), for prime p.4 Conclusions and Open ProblemsWe have presented several new upward separation results contrasting recently discoveredresults about some promise classes that fail to have upward separation in all relativizedworlds. As an immediate consequence, this, combined with the fact that equality of classesobeys standard upward translation, yields relativizations separating any two classes thatdi�er in their property of displaying or defying upward separation, e.g., BPPA 6= �PA,FewPA 6= ZPPA, etc. where A is the oracle constructed in [HJ93]. More precisely, the proofof, e.g., (9A) [BPPA 6= �PA] is as follows: Suppose BPPB = �PB for all oracles B. Then,by standard padding arguments, BPEB = �EB for all oracles B. But there exists an oracleA (constructed in [HJ93]) such that BPEA = �EA = EA and yet BPPA = �PA contains6

sparse sets not in PA, which contradicts that, by the relativized version of Corollary 3.3,�PA�PA lacks sparse sets if �EA = EA. Observe also that Corollary 3.3 adds \FewE 6= E"to Allender and Rubinstein's [AR88] list of characterizations of the existence of sparse sets inP that are not P-printable [HY84], a notion arising in the studies of generalized Kolmogorovcomplexity and data compression.In particular, we have invalidated the conjecture that a class must not be de�ned ina promise-like way to possess upward separation by giving the counterexamples of FewP,SPP, and LWPP. However, our technique does not seem to apply to the promise classesUP or NP\ coNP, and neither does it seem to apply to the non-promise classes PP or C=P.Whereas Theorem 3.2 immediately gives upward separation results for some exotic classessuch as Few � PP or Few � C=P that are trivially closed under the Few operator, it doesnot apply to PP or C=P itself, as these classes are unlikely to satisfy the assumption of thetheorem. For instance, supposing PP were closed under the Few operator, then the closureof PP under truth-table reductions [FR91] implies PPP � FewPPP = Few �PP = PP, thussettling the major open question of whether PP is closed under Turing reductions. Likewise,Few �UP = UP is equivalent to FewP = UP, another important open problem.Regarding PP, all we can prove is the following weak result: If BPP�P contains sparsesets, then PE 6= E. For proving the contrapositive, consider any sparse set S 2 BPP. SinceBPP is low for PP [KST+93], we have BLS(S) 2 Few � BPP � PPBPP = PP. Then, asin the proof of Theorem 3.2, the hypothesis PE = E implies that S 2 P. Clearly, thisapplies to every class that is low for PP. Regarding C=P, we suspect that (unless closedunder complementation) it resembles coNP in that it also fails to robustly have upwardseparation, as is suggested by the fact that their co-classes, coC=P and NP, possess thisproperty jointly. AcknowledgmentWe thank Lane Hemaspaandra for suggesting the investigation of upward separation forpromise classes, and for many interesting and helpful discussions and careful proofreading.References[All86] E. Allender. The complexity of sparse sets in P. In Proceedings of the Conferenceon Structure in Complexity Theory , A. Selman, ed., Springer-Verlag, 1986, 1{11.[All91] E. Allender. Limitations of the upward separation technique. MathematicalSystems Theory 24(1), 1991, 53{67.7

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