Optimal Proof Systems and Sparse Sets

Transcription

1 DIMACS Techical Report September 1999 Optimal Proof Systems ad Sparse Sets by Harry Buhrma CWI Steve Feer 1 Uiversity of South Carolia feer@cs.sc.edu Lace Fortow 2,3 Uiversity of Chicago ad NEC Research fortow@research.j.ec.com Dieter va Melkebeek 4,5 Uiversity of Chicago ad DIMACS dieter@dimacs.rutgers.edu 1 Supported i part by NSF grats CCR ad CCR Permaet Member 3 Supported i part by NSF grat CCR Postdoc 5 Supported i part by NSF grat CCR DIMACS is a partership of Rutgers Uiversity, Priceto Uiversity, AT&T Labs-Research, Bell Labs, Telcordia Techologies (formerly Bellcore) ad NEC Research Istitute. DIMACS is a NSF Sciece ad Techology Ceter, fuded uder cotract STC ; ad also receives support from the New Jersey Commissio o Sciece ad Techology.

2 ABSTRACT We exhibit a relativized world where NP SPARSE has o complete sets. This gives the first relativized world where o optimal proof systems exist. We also examie uder what reductios NP SPARSE ca have complete sets. We show a close coectio betwee these issues ad reductios from sparse to tally sets. We also cosider the questio as to whether the NP SPARSE laguages have a computable eumeratio.

3 1 Itroductio Computer scietists study lower bouds i proof complexity with the ultimate hope of actual complexity class separatio. Cook ad Reckhow [CR79] formalize this approach. They create a geeral otio of a proof system ad show that polyomial-size proof systems exist if ad oly if NP = conp. Cook ad Reckhow also ask about the possibility of whether optimal proof systems exist. Iformally a optimal proof system would have proofs which are o more tha polyomially loger tha ay other proof system. A optimal proof system would play a role similar to NP-complete sets. There exists a polyomial-time algorithm for Satisfiability if ad oly if P = NP. Likewise, if we have a optimal proof system, the this system would have polyomial-size proofs if ad oly if NP = conp. The existece of optimal proof systems remaied a iterestig ope questio. No oe could exhibit such a system except uder various urealistic assumptios [KP89, MT98]. Nor has ayoe exhibited a relativized world where optimal proof systems do ot exist. We costruct such a world by buildig the first oracle relative to which NP SPARSE does ot have complete sets. Messer ad Torá [MT98] give a relativizable proof that if a optimal proof system exists tha NP SPARSE does have complete sets. We also cosider whether NP SPARSE-complete sets exist uder other more geeral reductios tha the stadard may-oe reductios. We show several results such as: There exists a relativized world where NP SPARSE has o disjuctive-truth-table complete sets. There exists a relativized world where NP SPARSE has o complete sets uder truth-table reductios usig o(/log ) queries. For ay costat c > 0, there exists a relativized world where NP SPARSE has o complete sets uder truth-table reductios usig o(/ log ) queries ad c log bits of advice. Uder a reasoable assumptio for all values of k > 0, NP SPARSE has a complete set uder cojuctive truth-table reductios that ask queries ad use O(log ) k log bits of advice. The techiques used for relativized results o NP SPARSE-complete sets also apply to the questio of reducig sparse sets to tally sets. We show several results alog these lies as well. Every sparse set S is reducible to some tally set T uder a 2-roud truth-table reductio askig O() queries. Let c be ay positive costat. There exists a sparse set S that does ot reduce to ay tally set T uder truth-table reductios usig o(/log ) queries eve with c log bits of advice.

4 2 Uder a reasoable assumptio for every sparse set S ad every positive costat k, there exists a tally set T ad a ctt-reductio from S to T that asks queries ad k log O(log ) bits of advice. We ca also have a 2-roud truth-table reductio usig k log queries ad o advice. We use the reasoable assumptios to deradomize some of our costructios usig techiques of Klivas ad Va Melkebeek [KvM99]. Uder these assumptios we get tight bouds as described above. We also examie how NP SPARSE compares with other promise classes such as UP ad BPP i particular lookig at whether NP SPARSE has a uiform eumeratio. The proofs i our paper heavily use techiques from Kolmogorov complexity. We recommed the book of Li ad Vitáyi [LV97] for a excellet treatmet of this subject. 1.1 Reductios ad Relativizatios We measure the relative power of sets usig reductios. I this paper all reductios will be computed by polyomial-time machies. We say a set A reduces to a set B if there exists a polyomial-time computable fuctio f such that for all strigs x, x is i A if ad oly if f(x) is i B. We also call this a m-reductio, m for may-oe. For more geeral reductios we eed to use oracle machies. The set A Turig-reduces to B if there is a polyomial-time oracle Turig machie M such that M B (x) accepts exactly whe x is i A. A tt-reductio (truth-table) requires that all queries be made before ay aswers are received. A 2-roud tt-reductio allows a secod set of queries to be made after the aswers from the first set of queries is kow. This ca be geeralized to k-roud tt-reductios but we will ot eed k > 2 i this paper. We ca thik of a (oe-roud) tt-reductio R as cosistig of two polyomial-time computable fuctios: Oe that creates a list of queries to make ad a evaluator that takes the iput ad the value of B o those queries ad either accepts or rejects. We use the otatio Q R (x) to deote the set of queries made by reductio R o iput x. For a set of iputs X, we let Q R (X) = x X Q R (x). A dtt-reductio (disjuctive-truth-table) meas that M B (x) accepts if ay of the queries it makes are i B. A ctt-reductio (cojuctive-truth-table) meas that M B (x) accepts if all of the queries it makes are i B. A q()-tt reductio is a tt-reductio that makes at most q() queries. A btt-reductio (bouded-truth-table) is a k-tt reductio for some fixed k. We say a laguage L is r-hard for a class C if every laguage i C r-reduces to L. If L also sits i C the we say L is r-complete for C. All the results metioed ad cited i this paper relativize, that is they hold if all machies ivolved ca access the same oracle. If we show that a statemet holds i a relativized world that meas that provig the egatio would require radically differet techiques. Please see the survey by Fortow [For94] for a further discussio o relativizatio.

5 3 1.2 Optimal Proof Systems A proof system is simply a polyomial-time fuctio whose rage is the set of tautological formulae, i.e., formulae that remai true for all assigmets. Cook ad Reckhow [CR79] developed this cocept to give a geeral proof system that geeralizes proof systems such as resolutio ad Frege proofs. They also give a alterate characterizatio of the NP versus conp questio: Theorem 1.1 (Cook-Reckhow) NP = conp if ad oly if there exists a proof system f ad a polyomial p such that for all tautologies φ, there is a y, y p( φ ) ad f(y) = φ. Cook ad Reckhow [CR79] also defied optimal ad p-optimal proof systems. Defiitio 1.2 A proof system g is optimal if for all proof systems f, there is a polyomial p such that for all x, there is a y such that y p( x ) ad g(y) = f(x). A proof system g is p-optimal if y ca be computed i polyomial time from x. Messer ad Torá [MT98] buildig o work of Krají cek ad Pudlák [KP89] show that if NEE = conee the optimal proof systems exist ad if NEE = EE the p-optimal proof systems exist. Here EE, double expoetial time, is equal to DTIME[2 O(2) ]. The class NEE is the odetermiistic versio of EE. Messer ad Torá [MT98] show cosequeces of the existece of optimal proof systems. Theorem 1.3 (Messer-Torá) If p-optimal proof systems exist the UP has complete sets. If optimal proof systems exist the NP SPARSE has complete sets. Hartmais ad Hemachadra [HH84] give a relativized world where UP does ot have complete sets. Sice all of the results metioed here relativize, Messer ad Torá get the followig corollary. Corollary 1.4 (Messer-Torá) There exists a relativized world where p-optimal proof systems do ot exist. However Messer ad Torá leave ope the questio as to whether a relativized world exists where there are o optimal proof systems. Combiig our relativized world where NP SPARSE has o complete sets with Theorem 1.3 aswers this questio i the positive. 1.3 Reducig SPARSE to TALLY A tally set is ay subset of 1. Give a set S, the cesus fuctio c S () is the umber of strigs of legth i S. A set S is sparse if the cesus fuctio is bouded by a polyomial.

6 4 I some sese both sparse sets ad tally sets cotai the same amout of iformatio but i sparse sets the iformatio may be harder to fid. Determiig for which kid of reductios SPARSE ca reduce to TALLY is a excitig research area. Book ad Ko [BK88] show that every sparse set tt-reduces to some tally set but there is some sparse set that does ot btt-reduce to ay tally set. Ko [Ko89] shows that there is a sparse set that does ot dtt-reduce to ay tally set. He left ope the cojuctive case. Buhrma, Hemaspaadra ad Logpré [BHL95] give the surprisig result that every sparse set ctt-reduces to some tally set. Later Saluja [Sal93] proves the same result usig slightly differet techiques. Schöig [Sch93] uses these ideas to show that SPARSE may-oe reduces to TALLY with radomized reductios. I particular he shows that for every sparse set S ad polyomial p there is a tally set T ad a probabilistic polyomial-time computable f such that If x is i S the f(x) is always i T. If x is ot i S the Pr[f(x) T] 1/p( x ). We say that S co-rp-reduces to T. Schöig otes that his reductio oly requires O(log ) radom bits. 1.4 Complete sets for NP SPARSE Hartmais ad Yesha [HY84] first looked at the questio as to whether NP SPARSE has complete sets. They show that there exists a tally set T that is Turig-complete for NP SPARSE. They also give a relativized world where there is o tally set that is m-complete for NP SPARSE. We should ote that NP TALLY has m-complete sets. Let M i be a eumeratio of polyomial-time odetermiistic machies ad cosider {1 i,,k M i (1 ) accepts i k steps}. Also there exists a set i D p SPARSE that is m-hard for NP SPARSE. The class D p cotais the sets that ca be writte as the differece of two NP sets. For the NP SPARSEhard laguage we eed to cosider the differece A B where: A = { x, 1 i, 1 k M i (x) accepts i k steps} B = { x, 1 i, 1 k M i accepts more tha k strigs of legth x i k steps} As a simple corollary we get that if NP = conp the NP SPARSE has complete sets. However the results metioed i Sectio 1.2 imply that oe oly eeds the assumptio of NEE = conee. Schöig [Sch93] otes that from his work metioed i Sectio 1.3 if the sparse set S is i NP the the correspodig tally set T is also i NP. Sice NP TALLY has

7 5 complete sets we get that NP SPARSE has a complete set uder co-rp-reductios. The same argumet applied to Buhrma-Hemaspaadra-Logpré shows that NP SPARSE has complete sets uder ctt-reductios. 2 NP SPARSE-Complete Sets I this sectio, we establish our mai result. Theorem 2.1 There exists a relativized world where NP SPARSE has o complete sets uder may-oe reductios. Proof: Let M i be a stadard eumeratio of odetermiistic polyomial-time Turig machies ad f i be a eumeratio of polyomial-time reductios where M i ad f i use at most time i. Let t(m) be the tower fuctio, i.e., t(0) = 1 ad t(m + 1) = 2 t(m). We will build a oracle A. For each i we will let L i (A) = {x There is some y, y = 2 x ad i, x, y A}. The idea of the proof is that for each i ad j, we will guaratee that either L(Mi A ) has more tha j elemets at some iput legth or L i (A) is sparse ad fj A does ot reduce L i (A) to L(Mi A). We start with the oracle A empty ad build it up i stages. At each stage m = i, j we will add strigs of the form i, x, y to A where x = = t(m) ad y = 2. For each stage m we will do oe of the followig: 1. Put more tha r j strigs ito L(M A i ) for some legth r, or 2. Make L i (A) Σ have exactly oe strig ad for some x i Σ, have x L i (A) f A j (x) L(MA i ). By the usual tower argumets we ca focus oly o the strigs i A of legth : Smaller strigs ca all be queried i polyomial-time; larger strigs are too log to be queried. Pick a strig z of legth 22 that is Kolmogorov radom coditioed o the costructio of A so far. Read off 2 strigs y x of legth 2 for each x i Σ. Cosider B = { i, x, y x x Σ }. If L(M B i ) has more tha r j strigs of ay legth r the we ca fulfill the requiremet for this stage by lettig A = B. So let us assume this is ot the case. Note that f B j (x) for x of legth caot query ay strig y w i B or we would have a shorter descriptio of z by describig y w by x ad the idex of the query made by f B j (x). Our fial oracle will be a subset of B so we ca just use f j as the reductio.

8 6 Suppose f j (x) = f j (w) for some x ad w of legth. We just let A cotai the sigle strig i, x, y x ad f j caot be a reductio. Let us ow assume that there is o such x ad w. So by coutig there must be some x Σ such that f j (x) L(M B i ). Let v = f j (x). We are ot doe yet sice L i (B) has too may strigs. Now let A agai cosist of the sigle strig i, x, y x. If we still have v L(M A i ) the we have ow fulfilled the requiremet. Otherwise it must be the case that M A i (v) accepts but M B i (v) rejects. Thus every acceptig path (ad i particular the lexicographically least) of M A i (v) must query some strig i B A. Sice we ca describe v by x this allows us a short descriptio of some y w give y x for w x which gives us a shorter descriptio of z, so this case caot happe. 2 Corollary 2.2 There exists a relativized world where optimal proof systems do ot exist. Proof: Messer ad Torá [MT98] give a relativizable proof that if optimal proof systems exist the NP SPARSE has complete sets. 2 3 More Powerful Reductios I the previous sectio, we costructed a relativized world where NP SPARSE has o complete sets uder m-reductios. We ow stregthe that costructio to more powerful reductios. Usig the same techiques as well as other oes, we will also obtai ew results o the reducibility of SPARSE to TALLY. 3.1 Relativized Worlds We start by extedig Theorem 2.1 to dtt-reductios. Theorem 3.1 There exists a relativized world where NP SPARSE has o dtt-complete sets. The proof is a improvemet of the proof of Theorem 2.1. I order to facilitate other improvemets ad extesios, we cast it i a slightly differet form. Proof: Let M i be a stadard eumeratio of odetermiistic polyomial-time Turig machies ad R j be a eumeratio of polyomial-time dtt-reductios where M i ad R i use at most time i. We will costruct a oracle A. For each i ad j we will guaratee that either L(Mi A) has more tha j elemets at some iput legth, or else L i (A) is sparse ad R A j does ot reduce L i (A) to L(Mi A ), where L i (A) = {x There is some y, y = 2 x ad i, x, y A}. We start with the oracle A empty ad build it up i stages. At each stage m = i, j we will add strigs of the form i, x, y to A where x = = t(m), y = 2, ad t deotes the tower fuctio. For sufficietly large i ad j, we will do oe of the followig:

9 7 1. Put more tha r j strigs ito L(M A i ) for some legth r, or 2. Make L i (A) Σ have exactly oe strig ad for some x i Σ, have x L i (A) Q R A j (x) L(M A i ) =. (1) By the usual tower argumets, for large i ad j later stages caot udo these achievemets ad we ca focus o the strigs coded i A of legth ad 2. More specifically we do the followig at stage m. Pick a strig z of legth 22 that is Kolmogorov radom give the oracle as costructed so far. Read off 2 strigs y x of legth 2 for each x i Σ ad cosider B = { i, x, y x x Σ }. If L(M B i ) has more tha r j strigs of ay legth r the we let A = B ad we are doe. If ot, we proceed as follows. We first ote that the reductio does ot deped o the oracle B. Claim 3.2 For ay strig x of legth, R B j (x) does ot make a oracle query about a strig i B. Otherwise, we could describe a strig i B usig + O(j log ) bits as the k-th oracle query (for some k j ) R j makes o iput x. Thus we would obtai a descriptio of z of legth less tha z. Our oracle at the ed of stage m will be a subset C of B with oe elemet. By claim 3.2 we ca just use R j as the reductio, which we deote simply as R j. Next we ote that there exists a small set U cotaiig every dtt-query that R j makes o a iput of legth ad that belogs to L(M C i ) for some such C B. Claim 3.3 There exists a set U of size at most j(j+1) such that for ay C B with C = 1, Q Rj (Σ ) L(M C i ) U. Without loss of geerality, we ca assume that U Q Rj (Σ ). I fact, U = Q Rj (Σ ) L(Mi B ) satisfies Claim 3.3: Because of the sparseess of L(Mi B ), U j r=0 r j j(j+1). Moreover, for ay x Σ, ay q Q Rj (x), ad ay C B with C = 1, if q L(Mi C ) the q L(Mi B ). Otherwise every acceptig path (ad i particular the lexicographically least) of Mi C o iput q must query some strig i B C. This allows us to describe a tuple i, w, y w give y x for some w x usig oly + O(ij log ) bits, amely, as the k-th oracle query (for some k ij ) which Mi C makes o the lexicographically first acceptig path give as iput the l-th dtt-query (for some l j ) of R j o iput x. This i tur gives us a shorter descriptio of z. We the argue as follows. Associate with every query q U a sigle strig x q such that q Q Rj (x q ). Let X deote the set of all x q s. Sice U is sparse, for large i ad j, there exists a strig w of legth outside of X. Pick such a strig w ad set A = { i, w, y w }. If there exists a strig x X satisfyig (1) the we are doe. If ot, the Q Rj (X) L(Mi A) =, as X L i(a) =. Sice Q Rj (X) covers all of U, by Claim 3.3, Q Rj (w) L(Mi A) =. However, w L i (A) so x = w satisfies equatio (1). 2

10 8 We ote that the proof of Theorem 3.1 works for ay subexpoetial desity boud. I particular, it yields a relativized world where the class of NP sets with o more tha 2 o(1) strigs of ay legth has o dtt-complete sets. We ca hadle polyomial-time tt-reductios with arbitrary evaluators provided the umber of queries remais i o(/log ). Theorem 3.4 There exists a relativized world where NP SPARSE has o complete sets uder o(/ log )-tt-reductios. Proof: The proof follows the lies of the proof of Theorem 3.1. We redefie L i (A) as L i (A) = {x There is some y, y = 2 x 2 ad i, x, y A}, ad we will allow up to strigs of legth i L i (A). Alterative 2 i the proof of Theorem 3.1 ow reads: 1 (j+1) 2 2. If R A j makes o more tha queries o iputs of legth, the make L log i(a) Σ have at most strigs, ad for some x i Σ, have x L i (A) R A j (x) rejects whe queryig L(M A i ), (2) where R 1, R 2,... deotes a eumeratio of polyomial-time tt-reductios. The strigs y x are of legth 2 2 each, ad their cocateatio z is of legth The rest of the proof beig the same as for Theorem 3.1, we oly describe how to costruct A i the case where L(M B i ) has o more tha rj strigs of ay legth r. Claim 3.2 still holds: Claim 3.5 For ay strig x of legth, R B j (x) does ot make a oracle query about a strig i B. Otherwise, we could describe a strig i B as the k-th query (for some k j ) which R j makes o iput x whe give the ifo it eeds about L(M B i ). Sice this takes o more tha 2 + O(j log ) bits, z would have a descriptio shorter tha itself. As our fial oracle will be a subset C of B, it suffices to cosider R j = R j as the reductio. We ow allow the sets C to be of size up to. Claim 3.3 also holds for them. Claim 3.6 There exists a set U of size at most j(j+1) such that for ay C B with C, Q Rj (Σ ) L(M C i ) U. The same argumet as for Claim 3.3 i the proof of Theorem 3.1 works but ow the descriptio of the strig i, w, y w B C takes 2 + O(ij log ) bits. 1 Suppose that R j makes o more tha queries o iputs of legth. The (j+1) 2 log there exists a large set X of iputs of legth o which R j asks the same set Y of queries i U.

11 9 Claim 3.7 There exists a set X Σ 1 of size ad a set Y of size at most that x X : Q Rj (x) U Y. (j+1) 2 such log The sets X ad Y ca be costructed greedily. Start out with X = Σ ad Y =, ad perform the followig step util Q Rj (X) U Y : Pick amog the elemets of (Q Rj (X) U) Y a most popular oe, i.e., a elemet y U Y such that y Q Rj (x) for the largest umber of x s i X. The add y to Y ad restrict X to those x X for which y Q Rj (x) or Q Rj (x) U Y. 1 The procedure halts after at most steps, so the size of Y is as claimed. I (j+1) 2 log every step the size of X is shruk by o more tha a factor of U, so the fial X satisfies X U 2 1 (j+1) 2 log ( j(j+1) ) 2 1 (j+1) 2 log = 2 1 j+1 for sufficietly large. This establishes Claim 3.7. For ay subset X of X, let C(X ) deote { i, x, y x x X }. By Claims 3.6 ad 3.7, we have that Q Rj (X) L(M C(X ) i ) Y for ay X X. Sice X > Y, there are more subsets X of X tha there are subsets of Y. It follows that there are two subsets X 1 ad X 2 of X, X 1 X 2, such that Q Rj (X) L(M C(X 1) i ) = Q Rj (X) L(M C(X 2) i ). This implies that for at least oe of A = C(X 1 ) or A = C(X 2 ), equatio (2) holds for some x X. 2 For sets of subexpoetial desity the proof of Theorem 3.4 yields a relativized world where the class of NP sets cotaiig o more tha 2 o(1) strigs of ay legth, has o complete sets uder tt-reductios of which the umber of queries is at most α for some α < 1. O the positive side, recall from Sectio 1.4 that NP SPARSE has complete sets uder ctt-reductios as well as uder co-rp-reductios. 3.2 SPARSE to TALLY The techiques used i the proofs of Theorems 2.1, 3.1, ad 3.4 also allow us to costruct a sparse set S that does ot reduce to ay tally set uder the type of reductios cosidered. As metioed i Sectio 1.3, such sets were already kow for m-reductios ad for dttreductios. For o(/log )-tt-reductios we provide the first costructio. Theorem 3.8 There exists a sparse set S that does ot o(/log )-tt-reduce to ay tally set. Proof Sketch: We costruct a similar oracle A as i the proof of Theorem 3.4. The set L(A) = {x There is some y, y = 2 x 2 ad x, y A} will be the sparse set S we are lookig for.

12 10 There ow is a stage m = j accordig to every tt-reductio R j, ad durig that stage we do the followig for = t(m): If R A 1 j asks o more tha queries o iputs of (j+1) 2 log legth, the make L(A) Σ have at most strigs i such a way that for ay tally set T there is a strig x of legth o which R j fails to reduce L(A) to T. We realize this goal i the same way as we realize alterative 2 i the proof of Theorem 3.4. The argumet there for reductios to sparse NP A sets oly relies o the followig property: O iputs of legth, the reductio does ot deped o the extesios of A cosidered, ad the queries of the reductio that are aswered positively all lie i a small set U which is idepedet of the oracle extesio. The proof of Theorem 3.4 shows that these coditios are met i the case of reductios to sparse NP A sets. I the case of (urelativized) reductios to tally sets, they are trivially met. Therefore, the costructio yields a sparse set L(A) which does ot o(/log )-tt reduce to ay tally set. 2 O the other side, O() queries suffice to reduce ay sparse set to a tally set. Previously, it was kow that SPARSE ctt- ad co-rp-reduces to TALLY (see Sectio 1.3). We give the first determiistic reductio for which the degree of the polyomial boudig the umber of queries does ot deped o the desity of the sparse set. Theorem 3.9 Every sparse set S is reducible to some tally set T uder a 2-roud ttreductio askig O() queries. Proof: Schöig [Sch93] shows that for ay costat k > 0 there exists a tally set T 1 ad a polyomial-time reductio R such that for ay strig x of ay legth x S Pr[R(x, ρ) T 1 ] = 1 x S Pr[R(x, ρ) T 1 ] < 1 k, (3) where the probabilities are uiform over strigs ρ of legth O(log ). By pickig idepedet samples ρ k log i, we have for ay x Σ : x S Pr[( i)r(x, ρ i ) T 1 ] = 1 x S Therefore, there exists a sequece ρ i, i = 1,..., Pr[( i)r(x, ρ i ) T 1 ] < ( 1 k) k log = 1 2., such that k log x Σ : x S ( i)r(x, ρ i ) T 1. (4) Sice each ρ i is of legth O(log ), we ca ecode them i a tally set T 2 from which we ca recover them usig O( log ) oadaptive queries. This way, we obtai a 2-roud k log tt-reductio from S to T 1 T 2 usig O() queries: The first roud determies the ρ i s, ad the secod roud applies (4). Sice T 1 T 2 m-reduces to a tally set T, we are doe. 2

13 11 I Sectio 4.1, we will show that uder a reasoable hypothesis we ca reduce the umber of queries i Theorem 3.9 from O() to for ay costat k > 0. See Corollary 4.3. k log We do ot kow whether the equivalet of Theorem 3.9 holds i the NP SPARSE settig: Does NP SPARSE have a complete set uder reductios askig O() queries? See Sectio 6 for a discussio. 4 Reductios With Advice Tight Results Our results i Sectio 3 poited out a differece i the power of reductios makig o(/log ) queries ad reductios makig O() queries. I this sectio we close the remaiig gap betwee o(/ log ) ad O() by cosiderig reductios that take some advice. The approach works for both the NP SPARSE settig ad the SPARSE-to-TALLY settig. 4.1 SPARSE to TALLY We first observe that Theorem 3.8 also holds whe we allow the reductio O(log ) bits of advice. Theorem 4.1 Let c be ay positive costat. There exists a sparse set S that does ot reduce to ay tally set T uder o(/log )-tt-reductios that take c log bits of advice. Proof Sketch: We make use of the same costructio as i the proof of Theorem 3.8. Whe dealig with legth, we divide Σ ito c itervals of equal legth ad put the itervals i oe-to-oe correspodece with the possible advice strigs of legth c log. We the apply the strategy of the proof of Theorem 3.8 o each iterval separately i order to diagoalize agaist the reductio R j with the correspodig advice. This will put at most strigs of legth ito S for every possible advice strig, hece at most c+1 strigs of legth i total. 2 Theorem 4.1 is essetially optimal uder a reasoable assumptio as the ext result shows. Theorem 4.2 Suppose there exists a set i DTIME[2 O() ] that requires circuits of size 2 Ω() eve whe the circuits have access to a oracle for SAT. The for all relativized worlds, every sparse set S ad every positive costat k, there exists a tally set T ad a ctt-reductio from S to T that asks k log queries ad O(log ) bits of advice. Proof: Let S be a sparse set. The costructio i the proof of Theorem 3.9 ca be see as a ctt-reductio of S to the tally set T 1 that makes queries ad gets O() bits as k log advice, amely the sequece of ρ k log i s, each of legth l() O(log ). We will ow show how the hypothesis of Theorem 4.2 allows us to reduce the required advice from O() to O(log ) bits.

14 12 The requiremet the ρ i s have to fulfill is coditio (4). By a slight chage i the parameters of the proof of Theorem 3.9 (amely, by replacig k by 2k i (3)), we ca guaratee that most sequeces ρ i actually satisfy (4). Sice the implicatio from left to right i (4) holds for ay choice of ρ i s, we really oly have to check x Σ : x S ( i)r(x, ρ i ) T 1. (5) Without loss of geerality, we ca assume that Q R (Σ ) T 1 = Q R (S Σ ) T 1, where Q R (X) = {R(x, ρ) x X ad ρ = l( x )}. Therefore, we ca replace (5) by the coditio x Σ : x S ( i)r(x, ρ i ) Q R (S Σ ). (6) Sice S is sparse, this coditio o the ρ i s ca be checked by a polyomial-size family of circuits with access to a oracle for SAT: The circuit has a eumeratio of the elemets of S Σ built i, ad oce a polyomial-time eumeratio of S Σ is available, (6) becomes a conp predicate. Uder the hypothesis of Theorem 4.2, Klivas ad Va Melkebeek [KvM99, Theorem 4.2] costruct a polyomial-time computable fuctio f mappig strigs of O(log ) bits to sequeces ρ i such that most of the iputs map to sequeces satisfyig (6). A explicit iput to f for which this holds, suffices as advice for our reductio from S to T = T 1. 2 Sice we ca ecode the advice i a tally set ad recover it from the tally set usig O(log ) queries, we obtai the followig i the termiology of Theorem 3.9. Corollary 4.3 Uder the same hypothesis as i Theorem 4.2, for ay costat k > 0 every sparse set S is reducible to some tally set T uder a 2-roud tt-reductio askig queries. k log 4.2 Relativized Worlds Our tight results about the reducibility of SPARSE to TALLY carry over to the settig of NP SPARSE. Theorem 4.4 For ay costat c > 0, there exists a relativized world where NP SPARSE has o complete sets uder o(/log )-tt reductios that take c log bits of advice. We also ote that Theorem 3.1 ca take up to ω(log ) bits of advice. Theorem 4.5 There exists a relativized world where NP SPARSE has o complete sets uder dtt-reductios that take ω(log ) bits of advice. O the positive side, we obtai: Theorem 4.6 Suppose there exists a set i DTIME[2 O() ] that requires circuits of size 2 Ω() eve whe the circuits have access to a oracle for SAT. The for all relativized worlds ad all values of k > 0, NP SPARSE has a complete set uder ctt-reductios that ask k log queries ad O(log ) bits of advice. Proof: Let A be a arbitrary oracle. Note that if the set S i Theorem 4.2 lies i NP A, the the set T also lies i NP A. Sice NP A TALLY has a m-complete set, the result follows. 2

15 13 5 NP SPARSE ad Other Promise Classes Iformally, a promise class has a restrictio o the set of allowable machies beyod the usual time ad space bouds. For example, UP cosists of laguages accepted by NP-machies with at most oe acceptig path. Other commo promise classes icluded NP conp, BPP (radomized polyomial time), BQP (quatum polyomial time) ad NP SPARSE. Nopromise classes have easy complete sets, for example: { i, x, 1 j M i (x) accepts i at most j steps} is complete for NP if M i are odetermiistic machies, but o such aalogue works for UP. We say that UP has a uiform eumeratio if there exists a computable fuctio φ such that for each i ad iput x, M φ(i) (x) uses time at most x i ad has at most oe acceptig path o every iput ad UP = i L(M φ(i) ). Uiform eumeratios for the other promise classes are similarly defied. It turs out that for most promise classes, havig a complete set ad a uiform eumeratio are equivalet. Hartmais ad Hemachadra [HH84] show this for UP ad their proof easily geeralizes to the other classes. We iclude a proof here for completeess. Theorem 5.1 (Hartmais-Hemachadra) The classes UP, NP conp, BPP ad BQP have complete sets uder may-oe reductios if ad oly if they have uiform eumeratios. Proof: We will give the proof for UP. The proofs for the other classes are similar. Suppose UP has a complete set L accepted by a UP machie M that rus i time k. Let f 1, f 2,... be a eumeratio of the polyomial-time computable fuctios such that f i uses at most i steps. Defie M φ( i,ik ) (x) to simply simulate M(f i (x)). Suppose UP has a uiform eumeratio via φ. We defie the set L as follows: L = { x, i, 1 k φ(i) outputs j i k steps ad M j (x) accepts i k steps} If A is i UP the A = L(M j ) where for some i, k ad l, φ(i) outputs j i k steps ad M j rus i time l. We defie the reductio f(x) = x, i, 1 max(k, x l). 2 For NP SPARSE either directio of the proof goes through. I the first part, if f i is ot hoest the M φ(i) may accept too may strigs. I the secod part, L might ot be sparse if we merge too may sparse sets with differet cesus fuctios. I fact despite Theorem 2.1, NP SPARSE has a uiform eumeratio (i all relativized worlds). Theorem 5.2 The class NP SPARSE has a uiform eumeratio. Proof: Defie M φ(i) (x) as follows: First see if for ay m log, M i accepts more tha m i strigs of legth m by tryig all possible computatio paths o all iputs of legth m. If

16 14 so the reject. Otherwise simulate M i (x). Note that this will oly eumerate sparse sets: If M i accepts more tha m i strigs of legth m for some m, L(M φ(i) ) will evetually become fiite. O the other had, if M i accepts o more tha m i strigs of legth m for every m, the L(M φ(i) ) = L(M i ). 2 I some sese Theorem 5.2 is a cheat. I the uiform eumeratio, all the sets are sparse but we caot be sure of the cesus fuctio at a give iput legth. To examie this case we exted the defiitio of uiform eumeratio. Defiitio 5.3 We say NP SPARSE has a uiform eumeratio with size bouds if there exists a computable fuctio φ such that NP SPARSE = i L(M φ(i) ), ad for all i ad, M φ(i) accepts at most i strigs of legth usig at most i time. We ca use Defiitio 5.3 to prove a result similar to Theorem 5.1 for NP SPARSE. Theorem 5.4 NP SPARSE has complete sets uder ivertible reductios if ad oly if NP SPARSE has a uiform eumeratio with size bouds. Proof: Suppose NP SPARSE has a complete set S uder ivertible reductios, that is for every NP SPARSE set A there are two polyomial-time computable fuctios f ad g such that for all x, x is i A exactly whe f(x) is i S, ad g(f(x)) = x. Suppose S has at most k strigs at each legth. Let f 1, f 2,... be a eumeratio of the polyomial-time fuctios such that f i uses time at most i. Let us defie M φ( i,j,i(k+1) ) as follows: O iput x, compute y = f i (x) ad accept if 1. f j (y) = x, ad 2. y is i S. Note that this machie ca accept o more tha i(k+1) strigs sice the two tests guaratee that we accept at most oe strig for every strig i S of legth at most i. Now suppose NP SPARSE has a uiform eumeratio with size bouds. We defie the complete set as follows: L = { x, 1 i, 1 k φ(i) outputs j i at most k steps, k x i, ad M j (x) accepts} The set L clearly belogs to NP. It is sparse because for ay fixed i, k ad, there ca be o more tha k strigs x of legth such that x, 1 i, 1 k L. If A is i NP SPARSE the for some i, j ad l, A = L(M j ), φ(i) outputs j i l steps ad M j rus i time x i. We defie the reductio f(x) = x, 1 i, 1 max(l, x i) which is easily ivertible. 2 The promise class NP SPARSE differs from the other classes i aother iterestig way. Cosider the questio as to whether there exists a laguage accepted by a odetermiistic machie usig time 3 which has at most oe acceptig path o each iput that is

17 15 ot accepted by ay such machie usig time 2. This remais a murky ope questio for UP ad the other usual promise classes. For NP SPARSE the situatio is quite differet as show by Seiferas, Fischer ad Meyer [SFM78] ad Žàk [Žàk83]. Theorem 5.5 (Seiferas-Fischer-Meyer,Žàk) Let t 1 ad t 2 be two time-costructible fuctios such that t 1 ( + 1) = o(t 2 ()). There exists a tally set accepted by a odetermiistic machie i time t 2 () but ot i time O(t 1 ()). 6 Ope Problems Several iterestig questios remai icludig the followig. Theorem 3.9 which shows that every sparse set reduces to a tally set usig O() queries does ot seem to give a correspodig result for NP SPARSE-complete sets. Is there a relativized world where NP SPARSE does ot have complete sets uder Turig reductios usig O() queries? If we ca costruct the ρ i s i the proof of Theorem 3.9 i polyomial time usig access to a set i NP conp, the aser is yes. However, the best we kow is to costruct them i polyomial time with oracle access to NP NP. Ca we reduce or elimiate the assumptio eeded for Theorem 4.2, Corollary 4.3, ad Theorem 4.6? If we kew how to costruct the ρ i s from the proof of Theorem 4.2 i polyomial time with O(log ) bits of advice, we could drop the assumptio. Does NP SPARSE havig m-complete sets imply NP SPARSE has a uiform eumeratio with size bouds? Ca we costruct i a relativized world a complete set for NP SPARSE that is ot complete uder ivertible reductios? Refereces [BHL95] H. Buhrma, E. Hemaspaadra, ad L. Logpré. SPARSE reduces cojuctively to TALLY. SIAM Joural o Computig, 24(3): , [BK88] R. Book ad K. Ko. O sets truth-table reducible to sparse sets. SIAM Joural o Computig, 17(5): , [CR79] S. Cook ad R. Reckhow. The relative efficiecy of propositioal proof systems. Joural of Symbolic Logic, 44:36 50, [For94] L. Fortow. The role of relativizatio i complexity theory. Bulleti of the Europea Associatio for Theoretical Computer Sciece, 52: , February 1994.

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