This is page 1Printer: Opaque thisSparse Sets versus ComplexityClassesJin-Yi CaiMitsunori OgiharaABSTRACT The study of sparse hard sets and sparse complete sets hasbeen a central research area in complexity theory for nearly two decades.Recently new results using unexpected techniques have been obtained.They provide new and easier proofs of old theorems, proofs of new the-orems that unify previously known results, resolutions of old conjectures,and connections to the fascinating world of randomization and derandom-ization. In this article we give an exposition of this vibrant research area.1 IntroductionComplexity theory is concerned with the quantitative limitation and powerof computation. During the past several decades computational complex-ity theory developed gradually from its initial awakening [Rab59, Yam62,HS65, Cob65] to the current edi�ce of a scienti�c discipline that is rich inbeautiful results, powerful techniques, fascinating research topics and con-jectures, deep connections to other mathematical subjects, and of criticalimportance to everyday computing.The building blocks of complexity theory are the various complexityclasses. A complexity class consists of computational problems that areclassi�ed according to their inherent computational costs to solve the prob-lems. There are various complexity measures, the most basic of which arecost in time and space. Complexity theory is concerned with various mea-sures of complexity and their inter-relationships, such as time and space,determinism and non-determinism, sequential versus parallel computation,and deterministic versus probabilistic computation.The most well known complexity classes are the space bounded classesPSPACE, DSPACE[logO(1) n], LOGSPACE; time bounded classes EXP,E, P; nondeterministic time classes NEXP, NE, NP; parallel complexityclasses NC1, NC2, etc; probabilistic classes ZPP, RP, BPP, PP; countingclasses #P, �P; and the polynomial time hierarchy PH de�ned in terms ofalternations �pi ;�pi ;�pi . We assume the reader is familiarwith most of theseclasses. We will also be concerned with various reducibilities �Cr , where Cis a complexity class, denoting the power available to the reduction such as

2 Jin-Yi Cai, Mitsunori OgiharaP or L (for LOGSPACE), and where r denotes a type of reducibility suchas many-one, or Turing, or bounded truth-table reducibilities. The poweravailable to the reduction should be somewhat weaker than the power ofthe class for which the reduction is applied. For instance, within NP, wecan consider reducibilities �Pm, �PT or �Pbtt. Within P, we can consider �Lm.We refer the readers to any standard textbook on de�nitions of complexityclasses and reducibilities mentioned here, for example [BDG88].The complexity classes mentioned above are all based on uniform com-plexity measures, where a Turing machine which has a code of a �xedlength will handle all inputs of increasing length. This is the usual con-cept of an algorithm which solves a computational problem consisting ofin�nitely many instances. Contrary to that, there is the important notionof non-uniform complexity, where, for instance, a boolean circuit of certainsize exists that solves a problem for all instances of size n. The asymptoticgrowth of the size of the minimal circuit for all instances of size n can betaken as a non-uniform measure of complexity of the problem. It is non-uniform because no uniform procedure is given to �nd such a circuit foreach length n.Sparse sets are intimately related to non-uniform complexity.A sparse set is a set with only a small number of strings per length. Moreprecisely, a sparse set has at most p(n), a polynomial number of strings upto length n. Thus such a set can only encode a small amount of information;it is thought of as languages with small description. Sparse sets have playedan important role in complexity theory; above all, with respect to variousreducibilities, we ask:Does a complexity class C have a sparse hard or complete set?What consequences follow from such an assumption?Generally speaking, as sparse sets are sets of low information content, tohave a sparse hard or complete set places severe limitations on a complexityclass. Alternatively, having a sparse hard or complete set can be viewedas having a strong handle on the power of the complexity class. Manyconsequences are known assuming the existence of sparse hard sets. For asurvey of earlier results, see [Mah89, HOW92, You92].The two questions raised here are fascinating because they link togetherboth uniform complexity, in terms of a complexity class and a reducibility,and non-uniform complexity, via a sparse set. The �PT -closure of sparsesets is known [BDG88] to be identical to the class of languages havingpolynomial-size circuits, and to P=poly|the class of languages decidable inpolynomial time with short advice, introduced by Karp and Lipton [KL82].The connection to non-uniform complexity, especially polynomial size cir-cuits, was the �rst primary motivation for the study of sparse sets. A fun-damental result of Karp and Lipton [KL82] states that if a sparse set isNP-hard under polynomial time Turing reduction �PT , i.e., every languagein NP has polynomial size circuits, then the polynomial time hierarchy

1. Sparse Sets versus Complexity Classes 3collapses to its second level.The second major motivation for the study on sparse sets was the iso-morphism conjectures of Berman-Hartmanis for NP [BH77], and the con-jectures for P by Hartmanis [Har78].In 1976, L. Berman and Hartmanis [BH77] showed that all NP-completelanguages known at the time (such as those found in [GJ79]) under �Pm-reductions are isomorphic under one-to-one, onto, P-computable, and P-invertible functions on ��. They went on to conjecture that this is true forall NP-complete languages under �Pm. As typical NP-complete languagessuch as SAT are all exponentially dense, it follows from the isomorphismconjecture that no sparse set can be NP-complete under �Pm. This becomesthe sparse non-completeness conjecture for NP by Berman and Hartmanis.This latter conjecture was resolved by Mahaney, who showed that this istrue if and only if NP 6= P [Mah82].While Karp-Lipton's theorem deals with �PT , and Mahaney's theoremsettles the case for �Pm, the case for polynomial time bounded truth-tablereducibility �Pbtt, which is intermediate between �Pm and �PT , was left openfor ten years. It was settled by Ogihara and Watanabe [OW91] in 1990.They showed that the existence of sparse �Pbtt-hard sets for NP impliesP = NP. This is a remarkable achievement, unifying many previous partialresults. Their proof introduced the notion of a left-setwhich is very e�ective.We will demonstrate its usefulness by giving easier proofs of earlier resultsof Mahaney [Mah82] and Fortune [For79].The issue of isomorphism for complete languages in P was also raised byHartmanis in [Har78]. Just as in the case of NP, he showed that all knownP-complete languages under logspace computable many-one reduction �Lmare isomorphic under one-to-one, onto, logspace computable, and logspaceinvertible functions on ��. Hartmanis then conjectured that this is true forall P-complete languages under �Lm-reductions. Again typical P-completelanguages such as CVP, the circuit value problem [Lad75], are all expo-nentially dense, it follows that no sparse set can be P-complete under �Lm.This is the Hartmanis conjecture on the non-existence of sparse completesets in P. A similar conjecture was also made by Hartmanis [Har78] for NL,the nondeterministic logspace.However, unlike in the NP case, for nearly two decades, Hartmanis's con-jecture for P remained open with little progress until very recently. In abreakthrough, Ogihara [Ogi95] showed that if sparse �Lm-hard sets for P ex-ist, then P is contained in poly-logarithmic space DSPACE[log2 n]. In orderto prove his theorem, Ogihara introduced an ingenious auxiliary languagebased on the P-complete language CVP, which has a certain built-in self-reducibility. His proof then proceeds by solving CVP in DSPACE[log2 n]with a clever searching procedure using the P-hard sparse set.This was improved by Cai and Sivakumar [CS95a]. The starting pointof their proof is Ogihara's auxiliary language. Their proof consists of threestages. First they showed that under the hypothesis that a sparse hard set

4 Jin-Yi Cai, Mitsunori Ogiharaexists for P under �Lm, then P can be simulated in randomized NC2. Itis interesting to note that the original problem speaks neither parallel norprobabilistic computations. Second with derandomization techniques using�nite �elds, they showed that the result can be improved to deterministicNC2. As NC2 � DSPACE[log2 n], this gives another proof of Ogihara'stheorem [Ogi95]. It should be noted that at this stage, there is still anunderlying randomized computation to be \simulated" deterministically.Finally, relying more on algebraic techniques, they improved the �nal re-sult from NC2 to NC1, thus resolving Hartmanis's conjecture: There areno sparse complete sets in P under �Lm unless P = LOGSPACE. The Hart-manis conjecture for NL was also resolved: There are no sparse completesets in NL under �Lm unless NL = LOGSPACE [CS95b].The Cai-Sivakumar proof is highly algebraic. Under the hypothesis thata sparse hard set exists for P, it is shown that the P-complete circuit valueproblem CVP can be solved in NC1, modulo the reduction. As NC1 �LOGSPACE, this shows that CVP can be solved in LOGSPACE. How-ever, the proof goes through computations in NC1|it does not provide astep by step automata theoretic \simulation" of P in LOGSPACE. It isinstructive to note that many concepts crucial in this proof, such as NC1,NC2, algebraic computations over �nite �elds, and various randomizationand derandomization techniques, were only studied long after, and thus notavailable at, the time the conjecture was proposed. It is a forceful demon-stration of the power of randomization, a point on which we will elaboratefurther in Section 6. With respect to the philosophical point raised there,namely whether randomization is truly necessary to computation of de-terministic problems, it is amusing to note that in the �nal stage of theCai-Sivakumar proof all traces of randomization disappeared: there is nounderlying randomized computation to be simulated. Randomization onlymakes its presence felt in its former shadow: one probably would neverhave arrived at the �nal proof had it not been for the randomization andderandomization in the earlier stages.In this article, we give a selective exposition of reseach in this area. Inaddition to some earlier important results, we focus on some recent tech-nical and conceptual advances that have brought signi�cant achievementsin the area. The choices made are subjective ones, and re ect the personaltastes by the authors. Due to page limitations, many �ne results in the areaare not discussed. The literature on this subject is vast, we refer readers tothe bibliography for additional research not covered here.

1. Sparse Sets versus Complexity Classes 52 Earlier results for Turing reductions2.1 Sparse sets and polynomial size circuitsWe will start with a theorem attributed to A. Meyer [BH77], seealso [BDG88, page 128].Theorem 2.1 A set is in the closure of a sparse set under polynomial timeTuring reducibility �PT if and only if A has polynomial size circuits.Thus, for instance, the proposition that SAT has polynomial size circuitsis identical to the assertion that NP is polynomial time Turing reducibleto a sparse set S.The proof of this equivalence is rather easy however, and we will sketchthe proof idea here: Suppose A has a family of Boolean circuits fCng,where size jCnj � p(n) for some polynomial p, and Cn determines thesection A=n = fx 2 A j jxj = ng. Then we can simply let the sparse setS, at length n, encode the circuit Cn in a canonical way. To show that Ais decidable in polynomial time given the oracle S, a deterministic Turingmachine upon input x, starts by extracting the circuit Cjxj, and then usesthe circuit to decide whether x 2 A.To prove the other direction we extend the proof that the class P hassmall circuits. Thus suppose A = L(MS), where the running time of M isbounded by nc, and S is sparse. In the proof for P, we lay down a grid ofnc by nc, indexed by time step i and cell location j. Each grid node will bea �nitary boolean gadget that determines, at time step i and cell locationj, what the machine does there. The boolean gadget takes input a booleanvariable indicating whether the read/write head is currently located atlocation j, and if so, the state q of the Turing machine and the current tapesymbol a. The gadget produces the output simulating the Turing machinemove at step i and location j. The output a�ects cell location j � 1, j andj+ 1, at time i+ 1. (This is in fact a LOGSPACE reduction from P to thecircuit value problem CVP [Lad75], which will be used later.)Now the only modi�cation for the machineM with oracle S is as follows:Let S=n be represented by a boolean formula sn in disjunctive normal form.We append the boolean formulae s`, for ` � nc to the circuit describedabove, and augment each local gadget to take an additional input thatindicates whether the machine enters its query state. An extra bit vectorat each time step i can be used to store the queried string. And the queryis answered by the formulae s`.A related result of Pippenger [Pip79] is the following: The class of lan-guages decidable in polynomial time with polynomial length advice is iden-tical to the class of languages having polynomial-size circuits, thus alsoidentical to the �PT -closure of sparse sets.

6 Jin-Yi Cai, Mitsunori Ogihara2.2 Karp-Lipton theoremWe now come to the beautiful theorem of Karp and Lipton [KL82] withcontributions by Sipser [Kar].Theorem 2.2 If NP has a sparse hard set S under polynomial time Turingreducibility �PT , then the polynomial time hierarchy collapses to its secondlevel �p2 = �p2.Note that the condition in the theorem is equivalent to the assumptionthat every language in NP has polynomial size circuits.By self-reducibility, given a boolean circuit for SAT, one can modifythe circuit so that, upon input f , it will either output NO, meaning f isunsatis�able, or output YES, with a satisfying assignment to f . This isaccomplished by the most basic use of self-reducibility: If the circuit on foutputs YES, then it must answer YES to at least one of the sub-formulaefx1=0 and fx1=1. Take the one with a YES answer and repeat the process,till a satisfying assignment is obtained. We record this idea as a lemma forfuture use.Lemma 2.1 Any decision algorithm for SAT(or circuit for SAT�n) canbe modi�ed in polynomial time to �nd a satisfying assignment for any sat-is�able formula (of size up to n).Now the proof of the Karp-Lipton theorem:Suppose a circuit family of polynomial size for SAT exists. By the aboveobservation, for all n, we assume a circuit of size nc exists, such that forevery f of size up to n, it will say NO when f 62 SAT, or YES with asatisfying assignment to f , when f 2 SAT. Let L be an arbitrary languagein �p2. We will design a �p2-machine M to accept L, thus �p2 � �p2. Thetheorem follows from that.By Cook's theorem [Coo71], there is a polynomial time function �(�; �),such that, L = fx j (8py) [�(x; y) 2 SAT]g;where 8py ranges over all y of length some polynomial in the length ofx, say jxjd. Note that all the boolean formulae �(x; y) produced here arebounded by a polynomial in jxj, say jxje.Upon input x, our �p2-machineM �rst guesses all boolean circuits of sizejxjce. For each guess C, M treats it as if it will solve SAT up to lengthjxje as described above. M then enters its universal stage, 8y, jyj = jxjd, itcomputes �(x; y) and tries to use C to decide its satis�ability, and acceptson this path if and only if C says YES, and it is veri�ed that the presumedsatisfying assignment output by C indeed satis�es �(x; y).We claim that this �p2-machine M accepts L. First let x 2 L. Sincecircuits for SAT of polynomial size exists, some guess will �nd such a circuit,and 8y, jyj = jxjd, it will say YES to �(x; y), and the satisfying assignmentwill be veri�ed. Hence M accepts x.

1. Sparse Sets versus Complexity Classes 7Next, assume M accepts x. Then for some guess C, it is veri�ed that�(x; y) are satis�able for all y, jyj = jxjd. (This is so, regardless whether a\correct" C was at hand.) Hence x 2 L.2.3 Long's extensionSuppose there is a boolean circuit of polynomial size for SAT, then with thiscircuit in hand, any NP problem can be solved \locally" for all instancesof small size, as though it belongs to P. One can modify the same circuitto solve a �p2 or �p2 problem \locally." Whether this is also true for sparseoracles is answered by the following theorem of Long [Lon82].Theorem 2.3 If there is a sparse set S, such that NP �PT S, thenPH = �p2 = �p2 �PT S:In other words, if NP has a sparse-hard set S under polynomial time Turingreducibility�PT , then the same sparse set S is hard for the entire polynomialtime hierarchy.Proof Let SAT be accepted by MS . Let L be an arbitrary language in�p2 as before, L = fx j (8y; jyj = jxjd) [�(x; y) 2 SAT]g:The idea is to try to get our hands on the sparse set S. If sparse set Sis a tally set, namely S � 1�, or otherwise P-printable, then it would havebeen easy. (This is the case, such as in the proof of Meyer's theorem, whenwe had a choice how to code the circuit in the sparse oracle.) In the generalcase, we may not be able to query all the right places in S, and obtain allthe strings of S up to a polynomial length. Instead, we will be content withobtaining all the strings of S that \matter" in the local decision problemfor the SAT instances �(x; y).Thus, we �x an x of length n. Initially set S = ;. With the current Sin place of S, we can form a circuit Cn, using M just as in the proof ofMeyer's theorem. This circuit will solve SAT for length n, if our current Sagrees with S up to a length which is an appropriate polynomial of n. Alsowe make the circuit always check any purported satisfying assignment, thusthe only error the circuit may make is to say NO to an f 2 SAT.We formulate the following NP question:Is there a y, with jyj = jxjd, such that �(x; y) 2 SAT, yet ourcircuit says NO?Clearly this is an NP question, which we can solve deterministically runningMS. If the answer is NO, then we are done: the current S serves just aswell as S for all instances �(x; y) that matter to us. If the answer is YES,

8 Jin-Yi Cai, Mitsunori Ogiharathen we can use self reducibility as in Lemma 2.1 to extract an explicit ysuch that �(x; y) 2 SAT and yet our circuit says NO.Now, with this y in hand we run MS on �(x; y) using the oracle S. Somequeried strings byM must be in S but not in the current S. Otherwise, thecurrent circuit using S would have agreed with MS on �(x; y). Add thosestrings in S �S to S. Now we can repeat the process using the updated S.This process of updating S to approximate S can only go on at mosta polynomial number of steps, since S always remains a subset of S, andeach time we add some string of polynomial length from S to S.Once S gives us a circuit correct for all instances of �(x; y), we cantreat the questions of whether \[�(x; y) 2 SAT]?" as a polynomial timecomputation, and thus (8y; jyj = jxjd) [�(x; y) 2 SAT] becomes a coNPquestion that we can compute using MS . 2Long's theorem has the following corollary.Corollary 2.1 If there exists a sparse set S 2 �p2 such that NP �PT S,then PH = �p2.Very recently, Bshouty et al. [BCG+95] (see also [KW95]) show thatpolynomial size circuits are learnable in ZPP with equivalence queries andwith additional queries to NP, from which it follows that polynomial sizecircuits for SAT is learnable in ZPPNP, and thus, sparse �PT -hard sets forNP collapse the polynomial time hierarchy to ZPPNP.Theorem 2.4 If sparse �PT -hard sets exist for NP, then PH = ZPPNP.We also mention that for �PT -completeness, Kadin [Kad89] show that PHcollapses to the class PNP[log], the class of languages decidable in polynomialtime with O(logn) queries to an NP oracle.Theorem 2.5 If sparse �PT -complete sets exist for NP, then PH =PNP[log].Improving the Karp-Lipton collapse to P = NP, or even NP = coNPappears hard. We note that the optimality of these results in terms oforacles have been proven [Hel86, IM89, Kad89].3 Earlier results for many-one reductions3.1 The isomorphism conjecture for NPAs we mentioned earlier, one of the major motivations for the study ofsparse sets is the isomorphism conjecture of Berman and Hartmanis [BH77].It says that all NP-complete languages under �Pm are isomorphic. Theconjecture will imply, among other things, NP 6= P.Berman and Hartmanis showed in 1976, that all known NP-completelanguages at the time under polynomial time many-one reduction �Pm are

1. Sparse Sets versus Complexity Classes 9isomorphic. This means that there is a polynomial time computable func-tion � from �� to ��, that is one-to-one, onto, and its inverse is alsopolynomial time computable, such that � maps one NP-complete languageto another. Their proof relies on the notion of a padding function, and isan adaptation of the classical Cantor-Bernstein theorem in set theory: ifthere are one-to-one functions from set A to set B, and vice versa, thenthere is a one-to-one correspondence between A and B.It was observed in [BH77] that the usual NP-complete languages L, suchas those found in [GJ79], all admit a padding function:� : �� � �� ! ��satisfying the following conditions: 1) jxj+ jyj< j�(x; y)j, 2) given x and y,�(x; y) is polynomial time computable, 3) given �(x; y), it is also polynomialtime computable to extract x and y uniquely, (this implies that � is one-to-one), and �nally 4) for all x and y, x 2 L , �(x; y) 2 A. For instance,in the case of SAT, one can append a score of dummy variables and clausesto a boolean formula and not change its satis�ability.The use of such a padding function is to make a reduction length in-creasing. Thus, for a given reduction � : A �Pm B, we can modify it to bex 7! �B(�(x); 1jxj), provided a padding function �B exists for B.Suppose now there are two length increasing reductions � : A �Pm Band � : B �Pm A. Berman and Hartmanis [BH77] show how to obtain apolynomial time isomorphism between A and B.Denote by � and � respectively the alphabets over which the languagesA and B are de�ned (� and � could be identical of course). De�ne �0 = ��and �0 = ��. De�ne �i = � (�i�1) and �i = �(�i�1) for i � 1. Clearly�0 � �1 � �2 � � � � and similarly �0 � �1 � �2 � � � �. Since the reductionsare one-to-one and length increasing, the containments are strict. Also bythe length increasing property, T1i=0 �i = ; and�� = 1[i=0[�i � �i+1]:A similar decomposition holds for ��.Given any z 2 ��, let iz be the largest i � 0 such that z 2 �i, i.e., z 2�iz ��iz+1. Because the reductions are length increasing, and polynomialtime invertible, we can decide iz in polynomial time. The same is true forany string in ��.Now de�ne a map � : �� ! ��. Given z 2 ��, if iz is even, then de�ne�(z) = �(z); if iz is odd, then let �(z) = ��1(z). Thus, we set up a one-to-one correspondence in which the following terms are term by term mappedto each other:

10 Jin-Yi Cai, Mitsunori Ogihara�� = [�0 ��1] [ [�1 � �2] [ [�2 � �3] [ [�3 � �4] [ : : : ;and �� = [�1 � �2] [ [�0 � �1] [ [�3 � �4] [ [�2 � �3] [ : : : :It can be easily veri�ed that the map � is an isomorphism from A to B.From a structural complexity point of view, the theorem of Berman andHartmanis and their isomorphism conjecture, if true, are esthetically pleas-ing. If we consider polynomial time isomorphisms as merely renaming ofinputs within polynomial time, then the theorem says that all these NP-complete problems in [GJ79], while varied and fascinating combinatorially,are essentially one and the same problem.However, there is now considerable doubt whether the Berman-Hartmanis isomorphism conjecture is true [JY85]. The isomorphism con-jecture is an active research topic with considerable work on the subject.(For the current status of the subject, see [FFK92, KMR90, You90].) Wewill not go into this any further in this article.3.2 Mahaney's theoremAs typical NP-complete languages such as SAT are all exponentially dense,and polynomial time isomorphisms cannot change the density from expo-nential to polynomial, it follows from the isomorphism conjecture that nosparse set can be NP-complete under �Pm. This becomes the sparse non-completeness conjecture for NP by Berman and Hartmanis. It was resolvedby the following de�nitive theorem of Mahaney [Mah82].Theorem 3.1 There are no sparse NP-complete (or NP-hard) languagesunder �Pm if and only if NP 6= P.Mahaney's theorem was the culmination of considerable earlier e�ortstoward the sparse non-completeness conjecture of Berman and Hartmanis.In particular we mention that P. Berman [Ber78] was the �rst to obtain apartial result in this direction. He showed that NP-hard sets over one-letteralphabet exist if and only if P = NP. This was subsumed by a result ofFortune [For79], who showed that sparse coNP-hard sets collapse NP to P.Note that if S � 1� is NP-hard, then the complement 1� � S of S within1� becomes a coNP-hard sparse set.Theorem 3.2 Sparse coNP-hard sets exist if and only if P = NP.While the Karp-Lipton theorem handles the case of polynomial timeTuring reductions, and Mahaney's theorem settles for the many-one case,there have been numerous attempts to try to bridge the gap between thetwo reductions. In particular, quite a few partial results toward the case

1. Sparse Sets versus Complexity Classes 11of bounded truth-table reduction were obtained [Yap83, Ukk83, Wat92,Yes83]. In 1990, Ogihara and Watanabe [OW91] settled this case com-pletely. We will discuss their proof in the next section in detail; here wewill give a simultaneous and simple proof of both Mahaney's theorem andFortune's theorem, using a tool introduced by Ogihara and Watanabe,called left-sets.Let L be a language in NP, let p be a polynomial and A be a set in Psuch that x 2 L () (9w : jwj = p(jxj)) [hx;wi 2 A]: (1.1)De�ne the left-set with respect to p and A, denoted by Left [p;A] as the setof all hx; yi; jyj � p(jxj), such that(9w : jwj = p(jxj); w � y) [hx;wi 2 A]: (1.2)Here y is a node and w is a leaf in the full binary tree of depth p(jxj),and > denotes the dictionary order; that is, a node u is smaller than v ifand only if either v is a descendant of u or lies to the right of u. Thus,hx; yi 2 Left [p;A] if and only if the right-most leaf w, jwj = p(jxj), suchthat hx;wi 2 A, either lies to the right of y or through the node y.Clearly, Left[p;A] is in NP and L �Pm Left[p;A], by the map x 7! hx; �i.The set Left[p;A] embodies a certain \totally ordered self-reducible struc-ture:" For every x and every pair y; y0 2 ��p(jxj) with y0 > y, it holdsthat: hx; y0i 2 Left[p;A]) hx; yi 2 Left [p;A]: (1.3)For readers who are familiar with the notion of a P-selective set, thereis more than a passing resemblance here. A set S is called P-selective, ifthere is a selector s, which is polynomial time computable, such that forany strings x and y, s(x; y) 2 fx; yg, and if fx; yg\S 6= ; then s(x; y) 2 S.In other words, the selector selects the \more likely" candidate among thetwo input strings for membership in S. Clearly for a P-selective set S thereis a (quasi-) ordering for membership in S, such that one can compareany two given strings in this ordering in polynomial time. For any NP-complete language, the left-set Left[p;A] can be viewed as introducing acertain \local selectivity" (1.3) while remaining NP-complete. 1Now we prove the theorems of Mahaney and Fortune using left-sets.First, for Mahaney's result, let us assume NP �Pm S, a sparse set. Thenfor SAT, its left-set Left [p;A] is �Pm-reducible to S via some reduction f ,1The genesis of the idea of a left set can be found in a paper by G. Miller [Mil76]where he de�ned the following set for a partial function fRf = f(x; y) j x 2 dom(f) and y � f(x)g:Similar notions were also used in the study on public-key cryptosystems by Groll-mann and Selman [GS88].

12 Jin-Yi Cai, Mitsunori Ogiharawhere A consists of pairs h�; �i such that � is a boolean formula, and � isa satisfying assignment for �. For any � and a partial assignment � thatassigns boolean values to the �rst k variables for some k � 0, we assumethe image of the reduction f(h�; � i) is bounded in length by a polynomialp(j�j). Since S is sparse, we can assume there are at most q(j�j) manystrings in S among f(h�; � i), where q(�) is another polynomial.Now we will visit the partial truth assignments � to � in a breadth-�rst-search starting with the root, where throughtout the search, the nodes wemaintain are all on the same level. When we reach a level with more thanq(j�j) many nodes, we start pruning. We compute for each � on this level, alabel f(h�; � i). Whenever there are two labels f(h�; � i) and f(h�; � 0i) thatare identical, we conclude that h�; � i 2 Left [p;A] if and only if h�; � 0i 2Left[p;A]. Suppose � is to the left of � 0, � < � 0. We observe that the right-most satisfying assignment, if one exists, must not pass through � . Thus, wecan safely eliminate the node � and its subtree from further consideration.After the elimination of all duplicate labeled nodes, if we end up with nomore than q(j�j) many nodes at that level, we go on to the next level.If, on the other hand, we still have more than q(j�j) many distinct labelsf(h�; � i), we conclude that one of them must be outside of S. Thus oneof the pairs h�; � i must be outside of Left [p;A]. However, since we have\local selectivity" (1.3), the right most � among these must be outside ofLeft[p;A], thus it can be safely eliminated. In any case, we end up withno more than q(j�j) many nodes at this level. If we have not reached theleaves, we extend each remaining node by its two children, to arrive at thenext level.Inductively the pruning process from one level to the next satis�es thefollowing property: If at any level, the right most satisfying assignmentexists and passes through one of the labeled nodes, then the same is truefor the next level of labeled nodes after the pruning.When the partial assignments are fully extended, we reach the leaves ofthe assignment tree. At the leaves we can directly check whether any ofthe assignments satis�es the boolean formula �. Note that followed fromthe induction, � 2 SAT if and only if the right most satisfying assignmentexists and is one of the leaves. This gives a polynomial time algorithm forSAT, assuming the existence of an NP-complete (or NP-hard) sparse set.This proves Mahaney's theorem 3.1.A similar argument works for Fortune's theorem. The only change onemakes is that when the number of distinct labels exceeds q(j�j) , we alreadyknow one of the strings is in Sc, and thus one of the pair h�; � i 2 Left[p;A].Hence � 2 SAT.

1. Sparse Sets versus Complexity Classes 134 Bounded truth table reduction of NPWe have seen that the left-sets introduced in [OW91] give new and uniformproofs of earlier results. Not only that, using the new technique Ogihara andWatanabe succeeded in proving the following theorem that uni�es manyprevious results:Theorem 4.1 If NP �Pbtt S, a sparse set, then NP = P.Suppose there is a sparse �Pbtt-hard set for NP. The left-set L(SAT) ofSAT is �Pk-tt-reducible to a sparse S via some function f for some k. Thismeans for every boolean formula � and a partial assignment � , f(h�; � i)produces a list of strings y1; y2; : : : ; yk, and a truth table � on k inputs,such that h�; � i 2 L(SAT) if and only if the characteristic sequence ofmembership of yi in S satis�es �:�(�S(y1); �S(y2); : : : ; �S(yk)) = 1:There are 22k many truth tables on k inputs. We will �rst consider a�xed truth table �. The theorems of Mahaney and Fortune are two specialcases for 1-truth table reductions. Furthermore, in both cases, a �xed 1-truth table was used: the identity function for Mahaney, and the negationfunction for Fortune. In general, however, a truth table reduction producesa truth table that is also dependent on the input.We will present a proof based on an idea of Homer and Longpr�e [HL94].We will try to inductively build a pruning strategy that prunes any givenset T of nodes at a �xed level of the assignment tree, given a �xed k-truthtable �. The k-tt strategy will utilize the (k � 1)-tt strategy. Each nodein T produces a k-tuple hy1; y2; : : : ; yki by f . We collect those k tupleswith an identical ith entry yi, say yi0, for any �xed i. Note that, given themembership �S(yi0), � is e�ectively reduced to a (k�1)-truth table on thissubset. Of course, we don't know �S(yi0). However, if our inductively built(k�1)-tt pruning strategy is oblivious, i.e., is independent of the particular�xed (k � 1)-truth table used, then we would be able to carry through.Let us �rst consider 1-truth tables. Suppose there are at most q(n) manystrings of S that can possibly appear among the y's, due to length and spar-sity. The pruning strategies we presented for the Mahaney and Fortune the-orems are not quite oblivious. The non-oblivious case is when the numberof distinct labels after the elimination of duplicate ones still exceeds q(n).However, suppose we modify it as follows. We will �rst visit all the givennodes T at this level from left to right, eliminating duplicate labeled nodesas before. If there are more than q(n) nodes left, v0 < v1 < : : : < vq(n), westop and then start to visit T from right to left. During this right-to-leftsweep we also eliminate duplicate labeled nodes in the same way. If weencounter vq(n) before seeing more than q(n) distinct labels, we stop, andpass to the next level with at most 2q(n)+ 1 nodes. This becomes our new

14 Jin-Yi Cai, Mitsunori Ogiharabound on the width of the pruning procedure. Suppose now the right-to-leftsweep �nds more than q(n) distinctly labeled nodes uq(n) < : : : < u1 < u0,before encountering vq(n). In this case, vq(n) still lies to the left of uq(n). Thekey observation is the following: For some i and j, where 0 � i; j � q(n),the reduction f must map both vi and uj out of S. Thus the 1-truth tabletakes the same value at vi and uj, regardless of what truth table is beingused, be it the identity function or the negation function, or whatever. Thisis the obliviousness we are looking for. Thus,h�; vii 2 L(SAT)() h�; uji 2 L(SAT):It follows that, it cannot be the case that the right most satisfying assign-ment passes through a node w, such that vi � w < uj. Thus all nodes w in Tsuch that vi � w < uj can be eliminated. Of course vi and uj are unknownto us. But we can safely eliminate all w 2 T such that vq(n) � w < uq(n)because vi � vq(n) and uq(n) � uj . So, we end up with at most 2q(n) + 1nodes again. Inductively from one level to the next in the assignment tree,we maintained that(*) If at any level, the right most satisfying assignment exists andpasses through one of the given nodes, then the same is trueafter the pruning.This completes the proof of the case for any �xed 1-truth table, and thepruning strategy is now oblivious of the particular truth table used.We are now ready to tackle the case with a �xed k-truth table �. Wewould like to arrive at an oblivious pruning strategy Sk that satis�es theabove inductive requirement (*), and leaves at most a polynomial numberof nodes pk(n) after the pruning. We already see that we can set p1(n) =2q(n)+1. In general, suppose an oblivious pruning strategy Sk�1 has beenfound that works for any �xed (k � 1)-truth table.Given any set of nodes T on any level. Let each node w in T be labeled bya k-tuple hy1(w); y2(w); : : : ; yk(w)i via f . For any w 2 T , any i, 1 � i � k,consider the subset Ti;w = fu 2 T j yi(u) = yi(w)g:Note that if yi(w) = yi(w0), then Ti;w = Ti;w0 , i.e., the set Ti;w depends onlyon i and yi(w), and not on w. Also note that depending on �S(yi(w)) = 0or 1, the truth table � induces one of two possible �xed (k�1)-truth tableson the subset Ti;w. Employing the pruning strategy Sk�1 we can prune thissubset down to size at most pk�1(n).What if the total number of remaining nodes after pruning for all w andi is still too large? We will show that, in this case, some node v remainingmust have all its yi(v) 62 S, 1 � i � k.Indeed, suppose for every v there exists an i, such that yi(v) 2 S. Thenwe claim there could be at most kq(n)pk�1(n) nodes left. This can be seen

1. Sparse Sets versus Complexity Classes 15as follows. Let eTi;w be the set of nodes remaining from Ti;w after applyingSk�1, and let eT be the set of nodes remaining from T after pruning for alli and all w. If v still remains, then v 2 eTi;v, where i is an index such thatyi(v) 2 S. Hence eT � [y2S [i:yi(v)=y eTi;v:We have used the fact that the set Ti;v only depends on i and yi(v). Sinceeach jeTi;vj � pk�1(n), there can be at most kq(n)pk�1(n) nodes left.Hence, if j eT j > kq(n)pk�1(n), then for some w 2 eT , yi(w) 62 S, for all1 � i � k.Let pk(n) = 2kq(n)pk�1(n) + 1. It is easily veri�ed thatpk(n) � k! � (2q(n) + 1)k:At this point we are at a similar position as in the 1-truth table case afterthe elimination of all duplicate labels. We will carry out two sweeps: �rst aleft-to-right sweep, and if more than kq(n)pk�1(n) many nodes remain, thena second right-to-left sweep. If we end up with at most pk(n) many nodestotal after the two sweeps, we are done. Otherwise those nodes betweenthe (kq(n)pk�1(n)+ 1)st nodes from the left, and the (kq(n)pk�1(n)+ 1)stnodes from the right can be safely eliminated. (The interval is inclusive onthe left and exclusive on the right.) This completes the description of thestrategy Sk.Finally to handle the situation where the truth tables are not �xed, wesimply note that there can be at most 22k distinct k-truth tables. Thuswe can carry out the above pruning strategy 22k times in parallel for each�xed k-truth table on any given level of the assignment tree. The theoremof Ogihara and Watanabe is proved.4.1 ExtensionsThe breakthrough by Ogihara and Watanabe has provoked a urry of re-sults about sparse hard/complete set problems [HL94, AHH+93, AKM92b,AKM92a, AA95, RR92, OL93] (see [HOW92] for a survey). Below we statethe best known results on polynomial time sparse hard sets for NP.Theorem 4.2 1. [AKM92b] NP is included in the �Pbtt-reducibility clo-sure of the languages that are �Pctt-reducible to sparse sets if and onlyif P = NP.2. [AKM92a] Sparse �Pdtt-hard sets for NP exist only if PH = �p2.3. [HL94] Sparse �PO(logn)-tt-hard sets for NP exist only if NP is in-cluded in DTIME[2O(log2 n)].

16 Jin-Yi Cai, Mitsunori OgiharaNote that Part 1 in the above subsumes the Ogihara-Watanabe theorem.We note here that very recently, this part has been slightly strengthend byAgrawal and Arvind [AA95]: they showed that NP � C ) P = NP holdsfor a circuit class C provably larger than the \�Pbtt of �Pctt" closure of sparsesets in Part 1.Does the left-set technique shed light on the sparse hard/complete setproblems for other counting classes? It is easy to see that the \breadth-�rst-search plus tree-pruning" method for the largest satisfying assignmentworks for UP and FewP, subclasses of NP. Also, it is easy to see that astatement similar to Theorem 2.2 holds for every complexity class havingcomplete sets with some reasonable self-reducible structure, and thus, ifsuch a complexity class C includes either NP or coNP, then sparse �Pbtt-hard sets for C collapses C to P. Generally speaking, for every countingclass in the sense of [GNW90], sparse �Pbtt-hard sets for the class collapse itwithin NPT coNP [OL93]. The story for modulo-based counting complex-ity classes such as �P is slightly di�erent, for it is not known whether theclass contain NP or coNP. Ogihara and Lozano [OL93] extend the notionof left-sets and show that sparse �Pbtt-hard sets for a modulo-based count-ing class collapse it to P. This framework allows one to obtain a clone ofTheorem 4.2 (1) for many counting classes.Theorem 4.3 For each C chosen fromPSPACE;UP;FewP;C=P;PP;Mod2P;Mod3P; � � � ;C is included in the �Pbtt-reducibility closure of the languages that are �Pctt-reducible to sparse sets if and only if P = C.5 The Hartmanis conjecture for PWe now come to the Hartmanis conjecture on the non-existence of sparsecomplete sets for P under logspace computable many-one reductions. Thisconjecture, unlike its NP analog, remained open for many years. In a break-through Ogihara showed that the existence of P-hard sparse sets impliesthat P � DSPACE[log2 n] [Ogi95]. Ogihara de�ned an ingenious auxiliarylanguage, which is P-complete and has a certain built-in self-reducibility.The proof then proceeds by simulation of P in DSPACE[log2 n] using theP-hard sparse set.This was followed by the decisive improvement of Cai and Sivaku-mar [CS95a] that resolved the conjecture for P. The Cai-Sivakumar proofuses algebra extensively. Parallel computation and randomization play acrucial role as well. The starting point is the language de�ned by Ogihara.This language enables one to set up linear equations to solve for CVP,the circuit value problem. The backbone of the proof is the formation andsolution of certain linear equation systems over �nite �elds.

1. Sparse Sets versus Complexity Classes 17It turns out to be natural to employ randomization here to form theequations. Then, it is possible to prove that the system has a su�cientlyhigh rank, with high probability. With a lower bound on the rank, onecan then appeal to the NC2 algorithm of Mulmuley [Mul87] to solve thelinear system. This gives a randomized NC2 algorithm for CVP. Thus,both randomization and parallel complexity class NC enters the picturequite naturally, even though no mention of either subjects is made in theoriginal conjecture.The next step in the Cai-Sivakumar proof is derandomization which im-plies that P = NC2, given a P-hard sparse set exists. Here the main tools are�nite �elds and polynomial size sample spaces. It is shown that a determin-istic construction D exists such that if we exhaustively sample from D, itis as good as if we are taking uniform samples from an exponentially largersample space. It follows that the probabilistic conclusions are now alwaystrue, thus getting rid of randomness in the proof. This result also providesan alternative proof of Ogihara's theorem, since NC2 � DSPACE[log2 n].Note that since the proof goes through the parallel class NC2, proving theinclusion in DSPACE[log2 n] does not involve any step-by-step automatatheoretic simulation of CVP.Finally, Cai and Sivakumar improved their proof further from NC2 downto NC1 to resolve Hartmanis's conjecture. We will start with the languageconstructed by Ogihara and the randomized NC2 algorithm. In the follow-ing, all NC classes are logspace uniform, meaning that given 1m, a logspacecomputation will produce the mth circuit for input length m.5.1 Ogihara's language and randomized NC2To begin with, suppose a sparse set S exists that is hard for P underlogspace many-one reductions. Recall that a typical P-complete languageis the circuit value problem CVP. In [Ogi95], Ogihara de�ned a set A closelyrelated to CVP: It consists of tuples hC; x; I; bi, where C is a boolean circuit,x is an input to C, I is a subset of the gates, and b is a bit (0 or 1), suchthat the sum mod 2 of the values of the gates chosen in I from C on inputx equals b, i.e., Mi2I gi(x) = b:Clearly,A 2 P and hence A �Lm S. Let f be a logspace computable functionsuch that for all x, x 2 A () f(x) 2 S. It is also obvious that CVP �Lm A.Our goal here is to show that A can be solved in randomized NC2. HereNC2 denotes the class of languages computed by uniform family of circuitsof polynomial size and O(log2 n) depth.We note that for any C; x; I, exactly one of the bits b = 0; 1 satis�esthe equation, and thus exactly one of f(hC; x; I; 0i) and f(hC; x; I; 1i) is astring in S. Moreover, suppose for two distinct subsets I and J and some

18 Jin-Yi Cai, Mitsunori Ogiharapair of bits b; b0, f(hC; x; I; bi) = f(hC; x; J; b0i), (we are not assuming thatthe image is in S). In this case, regardless of whether Li2I gi(x) = b andLi2J gi(x) = b0 are true or not, they hold or fail simultaneously. Thus wehave an equation mod 2 on the values of the gates of C on input x, namelyMi2I4J gi(x) = b� b0; (1.4)and I 4 J 6= ;.Fix any C and x, let n denote the number of nodes in C (including theinputs, output, and the interior gates). Let p(n) be a polynomial boundon the number of strings in S among f(hC; x; I; bi), (over all I and b).Thus, some string w 2 S must be mapped on by at least 2n=p(n) manysubsets I, more precisely, by the tuple hC; x; I; bIi, where bI is the \rightvalue" bI =Li2I gi(x). As described above, any two such I give rise to anequation mod 2 on the values of the gates of C on input x. The idea nowis to choose polynomially many random subsets I 2 f0; 1gn and computef(hC; x; I; 0i) and f(hC; x; I; 1i), collecting as many equations as possible.The following lemma ensures that this process gives us a system of linearequations of su�ciently high rank, even if we restrict attention to a single\popular" w 2 S which appears for at least 2n=p(n) many subsets I.A probabilistic lemmaLet B = f0; 1gn denote the n-dimensional binary cube. With respect to the�nite �eld of two elements GF(2) = Z2, B is a vector space of dimensionn. We note that when we obtain an equation of the form (1.4), it is on thesymmetric di�erence of two subsets I and J , which are found to have thesame reduction value. In terms of the vector space, the coe�cient vectorof the equation (1.4) is the di�erence of the two boolean vectors in B thatrepresent I and J . Intuitively, �x any \popular" w 2 S which appears formany subsets I, we can expect a good many randomly sampled points to bemapped to w, which in turn give us equations with the coe�cient vectorsbeing the di�erences of such pairs.Let T � B be an arbitrary subset of the cube. We ask the followingquestion: If we uniformly and independently pick a sequence of m pointsin B, what can we say about the probability distribution of the dimensionof the a�ne span of those points picked from T as a function of m, n,and jT j? Here the a�ne span of a set T 0 is the smallest dimensional a�nesubspace that contains T 0. It is a translation of a linear subspace which isgenerated by all the di�erences of pairs of vectors from T 0. The dimensionof the a�ne span of the set of points picked by the random process whichhave the same reduction value w is a lower bound on the rank of our linearequation system.Lemma 5.1 Suppose jT j � 2n=k, where k = nO(1), then for m = 2kn2 +n = nO(1), if we uniformly and independently pick a sequence of m points

1. Sparse Sets versus Complexity Classes 19in B, the probability that the dimension of the a�ne span of the points fromT is less than n� log2 k is at most e�n2+O(n logn).We omit the proof here. The interested reader can �nd it in [CS95a].The randomized NC2 algorithmBy the above lemma, if in parallel we try polynomially many uniformlyand independently chosen I, with high probability we will obtain a systemof linear equations with rank de�ciency at most log2 p(n). We now describehow we can use these to determine in NC2 the outputs of all the gates ofC on input x.Without loss of generality, let the rank of the system be n � log2 p(n),and let m(= nO(1)) denote the number of equations we have. Denote theequations byE1; : : : ; Em, and for i � 1, call an equationEi useful if the rankrk(E1; : : : ; Ei) > rk(E1; : : : ; Ei�1). Clearly the number of useful equationsis n� log2 p(n). Mulmuley [Mul87] gives an algorithm to compute the rankof an ` � n matrix, which, for ` = nO(1), can be implemented by a circuitof depth O(log2 n) and size nO(1). For 1 � i � m, we compute in parallelrk(E1; : : : ; Ei), and identify all the useful equations. Now we have n �log2 p(n) equations in n variables, with rank n � log2 p(n). We apply thesame process to the columns, and identify the (n � log2 p(n))-many usefulcolumns. We rename the variables so that the �rst n � log2 p(n) columnsare all useful. For each of the p(n) possible assignments to the last log2 p(n)variables, we create in parallel a system of n � log2 p(n) equations as an(n � log2 p(n)) � (n � log2 p(n)) matrix. Each one of these can be solvedin log2 n depth and poly(n) size using the algorithm due to Borodin, etal. [BvzGH82]. For each potential solution we get for the gates of the circuitC on input x, we can check its validity using the local information aboutthe circuit C and input x, such as xi = 0, or xi = 1, or gj(x) = gk(x)^g`(x),etc. There will be a unique solution that passes all such tests and we will�nd the output of C(x) in particular. We have proved:Theorem 5.1 If there is a sparse set that is hard for P under logspacemany-one reductions, then P � Randomized NC2.5.2 Deterministic constructionAs before we have B = f0; 1gn = Zn2 considered as an n-dimensional vectorspace over the �nite �eld Z2. For each I 2 B, let bI = Li2I gi(x) be the\right value." Then the string w = f(hC; x; I; bIi) 2 S and this w is calledthe color of I. The presumed reduction to the sparse set S gives a coloringof B with at most p(n) colors. Let D � B be a subset of B of cardinalitybounded by a certain polynomial in n. The coloring of B induces a coloringof D, thus D is the union of at most p(n) many color classes:D = C1 [C2 [ : : :[Cp(n):

20 Jin-Yi Cai, Mitsunori OgiharaLet the a�ne span ofCi be denoted by Li+di, where Li is a linear subspace,and di is a displacement vector. Let L = L1+L2+ : : :+Lp(n) be the sum ofthe linear subspaces. We call L the span of the color classes. Li is spannedby di�erences of vectors in Ci. For some spanning set of vectors of Li, eachvector in the set gives us an equation mod 2 of the values of the gates of Cwith the given input. If we collect a generating set of vectors for each Li,together they span L. Thus, if we can construct a set D with polynomialsize and with dimL � n�O(logn) (irrespective of the coloring), we wouldhave succeeded in derandomizing the construction of the last section. Thatis, by sampling exhaustively in D, we would have obtained a system oflinear equations of rank � n�O(logn).We claim that the above task can be accomplished as follows: given p(n),construct a set D of polynomial size such that for any linear subspaceM of B with dimM < n � log2 p(n), and any p(n) displacement vectorsb1; : : : ; bp(n) 2 B, the union of the p(n) a�ne subspaces Sp(n)i=1 (M +bi) doesnot cover the set D. For if so, then no matter what the induced coloring onD is, the span of the color classes L must be of dimension � n� log2 p(n),simply because the union of at most p(n) a�ne subspaces Sp(n)i=1 (L + di)does cover D: p(n)[i=1 (L+ di) � p(n)[i=1 (Li + di) � D:Let k = 1 + log2 p(n) = O(logn). Without loss of generality, we mayassume such a linear subspace M has dimension exactly = n�k. Any suchM can be speci�ed as the null space of a linear system of equationsai1x1 + ai2x2 + : : :+ ainxn = 0;where i = 1; : : : ; k, and the k vectors f(ai1; ai2; : : : ; ain) j i = 1; : : : ; kg areindependent vectors in B over Z2.Let m = 2k+log2 n+1 = 2 log2 p(n)+ log2 n+3 = O(logn). The Galois�eld F = GF(2m) has a vector space structure over GF(2) of dimensionm.Choose any basis fe1; : : : ; emg, then for u =Pmi=1 uiei and v =Pmi=1 vieiin F, we can de�ne an inner product by lettinghu; vi = mXi=1 uivi;and doing all arithmetic over Z2.The (multi-) set D is de�ned as follows:D = f(h1; vi; hu; vi; : : : ; hun�1; vi) j u; v 2 Fg:Note that jDj = 22m = nO(1). Now consider any non-zero vector a =(a0; a1; : : : ; an�1) 2 B and any b 2 Z2. We claim that the a�ne hyperplane

1. Sparse Sets versus Complexity Classes 21Pn�1i=0 aixi = b cuts D into roughly two equal parts. To show that, letus estimate the size of the intersection of D with the a�ne hyperplanePn�1i=0 aixi = 0.Since the inner product h�; �i is bilinear over Z2 we haven�1Xi=0 aihui; vi = hn�1Xi=0 aiui; vi:Let qa(X) denote the polynomialPn�1i=0 aiXi 2 F[X]. If u is a root of thepolynomial qa(X), then clearly the inner product hPn�1i=0 aiui; vi is equalto 0. Now suppose u 2 F is not a root of qa(X), then Pn�1i=0 aiui = qa(u)is a non-zero element in F. It is easy to see that for any non-zero w 2 F,Prv2F[hw; vi = 0] = 1=2:Thus, Pru;v2F "n�1Xi=0 aihui; vi = 0#= Pru2F[u is a root of qa(X)] + Pru2F[u is not a root of qa(X)] � 12 :But qa(X) is a non-zero polynomial of degree at most n� 1, thusPru2F[u is a root of qa(X)] � n� 12m :Collecting terms, we have�����Pru;v2F "n�1Xi=0 aihui; vi = 0#� 12 ����� � n� 12m+1 :In particular, if m > log2 n, both a�ne hyperplanes Pn�1i=0 aixi = 0 andPn�1i=0 aixi = 1 must intersect our set D.In general, consider any k linearly independent equationsPn�1j=0 aijxj =bi, where aij; bi 2 Z2, and i = 1; : : : ; k. Denote this a�ne space by �.Denote the point in D speci�ed by u; v as D(u; v). Generalizing the ideasabove, we can show that����Pru;v2F[D(u; v) 2 �]� 12k ���� < n2m�k ;which by our choice of m and k is bounded above by 1=2k+1. Thus, inparticular, Pru;v2F[D(u; v) 2 �] > 0:

22 Jin-Yi Cai, Mitsunori OgiharaOther than linear independence, the coe�cient vectors and the vector(b1; : : : ; bk) on the right hand side in the de�nition of � are arbitrary; thetotal number of the b vectors is 2k = 2p(n) > p(n), and it follows that nolinear subspace M of dimension < n � log2 p(n) can cover the set D withsome p(n) displacements.Theorem 5.2 If there is a sparse set S which is hard for P under NC2many-one reductions, then P = NC2.5.3 The Finale: NC1 SimulationThe ideas discussed above paved way for the �nal assault|an optimalsimulation in NC1, circuits of polynomial size and logarithmic depth.This is really quite fortunate. Note that the proof ideas so far are basicallyto form a certain linear equation system and solve it. After some ten years ofcontinued progress, starting with the pioneer work of Csanky, culminatingin the remarkable achievement of Mulmuley, it was shown ultimately thatmany tasks in linear algebra such as rank, determinant, inverse, etc. over ar-bitrary characteristic can be done in NC2 [Csa76, BvzGH82, Chi85, Mul87].There is no evidence that all these heavy duty computations can be accom-plished in NC1. On the contrary, NC1 is considered rather weak; it was thecelebrated AKS sorting network [AKS83] that �nally puts sorting in NC1.It appears that it would be overly optimistic to hope to do all that linearalgebra in NC1.But Cai and Sivakumar succeeded in proving the following strong result:Theorem 5.3 If a sparse set S is hard for P under many-one reductions,then the P-complete circuit-value problem can be solved by a logspace-uniform family of polynomial size, logarithmic depth circuits that makepolynomially many parallel calls to the reduction.That is, modulo the complexity of the reduction to the sparse set, theresulting algorithm can be implemented by a uniform NC1 circuit of poly-nomial size and logarithmic depth. It follows that if the reduction itselfis computable in logspace-uniform NC1, then P equals logspace-uniformNC1; and, for the Hartmanis conjecture, if there is a logspace computablemany-one reduction from P to a sparse set S, then P = LOGSPACE.Proof It is known that the polynomial X2�3` + X3` + 1 2 Z2[X] isan irreducible polynomial over Z2 for all ` � 0 [vL91]. In the following,by a �nite �eld GF(2m), where m = 2 � 3`, we refer explicitly to the �eldZ2[X]=(X2�3` +X3` + 1).Let S be a sparse set hard for P under logspace-computable many-onereductions. As before, we will consider a re�nement of the circuit-value

1. Sparse Sets versus Complexity Classes 23problem. De�neL = (hC; x; 1m; u; vi j m = 2 � 3`; u; v 2 GF(2m); n�1Xi=0 uigi = v) ;where C is a boolean circuit and x is an input to C, and where g0; : : : ; gn�1are 0-1 variables that denote the values of the gates of C on input x. Hereexponentiation and summation are carried out in the �nite �eld GF(2m).It is easy to see that L 2 P, since all the required �eld arithmetic involvedin checking Puigi = v can be performed in polynomial time.Clearly jhC; x; 1m; u; vij is bounded polynomially in n and m. If f is alogspace-computable function that reduces L to S, the bound on the lengthof queries made by f on inputs of length jhC; x; 1m; u; vij is some polynomialq(n;m). Let p(n;m) be a polynomial that bounds the number of strings inS of length at most q(n;m). We will choose the smallestm of the form 2 �3`such that 2m=p(n;m) � n. It is clear that m = O(logn). Let F denote the�nite extension GF(2m) of GF(2).Facts.We �rst collect some facts about implementing the basic operationsof F. For each operation, the number of processors needed is at most nO(1).(1) Finding a primitive element ! that generates the multiplicative groupF� of F can be done in logspace by exhaustive search.(2) Adding two elements y1; y2 2 F is just the bitwise exclusive-or of therepresentations of y1 and y2, and can be done in depth O(1). AddingnO(1)-many elements can be done in depth O(logn).(3) Using logarithmic space, it is also possible to build the multiplicationtable for F, so multiplying two elements of F can be done by a circuitof depth O(log logn) and size (logn)O(1).(4) Raising the generator ! to any power i < 2m, or computing thediscrete logarithm of any element with respect to !, can be doneby table lookup in depth O(log logn). The tables themselves can beprecomputed using O(logn) space.(5) Multiplying k = nO(1) elements of F can be done in O(logn) depth.The idea is to use the discrete logarithms of the k elements withrespect to the generator !, and convert multiplications to additionsof k O(logn)-bit integers (modulo 2m � 1), which can be done inO(logn) depth using the folklore 3-to-2 trick.Our parallel algorithm for CVP begins by computing f(hC; x; u; vi) forall u; v 2 F. For every non-zero u 2 F, there is a unique element vu 2F such that hC; x; u; vui 2 L, and therefore f(hC; x; u; vui) 2 S. Since2m=p(n;m) � n, there is at least one string w 2 S such that the number

24 Jin-Yi Cai, Mitsunori Ogiharaof u satisfying f(hC; x; u; vui) = w is at least n. Of course, there could bemany such w (not necessarily in S), and we don't know which w is a stringin S. To handle this, we will assume that every w that has � n preimagesis a string in S, and attempt to solve for the gi's. As long as there is atleast one w 2 S that has � n preimages, one of the assumptions must becorrect, and we will have the correct solution. Since we know the details ofthe circuit C, the solutions can be veri�ed, and the incorrect ones weededout.Assume, therefore, without loss of generality, that w 2 S has � n preim-ages. Let u1; u2; : : : ; un denote n of them, and let v1; v2; : : : ; vn denote thecorresponding vu's. The equations1g0 + ujg1 + u2jg2 + : : :+ un�1j gn�1 = vj ; j = 1; 2; : : : ; nform an inhomogeneous system of linear equations, where the coe�cientsuij form a Vandermonde matrix. Since the uj's are distinct elements of the�eld F, the system has full rank.It remains to show how to solve this system of equations in NC1. Whilesolving general linear equation systems seems to require NC2, we will arriveat our NC1 solution via closed formulae.Using properties of the Vandermonde system one can show the followingclosed formula:gj = nXi=1(�1)1+i viQk 6=i(uk � ui)Pn�j�1(u1; : : : ; bui; : : : ; un);for j = 1; 2; : : :; n. Here bui denotes that ui is missing from the listu1; : : : ; un, and Pk denotes the k-th elementary symmetric polynomial, de-�ned as follows:P0(y1; : : : ; y`) = 1; Pk(y1; : : : ; y`) = XI�[`]kIk=k Yi2I yi; k > 0:By Facts (3) and (5), computing vi=(Qk 6=i(uk � ui)) in NC1 isfairly straightforward. Hence it su�ces to show that the polynomialsPk(u1; : : : ; bui; : : : ; un) are computable in logspace-uniform NC1. A folkloretheorem indicates that this can be done in non-uniform NC1. For our ap-plication, however, the uniformity is crucial.It is easy to see that for any y1; : : : ; y` 2 F, Pk(y1; : : : ; y`) is equal toPk(y1; y2; : : : ; y`; 0; 0; : : : ; 0) for any number of extra zeroes. Let r = kF�k,the number of elements in the multiplicative group of F. We will give anNC1 algorithm to compute the elementary symmetric polynomials of relements, not necessarily distinct, from the �nite �eld F. By appendingr � ` zeroes, we can then compute Pk(y1; y2; : : : ; y`).For each k; 0 < k � r, the value of the elementary symmetric polynomialPk(y1; y2; : : : ; yr) is the coe�cient of Xr�k in h(X) def= Qri=1(X +yi)�Xr .

1. Sparse Sets versus Complexity Classes 25Note that, given any � 2 F, h(�) can be evaluated in NC1, by Facts (2)and (5).If we write h(X) as Pr�1i=0 aiXi, the coe�cient ai = Pr�i(y1; : : : ; yr)for 0 � i < r. The idea now is to choose �'s carefully from F, computeh(�) and compute the coe�cients ai by interpolation. If we choose ! tobe a primitive element of order r in F�, the powers of !, namely 1 =!0; !1; !2; : : : ; !r�1, run through the elements of F�. For 0 � i < r, letbi = h(!i). The relationship between the pointwise values (bi's) and thecoe�cients (ai's) of h(X) can be written as:0BBB@ b0b1...br�1 1CCCA = 0BBB@ 1 !0 !0�2 : : : !0�(r�1)1 !1 !1�2 : : : !1�(r�1)... ... ... : : : ...1 !r�1 !(r�1)�2 : : : !(r�1)�(r�1) 1CCCA0BBB@ a0a1...ar�1 1CCCAThe matrix is the Discrete Fourier Transform matrix, and is a Vander-mondematrix. Since the powers of ! are all distinct, is invertible, and onecan compute the coe�cients ai by (a0; : : : ; ar�1)T = �1(b0; : : : ; br�1)T .The crucial advantage over the earlier Vandermonde system is that withthis particular choice of , the matrix �1 has a simple explicit form:�1ij = 1=(ij) = !�ij :Computing the coe�cients of h(X) is now simply a matrix-vector multi-plication. Theorem 5.3 is proven. 2Corollary 5.1 If there is a sparse set S that is hard for P under logspace-computable many-one reductions, then P = LOGSPACE.Corollary 5.2 If there is a sparse set S that is hard for P under many-onereductions computable in logspace-uniform NC1, then P equals logspace-uniform NC1.The algebraic approach presented here is rather powerful. Many addi-tional results have been obtained including the Hartmanis conjecture forNL, probabilistic and other weaker reductions, and logspace bounded truthtable reductions for P [CS95b, CNS95, vM95].6 ConclusionsWe have seen that the study of sparse sets shows great vitality. Signi�cantprogress is being made which provides unifying perspectives to known andnew results, and reaches out to new territory with unexpected links toparallel computation and randomization. In particular, the proof on the

26 Jin-Yi Cai, Mitsunori OgiharaHartmanis Conjecture for P has reached a new level of sophistication, onemight wonder if there is a more elementary proof.In the last ten years, there is a growing realization that a source of ran-domness can be e�ectively viewed as a computational resource. Whetherthis view is a Platonic truth, i.e., whether randomness in computation can-not be ultimately eliminated without any loss in e�ciency, is unresolved.The philosophical point is both fascinating and troubling. One might rea-sonably ask:Why couldn't one �nd deterministically a solution to a problem,which has nothing probabilistic in its de�nition, as e�ciently asa probabilistic procedure would?Despite this philosophical reservation, the enormous power and e�ective-ness using randomization cannot be denied. Solutions that are otherwiseunavailable are frequently found with the help of randomness. Sometimesthe use of randomness can be eliminated later, sometimes not. And evenwhen one can eventually eliminate randomness, frequently one does that viaa process called derandomization whereby a pseudorandom source, whichis generated deterministically, is substituted for a true random source, thus\mimicking" randomness. To paraphrase Eugene Wigner, there is over-whelming empirical evidence for the unreasonable e�ectiveness of a cointoss in computation, which leaves no doubt that randomization is an ex-tremely valuable tool in algorithm design and complexity theory.The impact of randomization in structural complexity theory in the pasthas not been perhaps as decisive as in some other areas of theoreticalcomputer science. It is hoped that the recent work on sparse sets will helpchange that perception. We believe that randomization will be increasinglyimportant in structural complexity theory in the future.7 References[AA95] M. Agrawal and V. Arvind. Reductions of self-reducible setsto depth-1 weighted threshold circuit classes, and sparse sets.In Proc. 10th Conf. on Structure in Complex. Theory, pages264{276. IEEE, June 1995.[AHH+93] V. Arvind, Y. Han, L. Hemachandra, J. K�obler, A. Lozano,M. Mundhenk, M. Ogiwara, U. Sch�oning, R. Silvestri, andT. Thierauf. Reductions to sets of low information content.In K. Ambos-spies, S. Homer, and U. Sch�oning, editors, Com-plexity Theory, pages 1{46. Cambridge University Press, 1993.[AKM92a] V. Arvind, J. K�obler, and M. Mundhenk. Lowness and thecomplexity of sparse and tally descriptions. In Proc. 3rd Int'lSymp. on Alg. and Comput., pages 249{258. Springer-VerlagLecture Notes in Comp. Sci. #650, December 1992.

1. Sparse Sets versus Complexity Classes 27[AKM92b] V. Arvind, J. K�obler, and M. Mundhenk. On bounded truth-table, conjunctive, and randomized reductions to sparse sets.In Proc. 12th Symp. on Found. of Software Tech. and Theoret.Comp. Sci., pages 140{151. Springer-Verlag Lecture Notes inComp. Sci. #652, 1992.[AKS83] M. Ajtai, J. Komlos, and E. Szemeredi. Sorting in c logn par-allel steps. Combinatorica, 3(1):1{19, 1983.[BCG+95] N. Bshouty, R. Cleve, R. Gavald�a, S. Kannan, and C. Tamon.Oracles and queries that are su�cient for exact learning. Tech.Report TR95-015, The Electronic Colloquium on Computa-tional Complexity, 1995.[BDG88] J. Balc�azar, J. D��az, and J. Gabarr�o. Structural Complex-ity I. EATCS Monographs in Theoretical Computer Science.Springer-Verlag, 1988.[Ber78] P. Berman. Relationship between density and deterministiccomplexity of NP-complete languages. In Proc. 5th Conf. onAutomata, Lang. and Prog., pages 63{71. Springer-Verlag Lec-ture Notes in Comp. Sci. #62, 1978.[BH77] L. Berman and J. Hartmanis. On isomorphisms and density ofNP and other complete sets. SIAM J. Comput., 6(2):305{322,1977.[BvzGH82] A. Borodin, J. von zur Gathen, and J. Hopcroft. Fast parallelmatrix and GCD computations. Inf. and Control, 52:241{256,1982.[Chi85] A. Chistov. Fast parallel calculation of the rank of matricesover a �eld of arbitrary characteristic. In Proc. 5th Int'l Conf.on Fundamentals of Comput. Theory, pages 63{69. Springer-Verlag Lecture Notes in Comp. Sci. 199, 1985.[CNS95] J. Cai, A. Naik, and D. Sivakumar. On the existence of hardsparse sets under weak reductions. Tech. Report 95-31, De-partment of Computer Science, State University of New Yorkat Bu�alo, Bu�alo, NY, July 1995.[Cob65] A. Cobham. The intrinsic computational di�culty of func-tions. In Y. Bar-Hellel, editor, Proc. 1964 Int'l Cong. for Logic,Methodology, and Phil. of Sci., pages 24{30. North Holland,Amsterdam, 1965.[Coo71] S. Cook. The complexity of theorem proving procedures. InProc. 3rd Symp. on Theory of Comput., pages 151{158. ACMPress, 1971.

28 Jin-Yi Cai, Mitsunori Ogihara[CS95a] J. Cai and D. Sivakumar. The resolution of a Hartmanis con-jecture. In Proc. 36th Symp. on Found. of Comp. Sci. IEEE,1995. To appear.[CS95b] J. Cai and D. Sivakumar. The resolution of a Hartmanis con-jecture for NL. Manuscript, 1995.[Csa76] L. Csanky. Fast parallel matrix inversion algorithms. SIAM J.Comput., 5(4):618{623, 1976.[FFK92] S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjec-ture holds relative to an oracle. In Proc. 33rd Symp. on Found.of Comp. Sci., pages 30{39. IEEE, October 1992.[For79] S. Fortune. A note on sparse complete sets. SIAM J. Comput.,8(3):431{433, 1979.[GJ79] M. Garey and D. Johnson. Computers and intractability: Aguide to the theory of NP-completeness. W. H. Freeman andCompany, 1979.[GNW90] T. Gundermann, N. Nasser, and G. Wechsung. A survey ofcounting classes. In Proc. 5th Conf. on Structure in Complex.Theory, pages 140{153. IEEE, July 1990.[GS88] J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM J. Comput., 17(2):309{335, 1988.[Har78] J. Hartmanis. On log-tape isomorphisms of complete sets.Theoret. Comput. Sci., 7(3):273{286, 1978.[Hel86] H. Heller. On relativized exponential and probabilistic com-plexity classes. Inf. and Control, 71(3):231{243, 1986.[HL94] S. Homer and L. Longpr�e. On reductions of NP sets to sparsesets. J. Comput. System Sci., 48(2):324{336, 1994.[HOW92] L. Hemachandra, M. Ogiwara, and O. Watanabe. How hardare sparse sets. In Proc. 7th Conf. on Structure in Complex.Theory, pages 222{238. IEEE, June 1992.[HS65] J. Hartmanis and R. Stearns. On the computational complex-ity of algorithms. Trans. Amer. Math. Soc., 117:285{306, 1965.[IM89] N. Immerman and S. Mahaney. Relativizing relativized com-putations. Theoret. Comput. Sci., 68(3):267{276, 1989.[JY85] D. Joseph and P. Young. Some remarks on witness functionsfor non-polynomial and non-complete sets in NP. Theoret.Comput. Sci., 39:225{237, 1985.

1. Sparse Sets versus Complexity Classes 29[Kad89] J. Kadin. PNP[log n] and sparse Turing-complete sets for NP.J. Comput. System Sci., 39(3):282{298, 1989.[Kar] R. Karp. Personal communication.[KL82] R. Karp and R. Lipton. Turing machines that take advice.L'enseignement Mathematique, 28(3/4):191{209, 1982.[KMR90] S. Kurtz, S. Mahaney, and J. Royer. The structure of completedegrees. In A. Selman, editor, Complexity Theory Retrospec-tive, pages 108{146. Springer-Verlag, 1990.[KW95] J. K�obler and O. Watanabe. New collapse consequences ofNP having small circuits. In Proc. 22nd Conf. on Automata,Lang. and Prog., pages 196{207. Springer-Verlag Lecture Notesin Comp. Sci. #944, July 1995.[Lad75] R. Ladner. The circuit value problem is log space complete forP. SIGACT News, 7(1):18{20, 1975.[Lon82] T. Long. A note on sparse oracles for NP. J. Comput. SystemSci., 24:224{232, 1982.[Mah82] S. Mahaney. Sparse complete sets for NP: Solution of a con-jecture of Berman and Hartmanis. J. Comput. System Sci.,25(2):130{143, 1982.[Mah89] S. Mahaney. The isomorphism conjecture and sparse sets. InJ. Hartmanis, editor, Computational Complexity Theory, pages18{46. American Mathematical Society Proceedings of Sym-posia in Applied Mathematics #38, 1989.[Mil76] G. Miller. Riemann's hypothesis and tests for primality. J.Comput. System Sci., 13:300{317, 1976.[Mul87] K. Mulmuley. A fast parallel algorithm to computer the rank ofa matrix over an arbitrary �eld. Combinatorica, 7(1):101{104,1987.[Ogi95] M. Ogihara. Sparse hard sets for P yield space-e�cient algo-rithms. In Proc. 36th Symp. on Found. of Comp. Sci. IEEE,1995. To appear.[OL93] M. Ogiwara and A. Lozano. Sparse hard sets for countingclasses. Theoret. Comput. Sci., 112(2):255{276, 1993.[OW91] M. Ogiwara and O. Watanabe. On polynomial time boundedtruth-table reducibility of NP sets to sparse sets. SIAM J.Comput., 20(3):471{483, 1991.

30 Jin-Yi Cai, Mitsunori Ogihara[Pip79] N. Pippenger. On simultaneous resource bound. In Proc.11th Symp. on Theory of Comput., pages 307{311. ACM Press,1979.[Rab59] M. Rabin. Speed of computation and classi�cation of recursivesets. In Proc. 3rd Convention Sci. Soc., pages 1{2, 1959.[RR92] D. Ranjan and P. Rohatgi. On randomized reductions to sparsesets. In Proc. 7th Conf. on Structure in Complex. Theory,pages 239{242. IEEE, June 1992.[Ukk83] E. Ukkonen. Two results on polynomial time truth-table re-ductions to sparse sets. SIAM J. Comput., 12(3):580{587,1983.[vL91] J. van Lint. Introduction to Coding Theory. Springer-Verlag,1991.[vM95] D. van Melkebeek. On truth-table reductions of P sets tosparse sets. Manuscript, 1995.[Wat92] O. Watanabe. On polynomial-time one-truth-table reducibilityto a sparse set. J. Comput. System Sci., 44:500{516, 1992.[Yam62] H. Yamada. Real time computation and recursive functionsnot real-time computable. IRE Trans. Electronics Computers,EC-11:753{760, 1962.[Yap83] C. Yap. Some consequences of non-uniform conditions on uni-form classes. Theoret. Comput. Sci., 26(3):287{300, 1983.[Yes83] Y. Yesha. On certain polynomial-time truth-table reducibil-ities of complete sets to sparse sets. SIAM J. Comput.,12(3):411{425, 1983.[You90] P. Young. Juris Hartmanis: fundamental contributions to iso-morphism problems. In A. Selman, editor, Complexity TheoryRetrospective, pages 28{58. Springer-Verlag, 1990.[You92] P. Young. How reductions to sparse sets collapse thepolynomial-time hierarchy: A primer. SIGACT News, 1992.Part I (#3, pages 107{117), Part II (#4, pages 83{94), andCorrigendum (#4, page 94).