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Almost-Everywhere Complexity Hierarchies for Nondeterministic Time

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Almost-Everywhere Complexity Hierarchies for Nondeterministic Time Almost-Everywhere Complexity Hierarchies for Nondeterministic Time y Eric Allender Department of Computer Science, Rutgers University New Brunswick, NJ, USA 08903 z Richard Beigel Department of Computer Science, Yale University 51 Prosp ect St. P.O.Box 2158, Yale Station New Haven, CT, USA 06520-2158 Ulrich Hertrampf Institut fur Informatik, Universitat Wurzburg D-8700 Wurzburg, Federal Republic of Germany x Steven Homer Department of Computer Science, Boston University Boston, MA, USA 02215 A preliminary version of this work app eared as [2]. y Supp orted in part by National Science Foundation grant CCR-9000045. Some of this researchwas p erformed while the author was a visiting professor at Institut fur Informatik, Universitat Wurzburg, D-8700 Wurzburg, Federal Republic of Germany. z Supp orted in part by the National Science Foundation grants CCR-8808949 and CCR-8958528. Work done in part while at the Johns Hopkins University. x Supp orted in part by National Science Foundation grants MIP-8608137 and CCR-8814339, Na- tional Security Agency grantMDA904-87-H, and a Fulbright-Hays research fellowship. Some of this researchwas p erformed while the author was a Guest Professor at Mathematisches Institut, Universitat Heidelb erg. 1 SUMMARY We present an a.e. complexity hierarchy for nondeterministic time, and show that it is essential ly the best result of this sort that can beproved using relativizable proof techniques. 1 Intro duction The hierarchy theorems for time and space are among the oldest and most basic results in complexity theory. One of the very rst pap ers in the eld of complexity theory 2 was [14], in whichitwas shown that if tn = oT n, then DTIMET n prop erly 1 contains DTIMEtn, for any time-constructible functions t and T . This result was 2 later improved by Hennie and Stearns [15 ], who proved a similar result, with \tn " replaced by\tn log tn". Still later, Co ok and Reckhow [8] and Furer [10] proved tighter results for RAM's and Turing machines with a xed numb er of tap es, resp ectively. The time hierarchy theorems proved in this series of pap ers may b e describ ed as \in nitely often" or \i.o." time hierarchies, b ecause they assert the existence of a set L 2 DTIMET n that requires more than time tn for in nitely many inputs. Note, however, that it may b e the case that for all n, memb ership in L for ninety-nine p ercent of the inputs of length n can b e decided in linear time. That is, the i.o. time hierarchy theorems assert the existence of sets that are hard in nitely often, although it is p ossible that the \hard inputs" for these sets form an extremely sparse set. However it turns out that much stronger time hierarchy theorems can b e proved. One can show the existence of sets L in DTIMET n that require more than time tn for al l large inputs. This notion is called \almost everywhere" or a.e. complexity, and has 1 A function T is time-constructible if there is a deterministic Turing machine that, for all inputs of length n, runs for exactly T n steps. For the purp oses of this pap er, if T is time-constructible, then T n n for all n. 2 b een investigated in [11, 12 , 25]. The following de nitions make this precise. De nitions: For anyTuring machine M , the partial function T : ! N is the time function M of M. T x is the numb er of steps that M uses on input x if this numb er exists; M T x is unde ned if M do es not halt on input x. M For functions f and g mapping either or N to N, wesay that f = ! g i , for all r>0 and for all but nitely many x,if f x is de ned, then f x >rgx. Wesay that an in nite set L is a.e. complex for DTIMEt i for all deterministic Turing machines M , LM =L = T x=! tjxj. M One of the a.e. time hierarchy theorems of [12 ] can now b e stated: Theorem 1 [12] If T is a time-constructible function and tn log tn=oT n, then there is a set in DTIMET n that is a.e. complex for DTIMEtn. Thus for deterministic time complexity classes, the a.e. hierarchy is just as tightas the i.o. hierarchy.However much less is known ab out hierarchies for probabilistic and nondeterministic time. The b est i.o. time hierarchy theorem for probabilisti c time that has app eared in the literature is due to Karpinski and Verb eek [16]; however they are only able to show that BPTIMEtn is prop erly contained in BPTIMET n when T n grows very much more quickly than tn. On the other hand, a recent oracle result of Fortnow and Sipser [9] shows that it is not p ossible to provea very tight time hierarchy theorem 3 for probabilistic time using relativizable techniques; they present an oracle according to which BPP = BPTIMEO n. Clearly the [9] result also shows that there is no tight a.e. time hierarchy for probabilistic time that holds for all oracles. An early i.o. hierarchy theorem for nondeterministic time was given by Co ok [7]. The strongest such theorem that is currently known is due to [23]. It is shown there that if t and T are time-constructible functions such that tn +1= oT n, then there is a set L in NTIMET n NTIMEtn. Furthermore, it is shown in [27, 17 ] that L can even b e chosen to b e a subset of 0 . When t and T are b ounded by p olynomials, this result is even tighter than the b est known results for deterministic time. However, when T and t are very large, the gap b etween tn and tn + 1 is also quite large, and thus the nondeterministic time hierarchy seems not to b e very tight. On the other hand, Racko and Seiferas show in [21] that the i.o. time hierarchy theorem for nondeterministic time cannot b e improved signi cantly using relativizable techniques. For example, they n+1 n A 2 A 2 = log n. Note on the = NTIME 2 present an oracle A such that NTIME 2 n n+1 A 2 A 2 other hand that for al l oracles A, DTIME 2 6= DTIME 2 = log n. Surprisingly, it seems that no results concerning an a.e. hierarchy for nondeterministic time have app eared. Geske, Huynh, and Seiferas explicitly raise the question of an a.e. hierarchy for nondeterministic time as an op en problem in [12]. This pap er essentially settles this question. We present an a.e. hierarchy theorem for nondeterministic time, and we present an oracle relative to which the given theorem cannot b e signi cantly improved. It is necessary to discuss the interpretation that should b e a orded our oracle con- structions, in light of the fact that comp elling examples of nonrelativizing pro of tech- niques do exist [3, 19 , 24 ]. We maintain that it is useful to know what sort of hierarchies 4 hold relativetoal l oracles. Note that many imp ortant complexity classes are de ned in terms of relativized nondeterministic computation; the levels of the p olynomial hierarchy are the most obvious examples. For example, although it is a standard fact that there are SAT 3 SAT 2 sets in DTIME n that are a.e. complex for DTIME n where the pro of of this fact makes no use of any prop erties of SAT, the results of this pap er show that there are Y 3 Y 2 oracles Y such that NTIME n contains no sets a.e. complex for NTIME n . Thus SAT 3 SAT 2 any pro of that NTIME n contains sets a.e. complex for NTIME n will have to make essential use of certain prop erties of the oracle SAT. A longer discussion along these lines may b e found in [1]. In this pap er we consider only time complexity.Aninvestigation of almost-everywhere complexity hierarchies for nondeterministic space has b een carried out by Geske and Kakihara [13]. They prove almost-everywhere hierarchy theorems for nondeterministic space that are quite similar to those for deterministic space. The organization of the pap er is as follows. In Section 2 we de ne precisely what we mean by almost everywhere complexity for nondeterministic time. In this section we also relate almost everywhere complexity to the concept of immunity. In Section 3we present our main results. In section 4 we turn to the questions of bi-immunity and co-immunity, which also are relevant to almost everywhere complexity. The results we prove concerning immunity, co-immunity and bi-immunity are the b est that can b e proved using relativizable pro of techniques. 5 2 A. E. Complexity and Immunity In order to talk ab out an a.e. complexity hierarchy for nondeterministic time, wemust rst agree on the notion of running time for nondeterministic Turing machines. For deterministic Turing machines, it is clear how to de ne the running time. However there are at least two de nitions that are commonly used in de ning the running time of a nondeterministic Turing machine: NTM M runs in time t on input x if every computation path of M on input x has length t. NTM M runs in time t on input x if [M accepts x implies there is an accepting computation path of length at most t].
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