The Power of Adaptiveness and Additional Queries in Random Self
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THE POWER OF ADAPTIVENESS AND ADDITIONAL QUERIES IN RANDOMSELFREDUCTIONS Joan Feigenbaum Lance Fortnow Carsten Lund and Daniel Spielman Abstract We study randomselfreductions from a structural com plexitytheoretic p oint of view Sp ecically we lo ok at relationships b etween adaptive and nonadaptive randomselfreductions We also lo ok at what happ ens to randomselfreductions if we restrict the numb er of queries they are allowed to make We show the following results There exist sets that are adaptively randomselfreducible but not nonadaptively randomselfreducible Under plausible assump tions there exist such sets in NP There exists a function that has a nonadaptive k n random selfreduction but do es not have an adaptive k nrandomself reduction For any countable class of functions C and any unb ounded func tion k n there exists a function that is nonadaptively k n uniformlyrandomselfreducibl e but is not in C pol y This should b e contrasted with Feigenbaum Kannan and Nisans theorem that all nonadaptively uniformlyrandomselfreducibl e sets are in NPpol y Key words Adaptiveness randomselfreducibili ty Sub ject classications Q Intro duction Informally a function f is randomselfreducible if the evaluation of f at any given instance x can b e reduced in p olynomial time to the evaluation of f at one or more random instances y i Randomselfreducible functions have many applications including average case complexity eg lower b ounds eg interactive pro of systems Feigenbaum et al and program checkers testers and correctors eg and cryptographic proto cols eg For a history and overview of the sub ject see In this pap er we study randomselfreductions from a structural complexi tytheoretic p oint of view In particular we analyze the relationships b etween adaptive and nonadaptive randomselfreducibility and the eect of changing the numb er of queries available to the randomselfreduction Our main results are There exists a set that is adaptively randomselfreducible but not non adaptively randomselfreducible We can make such a set sparse and recognizable in slightly more than p olynomial space If b oundederror probabilistic tripleexp onential time do es not contain nondeterministic tripleexp onential time then there exists a set in NP that is adaptively randomselfreducible but not nonadaptively random selfreducible For any p olynomially b ounded k n there exists a function that is non adaptively k n randomselfreducible but not adaptively k nran domselfreducible For any countable class of functions C and any unb ounded function k n there is a function that is nonadaptively k nuniformlyrandomself reducible but not in C pol y For example there is a nonadaptively k n uniformlyrandomselfreducible function that is not in FPSPACEpol y for any unb ounded k n This should b e contrasted with Feigenbaum Kannan and Nisans theorem that all nonadaptively uniformlyrandom selfreducible sets are in NPpol y Denitions and Notation Throughout this pap er x is the input to a randomized reduction n is the w n size of x and r is a random variable distributed uniformly on f g where w is some p olynomially b ounded function of n The parameter k n is also a p olynomially b ounded function of n Definition A nonadaptive k nrandomselfreduction for f is a pair of p olynomialtime computable functions and with the following prop erties Randomselfreducibil ity n For all n and all x f g f x x r f x r f k n x r with probability at least For all x and x such that jx j jx j and for all i i k n i x r and i x r are identically distributed Note that the random queries x r k n x r are in general de p endent Observation Szegedy see We can assume without loss of gen erality that in a nonadaptive randomselfreduction all of the random variables x r k n x r are identically distributed The word nonadaptive in Denition refers to the fact that all of the random queries are pro duced in one round that is i x r do es not dep end on f j x r for j i The next denition covers the case in which queries are pro duced in multiple roundsthat is the reduction follows an adaptive strategy Definition An adaptive k nrandomselfreduction for f is a p olynomi altime oracle Turing machine M with the following prop erties n For all n and all x f g f f x M x r with probability at least f For all x and r M x r asks at most k n queries to the f oracle The f f x r queries are denoted q x r q k n f For all x and x such that jx j jx j and all i i k n q x r i f and q x r are identically distributed Note that we do not require that i 0 0 f f q x r and q x r b e identically distributed for f f i i We say that a function f is randomselfreducible if it has a nonadaptive or adaptive k nrandomselfreduction for some p olynomially b ounded function k n A language L is randomselfreducible if the characteristic function for Feigenbaum et al L is randomselfreducible We use the abbreviation rsr for b oth random selfreducible and randomselfreduction pn For b oth adaptive and nonadaptive rsrs we can replace by for any p olynomial pn at the cost of increasing k n by a p olynomial factor We can achieve this b ound by using the standard technique of running indep endent copies of the rsr and taking a plurality vote See for details Prop erty in Denition and prop erty in Denition are referred to as the instancehiding property This means that any one query of the rsr reveals nothing ab out the instance x except its size It should b e noted that the instance x might b e computable from a set of two queries Note that the instancehiding prop erty need not hold for a cheating oracle f f ie an adaptive oracle machine could in round i leak something b esides the size of the instance if it is given wrong answers to one or more of its queries in rounds through i An rsr is oblivious if its queries only dep end on the input sizethat is not only are the distributions of queries the same for all inputs of the same size which is true of any rsr but moreover the reduction ignores the input until after all queries have b een made It is straightforward to prove that obliviously rsr sets are in Ppol y An rsr is uniform if for all inputs x each jxj query is uniformly distributed over f g We call such a reduction a k n uniformlyrandomselfreduction abbreviated k nursr and stress that the word uniform describ es the distribution of random queries ie it is not meant to distinguish b etween reduction pro cedures that take advice and those that do not Uniformlyrandomselfreductions are the typ e of rsrs studied in Let BPE b e the class of languages recognizable in b oundederror proba cn bilistic exp onential time ie the union of BPTIME for all constants c Similarly let BPEE and BPEEE denote the languages recognizable in b ounded error probabilistic doubleexp onential and tripleexp onential time The non deterministic time classes NE NEE and NEEE are dened analogously n Let n b e any unb ounded nondecreasing function such that n is time constructible eg n log n Adaptiveness versus Nonadaptiveness In this section we study the dierence b etween adaptive and nonadaptive rsrs We nd sets of relatively low complexity that are adaptively rsr but not nonadaptively rsr Randomselfreducibil ity It is tempting to conjecture that any sparse adaptively rsr set is also non adaptively rsr in the hop e that one could just lo ok for all the elements of that set However the lo cation of those elements could make that pro cess dicult In fact we show that our main results also hold for sparse sets Theorem For any k n n there exists a sparse set L such that L n is in DSPACEn and has an oblivious adaptive k nrsr but for which any k n nonadaptive k nrsr will err with probability at least k n for innitely many inputs Proof First we will describ e how to construct a nonsparse L From there the construction of a sparse L is straightforward In this pro of there will b e n a corresp ondence b etween strings of length n and the integers th ie the i string in the lexicographic ordering corresp onds to i A stepfunction B is the language nms n B fx f g js xg nms n where s m ie m is a b ound on how big s can b e Hence B nms th consists of all nbit strings from the s such string on We call s the step and m the range n Note that any function on f g whose restriction to f g is equal to B for some m has an oblivious adaptive dlog m ersr This is b ecause nms it takes dlog m e queries to nd the step using binary search and once the step is known it is simple to determine memb ership in B nms The language L will b e a union of stepfunctions one B for each input nms k n length n where the range m will always b e equal to and the steps will b e determined by diagonalization against nonadaptive k nrsrs The diagonalization is done as follows Let M M b e an enumeration of all the nonadaptive k nrsrs in which every k nrsr o ccurs innitely many times Assume without loss of generality that M s running time is b ounded i i by n We diagonalize against M on inputs of length n where n is the least i i i n i integer such that n i This choice is made in order to ensure that n i i is greater than the running time of M on inputs of length n i i Since M asks at most k n queries there must b e an x x m such i that the probability that one of M s k n queries is equal to x is at most i k nm Lo ok at M s b ehavior on the rst such input x with oracles B i n mx i and B Let n mx i B n