Results on ResourceBounded Measure Harry Buhrman and Stephen Fenner and Lance Fortnow 1 Centrum vo or Wiskunde en Informatica 2 University of Southern Maine 3 CWI The University of Chicago Abstract We construct an oracle relative to which NP has pmeasure p but D has measure in EXP This gives a strong relativized negative answer to a question p osed by Lutz Lut Secondly we give strong evidence that BPP is small We show that BPP has pmeasure unless EXP MA and thus the p olynomialtime hierarchy collapses This con trasts with the work of Regan et al RSC where it is shown that Ppol y do es not have pmeasure if exp onentially strong pseudorandom generators exist Intro duction Since the intro duction of resourceb ounded measure by Lutz Lut many re searchers investigated the size measure of complexity classes in exp onential time EXP A particular p oint of interest is the hypothesis that NP do es not have pmeasure Recent results have shown that many reasonable conjectures in computational complexity theory follow from the hyp othesis that NP is not small ie NP and hence it seems to b e a plausible scientic hyp oth p esis LM Lut P He shows In Lut Lutz shows that if NP then BPP is low for p P that this even follows from the seemingly weaker hyp othesis that p He asks whether the latter assumption is weaker or equivalent to NP p In this pap er we show that relative to some oracle the two assumptions are not equivalent p We show a relativized world where D EXP whereas NP has no Pbi immune sets This immediately implies via a result of Mayordomo Maya p P that in this relativized world NP has pmeasure and D and hence has measure in EXP and thus do es not have pmeasure or even p measure URL httpwwwcwinlcwip eopleHarryBuhrmanhtml Email buhrmancwinl Partially supp orted by the Dutch foundation for scientic research NWO by SION pro ject and by the Europ ean Union through Neuro COLT ESPRIT Working Group Nr and HCM grant nr ERBPL URL httpwwwcsusmmaineedufenner Email fennercsusmmaineedu Partially supp orted by NSF grant CCR URL httpwwwcsuchicagoedufortnow Email fortnowcsuchicagoedu Sup p orted in part by NSF grant CCR the Dutch Foundation for Scientic Research NWO and a Fulbright Scholar award NP This shows in a very strong way that relativized measure for NP and P NP NP dier NP whereas P Here P is the class of sets p p recognized by p olynomial time Turing machines that are allowed two queries to NP an NP oracle We show that our results cannot b e improved to P Secondly we investigate the p ossibility that BPP do es not have pmeasure Intuitively BPP is a feasible complexity class close to P and therefore it should b e the case that BPP is small We give very strong evidence supp orting this intuition We show that BPP unless EXP MA and thus the p p olynomialtime hierarchy collapses Since BPP Ppol y our result contrasts with the one by Regan Sivakumar and Cai RSC where it is shown that Ppol y unless exp onentially p strong pseudorandom generators do not exist Preliminaries We let f g and identify strings in with natural numb ers via the usual binary representation We x N N to b e a standard enumeration of all nondeterministic p olynomialtime oracle Turing machines NOTMs where i for each i and input of length n N runs in time n for all oracles All our i machines run using symb ols and blanks Fix a deterministic oracle TM M A p which accepts some standard complete language for EXP for all A m n We may assume that M runs in time We let h i b e the standard pairing p function and we note that x y hx y i for all x y A set is in D if it can b e expressed as the dierence of two sets in NP The notations R Q R and Q denote the real numb ers the rational num b ers the p ositive real numb ers and the p ositive rational numb ers resp ectively Resource Bounded Measure Classical Leb esque measure is an unusable to ol in complexity classes As these classes are all countable everything we dene in such a class has measure Yet we might wish to have a notion of abundance and randomness in complexity classes Lutz Lut Lut intro duced the notion of resource bounded measure and gave a to ol to talk ab out these notions inside complexity classes Denition A martingale d is a function from to R with the prop erty that dw dw dw for every w Denition A pmartingale is a martingale d Q that is p olynomial time computable Denition A martingale d succeeds on a language A if lim sup d n A n7!1 1 We write S d fA j d succeeds on Ag Denition Let X b e a class of languages X has pmeasure X i there exists a pmartingale d such that p 1 X S d X has pmeasure X i X p p X has pmeasure in EXP X jEXP i X EXP p p X has pmeasure in EXP X jEXP i X EXP p p One often denes measure in EXP using p measure where the martingale O (1) log n can use time All of our results also hold in this weaker mo del NP Measure of NP versus Measure of P In this section we concentrate on the question p osed by Lutz Lut We show NP that relative to some oracle NP do es not imply that P We p p do this in a very strong way by constructing an oracle such that NP do es not p contain Pbiimmune sets and D EXP Theorem There exists an oracle A such that relative to A NP has no p Pbiimmune sets and D EXP p Proof We will co de EXP into D on one side of the oracle and prevent Pbi immunity on the other ie strings in fx j x g will b e used to co de p EXP into D while strings in fx j x g will co de the information to nd an innite subset of each NP set or its complement Some diagonalization will also b e necessary to force certain NP computations To mix co ding with diagonalization we employ a simplied version of the NP trick used to construct an oracle for P NEXP BT FF For each x we A reserve two p otential regionsleft and right in which to co de M x only one of which will actually b e used To co de correctly in a region we must let exactly one string in the region enter A We will co de in the left region unless we have to diagonalize against some NP machine which may necessitate adding several strings of the left region to A If this happ ens we scrap the left region and co de in the right region but we can do this only if our diagonalization hasnt already put strings of the right region into A We now pro ceed with the formal treatment For every x with jxj n and b we call s an x b left coding string resp ectively an x b right coding string if s xy b resp ectively s xy b for some y of length n We identify left and right with and resp ectively We build the oracle A in stages each successive stage extending a nite p ortion of As characteristic function If is some partial characteristic function N an oracle machine and x then the computation N x is dened as usual except that when N makes any query outside domain it is answered negatively As is customary we regard as a set of ordered pairs If is another characteristic function we write to mean that extends Finally dene the tower of s function tn for n by t tn tn Stage End Stage Stage n We are given Set n n Forcing an NP computation If n tk for any k then set right if s for some x b left co ding string s with jxj n d n left otherwise and go to step Otherwise let n tk for some k hi j i If there exists a minimal such that b oth n a N has an accepting path in which all queries are in domain and i b for no x with jxj n and no x b rightco ding string s do es s i n then set f g and set d right note that is only dened on n i i n strings no longer than n Otherwise set f g and set d left n Preserving computations of M For all x of length n run M x and ex tend with just enough s to cover all queries made by M x not in domain Coding computations of M For all x of length n let y b e the lexicographically least string if one exists such that jy j n and neither the x d co ding string nor the x d co ding string corresp onding to n n y is in domain If M accepts set fxy d g otherwise set n fxy d g n Set to b e extended with just enough s to cover all remaining x b d n co ding strings for all b d fleft rightg and x of length n End Stage S Let A b e such that extends for all n x for any x n A n A n B For any B dene the language L by if either B contains an x rightco ding string or B B contains no x dco ding strings for any d fleft rightg L x otherwise B A A B p We now show that L x M x for all x and Clearly L coD A A A p p EXP D hence coD Pick an n large enough and x an input x of length n In Step of Stage n n n such a y must exist there are at most x b dco ding strings queried by M on inputs of length n b ecause of the running time of M and 2 log n log n less than n n total strings queried by the N in Step of Stages i n through n Thus there are less than x b dco ding strings in domain at Step of Stage n The fact that A A M x L x is now easily seen rst we observe that no x b rightco ding string for any b gets into A in Steps or of any stage Thus we have two cases d left For any b and d fleft rightg the only x b dco ding string n that ever enters A do es so in Step of Stage n This unique string is an A x left co ding string if M x accepts and is otherwise an x left co ding string thus is satised d right Exactly one x b rightco ding string enters A It is an x right n A co ding string i M x accepts Again is satised A A It remains to