Derandomizing from Random Strings

Derandomizing from Random S trings Harry Buhrman ∗ CWI and Univ ersit y of Amsterdam buhrman@cwi .nl Lance F ortno w † North wes t ern Univ ersity fortnow@nor thwestern. edu Mic hal Ko uc k´ y ‡ Institute of Mathematics, AS CR koucky@math .cas.cz Bruno Loff § CWI bruno.loff@ gmail.com Abstract In this pap er w e show t hat BPP is truth-table reducible to the set o f Kolmogor ov random str ings R K . It was previous ly known that PSP A CE , and hence BPP is T uring- reducible to R K . The earlier pro of relied on the adaptivity of the T uring- reduction to find a Ko lmogorov-rando m string of p olynomia l length using the set R K as oracle. Our new non-adaptive result relies o n a new fundamental fact a bo ut the set R K , namely each initial s e gment of the characteris tic seq ue nc e of R K is not compr essible by recursive means. As a partial co n verse to our cla im we show that s trings o f high Kolmogorov-complexity when used as advice are not muc h more useful than randomly chosen strings. 1 In tro d uction Kolmogoro v complexit y stud ies the amount of r andomness in a string by the smallest program that can generate it. The most rand om strings are those we cannot compress at all making the set R K = { x | K ( x ) ≥ | x |} of Kolmogoro v random s tr ings worth y of close analysis. Allender et al. [ABK + 02] show ed the surp rising compu tational p o wer of R K includ- ing that p olynomial time adaptiv e (T uring) access to R K enables one to d o PS P A CE- computations: P S P A CE ⊆ P R K . One o f the i ngredien ts in the proof sho ws h o w on inpu t 0 n one can in p olynomial time with adaptive access to R K generate a p olynomially long Kolmogoro v random string. With non-adap tive access it is only p ossible to generate in p olynomial time a random strin g of length at most O (log n ). ∗ P artially supp orted by an N WO VICI gran t. † Supp orted in part by NS F grants CCF-0829754 and DMS-0652521. ‡ P artially supp orted by pro ject No. 1M00216208 08 of M ˇ SMT ˇ CR and I nstitutional R esearc h Plan No. A V0Z1019050 3. § Supp orted by a Po rtuguese science FCT grant. 1 In an attempt to char acterize PSP ACE as th e class of sets reducible to R K , Allender, Buhrman and Kouc k´ y [ABK06] n oticed that this question dep en ds on the c hoice of unive r sal m ac hine used in the d efi nition of the notion of Kolmogoro v complexity . They also started a systematic stud y of weak er and non-adaptiv e access to R K . T hey sh o wed for example that P = REC ∩ \ U { A | A ≤ p dtt R K U } . This result and the fact that with non-adaptive access to R K in general only logarithmi- cally small strings can b e f ound seems to suggest that adaptive acce ss to R K is needed in order to b e u seful. Our first result p ro ves this in tuition false: W e show that p olynomial time non- adaptiv e access to R K can b e used to d erandomize any BPP computation. In order to derandomize a BPP computation one needs a (pseudo)random string of p olynomial size. As ment ioned b efore one can only obtain short, O (log n ) sized, r an d om strings fr om R K . Instead we show that the c haracteristic sequence formed by the str ings of length c log n , R = c log n K , itself a s trings of length n c , is complex enough to figur e as a hard fun ction in the hard ness v ersu s r an d omness framew ork of Impagliazzo and Wigderson [IW97]. Th is w ay w e construct a ps eudorandom generator that is str ong enough to derandomize BPP . In p articular we sho w that for ev ery time b ound t , there is a constant c su c h that R K 6∈ i.o.-DTIME( t ) / 2 n − c . This is in stark con trast with the time-unboun ded ca se where only n bits of advice are necessary [Bar68]. As a consequence w e giv e an alter- nativ e p ro of of the existence of an r.e. set A , due to Barzdin [Bar68], suc h that for all time b ounds t , there exists c t suc h that K t ( n ) ( A 1: n | n ) ≥ n/c t . W e simply tak e for A the complement of R K . Barzdin also show ed th at this lo wer b ound is optimal for r.e. sets. He nce the constan t d ep ending on th e time-b ound in our Theorem 4 is optimal. Next we try to establish whether we can charact erize BPP as th e class of sets that non-adaptiv ely reduce to R K . One can view the tr uth-table reduction to R K as a computation with ad v ice of K t ( n ) complexit y Ω( n ). W e can sho w that for sets in EXP and t ( n ) ∈ 2 n Ω(1) , p olynomial-time computation with p olynomial (exp onen tial, resp.) size advice of K t ( n ) complexit y n − O (log n ) ( n − O (log log n ), resp.) can b e simulat ed b y b ound ed error pr obabilistic mac hine w ith almost linear size advice. F or p addable sets that are complete for NP , P # P , PSP A C E, or EXP we d o not eve n n eed the linear size advice. Hence, advice of h igh K t ( n ) complexit y is no b etter than a truly random string. Summarizing our results: • F or ev ery computable time b oun d t ther e is a constant c (dep end ing on t ) su ch that R K 6∈ i.o.-DTIME( t ) / 2 n − c . • The complemen t of R K is a natural examp le of an computably en umerable set whose c haracteristic sequence has high time b oun ded K olmorogo v complexity for ev ery n . 2 • BPP is truth-table reducible to R K . • A p oly- up-to exp onen tial-size advice th at has ve r y large K t ( n ) complexit y can b e replaced b y O ( n log n ) bit advice and true r andomness. 2 Preliminaries W e remind the r eader of some of the definitions we u se. Let M b e a T urin g machine. F or an y string x ∈ { 0 , 1 } ∗ , the Kolmogoro v complexit y of x relativ e to M is K M ( x ) = min { | p | | p ∈ { 0 , 1 } ∗ & M ( p ) = x } , where | p | denotes the length of string p . It is w ell known that for a universal T uring mac h ine U and an y other mac hine M there is a constan t c M suc h that for all strin gs x , K U ( x ) ≤ K M ( x ) + c M . F or the rest of the p ap er w e will fix some un iv ersal T urin g mac hine U and we will measure Kolmogro v complexit y relativ e to that U . Thus, w e will not write the su bscript U explicitly . W e defi n e K t ( x ) = min { | p | | U ( p ) = x and U ( p ) uses at most t ( | x | ) steps } . Unlik e traditional computational complexity the time b ound is a function of th e length of the output of U . A string x is said to b e Kolmo gor ov-r andom if K ( x ) ≥ | x | . Th e set of Kolmogoro v- random str ings is denoted by R K = { x ∈ { 0 , 1 } ∗ | K ( x ) ≥ | x |} . F or an in teger n and set A ⊆ { 0 , 1 } ∗ , A = n = A ∩ { 0 , 1 } n . The follo wing w ell kn o wn claim can b e prov en by considering the Kolmogoro v complexit y of | R = n K | (see [L V08]). Prop osition 1 Ther e is a c onstant d such that for al l n , | R = n K | ≥ 2 n /d . W e also use computation with advice. W e deviate sligh tly fr om the usu al d efi nition of computation with advice in the wa y ho w we express and measure the ru n ning time. F or an advic e fu nction α : N → { 0 , 1 } ∗ , w e say that L ∈ P /α if there is a T urin g machine M su c h that for every x ∈ { 0 , 1 } ∗ , M ( x, α ( | x | )) runs in time p olynomial in the length of x and M ( x, α ( | x | )) accepts iff x ∈ L . W e assume that M h as random access to its input so the length of α ( n ) can gro w faster than any p olynomial in n . Similarly , we define EXP /α where we allo w the mac hine M to run in exp onen tial time in length of x on the input ( x, α ( | x | )). F ur thermore, we are in terested not only in Bo olean languages (decision pr oblems) but also in fu nctions, so w e naturally extend b oth definitions also to computation w ith advice of functions. T yp ically we are interested in the amount of advice that w e n eed for inputs of length n so for f : N → N , C /f is the union of all C /α for α satisfying | α ( n ) | ≤ f ( n ). Let L b e a language and C b e a language class. W e say th at L ∈ i . o . − C if there exists a language L ′ ∈ C such that for infin itely many n , L = n = L ′ = n . F or a T ur ing mac hine M , w e s a y L ∈ i.o.-M /f if there is some advice function α with | α ( n ) | ≤ f ( n ) suc h that for infinitely many n , L = n = { x ∈ Σ n | M ( x, α ( | x | )) accepts } . W e say that a set A p olynomial-time T u ring red uces to a set B , if there is an oracle mac hine M that on inpu t x ru ns in p olynomial time and w ith oracle B decides whether x ∈ A . If M asks its questions non-adaptively , i.e., eac h oracle question d o es n ot d ep end 3 on the answers to the previous oracle qu estions, we sa y th at A p olynomial-time truth- table redu ces to B ( A ≤ p tt B ). Moreo v er, A ≤ p dtt B if mac hin e M outp u ts as its answer the d isjunction of the oracle answe rs. Similarly , A ≤ p ctt B for the conjun ction of th e answ er s . 3 High circuit complexit y of R K In this section we prov e that th e c h aracteristic s equ ence of R K has h igh circuit com- plexit y almost ev erywhere. W e will first pro ve the follo win g lemma. Lemma 2 F or every total T uring machine M ther e is a c onstant c M such that R K is not in i.o.-M / 2 n − c M . There is a (n on-total) T u ring machine M suc h that R K is in M /n + 1 where th e advice is the num b er of strings in R = n K . Simp ly fin d all th e n on-random strings of length n . Th is machine will fail to halt if the advice underestimates the num b er of rand om strings. Pr o of of L emma 2. Supp ose the theorem is false. Fix a total mac hine M . W e hav e that, ( x, α ) ∈ L ( M ) if and only if x ∈ R K , for some advice α of length k ≤ 2 n − c M and ev ery x of some large enough length n . By padding the advice w e can assu me k = 2 n − c M . W e will set c M later in order to get a con tradiction. Let R β = { x ∈ Σ n | ( x, β ) ∈ L ( M ) } . By Prop osition 1 for some constant d , | R α | ≥ 2 n /d so we kno w th at if | R β | < 2 n /d then β 6 = α . W e call β go o d if | R β | ≥ 2 n /d . Fix a go o d β and c h o ose x 1 , . . . , x m at random. The prob ab ility that all the x i are not in R β is at most (1 − 1 /d ) m < 2 − m/d . Th ere are 2 k advice strings β of length k so if 2 − m/d ≤ 2 − k then there is a sequence x 1 , . . . , x m suc h that for every go o d β of length k there is an i such th at x i ∈ R β . W e can computably searc h all such sequences so let x 1 , . . . , x m b e the lexicographi- cally least sequence such th at for eac h go o d β of length k , there is s ome x i ∈ R β . T h is also means x i ∈ R α for some i so for one of th e x i w e hav e K ( x i ) ≥ n . Fix m = 2 n − a for a constan t a to b e c hosen later. W e can describ e x i b y n − a + b log a bits for some constan t b : n − a bits to d escrib e i , O (log a ) bits to reco v er n and a constan t num b er of additional bits to describ e k , M , d and the algorithm ab ov e for fi n ding x 1 , . . . , x m . If w e pic k a s uc h th at a > b log a we con tradict the fact that K ( x i ) ≥ n . If w e p ick c M ≥ a + log d we th en hav e 2 n − a ≥ 2 n − c M d , m > k d and 2 − m/d ≤ 2 − k completing our con tradiction. ✷ In order to get our statemen t ab out time b ounded advice classes we instan tiate Lemma 2 with univ ersal machines U t that run in time t , use the first part of th eir advice, in pr efix fr ee form, as a co d e for a mac hin e that runs in time t and has the second part of the advice for U t as its advice. T he follo win g is a direct consequ ence of Lemma 2. 4 Lemma 3 F or every c omputable time b ound t and universal advic e machine U t ther e is a c onstant c t such that R K is not in i.o.-U t / 2 n − c t . W e are no w ready to pro ve the main theorem from this section. Theorem 4 F or e very c omputable time b ound t ther e i s a c onstant d t such that R K is not in i.o.-DTIME ( t ) / 2 n − d t . Pr o of. Supp ose th e theorem is false, that is th ere is a time b oun d t s u c h that for ev ery d there is a mac hine M d that run s in time t s uc h that R k ∈ i.o.-M d / 2 n − d . Set t ′ = t log t and let c t ′ b e the constant that comes out of Lemma 3 wh en in stan tiated with time b ound t ′ . S et d = c t ′ + 1 and let the co de of mac hine M d from the (false) assumption h a ve size e . So we ha ve that R k ∈ i.o.-M d / 2 n − d . Th is in turn imp lies that R K ∈ i.o.-U t ′ / 2 n − d + e + 2 log e , whic h implies that R K ∈ i.o.-U t ′ / 2 n − c t ′ a contradictio n with Lemma 3. The last step is tr u e b ecause the universal m achine runnin g for at most time t ′ = t log t , can sim ulate M d , who runs in time t . ✷ As an immediate corollary we get an alternativ e, m ore natural candidate f or Barzdin’s computably enumerable set that has h igh resource b ounded Kolomoro v complexity , namely the set of compressib le str in gs. Corollary 5 F or eve ry c omputable time b ound t ther e is a c onstant c such that K t ( R k (1 : n ) | n ) ≥ n/c Barzdin [Bar68] also sho wed that this lo w er b oun d is optimal. Th at is the dep end ence of c on the time b ound t is needed for the c h aracteristic sequ ence of eve ry r.e. s et. Hence this dep ence is also n ecessary in our Theorem 4. 4 BPP truth-table reduces to R k In this section w e in v estigate wh at languages are reducible to R k . W e start with the follo wing th eorem w hic h one can prov e u sing n o wa da ys standard derandomization tec h- niques. Theorem 6 L et α : { 0 } ∗ → { 0 , 1 } ∗ b e a length pr eserving function and δ > 0 b e a c onstant. If α (0 n ) 6∈ i.o.-EXP /n δ then for every A ∈ BPP ther e exists d > 0 such that A ∈ P /α (0 n d ) . Pr o of. α (0 n ) 6∈ i.o.-EXP /n δ implies that wh en α (0 n ) is interpreted as a truth -table of a function f α (0 n ) : { 0 , 1 } log n → { 0 , 1 } , f α (0 n ) do es not hav e b o olean circuits of size n δ/ 3 for all n large enough. It is kno wn that su c h a fu nction can b e used to build the Im p agliazz o- Wigderson pseudorandom generator [IW97] w hic h can b e us ed to d erandomize b o olean 5 circuits of size n δ ′ for some δ ′ > 0 (see [IW97, K vM99, ABK + 02]). Hence, b ounded-error probabilistic compu tation runn ing in time n ℓ can b e d erandomized in p olynomial time giv en access to α (0 n 2 ℓ/δ ′ ). ✷ F rom Theorem 4 and the ab ov e Theorem w e obtain the follo wing corollary . Corollary 7 BPP ≤ p tt R K . Pr o of. Let α (0 n ) b e the truth -table of R K on strings of length ⌊ log n ⌋ pad d ed by zeros to the length of n . By Th eorem 4, α (0 n ) 6∈ i.o.-EXP / ( n/c ) for some c > 0. Consider an y A ∈ BPP. By Th eorem 6 for some d , A ∈ P /α (0 n d ). T h e claim f ollo ws by noting that a truth -table reduction to R k ma y qu ery the mem b ership of all the strin gs of length ⌊ log n d ⌋ to construct α (0 n d ) and then run the P /α (0 n d ) algorithm for A . ✷ Our goal would b e to sh o w that using R K as a source of rand omness is the only wa y to m ake use of it. Ideally we would lik e to sh o w that any recursiv e set that is truth-table reducible to R K m u st b e in BPP. W e fall short of su c h a goal. Ho we v er we can sh o w the follo wing claim. Theorem 8 L et α : { 0 } ∗ → { 0 , 1 } ∗ b e a length p r eserving function and c > 0 b e a c onstant. If α (0 n ) 6∈ i.o.-EXP /n − c log n then for ev ery A ∈ EXP if A ∈ P /α (0 n d ) for some d > 0 then A ∈ BPP /O ( n log n ) . This theorem sa ys that Kolmogoro v random advice of p olynomial size can b e replaced b y almost linear size adv ice and true r an d omness. W e come short of proving a con v erse of the ab o v e corollary in t wo resp ects. First, the advice is supp osed to mo del the initial segmen t of th e c haracteristic sequence of R K whic h the truth-table can access. Ho we v er, b y pro vidin g only p olynomial size advice w e restrict the hypothetical truth - table red u ction to query strings of only logarithmic length. Sec ond, the rand omness that we require from the initial segment is m uc h stronger than what one can pro ve and what is in fact tru e for the in itial segmen t of the c haracteristic s equence of R K . One can d eal with the firs t issue as is sh o wn b y Theorem 9 but we do n ot kn ow ho w to deal with the second one. Pr o of. Let M b e a p olynomial time T ur ing mac hine and A ∈ EXP b e a set suc h that A ( x ) = M ( x, α ( | x | d )). W e claim that for all n large enough there is a non-negligible fraction of advice strings r of size n d that could b e used in place of α ( n d ) more precisely: Pr r ∈{ 0 , 1 } n d [ ∀ x, x ∈ A ⇐ ⇒ M ( x, r ) = 1] > 1 n cd . T o prov e the claim consider th e set G = { r ∈ { 0 , 1 } n d ; ∀ x ∈ { 0 , 1 } n , x ∈ A ⇐ ⇒ M ( x, r ) = 1 } . Clearly , G ∈ EXP and α (0 n d ) ∈ G . If | G = n d | ≤ 2 n d /n cd then α (0 n d ) can b e compu ted in exp onential time fr om its index in the set G = n d of length n d − cd log n . Since α (0 n d ) 6∈ i.o.-EXP /n d − cd log n this cannot happ en infinitely often. 6 No w w e present an algorithm that on inp ut x samples from G using only O ( n log n ) bits of advice (in fact O (log n ) en tries from the tr u th table of A ) and outputs A ( x ) with high probabilit y . Consider the follo wing algorithm: 1. Giv en an input x of length n , and an advice string x 1 , A ( x 1 ) , ..., x k , A ( x k ), 2. sample at most 2 n cd strings of length n d unt il the first s tr ing r is found suc h that M ( x i , r ) = A ( x i ) for all i ∈ { 1 , . . . , k } . 3. If w e fi nd r consistent w ith the advice then output M ( x, r ) otherwise output 0. F or all n large enough the probabilit y that the second step d o es not find r compatible with the advice is upp er-b ound ed by the pr obabilit y that w e do n ot sample an y string from G whic h is at most (1 − 1 n cd ) 2 n cd < e − 2 < 1 / 6. It suffices to sho w that w e can find an advice sequence suc h th at for at least 5 / 6- fraction of the r ’s compatible with the advice M ( x, r ) = A ( x ). F or giv en n , we will find the advice b y prnn ing iterativ ely the s et of b ad random strings B = { 0 , 1 } n d \ G . Let i = 0 , 1 , . . . , 2 cd log 6 / 5 n . Set B 0 = B . If there is a string x ∈ { 0 , 1 } n suc h that for at least 1 / 6 of r ∈ B i , M ( x, r ) 6 = A ( x ), then set x i +1 = x and B i +1 = B i ∩ { r ∈ { 0 , 1 } n d | M ( x i +1 , r ) = A ( x i +1 ) } . If ther e is n o such s tring x then stop and the x i ’s obtained so f ar will f orm our advice. Not ice, if we stop for some i < 2 cd log 6 / 5 n then for all x ∈ { 0 , 1 } n , Pr r ∈B i [ M ( x, r ) 6 = A ( x )] < 1 / 6. Hence, any r found by the algorithm to b e compatible with the advice will giv e the correct answ er f or a giv en inpu t with probabilit y at least 5 / 6. On the other hand, if we stop building the advice at i = 2 cd log 6 / 5 n then | B 3 cd log 6 / 5 n | ≤ 2 n d · (5 / 6) 2 cd log 6 / 5 n ≤ | G = n d | /n cd . Hence, any string r found by the algorithm to b e compatible with the advice x 1 , A ( x 1 ) , ..., x i , A ( x i ) will come from G with goo d pr obabilit y , i.e., w ith probabilit y > 5 / 6 for n large enough. ✷ The follo w ing th eorem can b e established by a similar argument. It again relies on the fact that a p olynomially large fraction of all advice str ings of length 2 n d m u st work w ell as an ad v ice. By a prun in g pro cedu r e similar to the pro of of Theorem 8 we can a vo id bad advice. In the BPP algorithm one d o es not hav e to explicitly guess the whole advice but only the part relev an t to the pruning advice and to the current input. Theorem 9 L et α : { 0 } ∗ → { 0 , 1 } ∗ b e a length p r eserving function and c > 0 b e a c onstant. If α (0 n ) 6∈ i.o.-EXP /n − c log log n then f or every A ∈ EXP if A ∈ P /α (0 2 n d ) for some d > 0 then A ∈ BPP /O ( n log n ) . W e sho w next that if the set A h as some su itable pr op erties we can disp ense with the linear advice all together and r ep lace it with only rand om bits. Th us for example if SA T ∈ P /α (0 n ) f or s ome computationally hard advice α (0 n ) then SA T ∈ BPP. Theorem 10 L et α : { 0 } ∗ → { 0 , 1 } ∗ b e a length pr eserving function and c > 0 b e a c onstant such that α (0 n ) 6∈ i.o.-EXP /n − c log n . L et A b e p add able and p olynomial-time 7 many-one-c omplete for a class C ∈ { NP , P # P , PSP A C E , EXP } . If A ∈ P /α (0 n d ) for some d > 0 then A ∈ BPP (and henc e C ⊆ BPP ). T o prov e the theorem we will need the notion of instance chec k ers. W e u se the definition of T revisan and V adh an [TV02]. Definition 11 An instance c h ec k er C for a b o ole an function f is a p olyno mial-time pr ob abilistic or acle machine whose output is in { 0 , 1 , fail } suc h that • for al l inputs x , Pr [ C f ( x ) = f ( x )] = 1 , and • for al l inputs x , and al l or acles f ′ , Pr[ C f ′ ( x ) 6∈ { f ( x ) , fail } ] ≤ 1 / 4 . It is immed iate that by linearly many rep etitions and taking the ma jorit y answer one can reduce the error of an instance c hec ker to 2 − n . V adh an and T revisan also state the follo wing claim: Theorem 12 ([BFL91],[L F KN92, Sha92]) E very pr oblem that is c ompl ete for EXP , PSP A CE or P # P has an instanc e che cker. Mor e over, ther e ar e EXP -c omplete pr oblems, PSP A CE -c omplete pr oblems, and P # P -c omplete pr oblems f or which the i nstanc e che cker C only makes or acle queries of length e xactly ℓ ( n ) on inputs of length n for some p oly- nomial ℓ ( n ) . Ho we v er, it is not kno wn wh ether NP has instance c heck ers. Pr o of of The or em 10. T o prov e the claim for P # P -, PSP ACE- and EXP-complete problems we use the instance c h ec k ers. W e u se the same notation as in the pr o of of Theorem 8, i.e., M is a T ur ing mac hine su ch that A ( x ) = M ( x, α ( | x | d )) and the set of go o d ad v ice is G = { r ∈ { 0 , 1 } n d ; ∀ x ∈ { 0 , 1 } n , x ∈ A ⇐ ⇒ M ( x, r ) = 1 } . W e kno w from the previous p r o of that | G = n d | ≥ 2 n d /n cd b ecause α (0 n ) 6∈ i.o.-EXP /n − c log n . Let C b e the instance chec k er for A whic h on inp ut of length n asks oracle queries of length only ℓ ( n ) and makes error on a wrong oracle at most 2 − n . The follo wing algorithm is a b ounded err or p olynomial time algorithm for A : 1. On inpu t x of length n , rep eat 2 n cd times (a) Pic k a random string r of length ( ℓ ( n )) d . (b) Run the instance c hec ker C on in put x and ans w er eac h of h is oracle queries y by M ( y , r ). (c) If C outpu ts fail con tinue with another iteration otherwise outp u t the output of C . 2. Outpu t 0. 8 Clearly , if we sample r ∈ G then the instance c heck er will provi de a correct answer and we stop. The algorithm can pro d uce a wrong answ er either if the instance chec k er alw a ys fails (so we nev er sample r ∈ G dur ing the iterations) or if th e instance chec k er giv es a wrong answ er . P r obabilit y of not samplin g go o d r is at most 1 / 6. T h e probab ility of getting a wrong answe r from the instance c h ec k er in any of the iterations is at most 2 n cd / 2 n . Thus the algorithm pro vid es th e correct ans w er w ith probabilit y at least 2 / 3. T o pr o v e the claim for NP-complete languages we show it f or the canonical examp le of SA T. The follo wing algorithm solv es SA T correctly with probabilit y at least 5 / 6: 1. On inpu t φ of length n , r ep eat 2 n cd times (a) Pic k a random string r of length n d . (b) If M ( φ, r ) = 1 then use the self-reducibilit y of S A T to find a pr esumably satisfying assignment a of φ while askin g queries ψ of s ize n and answe ring them according to M ( ψ , r ). I f th e assignmen t a indeed satisfies φ then output 1 otherwise con tin ue with another iteration. 2. Outpu t 0. Clearly , if φ is satisfiable w e will answe r 1 with probabilit y at least 5 / 6. If φ is not satisfiable w e will alw a ys answ er 0. ✷ 5 Op en Problems W e ha v e sh o wn that the set R K cannot b e compr essed using a computable algorithm and u s ed this fact to reduce BPP non-adaptivel y to R K . W e conjecture that ev ery computable set that non-adaptivel y reduces in p olynomial-time to R K sits in BPP and ha ve sho wn a n u mb er of p artial r esults in th at dir ections. The classification of languages that p olynomial-time adaptiv ely reduce to R K also remains op en. Can w e c haracterize PSP ACE this wa y? References [ABK + 02] E. Allender, H. Buhrman, M. 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