L. Adleman. Two theorems on random polynomial time. FOCS 78.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....some finite level. Thus, if the polynomial hierarchy is strict, then each of these statements is false. Parts 1) 2) 3) and 5) of this corollary are immediate from Theorems 2.3, 3.1, 3.2, 3.3, and 3.4, and Propositions 2.5 and 2.6. Part 4) is an implication of part 3) using methods of Adleman [37] and Karp and Lipton [38] Part 6) is proved from part 4) as follows. The decision version of circuit minimization has instances where is a circuit and , and the question is whether there is a circuit equivalent to with . This problem has a form: there exists a such that and, for all inputs ....

....words over the alphabet . In particular, we are interested in bit codewords (where ) that contain exactly one occurrence of the symbol ; denote this set of words containing One Two by For those readers familiar with complexity theory, the argument follows. Using that 6 MBE D, it follows as in [37] that if MBE D2 P 6 P poly. Hemaspaandra et al. 39, Theorem 11] have observed that a result of Karp and Lipton [38] relativizes; in particular, if 6 P poly then 6 =6 for all k 3. As will become clear shortly, the symbol serves as a synchronization symbol ; for example, the DIFs we ....

L. Adleman, "Two theorems on random polynomial time," in Proc. 19th IEEE Symp. Foundations of Computer Science, 1978, pp. 75--83.

....Turing machines that take polynomial length advice, or alternatively by polynomial sized circuits. If A is not weakly coherent then A is called strongly incoherent. Recall that BPP=poly = P=poly, because randomness can be incorporated into nonuniform advice using standard techniques [2]. Hence, there is no need to consider weak probabilistic examiners, because, for a fixed oracle, they are equivalent to weak deterministic examiners. In particular, for a fixed oracle, every BPP examiner is a P=poly examiner, and so every coherent set is weakly coherent. A tally set is a subset ....

L. Adleman, Two Theorems on Random Polynomial Time, in Proc. of the 19th Symposium on Foundations of Computer Science (1978), IEEE, 75--83.

....so the algorithm correctly outputs 1. If F has no solutions then the algorithm cannot find one, so the algorithm correctly outputs 0. ii. If UniqueSAT and UnSAT are p separable, then obviously P = UP. iii. If UniqueSAT and UnSAT are p separable, then R = NP [56] iv. If R = NP then NP P=poly [1], so the polynomial time hierarchy collapses [61] Corollary 10. All NP hard sets (under tt reductions) are p superterse relative to almost all oracles B. Proof: The preceding results relativize. R relative to almost all oracles [20] It is also known that P 6= UP for ....

L. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science, pages 75-- 83, 1978.

....; ffl 2 : Z FPCP c;s [r; f; l] 1 Gamma (1 ffl 1 ) Delta (1 Gamma c) c Gamma ffl 1 , s = 1 ffl 2 ) Delta s and r = fl maxf Gamma log 2 (ffl 1 (1 Gamma c) log 2 (l) Gamma log 2 (ffl 2 s)g. Proof: The proof is reminiscent of Adleman s proof that RP P=poly [Ad]. Suppose we are given a pcp system for which we want to reduce the randomness complexity. The idea is that it suffices to choose the random pad for the verifier out of a relatively small set of possibilities (instead than from all possibilities) Furthermore, most small sets (i.e. sets of size ....

L. Adleman. Two theorems on random polynomial time. Proceedings of the 19th Symposium on Foundations of Computer Science, IEEE, 1978, pp. 75--83.

....= I #Q) 1 is finite. Furthermore Q # v is equal to the limit as # tends to 1 of (I #Q) 1 v. The two functions above map matrices to matrices, and by specifying a particular entry [i, j] of the output, we may view them as mapping matrices to real numbers (and in our case the range will be [0, 1]) For any real valued function f on some domain D, we define the threshold language L f associated to f to be the set of inputs x D such that f(x) 1 2. We will consider three versions of the membership problem for this language, the exact, approximate and onesided versions. In each we ....

....method to show that if we omit the requirement that the generator be computable, and take, for each S,2 to be a random function mapping c 1 S bits, then the result would fool all probabilistic computations running in space S. This fact can be viewed as a rough analog to Adleman s result ([1]) that BPP can be computed by non uniform polynomial circuits. Of course, we do not now know how to construct e#ciently computable generators that achieve the optimal simulation (otherwise, this would be quite a different paper ) The crux of the problem, then, is to find a PRGE whose seed length ....

L. Adleman. Two theorems on random polynomial time. In 19th IEEE Symposium on Foundations of Computer Science, pages 75--83, 1978.

....direct construction of an O(log n) pebble, n state deterministic JAG for directed st connectivity. More interestingly, they also prove a lower bound of Omega# log n= log log n) on the space required by JAGs solving this problem, nearly matching the upper bound. Standard techniques (Adleman [1], Aleliunas et al. 2] extend this result to any randomized JAG whose time bound is at most exponential in its space bound. Berman and Simon [11] extend this space lower bound to probabilistic JAGs with even larger time bounds, namely exponential in log n. In this paper we use a variant of ....

L. M. Adleman. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science, pages 75--83, Ann Arbor, MI, Oct. 1978. IEEE.

....1 Gamma fl with an O(log(1=fl) slowdown in efficiency. As a consequence, in a nonuniform model of computation (e.g. circuits) we can set fl = 1=jXj and then fix the coin tosses of A to obtain a deterministic procedure solving the same problem with only a O(log jXj) slowdown in efficiency (as in [Adl78] We are now ready to define codes that are nice for our purpose. These codes are parameterizd by two parameters: an integer k that counts (roughly) the length of the message to be encoded; and a positive real number that is related to the fraction of error from which we expect to be able ....

Leonard Adleman. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science, pages 75--83, Ann Arbor, Michigan, 16--18 October 1978. IEEE.

.... a polynomially sized advice that depends only on the input size 7 : Corollary 17 For every finite field F q the following holds: 7 Since our reduction achieves exponentially small error probability, hardness under non uniform reductions also follows from general results about derandomization [1]. However, the ad hoc derandomization method we just described is more efficient and intuitive. ffl For any ae 1=2, and fl 1 GAPRNC (ae) fl;q is hard for NP under non uniform deterministic polynomial reductions. ffl For every real fl 1, GAPDIST fl;q is hard for NP under non uniform ....

L. Adleman, "Two Theorems on Random Polynomial Time", in Proc. 19th Symposium on Foundations of Computer Science 1978, pp. 75--83.

....# (0, 1 3) we have: # # (#) # C#(#)log 1 # , where C is a constant independent of # and #. 6 Proof. Let P (U 1 , U n ) be a polynomial that 1 3 approximates #. Defining Q 0 (U 1 , U n ) 1 5 (3P (U 1 , U n ) 1) wehave Q 0 (u) # [0, 2 5 ]for u # # 1 (0) and Q 1 (u) # [ 3 5 , 1] for u # # 1 (1) Let w(z) denote the univariate polynomial w(z) 3z 2 2z 3 . This is the unique cubic polynomial having no constant or linear term that satisfies w(1 z) 1 w(z) It is routine to show that for any z # [0, 2 5 ]wehave w(z) # [0,z 1.1 )andw(1 z) # (1 ....

....no constant or linear term that satisfies w(1 z) 1 w(z) It is routine to show that for any z # [0, 2 5 ]wehave w(z) # [0,z 1.1 )andw(1 z) # (1 z 1. 1 , 1] For i # 0, define the polynomial Q i 1 = w(Q i ) Then it follows by induction that Q i (u 1 , u n ) # [0, 1] for all (u 1 , u n ) # B n and also Q i # i approximates # with # i = 2 5) 1.1 i , and that deg(Q i ) 3 i deg(P ) We can then choose m of size O(log log # 1 ) so that #m # # , and deg(Q i ) # C deg(P)log 1 # , for some absolute constant C . ## Formu l ae are defined in the ....

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L. Adleman, `Two theorems on random polynomial time', Proc. 19th IEEE Symposium on Foundations of Computer Science, 1978, 75--83.

....Min Max Theorem that shows that each player has a near optimal mixed strategy that plays uniformly from a multiset of size logarithmic in the number of pure strategies available to the opponent. The proof is a surprisingly simple probabilistic argument similar to circuit derandomization techniques [1, 15]. However, the central nature of games in theory suggests that this simple result may have far reaching consequences. This result was obtained independently by Althofer [3] Strengthening the connection between randomized and distributional complexities. This connection was first established by ....

....have linear size encodings. Thus, the stronger direction holds (approximately) for complexity measures that allow program size to grow linearly with input size. This includes most measures of circuit complexity. Note that this application is similar to known circuit derandomization techniques [1, 15]. However, the theorem has many other applications. For instance, by applying it to the program input game for the input player we show that there are hard distributions that can be generated by small circuits. Anti checkers and circuits that generate hard random instances. We give applications ....

Leonard M. Adleman. Two theorems on random polynomial time. In Proc. of the 19th IEEE Annual Symp. on Foundation of Computer Science, pages 75--83, 1978.

....of P as a sequence of random numbers ae 1 ; ae R in the interval [1; qR] where ae i is the random number chosen for the vertex with id (j Gamma 1)R i. Since the error probability is less than the reciprocal of the number of graphs in I j , it follows by a standard argument (e.g. [1]) that there must be a particular choice ae 1 ; ae R such that P is correct on all of I j when this particular random choice is made. We can then obtain a deterministic algorithm A that works on I j . This algorithm first chooses the random number ae i Gamma(j Gamma1)R at the vertex ....

.... 1) d 2 coloring and then show how to reduce the number of colors to two using additional steps. 14 Consider first the case of a d regular graph where d is odd and d 3. For a vertex v let id(v) be the id number assigned to v. We denote the color of a vertex v by a vector C v = hC v [0] C v [1]; C v [d 1]i where each component is in f1; d 1g. The following procedure is used at vertex v: 1. Get id(w) for all neighbors w of v. Sort the set of id s of neighbors including id(v) Let r v (w) denote the rank of id(w) among the neighborhood of v (where the neighborhood of ....

L. Adleman, Two theorems on random polynomial time, in Proc. 19th IEEE Symposium on Foundations of Computer Science, 1978, pp. 75--83.

....such that (C N; x) i = y i for at least ( 1 2 )N 0 of the positions i, 1 6 i 6 N 0 , there exists an index j such that for any position i, 1 6 i 6 N , M y j (i) outputs x i with high probability. Each machine M j runs in time polynomial in log N and 1= By Adleman s argument [Adl78] we can transform the randomized procedures in the second bullet of Theorem 3.15 into circuits. This yields the following corollary. Corollary 3.16 There exists a constant c such that for any N 2 N, x 2 f0; 1g N and y 2 f0; 1g N 0 satisfying (C N; x) i = y i for at least ( 1 2 )N ....

L. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, pages 75-83. IEEE, 1978.

....increased to 1 with an O(log(1= slowdown in eciency. As a consequence, in a nonuniform model of computation (e.g. circuits) we can set = 1=jX j and then x the coin tosses of A to obtain a deterministic procedure solving the same problem with only a O(log jX j) slowdown in eciency (as in [Adl78] We are now ready to de ne codes that are nice for our purpose. These codes are parameterizd by two parameters: an integer k that counts (roughly) the length of the message to be encoded; and a positive real number that is related to the fraction of error from which we expect to be able to ....

Leonard Adleman. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science, pages 75-83, Ann Arbor, Michigan, 16-18 October 1978. IEEE.

....characterisations hold: i) P=poly is precisely the class of languages recognized by (possibly nonuniform) families of polynomial size circuits. ii) NP=poly is precisely the class of languages generated by (possibly on uniform) families of polynomial size circuits. A classic result by Adleman [2] shows that P=poly contains the class RP consisting of the languages recognized by polynomial time bounded probabilistic Turing machines with one sided error. More generally it can be even shown that BPP P=poly, where BPP is the class of languages accepted by polynomial time bounded ....

L. Adleman. Two theorems on random polynomial time, in: Proc. 19th Ann. IEEE Symp. Foundations of Computer Science (FOCS'78), 1978, pp. 75-83.

....In fact, it is reasonable to conjecture that these classes are not contained in P . It is interesting to note that, analogous to the case of randomized polynomial time computation, all of the above mentioned classes can be simulated 3 non Gamma uniformly in Logspace (using the techniques of [Adl78]) i.e. we have even for the largest class, BP L, that BP L ae L=poly. In this paper we thus only discuss uniform simulations and classes. In both definitions of access to randomness the obvious relations hold: ZPL ae RL ae BPL and ZP L ae R L ae BP L, and also in both cases we have that ZPL ....

L. Adleman. Two theorems on random polynomial time. In 19 th Annual Symposium on Foundations of Computer Science, Ann Arbor, Michigan, pages 75--83, 1978.

....in S(f) need to be considered in our search for a good point in Omega Gamma Moreover, if it easy to recognize whether a point x in S(f) is good for a particular input, then it suffices to search any subset of S(f) which is guaranteed to contain a good point for each possible input. Adleman [2] shows that for any distribution f supporting an algorithm in RP, there exists a space S S(f) of polynomial size that contains a good point for every possible input. The proof of this fact is not constructive, and therefore cannot be used for de randomizing algorithms. A common technique for ....

L. Adleman, "Two theorems on random polynomial time," in: Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Angeles, 1978, pp. 75--83.

....high reliability can be built from (so called crummy) relays with arbitrarily poor reliability. The amplification method was next used by several researchers to show that particular Boolean functions have small Boolean circuits and formulas. Bennett and Gill [3] extending a result of Adleman [1], used it to show that every language in the complexity class BPP has polynomial size circuits. Ajtai and Ben Or [2] used the amplification method to show that probabilistic constant depth circuits can be simulated by deterministic constant depth circuits with only a polynomial increase in size. ....

....paper (and of [6] appeared in [5] 2. Preliminaries. Following Moore and Shannon [14] and Boppana [4] we introduce the following two definitions Definition 2. 1 (amplification functions) Given a Boolean function f : 0, 1 n # 0, 1 , we define its multivariate amplification function f :[0,1] n # [0, 1] as follows: f(p 1 ,p 2 , p n ) Pr[f(x 1 , x 2 , x n) 1 ] where x 1 , x 2 , x n are independent random variables and x i assumes the value 1 with probability p i and the value 0 with probability 1 p i . We define the univariate amplification function of f to be ....

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L. Adleman, Two theorems on random polynomial time, in Proc. 19th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1978, pp. 75--83.

....is not a satisfying assignment for OE x , then the value (x; y) is output. Otherwise the process is repeated for another random u . Repeating the process a polynomial number of times, the probability that the algorithm succeeds is exponentially close to 1. By the well known technique of Adleman [1], the above non uniform probabilistic algorithm can be shown to have polynomial size circuits. Namely, since for a given C , a random chosen u succeeds with high probability, there must be a small set of values for u , such that for all C of a given size, at least one of the u values succeeds. In ....

L. Adleman, Two theorems on random polynomial time, in Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science, 1978, pp. 75--83.

....: r t ) chosen uniformly at random from K t . For n independent trials r 1 ; r n we obtain Pr[p F (r 1 ) Delta Delta Delta = pF (r n ) 0] 1=c n ; 24 if val B (F ) 1. As there are at most c n tensor formulas of length n a well known counting argument (see, e.g. [1]) shows that there are assignments r 1 ; r n such that for every tensor formula F of length n, val B (F ) 0 if and only if pF (r 1 ) Delta Delta Delta = pF (r n ) 0; where pF (z) 2 K[z 1 ; z t ] is the polynomial assigned to F . The key observation is that pF (r) ....

L. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th Foundations of Computer Science, pages 75--83. IEEE Computer Society Press, 1978.

.... easily be modified to compute the 2 n=2 functions h(b 1 ; b n=2 ) for b 1 ; b n=2 2 IB by simply taking the AND gate that computes g(b 1 ; b n=2 ) x n=2 ; x n ) and giving it extra inputs from x 1 [b 1 ] x n=2 [b n=2 ] where x i [0] denotes x i , x i [1] denotes x i , x i [0] denotes x i , and x i [1] denotes x i ) The resulting circuit still has depth 2 and size 2 n=2 1 . Finally, we note that f(x 1 ; x n ) h(0; 0 z n=2 ) x 1 ; x n ) Delta Delta Delta h(1; 1 z n=2 ) x 1 ; x n ....

.... h(b 1 ; b n=2 ) for b 1 ; b n=2 2 IB by simply taking the AND gate that computes g(b 1 ; b n=2 ) x n=2 ; x n ) and giving it extra inputs from x 1 [b 1 ] x n=2 [b n=2 ] where x i [0] denotes x i , x i [1] denotes x i , x i [0] denotes x i , and x i [1] denotes x i ) The resulting circuit still has depth 2 and size 2 n=2 1 . Finally, we note that f(x 1 ; x n ) h(0; 0 z n=2 ) x 1 ; x n ) Delta Delta Delta h(1; 1 z n=2 ) x 1 ; x n ) and f can therefore be computed by a ....

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L. Adleman. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science, pages 75--83. IEEE Computer Society Press, 1978.

....and terminology used in this introduction. This refined the downward separation result E ae 6= ESPACE = P ae 6= PSPACE of Book [Boo74] and also led immediately to the upward separation result P ae 6= BPP = E ae 6= ESPACE (1. 1) of Hartmanis and Yesha [HY84] Work of Gill [Gil77] Adleman [Adl78], and Bennett and Gill [BG81] had already established that BPP is contained in P Poly PSPACE. It is reasonable to conjecture that BPP is in fact a proper subset of P Poly PSPACE, and hence that the P ae 6= BPP hypothesis might yield a stronger conclusion than the separation of E from ....

L. Adleman, Two theorems on random polynomial time, Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, 1978, pp. 75--83.

.... 0 and fl(n) 2 log (1 Gammaffl) n , GapDist fl;q is hard for NP under non uniform deterministic quasi polynomial reductions. 8 Since our reduction achieves exponentially small error probability, hardness under non uniform reductions also follows from general results about derandomization [1]. However, the ad hoc derandomization method we just described is more efficient and intuitive. We notice also that a uniform deterministic construction satisfying the properties of Lemma 8 would immediately give a proper NP hardness result (i.e. hardness under deterministic Karp reductions) for ....

L. Adleman, "Two Theorems on Random Polynomial Time", in Proc. 19th Symposium on Foundations of Computer Science 1978, pp. 75--83.

....independent trials of this test in parallel and takes the OR of the trials. If x is a multiple of p j , all of the tests reject, whereas if x is not a multiple of p j , then with probability at least 1 Gamma 1=2 (n) at least one of the tests will accept. Now, as in the standard argument of [Adl78], there must be at least one sequence of probabilistic inputs for the circuit having the property that, for all n bit inputs x, the OR of the n 4 tests is equal to :mult(x; j) This can be seen to be an AC 0 T reduction from mult to Primes, since by Proposition 3 (parts 2 and 3) x py can ....

L. Adleman, Two theorems on random polynomial time, in "Proc. 19th IEEE Symposium on Foundations of Computer Science", 1978, 75--83.

....set in NP, then P = NP. Next, we strengthen hypothesis A to obtain a stronger collapse of the polynomial hierarchy than the collapse known by Toda s theorem [Tod91] That is, recall that RP = NP implies a collapse of the polynomial hierarchy to ZPP NP because RP has polynomial size circuits [Adl78, KL80, KW94]. In the following theorem we obtain a collapse of the polynomial hierarchy to P NP . A standard left cut is a special kind of a p selective set and is defined in the next section. Theorem. If there exists a standard left cut L in NP such that SAT P tt L, then the polynomial hierarchy ....

L. Adleman. Two theorems on random polynomial time. In Proceedings of 19th IEEE Symposium on Foundations of Computer Science, pages 75--83, 1978.

....construction of an O(log n) pebble, n O(1) state deterministic JAG for directed st connectivity. More interestingly, they also prove a lower bound of Omega Gamma 29 2 n= log log n) on the space required by JAGs solving this problem, nearly matching the upper bound. Standard techniques (Adleman [1], Aleliunas et al. 2] extend this result to any randomized JAG whose time bound is at most exponential in its space bound. Berman and Simon [11] extend this space lower bound to probabilistic JAGs with even larger time bounds, namely exponential in (log n) O(1) In this paper we use variants ....

L. M. Adleman. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science, pages 75--83, Ann Arbor, MI, Oct. 1978. IEEE.

....T on input 1 n . Thus, f in fact is computable in FP #P 1 [1] 1 and so, by our supposition, in polynomial time. Since L is in MOD k P=poly with polynomial time computable advice, it follows that L 2 MOD k P. Hence, PH MOD k P. In order to prove the second part, notice that BPP is in P=poly [Adl78] with an advice computable in (the function analog of) PH [Sip83,Lau83] and that PH P #P[1] by Toda s Theorem [Tod91] An argument similar to the above shows that P = BPP. Now we show that the conditions of Theorem 3.4 in fact are equivalent to the two conditions stated in either part of ....

L. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, pages 75--83, 1978.

....by Cole and Vishkin [7] one can achieve a running time of O(lg n= lg lg n) using an optimal number of processors. This time was shown to be the best possible with any polynomial number of processors by Beame and Hastad [3] even if a randomized algorithm is sought (using a theorem by Adleman [1]) Interestingly, this lower bound does not hold for the approximation version of the parallel prefix sums problem, however, and, as we show in this paper, there are several applications where one needs only an approximate prefix sums sequence. Specifically, given some ffl 0, the ffl approximate ....

....we review below. 1.2 Related previous work. As mentioned above, the parallel prefix problem can be solved exactly in O(lg n= lg lg n) time using an optimal number of processors [7] and this is the fastest time possible using a polynomial number of processors [3] even in a randomized setting [1]. Perhaps because of this lower bound result, research on near constant time parallel algorithms abandoned using the prefix sums problem Parallel Prefix Sums Approximation 3 as a subroutine, in spite of its widely recognized usefulness in polylogarithmic parallel algorithms. Several problems ....

Leonard M. Adleman. Two theorems on random polynomial time. In Proc. of the 19th IEEE Annual Symp. on Foundation of Computer Science, pages 75-- 83, 1978.

....) so the algorithm correctly outputs 1. If F has no solutions then the algorithm cannot find one, so the algorithm correctly outputs 0. ii. If UniqueSAT and UnSAT are p separable, then obviously P = UP. iii. If UniqueSAT and UnSAT are p separable, then R = NP [56] iv. If R = NP then NP P=poly [1], so the polynomial time hierarchy collapses [61] Corollary 10. All NP B hard sets (under p tt reductions) are p B superterse relative to almost all oracles B. Proof: The preceding results relativize. R B 6= NP B relative to almost all oracles [20] It is also known that P B 6= ....

L. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science, pages 75-- 83, 1978.

....an oracle gate (with an arbitrary number of inputs) outputs 1 if and only if the 0 1 string at its inputs is a member of B. We call such circuits oracle circuits in the following. This relativized circuit model has been introduced by Wilson [43] Using an argument given in [8] see also [1]) it can be seen that A BPP T B implies A P=poly T B. Definition 2.7 Let C be any of the reducibilities defined above. If a set A has the property that for every L 2 NP it holds L C A, then A is called C hard for NP. If additionally A 2 NP holds, then A is called C complete ....

L. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th IEEE Symposium on the Foundations of Computer Science, 75--83. IEEE Computer Society Press, 1978.

....that our construction will exploit techniques from [18] and [3] but in a different manner. In general, not every result for RP and BPP translates easily (or at all) to a result on the class IP. Notable examples are results such as BPP equals almost P [13] and BPP is contained in non uniform P [1]. The IP counterpart of the first was open for several years and finally proved by Nisan and Wigderson [39] while the IP analogue of the second (i.e. IP is contained in non uniform NP) is not believed to be true. 1.5 A Result About Sampling Our results use properties of low independence ....

L. Adleman. Two Theorems on Random Polynomial Time. Proceedings of the 19th Annual IEEE Symposium on the Foundations of Computer Science, IEEE (1978).

....Turing machines that take polynomial length advice, or alternatively by polynomial sized circuits. If A is not weakly coherent then A is called strongly incoherent. Recall that BPP=poly = P=poly, because randomness can be incorporated into nonuniform advice using standard techniques [2]. Hence, there is no need to consider weak probabilistic examiners, because, for a fixed oracle, they are equivalent to weak deterministic examiners. In particular, for a fixed oracle, every BPP examiner is a P=poly examiner, and so every coherent set is weakly coherent. A tally set is a subset of ....

L. Adleman, Two Theorems on Random Polynomial Time, in Proc. of the 19th Symposium on Foundations of Computer Science (1978), IEEE, 75--83.

....might we use a random number generator to make this equiprobable selection 1. Get someone else to solve the problem: Perhaps the random number generator takes an integer d, and returns a uniformly chosen integer between 0 and d 0 1. 2. Generate random floating point numbers: Divide the interval [0,1] into d equal subintervals, generate a random number between 0 and 1, and choose a vertex accordingly. Because of finite precision, it is not possible to do this exactly, so the resulting assignments are not quite equiprobable. It might be possible to bound the deviations from equiprobable ....

....PSIZE. Lecture 11 BPP PSIZE; Pseudorandom Generators February 6, 1991 Notes: Rakesh Kumar Sinha 11.1. Relating Circuits and Probabilistic Complexity Classes Theorem 10.1 showed that the class P is contained in PSIZE the class of languages accepted by polynomial size circuits. Adleman [1] extended this result by showing that the class RP , which is a superset of P, is also contained in PSIZE . Since we have natural examples of problems in RP not known to be in P (for example, the set of composite integers [38] Adleman s result was very surprising: it says such problems can be ....

L. Adleman. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science, pages 75--83, Ann Arbor, MI, Oct. 1978. IEEE.

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L. Adleman. Two theorems on random polynomial time. FOCS 78.

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Leonard M. Adleman. Two theorems on random polynomial time. In proceedings of FOCS '78, pages 75--83, 1978.

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L. Adleman. Two theorems on random polynomial time. FOCS 78.

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L. Adleman. Two theorems on random polynomial time. In Proceedings of the Nineteenth Annual IEEE Symposium on Foundations of Computer Science, pages 75-83, 1978.

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L. Adleman. Two theorems on random polynomial time. Proceedings of the Nineteenth Annual Symposium on the Foundations of Computer Science, IEEE, 1978.

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L. Adleman, Two theorems on random polynomial time,in"Proc. 19th IEEE Symposium on Foundations of Computer Science", 1978, 75--83.

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L. Adleman, Two theorems on random polynomial time, in Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science, 1978, pp. 75--83.

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L. Adleman. Two theorems on random polynomial time. Proceedings of the Nineteenth Annual Symposium on the Foundations of Computer Science, IEEE, 1978.

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L. Adleman. Two theorems on random polynomial time. In 10th FOCS, pages 75--83, 1978.

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L. Adleman, "Two theorems on random polynomial time," in: Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Angeles, 1978, pp. 75--83.

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L. Adleman. Two theorems on random polynomial time. Proceedings of the Nineteenth Annual Symposium on the Foundations of Computer Science, IEEE, 1978.

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Leonard Adleman. Two theorems on random polynomial time. In Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, pages 75-- 83, 1978.

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L. Adleman. Two theorems on random polynomial time. Proceedings of the Nineteenth Annual Symposium on the Foundations of Computer Science, IEEE, 1978.

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Adleman, L. (1978). Two theorems on random polynomial time. Proc. of 19th Symposium on Foundations of Computer Science (FOCS), 75--83. 27

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L. Adleman, \Two Theorems on Random Polynomial Time", in Proc. 19th Symposium on Foundations of Computer Science 1978, pp. 75-83.

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L. Adleman, Two theorems on random polynomial time, 19th Symp. Found. of Comp. Sci. (1978), 75-83.

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L. M. Adleman, "Two theorems on random polynomial time", Proc. 19th Annual Symposium on Foundations of Computer Science, IEEE, New York, 1978, 75--83.

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L. Adleman. Two theorems on random polynomial time. In Proc. 19th Annual IEEE Symposium on Foundations of Computer Science, pages 75--83, 1978.

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