J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675-695, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....#P function as an oracle. Hence, that in terms of complexity, hard functions in #P lie above any problem in the polynomial time hierarchy. In 1994, Beigel, Reingold and Spielman [BRS95] proved that PP is closed under union. This result solved a longstanding open problem in this area, rst posed by Gill in 1977 [Gil77] in the initial paper on probabilistic classes. It implies that PP is also closed under intersection Those interested in further exploring counting classes and the power of counting in complexity theory should consult the papers of Sch oning [Sch90] and Fortnow [For97] 6 Probabilistic ....

....an oracle. Hence, that in terms of complexity, hard functions in #P lie above any problem in the polynomial time hierarchy. In 1994, Beigel, Reingold and Spielman [BRS95] proved that PP is closed under union. This result solved a longstanding open problem in this area, rst posed by Gill in 1977 [Gil77] in the initial paper on probabilistic classes. It implies that PP is also closed under intersection Those interested in further exploring counting classes and the power of counting in complexity theory should consult the papers of Sch oning [Sch90] and Fortnow [For97] 6 Probabilistic ....

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J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675-695, 1977.

....polynomial time case. In order to throw light upon that problem it is meaningful to analyze intermediate concepts like unambiguity or fewness. On the side of classes of sets these concepts have lead to various complexity classes located between P and NP such as UP [Val76] FewP [All86, AR88] or R [Gil77] VPP in Gill s notion) However, the same questions about function classes kept widely unnoticed, function classes between FP and #P [Val79a, Val79b] were not often considered (as an exception: HV95] Trying to close the gap a suitable machine concept will be developed. Under polynomial ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675--695, 1977. 16

....of some complexity classes of sets, already existing in the literature, that we will use in this paper. 22] UP is the class of all sets L such that c L 2 #P. 3, 14] NP is the class of all sets L for which there exists a function f 2 #P such that x 2 L , f(x) 0 for all x 2 . [20, 5] PP is the class of all sets L for which there exist functions f 2 #P and g 2 FP such that x 2 L , f(x) g(x) for all x 2 . 18, 4] SPP is the class of all sets L such that c L 2 #P FP. 16, 21] PH = NP [ NP NP [ NP NP NP [ The following results are well known or easy ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675-695, 1977.

....defined later in the thesis. See the survey article [For97] and references therein for more details. Among counting complexity classes, considerable research has gone into understanding the structure of the class PP (Probabilistic polynomial time) This class was originally defined by J. Gill [Gil77] and independently by J. Simon [Sim75] The class PP is very closely related to the class #P. Indeed, it is easy to show that the closure of PP and #P under polynomial time Turing reductions coincide. PP is computationally a hard class; the class NP is contained in PP. The hardness of PP was ....

....a hard class; the class NP is contained in PP. The hardness of PP was further established by a celebrated result due to S. Toda [Tod91] He showed that the entire polynomial time hierarchy is contained in P PP . PP also enjoys many nice closure properties. It is closed under complementation [Gil77] The question posed by Gill in his seminal work [Gil77] whether PP is closed under intersection (or union) was settled in the affirmative by Beigel et al. in [BRS95] The techniques used in [BRS95] were extended by Fortnow et al. FR96] to show that PP is also closed under polynomial time ....

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J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675--695, 1977.

....Barcelona, Spain 1 Introduction The intractability of the complexity class NP has motivated the study of subclasses that arise when certain restrictions on the definition of NP are imposed. For example, the study of sparse sets in NP [Ma82] the study of the probabilistic classes whithin NP [Gi77], and the study of low sets in NP for the classes in the polynomial time hierarchy [Sc83] have been three main research streams in the area of complexity theory, and have clarified many structural aspects of the class NP. In this paper we study two different ways to restrict the power of NP: We ....

....many queries. Next we define the complexity classes PP, C = P and PhiP that are also defined considering the number of computation paths of a nondeterministic machine, but in this case the number of paths is not necessarily polynomially bounded. These classes were first introduced in [Gi77], Wa86] and [PaZa83] respectively. Definition 2.4: A language L is in the class PP if there is a nondeterministic polynomial time machine M and a function f 2 FP such that for every x 2 Sigma , x 2 L ( accM (x) f(x) PP is called CP in the notation of [Wa86] This notation can be ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journ. Comput. 6 (1977): 675--695.

...., where we view M s nondeterministic choices as outcomes of a fair coin toss. Thus, every input is accepted with a certain probability. We say that the language accepted by M consists of all of its inputs which are accepted with probability greater than one half. The class PP, introduced by Gill [7] in 1977, is defined to consist of all languages accepted by such machines running in polynomial time. It has been observed (see e.g. 19] that PP can also be characterized in another way using Valiant s class #P [22] This class consists of those functions counting the number of accepting paths ....

J. Gill, Computational complexity of probabilistic complexity classes; SIAM Journal on Computing 6 (1977), pp. 675--695.

....N denotes the set of nonnegative integers. Throughout the paper, the base of log is 2. The textbooks [9, 10, 25, 31, 33] can be consulted for the standard notation used in the paper and for basic results in complexity theory. For definitions of probabilistic complexity classes like ZPP, see also [15]. An NP machine M is a polynomial time nondeterministic Turing machine. We assume that each computation path of M on a given input x either accepts, rejects, or outputs . M accepts on input x, if M performs at least one accepting computation, otherwise M rejects x. M strongly accepts (strongly ....

J. Gill, Computational complexity of probabilistic complexity classes, SIAM J. Comput., 6 (1977), pp. 675--695.

...., where we view M s nondeterministic choices as outcomes of a fair coin toss. Thus, every input is accepted with a certain probability. We say that the language accepted by M consists of all of its inputs which are accepted with probability greater than one half. The class PP, introduced by Gill [8] in 1977, is defined to consist of all languages accepted by such machines running in polynomial time. It has been observed (see e.g. 20] that PP can also be characterized in another way using Valiant s class #P [23] This class consists of those functions counting the number of accepting paths ....

J. Gill, Computational complexity of probabilistic complexity classes; SIAM Journal on Computing 6 (1977), pp. 675--695.

....to be familiar with standard complexity theory notions. Our notation here is standard; see e.g. 1, 2, 13] We presuppose familiarity with the classes of the alternating polynomial time hierarchy [17, 5] the alternating logarithmic time hierarchy [5, 16] and the probabilistic classes BPP and PP [6]. We use the following general computation models: An m valued locally defined acceptance type is a set F of functions from f0; m Gamma 1g r into f0; m Gamma 1g for some r 2 f2; 3; 4; g. Every locally definable acceptance scheme F corresponds to a complexity class, ....

J. Gill, Computational complexity of probabilistic complexity classes; SIAM Journal on Computing 6 (1977), pp. 675--695.

....oracle answer in one time step. The oracle answer replaces the query on the oracle tape. Let FCH = def S i0 i # P. The counting hierarchy (of sets) 25] is the hierarchy CH = def PP[PP PP [PP PP PP [ where PP denotes Gill s class of probabilistically polynomial time decidable sets [9]. The class UP [22] consists of all sets whose characteristic function is in # P. The class SPP [8] also known under the name XP [16] consists of all sets whose characteristic function is in Gap P. It is known that Gap P SPP = Gap P, which makes SPP low for all so called gap definable classes; ....

J. Gill, Computational complexity of probabilistic complexity classes; SIAM Journal on Computing 6 (1977) 675--695.

....GapP functions. For this paper we consider the following classes. De nition 2.3 The class PP consists of those languages L such that for some GapP function f and all x in Sigma , ffl If x is in L then f(x) 0. ffl If x is not in L then f(x) 0. The class PP was rst de ned by Gill [Gil77] as probabilistic polynomial time with unbounded error. De nition 2.3 rst given by Fenner, Fortnow and Kurtz [FFK94] makes the class considerably easier to work with. De nition 2.4 The class LWPP consists of those languages L such that for some GapP function f a polynomial time computable ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675695, 1977.

....such as those we listed above, even deserve simpler characterization. We call a class simply gap definable if it is gap definable and A, R in Definition 2.4 depend on gap M (x) only. Using a proposition in [FFK94] the classes PP, C=P, and Mod k P (for k 2) can be redefined as the following: 1. Gil77] Sim75] L 2 PP ( 9g 2 GapP) 8x) x 2 L g(x) 0] 2. Sim75] Wag86] L 2 C=P ( 9g 2 GapP) 8x) x 2 L g(x) 0] 3. CH90] Her90] BG92] L 2 Mod k P ( 9g 2 GapP) 8x) x 2 L g(x) 6j 0 mod k] Every gap definable class is countable, but the converse does not hold. However, ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675--695, 1977.

....languages accepted by NP machines that never have more than one accepting path. In terms of #P functions: Classification 4.2 The class UP consists of those languages L such that for some #P function f and all x in Sigma ffl If x is in L then f(x) 1. ffl If x is not in L then f(x) 0. Gill [Gil77] defined the class PP ( Probabilistic Polynomial Time ) as the set of languages L with probabilistic polynomial time Turing machines M where x is in L if the probability of M(x) accepting is greater than one half. If one considers M as a nondeterministic machine, this means that the accepting ....

....a version of Lemma 4.21 using a different f . We find it simpler to use the function f given by Yao [Yao90] See Beigel and Tarui [BT94] for an in depth look at these modulus amplifying polynomials. 4. 4 Closure Properties of PP In his original paper on probabilistic complexity classes, Gill [Gil77] showed that PP is closed under complement (Corollary 4.13) Gill left open the question as to whether PP is closed under union. Note that by DeMorgan s Law, since PP is closed under complement, closure under union and closure under intersection are equivalent questions. This question remained ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675--695, 1977.

....familiar with the common complexity classes such as P, NP, BPP, PH, PSPACE, EXP and NEXP. Definition2. Val79] #P df = f#M j M is a CMg. FFK91] GapP df = fgap M j M is a CMg where gap M df = #M Gamma #M : We use FP to denote the classes of all poly time computable functions. [Gil77] [Sim75] PP is the class of all languages L such that there exists a CM M and an FP function f such that, for all x, x 2 L ( #M (x) f(x) An equivalent condition is, there exists a CM M such that for all x, x 2 L ( gap M (x) 0. RS92] GKT92] MP, or MidBitP, is the class of all ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675--695, 1977.

....xn ) def x k ) These latter operations are included for technical reasons to obtain smoother classes. UP [35] is the class of all sets whose characteristic functions are in #P. SPP [9] is the class of all sets whose characteristic functions are in GapP. We will also refer to Gill s class PP [11], and Papadimitriou and Zachos s class Phi P [29] and its generalization to the classes MOD k P [4] The counting hierarchy (of sets) 42] is the hierarchy CH = def PP [ PP PP [ PP PP PP [ We consider the following general definition of counting classes beyond #P. For any class C, let ....

J. Gill, Computational complexity of probabilistic complexity classes; SIAM Journal on Computing 6 (1977), pp. 675--695.

....N denotes the set of non negative integers. Throughout the paper, the base of log is 2. The textbooks [9, 10, 25, 31, 33] can be consulted for the standard notations used in the paper and for basic results in complexity theory. For definitions of probabilistic complexity classes like ZPP see also [15]. An NP machine M is a polynomial time nondeterministic Turing machine. We assume that each computation path of M on a given input x either accepts, rejects, or outputs . M accepts on input x, if M performs at least one accepting computation, otherwise M rejects x. M strongly accepts (strongly ....

J. Gill, Computational complexity of probabilistic complexity classes, SIAM Journal on Computing, 6 (1977), pp. 675--695.

.... where the string w is chosen uniformly at random from the set Sigma q(jxj) Observe that for every set B, R p c (B) R co rp m (B) and R co rp m (R co rp m (B) R co rp m (B) RTIME(t(n) denotes the class of sets A accepted by O(t(n) time bounded randomized Turing machines (cf. Gi77] that have zero error probability for inputs not in A (and error probability at most 1=2 for instances in A) RP = RTIME(n O(1) For further notations we refer to [BDG] 3 Reducibilities and oracle machine properties In this section we investigate how the restricted truth table ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675-695, 1977.

....GapP functions. For this paper we consider the following classes. Definition 2.3 The class PP consists of those languages L such that for some GapP function f and all x in Sigma , ffl If x is in L then f(x) 0. ffl If x is not in L then f(x) 0. The class PP was first defined by Gill [Gil77] as probabilistic polynomial time with unbounded error. Definition 2.3 first given by Fenner, Fortnow and Kurtz [FFK94] makes the class considerably easier to work with. Definition 2.4 The class LWPP consists of those languages L such that for some GapP function f , and some ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675--695, 1977.

....and will be CM s unless stated otherwise. We now define some of the usual counting classes. These are not always the original definitions, but can easily be shown to be equivalent to them. See [4] for more details. Definition 2.3 ffl (Valiant [29] #P df = f#M j M is a CMg. ffl (Gill [9]) PP is the class of all languages L such that there exists M and an FP function f such that, for all x, x 2 L ( #M (x) f(x) The function f is the threshold of M . ffl (Wagner [30] C=P is the class of all languages L such that there exists M and an FP function f such that, for all x, x 2 ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6:675--695, 1977.

....that uses only logarithmically many random bits. Hence, ModP and AmpMP are not closed under polynomial time conjunctive reductions unless the counting hierarchy collapses. 1 Introduction The study of counting classes has been a major research stream in structural complexity theory since Gill [Gi77] introduced the probabilistic class PP (for formal definitions see the next section) Simon [Sim75] characterized PP as a counting (more precisely, threshold) class, and Wagner [Wag86] generalized PP to the classes of the counting hierarchy CH by introducing the counting operator C. As a variant ....

J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing 6, 675-695, 1977.

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