Jose Luis Balcazar, Josep Daz, and Joaquim Gabarro. Structural complexity II, volume 22 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....u 2 N with respect to j is a uniformly semilinear set in N . 2.4 Some Complexity Issues Since we are mainly concerned with the computational complexity of problems, it is necessary to speak about how we measure complexity. We consider the standard multitape Turing machine model (see, e.g. [2]) For space bounds, we only count, as is usual, the space used on the work tapes, and we do not take into account the space used on the write only output tape (which may be exponentially larger) We state complexity bounds as worst case bounds in terms of the input size, which is the number of ....

....to refer to the class of problems that can be decided by (multi tape) Turing machines using an amount of work space that is polynomial in the size of the input. PSPACE is a very fundamental and (with respect to variations of the machine model) very robust complexity class. For more details, see [2]. 3 Basic Results In this section, we are going to review several very basic and fundamental complexity results for the structures we have presented in the previous section. Arguably one of the most central problems for almost all of these structures turns out to be the uniform word problem for ....

Jos'e Luis Balc'azar, Josep D'iaz, and Joaquim Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo-Hong Kong, 1988.

....N 7 N 2. A f0 : k 2 Ng and 0 2 A can be determined in time O(f(k 1) Moreover, in addition to [1] we require that 3. 8r 2 N) a.e. n) f(n) f(n 1) A string w is relevant if w = 0 for some k. Because of the time constructibility of f , by a standard argument ([5]) we obtain Fact 3.4 The set of relevant strings is in PTIME. Polynomial time 1 tt reducibility is a reducibility of more technical interest. Here is one application of the notion, due to Ambos Spies. Theorem 3.5 ( 1] Suppose A is super sparse. Then the polynomial time Turing degree of any set ....

....is closely related to C : NP CoNP ; and use the rest of NP to represent I(B) A similar idea was used in the proof of Theorem 3.2. Let the variables R; S range over C . We use the concept of oracle nondeterministic Turing machine (oracle NTM) which is described in Balcazar e.a. [5]. Outline of the proof. The construction of U extends Baker e.a. 4] As a parameter, we determine a set Q 2 NP Gamma C , where for some polynomial time S f0g Q = fw 2 S : 9v 2 U jvj = jwjg: 9) Then we let B = B(Q) where B(Q) fQ R : R 2 C g; 10) R(Q) fR 2 C : R ....

J. L. Balc'azar, J. D ' iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer--Verlag, Heidelberg, 1988.

....it is closed downwards under polynomial time parity reductions. Gill [5] showed that NP is contained in PP. Russo [15] showed that the class PP is closed under symmetric difference. Using this observation, Papadimitriou and Yannakakis [13] showed that D PP, and Balc azar, D iaz, and Gabarr o [1] showed that the Boolean hierarchy is contained in PP. In this paper we extend Russo s approach, by showing that PP is closed under polynomial time truthtable reductions in which the truth table implements the parity operation. This closure property immedi A preliminary version of this paper ....

....Research performed in part while at Columbia University and while visiting Jena. Sektion Mathematik, Friedrich Schiller Universitat, Jena, German Democratic Republic. This improved on a result by Papadimitriou and Zachos, who showed that the Boolean hierarchy is contained in P #P[1] [14] ately implies that several classes of interest are contained in PP. Definition 1. A set A is polynomial time parity reducible to B (denoted A parity B) if A tt B via a truth table that tests whether an odd number of its inputs belong to B. Theorem 2. PP is closed under parity ....

J. L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, New York, 1988.

....where we distinguish between serial queries (as in a Turing reduction) and parallel queries (as in a truth table reduction) In [11] serial queries are called adaptive, and parallel queries are called nonadaptive. For additional background, see [1, 2, 8, 5] We adopt the following notation from [3]: Notation 1 Let A be a set, C a class of sets, and r a reduction. r is the class of all sets that are polynomial time r reducible to A. r is the class of all functions that are polynomial time r reducible to A. r = A2C P r . r = A2C PF r . We write k tt to denote truth table ....

....of all sets that are polynomial time r reducible to A. r is the class of all functions that are polynomial time r reducible to A. r = A2C P r . r = A2C PF r . We write k tt to denote truth table reducibility of norm k and k T to denote Turing reducibility with only k queries. See [3] for definitions of standard concepts in complexity theory. Thus, P k tt is the class of all sets that are accepted by a polynomial time algorithm that makes k parallel queries to A, and P k T is the class of all sets that are accepted by a polynomial time algorithm that makes k serial queries ....

J. L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, New York, 1988.

.... L ) Gap(N;X) 0: Gill [13] shows that NTIME(t(n) PrTIME(t(n) It should be noted that in our definition of PrTIME(t(n) all accepting (rejecting) paths are counted equally, regardless of length, as in [31] Other definitions of PrTIME(t(n) either insist that all paths have the same length [3] or weight the paths according to length [13] For time constructible t(n) these definitions are equivalent. Definition 3. PP = PrTIME(n ) For any nondeterministic Turing machine N , let the complement machine, denoted N , be the machine which runs N and then rejects if N accepts and accepts ....

J. L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, New York, 1988.

....in a Turing reduction) and parallel queries (as in a truth table reduction) In [23] serial queries are called adaptive, and parallel queries are called nonadaptive. For additional background, see [3, 4, 13, 9] Henceforth we write n to denote input length. We adopt the following notation from [6]: Notation 1. Let A be a set, C a class of sets, and r a reduction. r is the class of all sets that are polynomial time r reducible to A. r is the class of all functions that are polynomial time r reducible to A. r = A2C P r . r = A2C PF r . We write q(n) tt to denote truth table ....

....sets that are polynomial time r reducible to A. r is the class of all functions that are polynomial time r reducible to A. r = A2C P r . r = A2C PF r . We write q(n) tt to denote truth table reducibility of norm q(n) and q(n) T to denote Turing reducibility with only q(n) queries. See [6] for definitions of standard concepts in complexity theory. For example, P q(n) tt is the class of all sets that are accepted by a polynomial time algorithm that makes q(n) parallel queries to A, and q(n) T is the class of all sets that are accepted by a polynomial time algorithm that makes q(n) ....

J. L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, New York, 1988.

....on k, rather than on the exact choice of elements in S; call this constant c k . Thus p is equal to the summation of c k times the kth elementary symmetric function. Since c k = 0 for k d, 5. Circuits We review some definitions concerning circuits. For complete coverage, we recommend [7]. The size of a circuit is the number of gates it contains (others have defined the size as the number of wires, which is at most the square of our notion of size) The depth of a circuit is the length of the longest path from the output to an input. A gate that computes a symmetric function is ....

....sign of a degree 2 polynomial. Proof: Let p(x 1 ; x 2n ) x 1 Delta Delta Delta x n ) x n 1 Delta Delta Delta x 2n ) This gives Russo s [47] theorem that PP is closed under symmetric difference. This has been extended to closure under polynomial time parity reductions [7, 17]. In Section 6 we mentioned that the AND of two MAJORITYs is not represented by the sign of a degree O(1) polynomial. However, Beigel, Reingold, and Spielman [19] proved that it is represented by the sign of a degree O(log n) polynomial. Their proof uses rational approximations to the ....

J. L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, New York, 1988.

....for time constructible bounds t. 2 Preliminaries We assume the standard de nitions of deterministic and nondeterministic Turing machines and of time and space bounded complexity classes [13] We also assume the de nitions of alternating Turing machines making a bounded number of alternations [2]. A computation of an alternating Turing machine on an input is any correct nite sequence of con gurations of the machine beginning with the initial con guration and ending with an accepting or rejecting con guration. The time taken by the computation is the length of the sequence of con ....

J.L.Balcazar, J.Diaz and J.Gabarro. Structural Complexity 2. Volume 22 of EATCS Monographs in Theoretical Computer Science, Springer Verlag, 1990.

....type r . For the most part we are interested in p T and p m reductions. As usual, 0 ; 1 ; 2 ; is a standard acceptable enumeration of the computable partial functions (as in [Soa87] Definition 2. 1 For any set A Sigma , we define, in the spirit of Balc azar, et al. [BDG88], K(A) fhe; x; 0 t i j N A e (x) has an accepting path of length tg: We call K(A) the NP jump of A. It is complete for NP A under (unrelativized) p m reductions. It is easy to check that K( Delta) lifts to a well defined operator on the p T degrees. We denote by K k ( Delta) ....

....the advice string. This definition is equivalent to the usual one for defining common classes such as P=poly, P=log, etc. The motivation for our definition comes from the observation that, in defining P=F for some function class F such that F 6 FP, the usual definition in terms of languages (see [BDG88]) allows the accepting machine not only to get advice from the advice string, but may also provide it with greater resource bounds, since the length of the advice string counts toward the length of the input. Thus, for example, an advice string consisting solely of a super polynomially long ....

J. L. Balcazar, J. D iaz, and J. Gabarro. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

....by deterministic and nondeterministic polynomial time Turing machines. We will use PH to denote the Polynomial Hierarchy and R to denote the class of languages recognized by probabilistic polynomial time Turing machines with bounded one sided error. We refer the reader to standard references [BDG88, BDG90, Sch85] in complexity theory for explanations on the relationships among these classes. An instance of the Graph Isomorphism (GI) problem is a pair of undi 2 2 rected graphs (G, H) Without loss of generality, the vertices of the graphs are indexed 1 through n. The pair (G, H) is an element of GI if ....

J. L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity II, volume 22 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1990.

....by deterministic and nondeterministic polynomial time Turing machines. We will use PH to denote the Polynomial Hierarchy and R to denote the class of languages recognized by probabilistic polynomial time Turing machines with bounded one sided error. We refer the reader to standard references [BDG88, BDG90, Sch85] in complexity theory for explanations on the relationships among these classes. An instance of the Graph Isomorphism (GI) problem is a pair of undi 2 2 rected graphs (G, H) Without loss of generality, the vertices of the graphs are indexed 1 through n. The pair (G, H) is an element of GI if ....

J. L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1988.

....by the reduction type r . For the most part we are interested in p T and p m reductions. As usual, 0 ; 1 ; 2 ; is a standard acceptable enumeration of the computable partial functions. Definition 2. 1 For any set A Sigma , we define, in the spirit of Balcazar, et al. [BDG88], K(A) fhe; x; 0 t i j N A e (x) has an accepting path of length tg: We call K(A) the NP jump of A. It is complete for NP A under (unrelativized) p m reductions. It is easy to check that K( Delta) lifts to a well defined operator on the p T degrees. We denote by K k ( Delta) ....

....the advice string. This definition is equivalent to the usual one for defining common classes such as P=poly, P=log, etc. The motivation for our definition comes from the observation that, in defining P=F for some function class F such that F 6 FP, the usual definition in terms of languages (see [BDG88]) allows the accepting machine not only to get advice from the advice string, but may also provide it with greater resource bounds, since the length of the advice string counts toward the length of the input. Thus, for example, an advice string consisting solely of a super polynomially long ....

J. L. Balcazar, J. D iaz, and J. Gabarro. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

.... f(x) y (or simply f y when the independent variable is clear from the context) is a shorthand for lim x 1 f(x) y Also, the reader should be familiar with sequential computational models like Turing machines or Random Access Machines [AHU74] and basic complexity theoretic definitions [GJ79, BDG88]. Asymptotic notations like O(n 2 ) o(1) n) 2 n ) and (n) will denote function classes but we will normally write f = n) instead of f 2 n) with the intended meaning that there exists a constant c such that f(n) cn for n sufficiently large. The reader is referred to Section 2.1 ....

....algorithmic properties of graphs. Finally Section 1.2.6 introduces all basic definitions and results in group and action theory that will be needed later, especially in Chapter 2. 1.2. 1 Problems Many computational problems can be viewed as the seeking of partial information about a relation (see [BDG88, Chapter 1] or [BC91] More specifically suppose is a finite alphabet and that problem instances and solutions are encoded as strings over . A relation SOL defines an association between problem instances and solutions. For every x 2 the set SOL(x) contains all the y 2 that ....

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J.L. Balcazar, J. Diaz, and J. Gabarro. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

....Then C either has p measure zero or else has p measure one. For the same reason as above, Corollary 1.4 is as powerful as Theorem 1.3. 2 Notation and Preliminaries Most of our complexity theoretic notation is standard. We refer the reader to the textbooks by Balcazar, Daz and Gabarro [3, 2], and by Papadimitriou [7] BPP denotes the complexity class of languages that can be decided by probabilistic polynomialtime Turing machines with bounded two sided error. EXP denotes # c 0 DT IME[2 n c ] and E denotes # c 0 DT IME[2 cn ] EXP contains both BPP and E . A ....

J. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1995.

....Then C either has p measure zero or else has p measure one. For the same reason as above, Corollary 1.4 is as powerful as Theorem 1.3. 2 Notation and Preliminaries Most of our complexity theoretic notation is standard. We refer the reader to the textbooks by Balcazar, Daz and Gabarro [3, 2], and by Papadimitriou [7] BPP denotes the complexity class of languages that can be decided by probabilistic polynomialtime Turing machines with bounded two sided error. EXP denotes # c 0 DT IME[2 n c ] and E denotes # c 0 DT IME[2 cn ] EXP contains both BPP and E . A ....

J. Balcazar, J. Daz, and J. Gabarro. Structural Complexity II, volume 22 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1990.

....6. Because of space limitations of this extended abstract, we omitted the proofs of some of our results. The interested reader can find them in the full version. 2 Notation Most of our complexity theoretic notation is standard. We refer the reader to the textbooks by Balcazar, Daz and Gabarro [BDG95, BDG90] and by Papadimitriou [Pap94] An oracle circuit D is a circuit with AND, OR and NOT gates as well as oracle gates, which compute membership of the string formed by their input bits to some unspecified oracle B. The function D B the circuit computes depends on the oracle B. For fixed ....

J. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1995.

....of space limitations of this extended abstract, we omitted the proofs of some of our results. The interested reader can find them in the full version. 2 Notation Most of our complexity theoretic notation is standard. We refer the reader to the textbooks by Balcazar, Daz and Gabarro [BDG95, BDG90] and by Papadimitriou [Pap94] An oracle circuit D is a circuit with AND, OR and NOT gates as well as oracle gates, which compute membership of the string formed by their input bits to some unspecified oracle B. The function D B the circuit computes depends on the oracle B. For fixed B, we ....

J. Balcazar, J. Daz, and J. Gabarro. Structural Complexity II, volume 22 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1990.

....set of all oracle computations running in time n i i for all oracles and inputs of length n. As usual, 0 ; 1 ; 2 ; is a standard acceptible enumeration of the computable partial functions. Definition 2. 1 For any set A Sigma , we define, in the spirit of B alcazar, et al. [BDG88], K(A) fhe; x; 0 t i j N A e (x) has an accepting path of length tg: We call K(A) the NP jump of A. It is complete for NP A under (unrelativized) p m reductions. It is easy to check that K( Delta) lifts to a well defined operator on the p T degrees. We denote by K k ( Delta) ....

....the advice string. This definition is equivalent to the usual one for defining common classes such as P=poly, P=log, etc. The motivation for our definition comes from the observation that, in defining P=F for some function class F such that F 6 FP, the usual definition in terms of languages (see [BDG88]) allows the accepting machine not only to get advice from the advice string, but may also provide it with greater resource bounds, since the length of the advice string counts toward the length of the input. Thus, for example, an advice string consisting solely of a superpolynomially long ....

J. L. Balcazar, J. D#az, and J. Gabarro. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. SpringerVerlag, 1988.

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Jose Luis Balcazar, Josep Daz, and Joaquim Gabarro. Structural complexity II, volume 22 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1988.

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Jose Luis Balcazar, Josep Daz, and Joaquim Gabarro. Structural complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1988.

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J. L. Balc'azar, J. D ' iaz, and J. Gabarr'o. Structural Complexity I,volume 11 of EATCS Monographs on Theoretical Computer Science. SpringerVerlag, 1988.

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J. L. Balc'azar, J. D ' iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. SpringerVerlag, 1988.

No context found.

J. L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I, volume 11 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

No context found.

J.L.Balcazar, J.Diaz and J.Gabarro. Structural Complexity 1. Volume 11 of EATCS Monographs in Theoretical Computer Science, Springer Verlag, 1988.

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