Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)

24-27 May 1996

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  • Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)

    Publication Year: 1996
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    Freely Available from IEEE
  • An isomorphism theorem for circuit complexity

    Publication Year: 1996, Page(s):2 - 11
    Cited by:  Papers (3)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (1030 KB)

    We show that all sets complete for NC/sup 1/ under AC/sup 0/ reductions are isomorphic under AC/sup 0/-computable isomorphisms. Although our proof does not generalize directly to other complexity classes, we do show that, for all complexity classes C closed under NC/sup 1/-computable many-one reductions, the sets complete for C under NC/sup 0/ reductions are all isomorphic under AC/sup 0/-computab... View full abstract»

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  • Nondeterministic NC/sup 1/ computation

    Publication Year: 1996, Page(s):12 - 21
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (961 KB)

    We define the counting classes NC/sup 1/, GapNC/sup 1/ PNC/sup 1/ and C/sub =/NC/sup 1/. We prove that Boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that NC/sup 1//spl sube/L and that C/sub =/NC/sup 1//spl sube/L... View full abstract»

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  • Parallel complexity hierarchies based on PRAMs and DLOGTLME-uniform circuits

    Publication Year: 1996, Page(s):24 - 32
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (794 KB)

    Unlike the case of logspace-uniform circuits, complexity hierarchies do exist for PRAMs and DLOGTIME-uniform circuits: (i) There exist a constant d and a language L such that L is recognizable in time dT(n) by some PRIORITY CRCW PRAM but is not recognizable in time T(n) by any PRIORITY CRCW PRAM if the number of processors is fixed. (ii) There exist constants c, d and a language L such that L is r... View full abstract»

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  • Collapsing oracle-tape hierarchies

    Publication Year: 1996, Page(s):33 - 42
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (996 KB)

    Other authors have shown that equipping a logspace oracle Turing machine with more than one oracle tape may result in an increased computational power. We are interested in the inverse problem: For which oracle classes C does the oracle-tape hierarchy collapse in the sense that logspace machines with a fixed number of oracle tapes cannot compute more than machines with a single oracle tape? Surpri... View full abstract»

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  • Extracting randomness: how and why. A survey

    Publication Year: 1996, Page(s):44 - 58
    Cited by:  Papers (22)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (1373 KB)

    Extractors are Boolean functions that allow, in some precise sense, extraction of randomness from somewhat random distributions. Extractors, and the closely related "Dispersers", exhibit some of the most "random-like" properties of explicitly constructed combinatorial structures. In turn, extractors and dispersers have many applications in "removing randomness" in various settings and in making ra... View full abstract»

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  • On coherence, random-self-reducibility, and self-correction

    Publication Year: 1996, Page(s):59 - 67
    Cited by:  Papers (1)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (814 KB)

    We address two questions about self-reducibility-the power of adaptiveness in examiners that take advice and the relationship between random-self-reducibility and self-correctability. We first show that adaptive examiners are more powerful than nonadaptive examiners, even if the nonadaptive ones are nonuniform. Blum et al. (1993) showed that every random-self-reducible function is self-correctable... View full abstract»

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  • Error reduction by parallel repetition-a negative result

    Publication Year: 1996, Page(s):70 - 76
    Cited by:  Papers (4)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (728 KB)

    We show that no fixed number of parallel repetitions suffices in order to reduce the error in two-prover one-round proof systems from one constant to another. Our results imply that the recent bounds proven by Ran Raz (1995), showing that the number of rounds that suffice is inversely proportional to the answer length, are nearly best possible. View full abstract»

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  • Deciding the Vapnik-Cervonenkis dimension is /spl Sigma//sub 3//sup p/-complete

    Publication Year: 1996, Page(s):77 - 80
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (371 KB)

    Linial et al. (1988) raised the question of how difficult the computation of the Vapnik-Cervonenkis dimension of a concept class over a finite universe is. Papadimitriou and Yannakakis (1993) obtained a first answer using matrix representations of concept classes. However, this approach does not capture classes having exponential size, like monomials, which are encountered in learning theory. We c... View full abstract»

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  • VC dimension in circuit complexity

    Publication Year: 1996, Page(s):81 - 85
    Cited by:  Papers (2)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (416 KB)

    The main result of this paper is a /spl Omega/(n/sup 1/4/) lower bound on the size of a sigmoidal circuit computing a specific AC/sub 2//sup 0/ function. This is the first lower bound for the computation model of sigmoidal circuits with unbounded weights. We also give upper and lower bounds for the same function in a few other computation models: circuits of AND/OR gates, threshold circuits, and c... View full abstract»

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  • Reducing P to a sparse set using a constant number of queries collapses P to L

    Publication Year: 1996, Page(s):88 - 96
    Cited by:  Papers (1)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (736 KB)

    We prove that there is no sparse hard set for P under logspace computable bounded truth-table reductions unless P=L. In case of reductions computable in NC/sup 1/, the collapse goes down to P=NC/sup 1/. We generalize this result by parameterizing the sparseness condition, the space bound and the number of queries of the reduction, apply the proof technique to NL and L, and extend all these theorem... View full abstract»

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  • L-printable sets

    Publication Year: 1996, Page(s):97 - 106
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (944 KB)

    Properties of L-printable sets are considered, and it is shown that two sets A and B that are L-printable and have similar density are L-isomorphic. L-printable sets are characterized as those sets L-isomorphic to tally sets in L, and as subsets of KS[k log n, k log n]. Several classes of L-printable sets are given, including sparse regular and context-free sets; a characterization of sparse regul... View full abstract»

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  • Hierarchies of circuit classes that are closed under complement

    Publication Year: 1996, Page(s):108 - 117
    Cited by:  Papers (2)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (876 KB)

    We examine three hierarchies of circuit classes and show they are closed under complementation. (1) The class of languages recognized by a family of polynomial size skew circuits with width O(w), are closed under complement. (2) The class of languages recognized by family of polynomial size circuits with width O(w) and polynomial tree-size, are closed under complement. (3) The class of languages r... View full abstract»

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  • Succinct representation, leaf languages, and projection reductions

    Publication Year: 1996, Page(s):118 - 126
    Cited by:  Papers (2)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (860 KB)

    The concepts of succinct problem representation, and of NP leaf languages, were developed to characterize complexity classes above polynomial time. Here, we work out a descriptive complexity approach to succinctly represented problems, and prove a strictly stronger version of the Conversion Lemma from Balcazar et al (1992) which allows iterated application. Moreover, we prove that for every proble... View full abstract»

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  • Randomization and derandomization in space-bounded computation

    Publication Year: 1996, Page(s):128 - 149
    Cited by:  Papers (12)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (2244 KB)

    This is a survey of spacebounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic space-bounded computations, and the application of such generators to obtain det... View full abstract»

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  • On the measure of two-dimensional regions with polynomial-time computable boundaries

    Publication Year: 1996, Page(s):150 - 159
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (839 KB)

    We study the computability of the Lebesgue measure of a two-dimensional region that has a polynomial-time computable boundary. It is shown that the two-dimensional measure of the boundary itself completely characterizes the computability of the measure of the interior region. Namely, if a polynomial-time computable, simple, closed curve has measure zero, then its interior region must have a comput... View full abstract»

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  • On positive P

    Publication Year: 1996, Page(s):162 - 170
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (764 KB)

    Continuing a line of research opened up by Grigni and Sipser (1992) and further pursued by Stewart (1994), we show that a wide variety of equivalent characterizations of P still remain equivalent when restricted to be positive. All these restrictions thus define the same class posP, a proper subclass of monP, the class of monotone problems in P. We also exhibit complete problems for posP under ver... View full abstract»

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  • A comparison of weak completeness notions

    Publication Year: 1996, Page(s):171 - 178
    Cited by:  Papers (2)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (744 KB)

    We compare the weak completeness notions for E in the sense of Lutz's resource-bounded measure theory (1992) with respect to the standard polynomial time reducibilities. Our results parallel results for classical completeness by Watanabe (1987) and others. We show that the weak completeness notions for 1-query reductions coincide: A set is weakly complete for E under 1-truth-table reducibility iff... View full abstract»

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  • The law of the iterated logarithm for random sequences

    Publication Year: 1996, Page(s):180 - 189
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (712 KB)

    The stochastic properties of p-random sequences are studied in this paper. It is shown that the law of the iterated logarithm holds for p-random sequences. This law gives a quantitative characterization of the density of p-random sets. When combined with the invariance property of p-random sequences, this law is also useful in proving that some complexity classes have p-measure 0. View full abstract»

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  • Truth-table closure and Turing closure of average polynomial time have different measures in EXP

    Publication Year: 1996, Page(s):190 - 195
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (546 KB)

    Let P/sub P-comp/ denote the sets that are solvable in polynomial time on average under every polynomial time computable distribution on the instances. In this paper we show that the truth-table closure of P/sub P-comp/ has measure 0 in EXP. Since, as we show, EXP is Turing reducible to P/sub P-comp/, the Turing closure has measure 1 in EXP and thus, P/sub P-comp/ is an example of a subclass of E ... View full abstract»

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  • Much ado about functions

    Publication Year: 1996, Page(s):198 - 212
    Cited by:  Papers (6)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (1316 KB)

    This paper surveys basic results on complexity classes of partial multivalued functions. We stress basic inclusion relations, interesting hierarchies, and results that demonstrate that hierarchies are extant. View full abstract»

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  • Inverting onto functions

    Publication Year: 1996, Page(s):213 - 222
    Cited by:  Papers (4)  |  Patents (1)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (806 KB)

    We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including: One can find accepting paths of nondeterministic polynomial-time Turing machines that accept /spl Sigma/*. Every total multivalued nondeterministic function has a polynomial-time computab... View full abstract»

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  • A note on p-selective sets and on adaptive versus nonadaptive queries to NP

    Publication Year: 1996, Page(s):224 - 232
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (804 KB)

    We study two properties of a complexity class C-whether there exists a truth-table hard p-selective language for C, and whether polynomially-many nonadaptive queries to C can be answered by making O(log n)-many adaptive queries to C (in symbols, is PF/sub tt//sup C//spl sube/PF/sup C/[O(log n)]). We show that if there exists an NP-hard p-selective set under truth-table reductions, then PF/sub tt//... View full abstract»

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  • Good degree bounds on Nullstellensatz refutations of the induction principle

    Publication Year: 1996, Page(s):233 - 242
    Cited by:  Papers (3)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (998 KB)

    This paper gives nearly optimal, logarithmic upper and lower bounds on the minimum degree of Nullstellensatz refutations (i.e., polynomials) of the propositional induction principle. View full abstract»

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  • The complexity of knowledge representation

    Publication Year: 1996, Page(s):244 - 248
    Cited by:  Papers (1)
    Request permission for commercial reuse | Click to expandAbstract | PDF file iconPDF (504 KB)

    Representing knowledge in forms appropriate for rapid common-sense reasoning is a challenging current problem in artificial intelligence. We review certain recent results which suggest that complexity theory has an important role to play in this field. View full abstract»

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