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Dagstuhl-Seminar-Report 82 Report on the Dagstuhl-Seminar ”Structure and Complexity” Organizers: Klaus Ambos-Spies (Heidelberg) Steven Homer (Boston) Uwe Sch¨oning (Ulm) February 14-18, 1994 The seminar ”Structure and Complexity” was the second Dagstuhl Sem- inar devoted to the structural aspects of Computational Complexity Theory. It was attented by 47 scientists who in 37 talks presented new results in this field. The following topics were among the main subjects covered by the talks: Kolmogorov complexity, resource bounded genericity and randomness, relativizations, descriptive complexity theory, and computational models. 1 PROGRAM Eric Allender Pseudorandom Sources and Oracles for BPP Klaus Ambos-Spies Genericity and Measure for Exponential Time Ingrid Biehl Definition and Existence of Super Complexity Cores Ronald V. Book On Collapsing the Polynomial-Time Hierarchy Bernd Borchert Predicate Classes and Promise Classes Harry Buhrman Separating PSPACE and EXP using autoreducibility Gerhard Buntrock On the Computational Power of Restricted Two-Pushdown Au- tomata Cristian Calude Information and Randomness – An Overview Anne Condon Finite State Automata with Nondeterministic and Probabilistic States Rod Downey The Density Problem for Parameterized Polynomial Reducibilities Mike Fellows The Natural Degrees of Parameterized Complexity: Recent Re- sults and Open Problems Stephen Fenner Inverting the Turing Jump in Complexity Theory Lance Fortnow The Role of Relativization in Complexity Theory 2 William Gasarch Frequency Computation and Bounded Queries Judy Goldsmith Limits on Nondeterminism and Upward Collapse Neil Immerman NP-Completeness via Syntax Birgit Jenner Logspace Leaf Languages Johannes K¨obler Nondeterministic Kolmogorov Complexity and Sparse Sets in NP Phokion G. Kolaitis How to Define a Linear Order on Finite Structures Martin Kummer Kolmogorov Complexity and its Relation to Instance Complexity Jack H. Lutz Feasible Martingales, Weak Completeness, and Strong Hypotheses Elvira Mayordomo Separating NP-Completeness Notions if NP is not Small Christoph Meinel Lower Bounds on the MODk- and MAJ-Communication Complex- ity of Various Graph-Accessability Problems Martin Mundhenk Reductions to Sparse Sets Noam Nisan Pseudorandomness for Network Algorithms Kenneth W. Regan The Block Move Model and Circuit Complexity R¨udiger Reischuk Average Case Complexity 3 Michel de Rougemont Interactive Complexity in Graph theory Rainer Schuler Bounding Prover Complexity in Interactive Proof Systems Thomas Schwentick On Monadic NP Alan Selman Computing Solutions Uniquely Collapses the Polynomial Hierarchy Thomas Thierauf On the Correlation of Symmetric Functions Leen Torenvliet The Resource Bounded Injury Method Paul Vitanyi Two Heads are better than Two Tapes Klaus W. Wagner Complexity Classes Defined by Symmetric Leaf Languages Jie Wang An Average-Case Complete Word Problem for Finitely Presented Groups Xizhong Zheng nk-Random Sets are Weakly Complete for Exponential Time 4 ABSTRACTS Pseudorandom Sources and Oracles for BPP Eric Allender Rutgers University (joint work with Martin Strauss, Rutgers University) Lutz defined the notion of a pseudorandom oracle for BPP, and showed that almost every set in ESPACE is a source in this sense. We improve this result, showing that almost every set in PSPACE is a source. Using a new notion of measure for P, we show that the set of sources for BPP does not have measure 1 in P, although there are sources for BPP in AC0. These results depend on our characterization of Pseudorandom Sources for BPP as exactly the Borel-normal binary sequences. Our main result shows that, for almost every set A in E, BPPA = EA. (This improves an earlier result of Lutz, showing that this property holds for almost every A ESPACE.) A similar construction can be carried out in ∈ PSPACE, although we do not know if it holds with measure 1. Notions of measure for classes such as PSPACE and DTIME(T (n)) for T (n) << 2n do not seem to relativize in a meaningful way. The talk concludes with speculation as to whether probabilistic methods can be applied in this way to yield fast simulations of BPP. Genericity and Measure for Exponential Time Klaus Ambos-Spies Universit¨at Heidelberg (joint work with C. Neis, Freiburg, and S. Terwijn, Amsterdam) We relate the genericity concepts of Ambos-Spies, Fleischhack and Huwig to Lutz’s resource bounded measure. We show that the class of nk-generic sets has p-measure 1. This allows us to simplify and extend some p-measure- 1-results. We illustrate this approach by proving a new small span theorem for p-k-tt-reductions. 5 Definition and Existence of Super Complexity Cores Ingrid Biehl Universit¨at des Saarlandes I define and study super complexity cores of languages L with respect to classes with L . For example let = and S be a super complexity C 6∈ C C P core of L. Then S is infinite and for all polynomials p IN[X] and all deterministic Turing machines M which output 1 on input∈x S L and which output 0 on input x S L it follows that the running∈ time∩ of M ∈ ∩ on all but finitely many inputs x S exceeds p( x ). I prove that for all non-empty, countable classes of languages∈ which| | are closed under finite C variation, finite union and under complement and for all languages L it follows that such a super complexity core of L with respect to exists.6∈ C Moreover one can show: Given a recursively enumerable class of recursiveC C languages and a recursive language L. If there is a super complexity core of L with respect to , then there exists a recursive super complexity core too. C On Collapsing the Polynomial-Time Hierarchy Ronald V. Book University of California at Santa Barbara Theorem: If for almost every oracle set A the polynomial-time hierarchy relative to A collapses, then the (unrelativized) polynomial-time hierarchy collapses. 6 Predicate Classes and Promise Classes Bernd Borchert Universit¨at Heidelberg Considering computation trees produced by polynomial time nondeter- ministic computations one can define a complexity class by any predicate on computation trees, such classes will be called predicate classes. It will be shown that these classes are exactly the principal ideals of the polynomial time many-one reducibility. Additionally, the set of classes – which will be called recursive promise classes – definable by recursive promise functions instead of predicates will be shown to be equal to the set of the recursively presentable ideals. Separating PSPACE and EXP using autoreducibility Harry Buhrman Universidad Politecnica de Catalunya We investigate the complete sets of all the levels of the Polynomial Time Hierarchy, PSPACE and EXP. We show that all the Turing complete sets for each level of the Polynomial Time Hierarchy as well as PSPACE are auto- reducible. On the other hand we give some evidence for the existence of a Turing complete non-auto-reducible set for EXP, for we prove that for every k there exists a Turing complete set in EXP that is not auto-reducible with nk adaptive queries. Improving this result to a non auto-reducible Turing complete set will (with the previous result) separate PSPACE from EXP. Since there are oracles where PSPACE = EXP, our results are in some sense optimal. Addendum: Recently we have been informed that the results on the auto- reducibility for PSPACE appear in ”On Being Incoherent without Being Very Hard,” by Richard Beigel and Joan Feigenbaum, journal Computational Complexity in 1992 (vol. 2, 1992, pp. 1-17). 7 On the Computational Power of Restricted Two-Pushdown Automata Gerhard Buntrock Universit¨at W¨urzburg We introduce a model of two pushdown automata without input tape (TPDA); i. e. the input is assumed to be in one of the pushdowns initially. With the concept of weight functions we define shrinking automata and prove that shrinking TPDA (sTPDA) characterize the growing context-sensitive languages. The fact that growing context-sensitive languages are contained in LOGCFL is sharpened to the containment in One-Way-LOGCFL. This is shown by using a characterization of One-Way-LOGCFL by oneway auxil- iary pushdown automata with polynomial time and logarithmic space bounds (OWauxPDA). On the other hand it is shown that deterministic sTPDA (sDTPDA) cannot be simulated by deterministic OWauxPDA. Surprisingly for unambiguous GCSL we can give a simulation with unambiguous OWaux- PDA. Information and Randomness – An Overview Cristian Calude University of Auckland We survey the core constructions and results in our forthcoming book Information and Randomness – An Algorithmic Perspective to be published by Springer-Verlag. philosophy of the book • computers and complexities • binary vs non-binary coding • a model for random strings • from random strings to random sequences • 8 topological methods in the study of randomness • open problems. • Finite State Automata with Nondeterministic and Probabilistic States Anne Condon University of Wisconsin-Madison (joint work with Lisa Hellerstein, Sam Pottle and Avi Wigderson) We study finite automata with both nondeterministic and random states (npfa’s). We restrict our attention to those npfa’s that accept their languages with a small probability of error and run in polynomial expected time. Equiv- alently, we study Arthur-Merlin games where Arthur is limited to polynomial time and constant space. Dwork and Stockmeyer asked whether these npfa’s accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa’s with a 1-way input head. We also show that if L is a nonregular language, then either L or its complement is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1’s in the “characteristic matrix” of L, namely the binary matrix with a row and column for each string of length at most n, where entry [x,y]=1 if and only if the string xy in L.
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