Genericity, Randomness, and Polynomial-Time Approximations

Architects of the information society: Thirty-five years of the laboratory for computer science at MIT: By Simson L. Garfinkel, edited by Hal Abelson. MIT Press, Cambridge, MA. (1999). 72 pages. $20.00

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what before-the-fact seemed to be) lower complexity levels. For example, for each k ≥ 2 it. is now known that if one query to σ p k is as powerful as two queries to σ p k (i.e., P Σ[1] = P Σ[2] ), then PH = σ p k . This article surveys the history, the results, and the method---the so-called easy-hard technique---of the just-mentioned nine-paper progression:1. J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing , 17(6):1263--1282, 1988. Erratum appears in the same journal. 20(2):404.2. K. Wagner. Number-of-query hierarchies. Technical Report 158, Institut für Mathematik. Universität Augsburg, Augsburg, Germany, October 1987.3. K. Wagner. Number-of-query hierarchies. Technical Report 4. Institut für Informatik. Universität Würzburg, Würzburg, Germany, February 1989.4. R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing . 25(2):340--354. 1996.5. R. Beigel. R. Chang. and M. Ogiwara. A relationship between difference hierarchies and relativized polynomial hierarchies. Mathematical Systems Theory . 26(3):293--310, 1993.6. E. Hemaspaandra, L. Hemaspaandra. and H. Hempel. An upward separation in the polynomial hierarchy. Technical Report Math/Inf/96/15, Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany, June 1996.7. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. A downward collapse within the polynomial hierarchy. SIAM Journal on Computing , 28(2):383--393, 1999.8. H. Buhrman and L. Fortnow. Two queries. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity . 13--19. IEEE Computer Society Press, June 1998.9. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Translating equality downwards. Technical Report TR-657. Department of Computer Science, University of Rochester, Rochester, NY. April 1997.

Discounting Infinite Games But How and Why?

The correspondence between partial metrics and semivaluations

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

Safe and unsafe polynomial time approximations were introduced by Meyer and Paterson [4] and Yesha [8], respectively. The question of which sets have optimal safe approximations was investigated by several authors (see e.g. [3,6,7]). Orponen et al. [5] showed that a problem has an optimal polynomial time approximation if and only if neither it nor its complement is p-levelable. Recently Duris and Rolim [2] considered the unsafe case and compared the existence of optimal polynomial time approximations for both cases. They left open the question, however, whether there are intractable sets with optimal unsafe approximations and whether there are sets with optimal unsafe approximations but without optimal safe approximations. Here we answer these questions affirmatively. Moreover, we consider a variant of Duris and Rolim's δ-levelability concept related to the nonexistence of optimal unsafe approximations.

Robert L. Rees, B.S., M.S., is senior systems analyst at the Laboratory of Computer Science, Massachusetts General Hospital, Boston, Mass

NP = PCP(logn, 1) and related results crucially depend upon the close connection between the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [30]. The strongest previously known connection for this test states that a function passes the test with probability δ for some δ > 7/8 iff the function has agreement ≈ δ with a polynomial of degree d. We present a new, and surprisingly strong, analysis which shows that the preceding statement is true for arbitrarily small δ, provided the field size is polynomially larger than d/δ. The analysis uses a version of Hilbert irreducibility, a tool of algebraic geometry. As a consequence we obtain an alternate construction for the following proof system: A constant prover 1-round proof system for NP languages in which the verifier uses O(logn) random bits, receives answers of size O(logn) bits, and has an error probability of at most 2− log 1− . Such a proof system, which implies the NP-hardness of approximating Set Cover to within Ω(logn) factors, has already been obtained by Raz and Safra [29]. Raz and Safra obtain their result by giving a strong analysis, in the sense described above, of a new low-degree test that they present. A second consequence of our analysis is a self tester/corrector for any buggy program that (supposedly) computes a polynomial over a finite field. If the program is correct only on δ fraction of inputs where δ = 1/ |F| 0.5, then the tester/corrector determines δ and generates O( 1 δ ) values for every input, such that one of them is the correct output. In fact, our results yield something stronger: Given the buggy program, we can construct O( 1 δ ) randomized programs such that one of them is correct on every input, with high probability. Such a strong selfcorrector is a useful tool in complexity theory with some applications known. ∗arora@cs.princeton.edu. Supported by an NSF CAREER award, an Alfred P. Sloan Fellowship, and a Packard Fellowship. †madhu@lcs.mit.edu. Laboratory for Computer Science, MIT, Cambridge, MA 02139. Part of this work was done when this author was at the IBM Thomas J. Watson Research Center.

Physically-based modeling: past, present, and future

Mobile programs can potentially be malicious. To protect itself, a host that receives such mobile programs from an untrusted party or via an untrusted network connection will want some kind of guarantee that the mobile code is not about to cause any damage. The traditional solution to this problem has been verification, by which the receiving host examines the mobile program to discover all its actions even before starting execution. Unfortunately, aside from consuming computing resources in itself, verification inhibits traditional compiler optimizations, making such verifiable mobile code much less efficient than native code. We have found an alternative solution by identifying a class of mobile-code representations in which malicious programs can simply not be encoded to begin with. In such an encoding, verification turns into an integral part of the decoding routine. Moreover, support for high-quality justin-time code generation can be provided. We present two such encodings, one based on highly effective compression of abstract syntax trees, and another based on a referencesafe and type-safe variant of Static Single Assignment form. Parts of this material were previously published under the title “Project transPROse: Reconciling Mobile-Code Security With Execution Efficiency” in The Second DARPA Information Survivability Conference and Exhibition (DISCEX II), Anaheim, California, June 2001, IEEE Computer Society Press, ISBN 0-7695-1212-7, pp. II.196–II.210. This research effort was sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory (AFRL), Air Force Materiel Command, USAF, under agreement number F30602-99-1-0536. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA, AFRL, or the U.S. Government.

Proceedings of the first conference on artificial intelligence applications: Published by IEEE Computer Society Press, 1109 Spring Street, Suite 300, Silver Spring, MD, USA

The best-performing on-line methods for estimating probabilities of symbols in a sequence (required for computing minimal codes) use context trees with either information-theoretic state selection or context-tree weighting. This paper derives de novo from Bayes' theorem, a novel technique for modeling sequences on-line with context trees, which we call "Bayesian state combining" or BSC. BSC is comparable in function to both information-theoretic state selection and context-tree weighting. However, it is a truly distinct alternative to either of these techniques, which like BSC, can be viewed as "dispatchers" of probability estimates from the set of competing, memoryless models represented by the context tree. The resulting technique handles sequences over m-ary input alphabets for arbitrary m and may employ any probability estimator applicable to context models (e.g., Laplace, Krichevsky-Trofimov, blending, and more generally, mixtures). In experiments that control other (256-ary) context-tree model features such as Markov order and probability estimators, we compare the performance of BSC and information-theoretic state selection. The background notation and concepts are reviewed, as required to understand the modeling problem and application of our result. The leading notion of the paper is derived, which dynamically maps certain states in context-models to a set of mutually exclusive hypotheses and their prior and posterior probabilities. The efficient sequential computation of the posterior probabilities of the hypotheses, which was made possible via a non-obvious application of the percolating description-length update mechanism introduced by Bunton (see Proceedings Data Compression Conference, IEEE Computer Society Press, 1997) is described. The preliminary empirical performance of the technique on the Calgary Corpus is presented, the relationship of BSC to information-theoretic state selection and context-tree weighting is discussed.

We show that Nechiporuk's method [I. Wegener, The Complexity of Boolean Functions, Teubner-Wiley, New York, 1987] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an $\Omega(n^2/\log^2 n)$ lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [A. Yao, Proceedings of 34th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 352--361] states that the majority function does not have a linear-size quantum formula. We also show that quantum formulas can be simulated by Boolean circuits of almost the same size.

Book review: Software inspection: an industry best practice. By David A. Wheeler, Bill Brykczynski and Reginald N. Meeson, Jr. Published by IEEE Computer Society Press, Los Alamitos, California, U.S.A., 1996. ISBN: 0 8186 7340 0, 293 pages. Price: U.S.A. $30.00 members/$38.00 non‐members. Including disk, soft cover.

No abstract available.