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Interactive Proof Systems and Alternating Time-Space Complexity
Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement. -
Maurice Pouzet References
Maurice Pouzet References [1] M. Kabil, M. Pouzet, I.G. Rosenberg, Free monoids and metric spaces, To the memory of Michel Deza, Europ. J. of Combinatorics, Online publi- cation complete: April 2, 2018, https://doi.org/10.1016/j.ejc.2018.02.008. arXiv: 1705.09750v1, 27 May 2017. [2] I.Chakir, M.Pouzet, A characterization of well-founded algebraic lattices, Contribution to Discrete Math. Vol 13, (1) (2018) 35-50. [3] K.Adaricheva, M.Pouzet, On scattered convex geometries, Order, (2017) 34:523{550. [4] C.Delhomm´e,M.Pouzet, Length of an intersection, Mathematical Logic Quarterly 63 (2017) no.3-4, 243-255. [5] C.DELHOMME,´ M.MIYAKAWA, M.POUZET, I.G.ROSENBERG, H.TATSUMI, Semirigid systems of three equivalence relations, J. Mult.-Valued Logic Soft Comput. 28 (2017), no. 4-5, 511-535. [6] C.Laflamme, M.Pouzet, R.Woodrow, Equimorphy- the case of chains, Archive for Mathematical Logic, 56 (2017), no. 7-8, 811{829. [7] C.Laflamme, M.Pouzet, N.Sauer, Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif, Ab- handlungen aus dem Mathematischen Seminar der Universitt Hamburg (Abh. Math. Sem. Univ. Hamburg) October 2017, Volume 87, Issue 2, pp 369-408. [8] D.Oudrar, M.Pouzet, Profile and hereditary classes of relational struc- tures, J. of MVLSC Volume 27, Number 5-6 (2016), pp. 475-500. [9] M.Pouzet, H.Si-Kaddour, Isomorphy up to complementation, Journal of Combinatorics, Vol. 7, No. 2 (2016), pp. 285-305. [10] C.Laflamme, M.Pouzet, N.Sauer, I.Zaguia, Orthogonal countable ordinals, Discrete Math. -
Polynomial Sequences of Binomial Type and Path Integrals
Preprint math/9808040, 1998 LEEDS-MATH-PURE-2001-31 Polynomial Sequences of Binomial Type and Path Integrals∗ Vladimir V. Kisil† August 8, 1998; Revised October 19, 2001 Abstract Polynomial sequences pn(x) of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express pn(x) as a path integral in the “phase space” N × [−π,π]. The Hamilto- ∞ ′ inφ nian is h(φ)= n=0 pn(0)/n! e and it produces a Schr¨odinger type equation for pn(x). This establishes a bridge between enumerative P combinatorics and quantum field theory. It also provides an algorithm for parallel quantum computations. Contents 1 Introduction on Polynomial Sequences of Binomial Type 2 2 Preliminaries on Path Integrals 4 3 Polynomial Sequences from Path Integrals 6 4 Some Applications 10 arXiv:math/9808040v2 [math.CO] 22 Oct 2001 ∗Partially supported by the grant YSU081025 of Renessance foundation (Ukraine). †On leave form the Odessa National University (Ukraine). Keywords and phrases. Feynman path integral, umbral calculus, polynomial sequence of binomial type, token, Schr¨odinger equation, propagator, wave function, cumulants, quantum computation. 2000 Mathematics Subject Classification. Primary: 05A40; Secondary: 05A15, 58D30, 81Q30, 81R30, 81S40. 1 Polynomial Sequences and Path Integrals 2 Under the inspiring guidance of Feynman, a short- hand way of expressing—and of thinking about—these quantities have been developed. Lewis H. Ryder [22, Chap. 5]. 1 Introduction on Polynomial Sequences of Binomial Type The umbral calculus [16, 18, 19, 21] in enumerative combinatorics uses poly- nomial sequences of binomial type as a principal ingredient. A polynomial sequence pn(x) is said to be of binomial type [18, p. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Computational Learning Theory: New Models and Algorithms
Computational Learning Theory: New Models and Algorithms by Robert Hal Sloan S.M. EECS, Massachusetts Institute of Technology (1986) B.S. Mathematics, Yale University (1983) Submitted to the Department- of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1989 @ Robert Hal Sloan, 1989. All rights reserved The author hereby grants to MIT permission to reproduce and to distribute copies of this thesis document in whole or in part. Signature of Author Department of Electrical Engineering and Computer Science May 23, 1989 Certified by Ronald L. Rivest Professor of Computer Science Thesis Supervisor Accepted by Arthur C. Smith Chairman, Departmental Committee on Graduate Students Abstract In the past several years, there has been a surge of interest in computational learning theory-the formal (as opposed to empirical) study of learning algorithms. One major cause for this interest was the model of probably approximately correct learning, or pac learning, introduced by Valiant in 1984. This thesis begins by presenting a new learning algorithm for a particular problem within that model: learning submodules of the free Z-module Zk. We prove that this algorithm achieves probable approximate correctness, and indeed, that it is within a log log factor of optimal in a related, but more stringent model of learning, on-line mistake bounded learning. We then proceed to examine the influence of noisy data on pac learning algorithms in general. Previously it has been shown that it is possible to tolerate large amounts of random classification noise, but only a very small amount of a very malicious sort of noise. -
• La Gestion Efficace De L’Énergie • Le Réseautage Planétaire • Les Processus Géophysiques • L’Économie Globale, La Sécurité Et La Stabilité
SupérieureS atiqueS athéM e M inaire D SéM The planet on which we live and the challenges that we face on this planet become increasingly complex as ecological, economic and social systems are large intertwined networks governed by dynamic processes and feedback loops. Mathematical models are indispensable in understanding and managing such systems since they provide insight into governing processes; they help predict future behavior; and they allow for risk-free evaluation of possible interventions. The goal of this thematic program is to tackle pressing and emerging challenges in population and ecosystem health, including understanding and controlling major transmissible diseases, optimizing and monitoring vaccination, predicting the impacts of climate change on invasive species, protecting biodiversity and managing ecosystems sustainably. This pan-Canadian program will bring together the international community of researchers who work on these topics in a series of workshops to foster exchange and stimulate cross-disciplinary research between all scientific areas involved, to discuss perspectives and directions for future advances in the field, including new models and methods and to foster tighter links between the research community, government agencies and policy makers. Three summer schools will introduce graduate students and postdoctoral fellows to the art of modeling living systems and to the latest tools and techniques to analyze these models. SCIENTIFIC COMMITTEE Jacques Bélair Mark Lewis (Montréal) Models and Methods in Ecology (Alberta) Frithjof Lutscher and Epidemiology Mathematical Modeling James Watmough (Ottawa) February 6-8, 2013 (UNB) of Indigenous Population Jianhong Wu CRM, Montréal (York) Organizers: Jacques Bélair (Montréal), Health Jianhong Wu (York) September 28-29, 2013 AISENSTADT CHAIRS BIRS Bryan Grenfell (Princeton), May 2013 Graphic Design: www.neograf.ca Simon A. -
RIEMANN's HYPOTHESIS 1. Gauss There Are 4 Prime Numbers Less
RIEMANN'S HYPOTHESIS BRIAN CONREY 1. Gauss There are 4 prime numbers less than 10; there are 25 primes less than 100; there are 168 primes less than 1000, and 1229 primes less than 10000. At what rate do the primes thin out? Today we use the notation π(x) to denote the number of primes less than or equal to x; so π(1000) = 168. Carl Friedrich Gauss in an 1849 letter to his former student Encke provided us with the answer to this question. Gauss described his work from 58 years earlier (when he was 15 or 16) where he came to the conclusion that the likelihood of a number n being prime, without knowing anything about it except its size, is 1 : log n Since log 10 = 2:303 ::: the means that about 1 in 16 seven digit numbers are prime and the 100 digit primes are spaced apart by about 230. Gauss came to his conclusion empirically: he kept statistics on how many primes there are in each sequence of 100 numbers all the way up to 3 million or so! He claimed that he could count the primes in a chiliad (a block of 1000) in 15 minutes! Thus we expect that 1 1 1 1 π(N) ≈ + + + ··· + : log 2 log 3 log 4 log N This is within a constant of Z N du li(N) = 0 log u so Gauss' conjecture may be expressed as π(N) = li(N) + E(N) Date: January 27, 2015. 1 2 BRIAN CONREY where E(N) is an error term. -
Fall 2016 Is Available in the Laboratory of Dr
RNA Society Newsletter Aug 2016 From the Desk of the President, Sarah Woodson Greetings to all! I always enjoy attending the annual meetings of the RNA Society, but this year’s meeting in Kyoto was a standout in my opinion. This marked the second time that the RNA meeting has been held in Kyoto as a joint meeting with the RNA Society of Japan. (The first time was in 2011). Particular thanks go to the local organizers Mikiko Siomi and Tom Suzuki who took care of many logistical details, and to all of the organizers, Mikiko, Tom, Utz Fischer, Wendy Gilbert, David Lilley and Erik Sontheimer, for putting together a truly exciting and stimulating scientific program. Of course, the real excitement in the annual RNA meetings comes from all of you who give the talks and present the posters. I always enjoy meeting old friends and colleagues, but the many new participants in this year’s meeting particularly encouraged me. (Continued on p2) In this issue : Desk of the President, Sarah Woodson 1 Highlights of RNA 2016 : Kyoto Japan 4 Annual Society Award Winners 4 Jr Scientist activities 9 Mentor Mentee Lunch 10 New initiatives 12 Desk of our CEO, James McSwiggen 15 New Volunteer Opportunities 16 Chair, Meetings Committee, Benoit Chabot 17 Desk of the Membership Chair, Kristian Baker 18 Thank you Volunteers! 20 Meeting Reports: RNA Sponsored Meetings 22 Upcoming Meetings of Interest 27 Employment 31 1 Although the graceful city of Kyoto and its cultural months. First, in May 2016, the RNA journal treasures beckoned from just beyond the convention instituted a uniform price for manuscript publication hall, the meeting itself held more than enough (see p 12) that simplifies the calculation of author excitement to keep ones attention! Both the quality fees and facilitates the use of color figures to and the “polish” of the scientific presentations were convey scientific information. -
Interactive Proofs
Interactive proofs April 12, 2014 [72] L´aszl´oBabai. Trading group theory for randomness. In Proc. 17th STOC, pages 421{429. ACM Press, 1985. doi:10.1145/22145.22192. [89] L´aszl´oBabai and Shlomo Moran. Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes. J. Comput. System Sci., 36(2):254{276, 1988. doi:10.1016/0022-0000(88)90028-1. [99] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic ex- ponential time has two-prover interactive protocols. In Proc. 31st FOCS, pages 16{25. IEEE Comp. Soc. Press, 1990. doi:10.1109/FSCS.1990.89520. See item 1991.108. [108] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic expo- nential time has two-prover interactive protocols. Comput. Complexity, 1 (1):3{40, 1991. doi:10.1007/BF01200056. Full version of 1990.99. [136] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. In Proc. 34th FOCS, pages 724{733, Palo Alto CA, 1993. IEEE Comp. Soc. Press. doi:10.1109/SFCS.1993.366815. Conference version of item 1997:160. [160] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. System Sci., 54(2):317{331, 1997. doi:10.1006/jcss.1997.1472. Full version of 1993.136. [111] L´aszl´oBabai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. In Proc. -
Four Results of Jon Kleinberg a Talk for St.Petersburg Mathematical Society
Four Results of Jon Kleinberg A Talk for St.Petersburg Mathematical Society Yury Lifshits Steklov Institute of Mathematics at St.Petersburg May 2007 1 / 43 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 / 43 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 2 / 43 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 2 / 43 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 2 / 43 Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams 2 / 43 Part I History of Nevanlinna Prize Career of Jon Kleinberg 3 / 43 Nevanlinna Prize The Rolf Nevanlinna Prize is awarded once every 4 years at the International Congress of Mathematicians, for outstanding contributions in Mathematical Aspects of Information Sciences including: 1 All mathematical aspects of computer science, including complexity theory, logic of programming languages, analysis of algorithms, cryptography, computer vision, pattern recognition, information processing and modelling of intelligence. -
Research Notices
AMERICAN MATHEMATICAL SOCIETY Research in Collegiate Mathematics Education. V Annie Selden, Tennessee Technological University, Cookeville, Ed Dubinsky, Kent State University, OH, Guershon Hare I, University of California San Diego, La jolla, and Fernando Hitt, C/NVESTAV, Mexico, Editors This volume presents state-of-the-art research on understanding, teaching, and learning mathematics at the post-secondary level. The articles are peer-reviewed for two major features: (I) advancing our understanding of collegiate mathematics education, and (2) readability by a wide audience of practicing mathematicians interested in issues affecting their students. This is not a collection of scholarly arcana, but a compilation of useful and informative research regarding how students think about and learn mathematics. This series is published in cooperation with the Mathematical Association of America. CBMS Issues in Mathematics Education, Volume 12; 2003; 206 pages; Softcover; ISBN 0-8218-3302-2; List $49;AII individuals $39; Order code CBMATH/12N044 MATHEMATICS EDUCATION Also of interest .. RESEARCH: AGul<lelbrthe Mathematics Education Research: Hothomatldan- A Guide for the Research Mathematician --lllll'tj.M...,.a.,-- Curtis McKnight, Andy Magid, and -- Teri J. Murphy, University of Oklahoma, Norman, and Michelynn McKnight, Norman, OK 2000; I 06 pages; Softcover; ISBN 0-8218-20 16-8; List $20;AII AMS members $16; Order code MERN044 Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom Graduate Student Edition Faculty -
Navigability of Small World Networks
Navigability of Small World Networks Pierre Fraigniaud CNRS and University Paris Sud http://www.lri.fr/~pierre Introduction Interaction Networks • Communication networks – Internet – Ad hoc and sensor networks • Societal networks – The Web – P2P networks (the unstructured ones) • Social network – Acquaintance – Mail exchanges • Biology (Interactome network), linguistics, etc. Dec. 19, 2006 HiPC'06 3 Common statistical properties • Low density • “Small world” properties: – Average distance between two nodes is small, typically O(log n) – The probability p that two distinct neighbors u1 and u2 of a same node v are neighbors is large. p = clustering coefficient • “Scale free” properties: – Heavy tailed probability distributions (e.g., of the degrees) Dec. 19, 2006 HiPC'06 4 Gaussian vs. Heavy tail Example : human sizes Example : salaries µ Dec. 19, 2006 HiPC'06 5 Power law loglog ppk prob{prob{ X=kX=k }} ≈≈ kk-α loglog kk Dec. 19, 2006 HiPC'06 6 Random graphs vs. Interaction networks • Random graphs: prob{e exists} ≈ log(n)/n – low clustering coefficient – Gaussian distribution of the degrees • Interaction networks – High clustering coefficient – Heavy tailed distribution of the degrees Dec. 19, 2006 HiPC'06 7 New problematic • Why these networks share these properties? • What model for – Performance analysis of these networks – Algorithm design for these networks • Impact of the measures? • This lecture addresses navigability Dec. 19, 2006 HiPC'06 8 Navigability Milgram Experiment • Source person s (e.g., in Wichita) • Target person t (e.g., in Cambridge) – Name, professional occupation, city of living, etc. • Letter transmitted via a chain of individuals related on a personal basis • Result: “six degrees of separation” Dec.