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Randomness and Computation
Randomness and Computation Rod Downey School of Mathematics and Statistics, Victoria University, PO Box 600, Wellington, New Zealand [email protected] Abstract This article examines work seeking to understand randomness using com- putational tools. The focus here will be how these studies interact with classical mathematics, and progress in the recent decade. A few representa- tive and easier proofs are given, but mainly we will refer to the literature. The article could be seen as a companion to, as well as focusing on develop- ments since, the paper “Calibrating Randomness” from 2006 which focused more on how randomness calibrations correlated to computational ones. 1 Introduction The great Russian mathematician Andrey Kolmogorov appears several times in this paper, First, around 1930, Kolmogorov and others founded the theory of probability, basing it on measure theory. Kolmogorov’s foundation does not seek to give any meaning to the notion of an individual object, such as a single real number or binary string, being random, but rather studies the expected values of random variables. As we learn at school, all strings of length n have the same probability of 2−n for a fair coin. A set consisting of a single real has probability zero. Thus there is no meaning we can ascribe to randomness of a single object. Yet we have a persistent intuition that certain strings of coin tosses are less random than others. The goal of the theory of algorithmic randomness is to give meaning to randomness content for individual objects. Quite aside from the intrinsic mathematical interest, the utility of this theory is that using such objects instead distributions might be significantly simpler and perhaps giving alternative insight into what randomness might mean in mathematics, and perhaps in nature. -
Part I Background
Cambridge University Press 978-0-521-19746-5 - Transitions and Trees: An Introduction to Structural Operational Semantics Hans Huttel Excerpt More information PART I BACKGROUND © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-19746-5 - Transitions and Trees: An Introduction to Structural Operational Semantics Hans Huttel Excerpt More information 1 A question of semantics The goal of this chapter is to give the reader a glimpse of the applications and problem areas that have motivated and to this day continue to inspire research in the important area of computer science known as programming language semantics. 1.1 Semantics is the study of meaning Programming language semantics is the study of mathematical models of and methods for describing and reasoning about the behaviour of programs. The word semantics has Greek roots1 and was first used in linguistics. Here, one distinguishes among syntax, the study of the structure of lan- guages, semantics, the study of meaning, and pragmatics, the study of the use of language. In computer science we make a similar distinction between syntax and se- mantics. The languages that we are interested in are programming languages in a very general sense. The ‘meaning’ of a program is its behaviour, and for this reason programming language semantics is the part of programming language theory devoted to the study of program behaviour. Programming language semantics is concerned only with purely internal aspects of program behaviour, namely what happens within a running pro- gram. Program semantics does not claim to be able to address other aspects of program behaviour – e.g. -
A Second Course in Formal Languages and Automata Theory Jeffrey Shallit Index More Information
Cambridge University Press 978-0-521-86572-2 - A Second Course in Formal Languages and Automata Theory Jeffrey Shallit Index More information Index 2DFA, 66 Berstel, J., 48, 106 2DPDA, 213 Biedl, T., 135 3-SAT, 21 Van Biesbrouck, M., 139 Birget, J.-C., 107 abelian cube, 47 blank symbol, B, 14 abelian square, 47, 137 Blum, N., 106 Ackerman, M., xi Boolean closure, 221 Adian, S. I., 43, 47, 48 Boolean grammar, 223 Aho, A. V., 173, 201, 223 Boolean matrices, 73, 197 Alces, A., 29 Boonyavatana, R., 139 alfalfa, 30, 34, 37 border, 35, 104 Allouche, J.-P., 48, 105 bordered, 222 alphabet, 1 bordered word, 35 unary, 1 Borges, M., xi always-halting TM, 209 Borwein, P., 48 ambiguous, 10 Boyd, D., 223 NFA, 102 Brady, A. H., 200 Angluin, D., 105 branch point, 113 angst Brandenburg, F.-J., 139 existential, 54 Brauer, W., 106 antisymmetric, 92 Brzozowski, J., 27, 106 assignment Bucher, W., 138 satisfying, 20 Buntrock, J., 200 automaton Burnside problem for groups, 42 pushdown, 11 Buss, J., xi, 223 synchronizing, 105 busy beaver problem, 183, 184, 200 two-way, 66 Axelrod, R. M., 105 calliope, 3 Cantor, D. C., 201 Bader, C., 138 Cantor set, 102 balanced word, 137 cardinality, 1 Bar-Hillel, Y., 201 census generating function, 133 Barkhouse, C., xi Cerny’s conjecture, 105 base alphabet, 4 Chaitin, G. J., 200 Berman, P., 200 characteristic sequence, 28 233 © Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86572-2 - A Second Course in Formal Languages and Automata Theory Jeffrey Shallit Index More information 234 Index chess, -
The Standard Model for Programming Languages: the Birth of A
The Standard Model for Programming Languages: The Birth of a Mathematical Theory of Computation Simone Martini Department of Computer Science and Engineering, University of Bologna, Italy INRIA, Sophia-Antipolis, Valbonne, France http://www.cs.unibo.it/~martini [email protected] Abstract Despite the insight of some of the pioneers (Turing, von Neumann, Curry, Böhm), programming the early computers was a matter of fiddling with small architecture-dependent details. Only in the sixties some form of “mathematical program development” will be in the agenda of some of the most influential players of that time. A “Mathematical Theory of Computation” is the name chosen by John McCarthy for his approach, which uses a class of recursively computable functions as an (extensional) model of a class of programs. It is the beginning of that grand endeavour to present programming as a mathematical activity, and reasoning on programs as a form of mathematical logic. An important part of this process is the standard model of programming languages – the informal assumption that the meaning of programs should be understood on an abstract machine with unbounded resources, and with true arithmetic. We present some crucial moments of this story, concluding with the emergence, in the seventies, of the need of more “intensional” semantics, like the sequential algorithms on concrete data structures. The paper is a small step of a larger project – reflecting and tracing the interaction between mathematical logic and programming (languages), identifying some -
Kolmogorov Complexity of Graphs John Hearn Harvey Mudd College
Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2006 Kolmogorov Complexity of Graphs John Hearn Harvey Mudd College Recommended Citation Hearn, John, "Kolmogorov Complexity of Graphs" (2006). HMC Senior Theses. 182. https://scholarship.claremont.edu/hmc_theses/182 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Applications of Kolmogorov Complexity to Graphs John Hearn Ran Libeskind-Hadas, Advisor Michael Orrison, Reader May, 2006 Department of Mathematics Copyright © 2006 John Hearn. The author grants Harvey Mudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copy- ing is by permission of the author. To disseminate otherwise or to republish re- quires written permission from the author. Abstract Kolmogorov complexity is a theory based on the premise that the com- plexity of a binary string can be measured by its compressibility; that is, a string’s complexity is the length of the shortest program that produces that string. We explore applications of this measure to graph theory. Contents Abstract iii Acknowledgments vii 1 Introductory Material 1 1.1 Definitions and Notation . 2 1.2 The Invariance Theorem . 4 1.3 The Incompressibility Theorem . 7 2 Graph Complexity and the Incompressibility Method 11 2.1 Complexity of Labeled Graphs . 12 2.2 The Incompressibility Method . -
Compressing Information Kolmogorov Complexity Optimal Decompression
Compressing information Optimal decompression algorithm Almost everybody now is familiar with compress- The definition of KU depends on U. For the trivial ÞÔ ÞÔ µ ´ µ ing/decompressing programs such as , , decompression algorithm U ´y y we have KU x Ö , , etc. A compressing program can x. One can try to find better decompression algo- be applied to any file and produces the “compressed rithms, where “better” means “giving smaller com- version” of that file. If we are lucky, the compressed plexities”. However, the number of short descrip- version is much shorter than the original one. How- tions is limited: There is less than 2n strings of length ever, no information is lost: the decompression pro- less than n. Therefore, for any fixed decompres- gram can be applied to the compressed version to get sion algorithm the number of words whose complex- n the original file. ity is less than n does not exceed 2 1. One may [Question: A software company advertises a com- conclude that there is no “optimal” decompression pressing program and claims that this program can algorithm because we can assign short descriptions compress any sufficiently long file to at most 90% of to some string only taking them away from other its original size. Would you buy this program?] strings. However, Kolmogorov made a simple but How compression works? A compression pro- crucial observation: there is asymptotically optimal gram tries to find some regularities in a file which decompression algorithm. allow to give a description of the file which is shorter than the file itself; the decompression program re- Definition 1 An algorithm U is asymptotically not µ 6 ´ µ · constructs the file using this description. -
The London University Atlas
The London University Atlas: A talk given at the Atlas 50th Anniversary Symposium on 5th December 2012. Dik Leatherdale. London >> Manchester London University was, indeed still is, a HUGE organisation even by the standards of this place. It’s a collegiate university the individual colleges spread across the capital from Queen Mary College in the east end to the baroque splendour of Royal Holloway in leafy Egham built, you may be interested to learn, upon the proceeds of Victorian patent medicines hardly any of which had any beneficial effect whatsoever. In the “build it yourself” era, two of the colleges took the plunge. At Imperial College, Keith Tocher, Sid Michaelson and Tony Brooker constructed the “Imperial College Computing Engine” using relays. With the benefit of hindsight, this looks like a dead end in development. But Imperial weren’t the only team to put their faith in relay technology. Neither were they the last. Yet amazingly, the legacy of ICCE lived on because of a chance lunchtime conversation years later between Tom Kilburn and Tony Brooker, the Atlas adder logical design was derived from that of ICCE. Over in the unlikely surroundings of Birkbeck College, Andrew Booth was building a series of low cost, low performance machines. His ambition was to build computers cheap enough for every college to be able to afford one. You might say he invented the minicomputer a decade before anybody noticed. His designs were taken up by the British Tabulating Machine company and became the ICT 1200 series; for a while the most popular computer in the UK. -
Current Issue of FACS FACTS
Issue 2021-2 July 2021 FACS A C T S The Newsletter of the Formal Aspects of Computing Science (FACS) Specialist Group ISSN 0950-1231 FACS FACTS Issue 2021-2 July 2021 About FACS FACTS FACS FACTS (ISSN: 0950-1231) is the newsletter of the BCS Specialist Group on Formal Aspects of Computing Science (FACS). FACS FACTS is distributed in electronic form to all FACS members. Submissions to FACS FACTS are always welcome. Please visit the newsletter area of the BCS FACS website for further details at: https://www.bcs.org/membership/member-communities/facs-formal-aspects- of-computing-science-group/newsletters/ Back issues of FACS FACTS are available for download from: https://www.bcs.org/membership/member-communities/facs-formal-aspects- of-computing-science-group/newsletters/back-issues-of-facs-facts/ The FACS FACTS Team Newsletter Editors Tim Denvir [email protected] Brian Monahan [email protected] Editorial Team: Jonathan Bowen, John Cooke, Tim Denvir, Brian Monahan, Margaret West. Contributors to this issue: Jonathan Bowen, Andrew Johnstone, Keith Lines, Brian Monahan, John Tucker, Glynn Winskel BCS-FACS websites BCS: http://www.bcs-facs.org LinkedIn: https://www.linkedin.com/groups/2427579/ Facebook: http://www.facebook.com/pages/BCS-FACS/120243984688255 Wikipedia: http://en.wikipedia.org/wiki/BCS-FACS If you have any questions about BCS-FACS, please send these to Jonathan Bowen at [email protected]. 2 FACS FACTS Issue 2021-2 July 2021 Editorial Dear readers, Welcome to the 2021-2 issue of the FACS FACTS Newsletter. A theme for this issue is suggested by the thought that it is just over 50 years since the birth of Domain Theory1. -
The Birth of a Mathematical Theory of Computation
The Standard Model for Programming Languages: The Birth of a Mathematical Theory of Computation Simone Martini Universit`adi Bologna and INRIA Sophia-Antipolis Bologna, 27 November 2020 Happy birthday, Maurizio! 1 / 58 This workshop: Recent Developments of the Design and Implementation of Programming Languages Well, not so recent: we go back exactly 60 years! It's more a revisionist's tale. 2 / 58 This workshop: Recent Developments of the Design and Implementation of Programming Languages Well, not so recent: we go back exactly 60 years! It's more a revisionist's tale. 3 / 58 viewpoints VDOI:10.1145/2542504 Thomas Haigh Historical Reflections HISTORY AND PHILOSOPHY OF LOGIC, 2015 Actually, Turing Vol. 36, No. 3, 205–228, http://dx.doi.org/10.1080/01445340.2015.1082050 Did Not Invent Edgar G. Daylight the Computer Separating the origins of computer science and technology.Towards a Historical Notion of ‘Turing—the Father of Computer Science’ viewpoints HE 100TH ANNIVERSARY of the birth of Alan Turing was cel- EDGAR G. DAYLIGHT ebrated in 2012. The com- viewpointsputing community threw its Utrecht University, The Netherlands biggest ever birthday party. [email protected]:10.1145/2658985V TMajor events were organized around the world, including conferences or festi- vals in Princeton, Cambridge, Manches- Viewpoint Received 14 January 2015 Accepted 3 March 2015 ter, and Israel. There was a concert in Seattle and an opera in Finland. Dutch In the popular imagination, the relevance of Turing’s theoretical ideas to people producing actual machines was and French researchers built small Tur- Why Did Computer ing Machines out of Lego Mindstorms significant and appreciated by everybody involved in computing from the moment he published his 1936 paper kits. -
The Discoveries of Continuations
LISP AND SYMBOLIC COMPUTATION: An InternationM JournM, 6, 233-248, 1993 @ 1993 Kluwer Academic Publishers - Manufactured in The Nett~eriands The Discoveries of Continuations JOHN C. REYNOLDS ( [email protected] ) School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3890 Keywords: Semantics, Continuation, Continuation-Passing Style Abstract. We give a brief account of the discoveries of continuations and related con- cepts by A. van Vv'ijngaarden, A. W. Mazurkiewicz, F. L. Morris, C. P. Wadsworth. J. H. Morris, M. J. Fischer, and S. K. Abdali. In the early history of continuations, basic concepts were independently discovered an extraordinary number of times. This was due less to poor communication among computer scientists than to the rich variety of set- tings in which continuations were found useful: They underlie a method of program transformation (into continuation-passing style), a style of def- initionM interpreter (defining one language by an interpreter written in another language), and a style of denotational semantics (in the sense of Scott and Strachey). In each of these settings, by representing "the mean- ing of the rest of the program" as a function or procedure, continnations provide an elegant description of a variety of language constructs, including call by value and goto statements. 1. The Background In the early 1960%, the appearance of Algol 60 [32, 33] inspired a fi~rment of research on the implementation and formal definition of programming languages. Several aspects of this research were critical precursors of the discovery of continuations. The ability in Algol 60 to jump out of blocks, or even procedure bod- ies, forced implementors to realize that the representation of a label must include a reference to an environment. -
The Folklore of Sorting Algorithms
IJCSI International Journal of Computer Science Issues, Vol. 4, No. 2, 2009 25 ISSN (Online): 1694-0784 ISSN (Print): 1694-0814 The Folklore of Sorting Algorithms Dr. Santosh Khamitkar1, Parag Bhalchandra2, Sakharam Lokhande 2, Nilesh Deshmukh2 School of Computational Sciences, Swami Ramanand Teerth Marathwada University, Nanded (MS ) 431605, India Email : 1 [email protected] , 2 [email protected] Abstract progress. Every researcher attempted optimization in past has found stuck to his / her observations regarding sorting The objective of this paper is to review the folklore knowledge experiments and produced his/ her own results. Since seen in research work devoted on synthesis, optimization, and these results were specific to software and hardware effectiveness of various sorting algorithms. We will examine environment therein, we found abundance in the sorting algorithms in the folklore lines and try to discover the complexity analysis which possibly could be thought as tradeoffs between folklore and theorems. Finally, the folklore the folklore knowledge. We tried to review this folklore knowledge on complexity values of the sorting algorithms will be considered, verified and subsequently converged in to knowledge in past research work and came to a conclusion theorems. that it was mainly devoted on synthesis, optimization, effectiveness in working styles of various sorting Key words: Folklore, Algorithm analysis, Sorting algorithm, algorithms. Computational Complexity notations. Our work is not mere commenting earlier work rather we are putting the knowledge embedded in sorting folklore 1. Introduction with some different and interesting view so that students, teachers and researchers can easily understand them. Folklore is the traditional beliefs, legend and customs, Synthesis, in terms of folklore and theorems, hereupon current among people. -
Chiasmic Rhetoric: Alan Turing Between Bodies and Words Patricia Fancher Clemson University, [email protected]
Clemson University TigerPrints All Dissertations Dissertations 8-2014 Chiasmic Rhetoric: Alan Turing Between Bodies and Words Patricia Fancher Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Part of the Rhetoric and Composition Commons Recommended Citation Fancher, Patricia, "Chiasmic Rhetoric: Alan Turing Between Bodies and Words" (2014). All Dissertations. 1285. https://tigerprints.clemson.edu/all_dissertations/1285 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. CHIASMIC RHETORIC: ALAN TURING BETWEEN BODIES AND WORDS A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctorate of Philosophy Rhetorics, Communication, and Information Design by Patricia Fancher August 2014 Accepted by: Dr. Steven B. Katz, Committee Chair Dr. Diane Perpich Dr. Cynthia Haynes Dr. D. Travers Scott DISSERTATION ABSTRACT This dissertation analyzes the life and writing of inventor and scientist Alan Turing in order to define and theorize chiasmic relations between bodies and texts. Chiasmic rhetoric, as I develop throughout the dissertation, is the dynamic processes between materials and discourses that interact to construct powerful rhetorical effect, shape bodies, and also compose new knowledges. Throughout the dissertation, I develop chiasmic rhetoric as intersecting bodies and discourse, dynamic and productive, and potentially destabilizing. Turing is an unusual figure for research on bodily rhetoric and embodied knowledge. He is often associated with disembodied knowledge and as his inventions are said to move intelligence towards greater abstraction and away from human bodies.