DOCSLIB.ORG

Variant Construction from Theoretical Foundation to Applications Variant Construction from Theoretical Foundation to Applications Jeffrey Zheng Editor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Variant Construction from Theoretical Foundation to Applications Variant Construction from Theoretical Foundation to Applications Jeffrey Zheng Editor Jeffrey Zheng Editor Variant Construction from Theoretical Foundation to Applications Variant Construction from Theoretical Foundation to Applications Jeffrey Zheng Editor Variant Construction from Theoretical Foundation to Applications 123 Editor Jeffrey Zheng School of Software Yunnan University Kunming, Yunnan, China ISBN 978-981-13-2281-5 ISBN 978-981-13-2282-2 (eBook) https://doi.org/10.1007/978-981-13-2282-2 Library of Congress Control Number: 2018958351 © The Editor(s) (if applicable) and The Author(s) 2019. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi- cation does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Dedicated to I-Ching—First Variant Construction Lan Z. Yin & Su M. Zheng—Mother & Father Qing S. Gao—Mentor on Parallel Sorting Algorithm & Computer Architecture Tasiyasu L. Kunii—Master on Meta Knowledge Bob Beaumont—Adviser on Optimization Ping Zhang—Wife Graduate School of USTC & UCAS—40-th Anniversary (1978–2018) Foreword Dr. Jeffrey Zheng was one of the first postgraduate students supervised by Prof. Qingshi Gao (Member, Chinese Academy of Sciences) at the Institute of Computing Technology, Chinese Academy of Sciences. I have known Dr. Zheng for 40 years since then. Building upon his postgraduate work (Parallel Sorting Algorithm and 0-1 Transformation), Dr. Zheng has made significant contribution to the field of Variant Construction, ranging from theoretical foundations to various applications. His research has been published at many academic journals and conferences. For the convenience of readers, Dr. Zheng compiled his representative works of 40 years into two monographs with complementary contents. I believe that professionals in related fields will find this book both an excellent reference and a source of inspi- ration. Other readers will enjoy this book as an introduction to topics of Variant Construction. I am very happy to recommend this book in the form of a foreword. Beijing, China Yunmei Dong April 2018 Professor, The Institute of Software Chinese Academy of Sciences Member, Chinese Academy of Sciences As head of the R&D team for Lenovo Chinese Systems, I am very pleased to see the research work of former colleague Dr. Jeffrey Zheng, which began 30 years ago with the “Smoothly Enlarging Chinese Font Algorithm of 0-1 logic operations” at the Institute of Computing Technology of the Chinese Science Academy. His most recent work “Variant Construction” is summarized as a professional monograph. I expect this new measurement system to be used efficiently for advanced crypto- graphic tests in modern cyberspace security. I am pleased to give this foreword. Beijing, China Guangnan Ni April 2018 Professor, The Institute of Computing Technology Chinese Academy of Sciences Member, Chinese Academy of Engineering vii viii Foreword Dr. Jeffrey Zheng and I were in the first group of postgraduates major in Computer Architecture at the Graduate School of the Chinese Academy of Sciences 40 years ago. Professor Qingshi Gao (Member, Chinese Academy of Sciences) supervised him in particular in the areas of parallel algorithm and computer architecture. Dr. Jeffrey Zheng is one of the few classmates who continue to works in basic research and advanced applications. It is great for Dr. Jeffrey Zheng to collect his research work in a monograph. Variant Measurement Technology could be used in the next generation of Quantum Cryptographic Communication Services. On the occasion of the 40th anniversary of the Graduate School of Chinese Academy of Sciences, I would like to express my good wishes as a classmate for this monograph in the foreword. Beijing, China Guojie Li April 2018 Professor, The Institute of Computing Technology Chinese Academy of Sciences Member, Chinese Academy of Engineering Preface Associated with the fast development of science and technology in the twenty-first century, the modern computer and communication system in optical fiber com- munication supporting the global Internet shows profound influence on society and economy. As a result, globalization has become an extremely important issue in social and economic systems. The Internet and optical fiber communication systems have revolutionized the geographic and communication patterns of the world, by creating an open era of integrated global Internet connectivity. Quantum key communication technology and quantum entanglement experiments on a quantum satellite represent typical examples of China’s world-leading science and technol- ogy from the perspective of frontier application research. The latest achievements of artificial intelligence, which is the lead of Alpha-Go, show the potential intelligence prospect of advanced technology based on deep learning, artificial neural networks, and knowledge-based support vector machine systems. Related achievements are very attractive, such as poetry robots, service robots, industrial robots, face recognition, gesture recognition, unmanned aerial vehicles, self-driving cars, and unmanned underwater vehicles. A list of military and civilian high-tech achieve- ments supports daily life with rich and colorful intelligent products. From the viewpoint of mathematics and logics, the foundation framework to design and simulate both modern computer systems and optical fiber communi- cation networks is dependent on the 0-1 logical system and representations of multiple bit states. For integrated circuits, the theoretical basis can be traced back to the 1930s. Shannon developed the Boolean algebra to design circuits establishing switch circuit theory, Turing proposed the Turing machine, and von Neumann established a modern computer architecture. After more than 50 years of devel- opment follows Moore’s Law: the observation that the number of transistors in a dense integrated circuit doubles approximately every 2 years. Optimization of very large-scale integrated circuit technology appears everywhere with evolution of magical functions. ix x Preface Looking ahead, the development of advanced science and technology is subject to the limitations of basic theory and applications on foundational supports. From the perspective of basic research, how we can extend this classical level is a very interesting issue and an extremely difficult research topic. Purpose of This Book After four decades of deep exploration on 0-1 logical systems, the authors expended vector 0-1 logical systems to establish a variant logic framework in 2010. After further research and development for one decade, three theoretical components were established: variant logic, variant measurement, and variant map. At the same time, various sample applications were investigated and developed. However, because most published papers are scattered in professional journals, conference proceedings, and academic books, it is difficult for other people to obtain com- prehensive information on the topic. In addition, each article may be focused on a specific issue, and it is difficult for readers to understand the whole structure from a few papers. We are going to organize relevant papers in this book, which will be the first book on variant construction with intrinsic logical connections on the selected papers. Selected papers are composed of different parts. Based on this architecture, different readers can easily access suitable content from specific chapters. The Need for a New Logic System In modern computer and communication systems, the theory of switch circuits uses multiple bits, states, and logic operations for state automata and combinatorial logic units to design and implement complex computing and communication systems. For solving linear equations with n variables as algebraic equation, Boolean equation or differential equation, it is useful to apply a matrix associated with a set of eigenvectors. Matrices and eigenvalues are valid to provide solutions on periodic problems of special basis
Load more

Recommended publications
  • Handbook Vatican 2014
    HANDBOOK OF THE WORLD CONGRESS ON THE SQUARE OF OPPOSITION IV Pontifical Lateran University, Vatican May 5-9, 2014 Edited by Jean-Yves Béziau and Katarzyna Gan-Krzywoszyńska www.square-of-opposition.org 1 2 Contents 1. Fourth World Congress on the Square of Opposition ..................................................... 7 1.1. The Square : a Central Object for Thought ..................................................................................7 1.2. Aim of the Congress.....................................................................................................................8 1.3. Scientific Committee ...................................................................................................................9 1.4. Organizing Committee .............................................................................................................. 10 2. Plenary Lectures ................................................................................................... 11 Gianfranco Basti "Scientia una contrariorum": Paraconsistency, Induction, and Formal Ontology ..... 11 Jean-Yves Béziau Square of Opposition: Past, Present and Future ....................................................... 12 Manuel Correia Machuca The Didactic and Theoretical Expositions of the Square of Opposition in Aristotelian logic ....................................................................................................................... 13 Rusty Jones Bivalence and Contradictory Pairs in Aristotle’s De Interpretatione ................................
    [Show full text]
  • Proceedings of the 30 Nordic Workshop on Programming Theory
    Proceedings of the 30th Nordic Workshop on Programming Theory Daniel S. Fava Einar B. Johnsen Olaf Owe Editors Report Nr. 485 ISBN 978-82-7368-450-9 Department of Informatics Faculty of Mathematics and Natural Sciences UNIVERSITY OF OSLO October 24, 2018 NWPT'18 Preface Preface This volume contains the extended abstracts of the talks presented at the 30th Nordic Workshop on Programming Theory held on October 24-26, 2018 in Oslo, Norway. The objective of Nordic Workshop on Programming Theory is to bring together researchers from the Nordic and Baltic countries interested in programming theory, in order to improve mutual contacts and co-operation. However, the workshop also attracts researchers outside this geographical area. In particular, it is targeted at early- stage researchers as a friendly meeting where one can present work in progress. Typical topics of the workshop include: • semantics of programming languages, • programming language design and programming methodology, • programming logics, • formal specification of programs, • program verification, • program construction, • tools for program verification and construction, • program transformation and refinement, • real-time and hybrid systems, • models of concurrency and distributed computing, • language-based security. This volume contains 29 extended abstracts of the presentations at the workshop including the abstracts of the three distinguished invited speakers. Prof. Andrei Sabelfeld, Chalmers University, Sweden Prof. Erika Abrah´am,RWTH´ Aachen University, Germany Prof. Peter Olveczky,¨ University of Oslo, Norway After the workshop selected papers will be invited, based on the quality and topic of their presentation at the workshop, for submission to a special issue of The Journal of Logic and Algebraic Methods in Programming.
    [Show full text]
  • Model Based Decision Support for Creation and Operation of Sustainable Infrastructure
    Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009 Model Based Decision Support for Creation And Operation of Sustainable Infrastructure Igor Nikolic Member IEEE, Gerard P.J. Dijkema Member IEEE, Emile Chappin Member IEEE, Chris Davis Member IEEE Faculty of Technology, Policy and Management Delft University of Technology Delft, Netherlands Abstract—This paper focuses on the use of system thinking, II. PARADIGM AND THEORY complex adaptive systems theory and modeling to help create and operate sustainable energy infrastructure. These are capital- A. Large Scale Socio-Technical Systems intensive, long-lived large scale socio-technical systems that evolve The global industrial system and the infrastructure systems over decades. To enable high-level decision support on climate that increasingly form its backbone can be seen as Large Scale policy, strategic investment decisions or transition management, λ we are evolving a simulation engine for Agent-Based simulation of Socio-Technical Systems [7], or -systems [8]. This is a class infrastructure development. The key functionality and capability of systems that span technical artifacts embedded in a social of this engine is its ability for playing out possible futures, to network. Large Scale Socio-Technical Systems include social identify development patterns and conditions for sustainability elements such as operating companies, investors, local and by exploring multiple scenarios. Two of the models completed national governments, regional development agencies, non- serve to illustrate the approach. These address the ecological governmental organizations, customers and institutions. These impact of bio-electricity and the long-term effect of CO2 policy on the electricity production technology portfolio.
    [Show full text]
  • Web Corrections-Vector Logic and Counterfactuals-Rev 2
    This article was published in: "Logic Journal of IGPL" Online version: doi:10.1093/jigpal/jzaa026, 2020 Vector logic allows counterfactual virtualization by The Square Root of NOT Eduardo Mizraji Group of Cognitive Systems Modeling, Biophysics and Systems Biology Section, Facultad de Ciencias, Universidad de la República Montevideo, Uruguay Abstract In this work we investigate the representation of counterfactual conditionals using the vector logic, a matrix-vectors formalism for logical functions and truth values. Inside this formalism, the counterfactuals can be transformed in complex matrices preprocessing an implication matrix with one of the square roots of NOT, a complex matrix. This mathematical approach puts in evidence the virtual character of the counterfactuals. This happens because this representation produces a valuation of a counterfactual that is the superposition of the two opposite truth values weighted, respectively, by two complex conjugated coefficients. This result shows that this procedure gives an uncertain evaluation projected on the complex domain. After this basic representation, the judgment of the plausibility of a given counterfactual allows us to shift the decision towards an acceptance or a refusal. This shift is the result of applying for a second time one of the two square roots of NOT. Keywords : Vector logic; Conditionals; Counterfactuals; Square Roots of NOT. Address: Dr. Eduardo Mizraji Sección Biofísica y Biología de Sistemas, Facultad de Ciencias, UdelaR Iguá 4225, Montevideo 11400, Uruguay e-mails: 1) [email protected], 2) [email protected] Phone-Fax: +598 25258629 1 1. Introduction The comprehension and further formalization of the conditionals have a long and complex history, full of controversies, famously described in an article of Lukasiewicz [20].
    [Show full text]
  • Can Neoclassical Economics Handle Complexity? the Fallacy of the Oil Spot Dynamic Magda Fontana
    Can Neoclassical Economics Handle Complexity? The Fallacy of the Oil Spot Dynamic Magda Fontana To cite this version: Magda Fontana. Can Neoclassical Economics Handle Complexity? The Fallacy of the Oil Spot Dynamic. Journal of Economic Behavior and Organization, Elsevier, 2010, 76 (3), pp.584. 10.1016/j.jebo.2010.08.010. hal-00911826 HAL Id: hal-00911826 https://hal.archives-ouvertes.fr/hal-00911826 Submitted on 30 Nov 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Accepted Manuscript Title: Can Neoclassical Economics Handle Complexity? The Fallacy of the Oil Spot Dynamic Author: Magda Fontana PII: S0167-2681(10)00163-0 DOI: doi:10.1016/j.jebo.2010.08.010 Reference: JEBO 2590 To appear in: Journal of Economic Behavior & Organization Received date: 17-9-2008 Revised date: 25-8-2010 Accepted date: 26-8-2010 Please cite this article as: Fontana, M., Can Neoclassical Economics Handle Complexity? The Fallacy of the Oil Spot Dynamic, Journal of Economic Behavior and Organization (2010), doi:10.1016/j.jebo.2010.08.010 This is a PDF file of an unedited manuscript that has been accepted for publication.
    [Show full text]
  • Metaphsyics of Knowledge.Pdf
    The Metaphysics of Knowledge This page intentionally left blank The Metaphysics of Knowledge Keith Hossack 1 1 Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York Keith Hossack 2007 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Hossack, Keith.
    [Show full text]
  • Complex Adaptive Systems and Agent Based Modeling of Industrial Network Growth
    Complex Adaptive Systems and Agent Based Modeling of Industrial Network Growth Chris Davis Igor Nikolic Faculty of Technology, Policy and Management Delft University of Technology The Netherlands wiki.tudelft.nl [email protected] Who am I, and why am I here ? Chris Davis ➲ Previously an engineer at Dell z Signal Integrity/Electrical Analysis MSc in Industrial Ecology Thesis on combining Life Cycle Analysis within Agent Based Modeling Currently doing my Doctoral research at TPM / TUDelft, section Energy and Industry I will talk to you about : ➲ Complex Adaptive Systems ➲ Industrial Networks ➲ Agent Based Modeling Take Home (meta)Message ➲ The world around is Complex, both the biosphere and the Technosphere ➲ Any activity within both must acknowledge this, or fail miserably ➲ Complex Adaptive Systems is a Meta-Theory, that gives you a world view and enables cross- disciplinary knowledge sharing. ➲ These are very powerful tools to understand / model the world, but are not mainstream yet. ➲ You don't need to understand everything now, but you will know how to find it, when you need it... ➲ Yes, you can fall asleep now... Defender – Attacker game ➲ Lets clear the tables, and stand in a big circle... ➲ Randomly select 2 people : z A: The Defender z B: The Attacker ➲ You are The Target ! ➲ Move so that you always keep The Defender between you and The Attacker ➲ Source, Eric Bonabeu, http://www.icosystem.com/game.htm STOP ! ➲ The Defender is now the Target ➲ You are The Defender ! ➲ Move so that you keep yourself between The Attacker and The Target. What the heck just happened ? Take Home Message ➲ Simple rules of individual behavior can lead to surprisingly coherent system level results.
    [Show full text]
  • The Application of Agent-Based Models to Marketing Science I Refer to As Virtual Markets
    VIRTUAL MARKETS: THE APPLICATION OF AGENT-BASED MODELING TO MARKETING SCIENCE by Roger A. Parker Submitted to the School of Marketing, Faculty of Business in fulfillment of the requirements for the degree of Doctor of Philosophy at the UNIVERSITY OF TECHNOLOGY SYDNEY September 2010 © 2010 Roger A. Parker. All rights reserved. Permission is herewith granted the University of Technology Sydney to circulate and to have copied for non-commercial purposes at its discretion, the above title upon the request of individuals and institutions. ii Certification of Authorship and Originality I certify that the work in this thesis has not previously been submitted for a degree nor has it been submitted as part of the requirements for a degree except as fully acknowledged within the text. I also certify that the thesis has been written by me. Any help that I have received in my research work and the preparation of the thesis itself has been acknowledged. In addition, I certify that all information sources and literature used are indicated in the thesis. Author:_______________________________________________________ Roger A. Parker, UTS ID #10247548. 26 September 2010 iii iv Dedication To Anita. Without her love and encouragement, this dissertation would not exist. And, quite literally, neither would I. v Abstract Advances in any science ultimately depend on the creation of instruments that can create observations from which theories can be hypothesized and tested. This dissertation proposes that significant advances in marketing science can be realized with the engagement of the advanced computational science technique of agent-based modeling. To support this proposition, the methodology is examined from first principles to concrete implementation.
    [Show full text]
  • Eigenlogic in the Spirit of George Boole
    Eigenlogic in the spirit of George Boole Zeno TOFFANO July 1, 2019 CentraleSupelec - Laboratoire des Signaux et Systèmes (L2S-UMR8506) - CNRS - Université Paris-Saclay 3 rue Joliot-Curie, F-91190 Gif-sur-Yvette, FRANCE zeno.toff[email protected] Abstract This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole’s theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will appear as commuting projection operators and the truth values, which take the binary values {0, 1}, are the respective eigenvalues. In this view the solution of a logical proposition resulting from the operation on a combination of arguments will appear as a selection where the outcome can only be one of the eigenvalues. In this way propositional logic can be formalized in linear algebra by using elective developments which correspond here to combinations of tensored elementary projection operators. The original and principal motivation of this work is for applications in the new field of quantum information, differences are outlined with more traditional quantum logic approaches. 1 Introduction The year 2015 celebrated discretely the 200th anniversary of the birth of George Boole (1815-1864). His visionary approach to logic has led to the formalization in simple mathematical language what was prior to him a language and philosophy oriented discipline.
    [Show full text]
  • A Transfer Function Model for Ternary Switching Logic Circuits
    A Transfer Function Model for Ternary Switching Logic Circuits Mitchell A. Thornton Southern Methodist University Dallas, Texas USA 75275–0122 Email: [email protected] Abstract an interconnection of symbols or gates whose function- ality is modeled by the operators within the algebraic Ternary switching functions are formulated as trans- structure. A common set of ternary logic network formations over vector spaces resulting in a character- elements are the MIN, MAX, and Literal Selection ization in the form of a transfer function. Ternary logic gates whose corresponding functionalities provide for constants are modeled as vectors, thus the transfer the specification of a functionally complete algebra functions are of the form of matrices that map vectors with constants. For conciseness we refer to the Literal representing logic network input values to correspond- Selection gate as a Ji gate where i denotes the polarity ing output vectors. Techniques for determination of the of the selected literal, i 2 f0; 1; 2g as defined in [2]. transfer matrix from a logic switching model or directly Here we use three-dimensional vectors to represent from a netlist are provided. The use of transfer matrices ternary logic values instead of the integers f0; 1; 2g for logic network simulation are then developed that and we model a ternary logic function or a ternary allow for multiple output responses to be obtained logic network by a characterizing transfer matrix. The through a single vector-matrix product calculation. transfer matrix transforms the logic network input vectors to corresponding output vectors. This linear algebraic model of a ternary logic network or function 1.
    [Show full text]
  • Philosophy in Reality: Scientific Discovery and Logical Recovery
    philosophies Article Philosophy in Reality: Scientific Discovery and Logical Recovery Joseph E. Brenner 1,2,* and Abir U. Igamberdiev 3 1 International Center for the Philosophy of Information, Xi’an Social Sciences University, Xi’an 710049, China 2 Chemin du Collège 1, 1865 Les Diablerets, Switzerland 3 Department of Biology, Memorial University of Newfoundland, St. John’s, NL A1B 3X9, Canada; [email protected] * Correspondence: [email protected] Received: 31 March 2019; Accepted: 6 May 2019; Published: 14 May 2019 Abstract: Three disciplines address the codified forms and rules of human thought and reasoning: logic, available since antiquity; dialectics as a process of logical reasoning; and semiotics which focuses on the epistemological properties of the extant domain. However, both the paradigmatic-historical model of knowledge and the logical-semiotic model of thought tend to incorrectly emphasize the separation and differences between the respective domains vs. their overlap and interactions. We propose a sublation of linguistic logics of objects and static forms by a dynamic logic of real physical-mental processes designated as the Logic in Reality (LIR). In our generalized logical theory, dialectics and semiotics are recovered from reductionist interpretations and reunited in a new synthetic paradigm centered on meaning and its communication. Our theory constitutes a meta-thesis composed of elements from science, logic and philosophy. We apply the theory to gain new insights into the structure and role of semiosis, information and communication and propose the concept of ‘ontolon’ to define the element of reasoning as a real dynamic process. It is part of a project within natural philosophy, which will address broader aspects of the dynamics of the growth of civilizations and their potential implications for the information society.
    [Show full text]
  • Complexity of Fixed-Size Bit-Vector Logics
    Noname manuscript No. (will be inserted by the editor) Complexity of Fixed-Size Bit-Vector Logics Gergely Kov´asznai ¨ Andreas Fr¨ohlich ¨ Armin Biere Received: date / Accepted: date Abstract Bit-precise reasoning is important for many practical applications of Satisfiability Modulo Theories (SMT). In recent years, efficient approaches for solving fixed-size bit-vector formulas have been developed. From the theo- retical point of view, only few results on the complexity of fixed-size bit-vector logics have been published. Some of these results only hold if unary encoding on the bit-width of bit-vectors is used. In our previous work [41], we have already shown that binary encoding adds more expressiveness to various fixed-size bit-vector logics with and without quantification. In a follow-up work [30], we then gave additional complexity results for several fragments of the quantifier-free case. In this paper, we revisit our complexity results from [30,41] and go into more detail when specifying the underlying logics and presenting the proofs. We give a better insight in where the additional expressiveness of binary en- coding comes from. In order to do this, we bring together our previous work and propose several new complexity results for new fragments and extensions of earlier bit-vector logics. We also discuss the expressiveness of various bit- vector operations in more detail. Altogether, we provide the currently most complete overview on the complexity of common bit-vector logics. This work is partially supported by FWF, NFN Grant S11408-N23
    [Show full text]