Automated Theorem Proving - Summary of Lecture 1
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Mathematical Logic Part One
Mathematical Logic Part One Question: How do we formalize the definitions and reasoning we use in our proofs? Where We're Going ● Propositional Logic (Today) ● Basic logical connectives. ● Truth tables. ● Logical equivalences. ● First-Order Logic (Wednesday/Friday) ● Reasoning about properties of multiple objects. Propositional Logic A proposition is a statement that is, by itself, either true or false. Some Sample Propositions ● Puppies are cuter than kittens. ● Kittens are cuter than puppies. ● Usain Bolt can outrun everyone in this room. ● CS103 is useful for cocktail parties. ● This is the last entry on this list. More Propositions ● They say time's supposed to heal ya. ● But I ain't done much healing. ● I'm in California dreaming about who we used to be. ● I've forgotten how it felt before the world fell at our feet. ● There's such a difference between us. Things That Aren't Propositions CommandsCommands cannotcannot bebe truetrue oror false.false. Things That Aren't Propositions QuestionsQuestions cannotcannot bebe truetrue oror false.false. Things That Aren't Propositions TheThe firstfirst halfhalf is is aa validvalid proposition.proposition. I am the walrus, goo goo g©joob JibberishJibberish cannotcannot bebe truetrue oror false. false. Propositional Logic ● Propositional logic is a mathematical system for reasoning about propositions and how they relate to one another. ● Every statement in propositional logic consists of propositional variables combined via propositional connectives. ● Each variable represents some proposition, such as “You liked it” or “You should have put a ring on it.” ● Connectives encode how propositions are related, such as “If you liked it, then you should have put a ring on it.” Propositional Variables ● Each proposition will be represented by a propositional variable. -
A Friendly Introduction to Mathematical Logic Christopher C
SUNY Geneseo KnightScholar Geneseo Authors Milne Library Publishing 1-1-2015 A Friendly Introduction to Mathematical Logic Christopher C. Leary SUNY Geneseo Lars Kristiansen Follow this and additional works at: https://knightscholar.geneseo.edu/geneseo-authors This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Recommended Citation Leary, Christopher C. and Kristiansen, Lars, "A Friendly Introduction to Mathematical Logic" (2015). Geneseo Authors. 6. https://knightscholar.geneseo.edu/geneseo-authors/6 This Book is brought to you for free and open access by the Milne Library Publishing at KnightScholar. It has been accepted for inclusion in Geneseo Authors by an authorized administrator of KnightScholar. For more information, please contact [email protected]. oduc Intr tio y n dl to n ie r F A Mathematical Logic 2nd Edition Christopher C. Leary Lars Kristiansen A Friendly Introduction to Mathematical Logic A Friendly Introduction to Mathematical Logic 2nd Edition Second printing With corrections and some renumbered exercises Christopher C. Leary State University of New York College at Geneseo Lars Kristiansen The University of Oslo Milne Library, SUNY Geneseo, Geneseo, NY c 2015 Christopher C. Leary and Lars Kristiansen ISBN: 978-1-942341-07-9 Milne Library SUNY Geneseo One College Circle Geneseo, NY 14454 Lars Kristiansen has received financial support from the Norwegian Non-fiction Literature Fund Contents Preface ix 1 Structures and Languages 1 1.1 Na¨ıvely . .3 1.2 Languages . .5 1.3 Terms and Formulas . .9 1.4 Induction . 13 1.5 Sentences . 19 1.6 Structures . 22 1.7 Truth in a Structure . -
A Concise Introduction to Mathematical Logic
Wolfgang Rautenberg A Concise Introduction to Mathematical Logic Textbook Third Edition Typeset and layout: The author Version from June 2009 corrections included Foreword by Lev Beklemishev, Moscow The field of mathematical logic—evolving around the notions of logical validity, provability, and computation—was created in the first half of the previous century by a cohort of brilliant mathematicians and philosophers such as Frege, Hilbert, Gödel, Turing, Tarski, Malcev, Gentzen, and some others. The development of this discipline is arguably among the highest achievements of science in the twentieth century: it expanded mathe- matics into a novel area of applications, subjected logical reasoning and computability to rigorous analysis, and eventually led to the creation of computers. The textbook by Professor Wolfgang Rautenberg is a well-written in- troduction to this beautiful and coherent subject. It contains classical material such as logical calculi, beginnings of model theory, and Gödel’s incompleteness theorems, as well as some topics motivated by applica- tions, such as a chapter on logic programming. The author has taken great care to make the exposition readable and concise; each section is accompanied by a good selection of exercises. A special word of praise is due for the author’s presentation of Gödel’s second incompleteness theorem, in which the author has succeeded in giving an accurate and simple proof of the derivability conditions and the provable Σ1-completeness, a technically difficult point that is usually omitted in textbooks of comparable level. This work can be recommended to all students who want to learn the foundations of mathematical logic. v Preface The third edition differs from the second mainly in that parts of the text have been elaborated upon in more detail. -
Smartbpm Vs. Eclipse IDE
In collaboration with SmartBPM vs. Eclipse IDE A study to compare the productivity of SmartBPM and Eclipse Java IDE Table of Contents 1. Executive Summary 1 2. About the Study 3 3. Why Eclipse IDE? 5 4. Assumptions 7 5. Team Composition 8 6. Measuring Productivity 10 6.1. Productivity Metrics Comparison and Interpretation 11 6.2. Analysis and Design 12 6.3. Development 12 6.4. Information Model 14 6.5. Integration 15 6.6. Generating Reports 16 6.7. Testing 17 6.8. Build 17 6.9. Deployment 18 6.10. Business Changes and Reverting the Changes 19 6.11. Process Flow 21 ® 7. Demonstrated Advantages of Pega SmartBPM over Java Eclipse IDE 23 8. Conclusion 25 Appendix A – About Pegasystems Inc. 26 Appendix B – About Capgemini 27 Appendix C – Time Comparison Data 28 Appendix D – Applications Developed 30 Financial Services | the way we see it 1. Executive Summary SmartBPM Suite®, a product of Pegasystems Inc., is a business process management (BPM) solution built on the PegaRULES® automated decision engine. SmartBPM provides process and practice rules in one integrated package, enabling users to develop, execute, manage, maintain, and grow their own decision-intensive BPM applications efficiently. The product is built in Java and uses an XML data structure, generates Java at runtime, and runs on leading J2EE application servers. Capgemini was engaged by Pegasystems Inc. to conduct a study comparing SmartBPM with Java Eclipse IDE across the entire application development lifecycle, from Analysis and Design through Deployment and Maintenance. Pegasystems Inc. asked Capgemini to design and develop applications for vehicle-insurance quote generation and homeowner-quote request, once with SmartBPM and once with Eclipse IDE. -
Contributing to the Eclipse IDE Project Principles, Plug-Ins and Gerrit Code Review
Contributing to the Eclipse IDE Project Principles, Plug-ins and Gerrit Code Review Contributing to the Eclipse IDE Project: Principles, Plug-ins and Gerrit Code Review Lars Vogel First edition Publication date 30.10.2015 Copyright © 2007 - 2015 vogella GmbH ALL RIGHTS RESERVED. This book contains material protected under International and Federal Copyright Laws and Treaties. Any unauthorized reprint or use of this material is prohibited. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without express written permission from the author. The vogella name and logo and related trade dress are trademarks of vogella GmbH. Eclipse is a trademark of the Eclipse Foundation, Inc. Oracle and Java are registered trademarks of Oracle and/or its affiliates. All other trademarks referenced in this book are trademarks of their respective firms. The publisher and author(s) assume no responsibility for errors or omissions or for damages resulting from the use of the information contained herein. To my fellow open source developers and users You guys make my work life so much more fun Free edition provided by vogella GmbH i ii Foreword ................................................................................................................. xi Preface .................................................................................................................. xiii 1. Welcome ....................................................................................................... -
A Survey on Theorem Provers in Formal Methods
1 A Survey on Theorem Provers in Formal Methods M. Saqib Nawaz, Moin Malik, Yi Li, Meng Sun and M. Ikram Ullah Lali Abstract—Mechanical reasoning is a key area of research that lies at the crossroads of mathematical logic and artificial intelligence. The main aim to develop mechanical reasoning systems (also known as theorem provers) was to enable mathematicians to prove theorems by computer programs. However, these tools evolved with time and now play vital role in the modeling and reasoning about complex and large-scale systems, especially safety-critical systems. Technically, mathematical formalisms and automated reasoning based-approaches are employed to perform inferences and to generate proofs in theorem provers. In literature, there is a shortage of comprehensive documents that can provide proper guidance about the preferences of theorem provers with respect to their designs, performances, logical frameworks, strengths, differences and their application areas. In this work, more than 40 theorem provers are studied in detail and compared to present a comprehensive analysis and evaluation of these tools. Theorem provers are investigated based on various parameters, which includes: implementation architecture, logic and calculus used, library support, level of automation, programming paradigm, programming language, differences and application areas. Index Terms—Mathematical logics, Reasoning, Interactive theorem provers, Automated theorem provers, Proof automation, Survey. F 1 INTRODUCTION Recent developments and evolution in Information and Com- events. However, there are some inherent limitations of these munication Technology (ICT) have made computing systems more techniques. A program can be only tested against the functional and more complex. The criteria that how much we can rely on requirements of the system, which may be not refined and may these systems is based on their correctness. -
Some Key Research Problems in Automated Theorem Proving for Hardware and Software Verification
Å RACË Rev. R. Acad. Cien. Serie A. Mat. VOL. 98 (1), 2004, pp. 181–195 Ciencias de la Computaci´on / Computational Sciences Some Key Research Problems in Automated Theorem Proving for Hardware and Software Verification Matt Kaufmann and J Strother Moore Abstract. This paper sketches the state of the art in the application of mechanical theorem provers to the verification of commercial computer hardware and software. While the paper focuses on the theorem proving system ACL2, developed by the two authors, it references much related work in formal methods. The paper is intended to satisfy the curiosity of readers interested in logic and artificial intelligence as to the role of mechanized theorem proving in hardware and software design today. In addition, it points out some of the key research topics in the area. These topics transcend any one particular theorem proving system. Algunos problemas claves de investigacion´ en la demostracion´ mecanica´ de teoremas para la verificacion´ de hardware y software Resumen. En este art´ıculo se resume el estado del arte de la aplicaci´on de demostradores mec´anicos de teoremas a la verificaci´on de hardware y software comerciales. Aunque el trabajo se centra en el sistema de demostraci´on de teoremas ACL2, que desarrollaron los autores, se citan trabajos en m´etodos formales muy relacionados. Este art´ıculo tiene por objetivo satisfacer la curiosidad de aquellos lectores que se interesan por la l´ogica y la inteligencia artificial en lo relativo al papel que tiene la demostraci´on mec´anica de teoremas en el dise˜no de software y hardware hoy en d´ıa.