LOGIC AND DISCRETE MATHEMATICS
|
|
- Karen Chase
- 5 years ago
- Views:
Transcription
1 LOGIC AND DISCRETE MATHEMATICS A Computer Science Perspective WINFRIED KARL GRASSMANN Department of Computer Science University of Saskatchewan JEAN-PAUL TREMBLAY Department of Computer Science University of Saskatchewan PRENTICE HALL, Upper Saddle River, New Jersey 07458
2 Preface xv Propositional Calculus Logical Arguments and Propositions Introduction Some Important Logical Arguments Propositions 4 Logical Connectives Introduction Negation Conjunction Disjunction Conditional Biconditional Further Remarks on Connectives 12 Compound Propositions Introduction Logical Expressions Analysis of Compound Propositions Precedence Rules Evaluation of Expressions and Truth Tables Examples of Compound Propositions 21 Tautologies and Contradictions Introduction 23 Tautologies 24 19
3 yi Contents Tautologies and Sound Reasoning Contradictions Important Types of Tautologies Logical Equivalences and Their Use Introduction Proving Logical Equivalences by Truth Tables Statement Algebra Removing Conditionals and Biconditionals Essential Laws for Statement Algebra Shortcuts for Manipulating Expressions Normal Forms Truth Tables and Disjunctive Normal Forms Conjunctive Normal Forms and Complementation Logical Implications and Derivations Introduction Logical Implications Soundness Proofs through Truth Tables Proofs Systems for Derivations The Deduction Theorem 52 Predicate Calculus Syntactic Components of Predicate Calculus Introduction 60.2 The Universe of Discourse 60.3 Predicates 61.4 Variables and Instantiations 63.5 Quantifiers 65.6 Restrictions of Quantifiers to Certain Groups Interpretations and Validity Introduction Interpretations Validity Invalid Expressions Proving Validity Derivations Introduction Universal Instantiation Universal Generalization Deduction Theorem and Universal Generalization Dropping the Universal Quantifiers Existential Generalization Existential Instantiation 88
4 vii 2.4 Logical Equivalences Introduction Basic Logical Equivalences Other Important Equivalences Equational Logic Introduction Equality Equality and Uniqueness Functions and Equational Logic Function Compositions Properties of Operators Identity and Zero Elements Derivations in Equational Logic Equational Logic in Practice Boolean Algebra 115 Induction and Recursion Induction on Natural Numbers Introduction Natural Numbers Mathematical Induction Induction for Proving Properties of Addition Changing the Induction Base Strong Induction Sums and Related Constructs Introduction Recursive Definitions of Sums and Products Identities Involving Sums Double Sums and Matrices Proof by Recursion Introduction Recursive Definitions Descending Sequences The Principle of Proofs by Recursion Structural Induction Applications of Recursion to Programming Introduction Programming as Function Composition Recursion in Programs Programs Involving Trees Recursive Functions Introduction Primitive Recursive Functions 168
5 viii Contents Programming and Primitive Recursion Minimalization 173 Prolog Basic Prolog 178 Introduction 178 Facts, Rules, and Queries 179 Derivations Involving Facts 181 Derivations Involving Rules 183 Instantiations and Unification 186 Backtracking 188 Resolution 190 Running and Testing Programs Introduction Prolog Compilers and Interpreters Consulting a Database Debugging and Tracing 196 Additional Features of Prolog Introduction Input and Output Structures Infix Notation Arithmetic Equality Predicates 201 Recursion Introduction Recursive Predicates Termination Loops and Prolog Lists Recursive Predicates Involving Lists Successive Refinement 213 Negation in Prolog Introduction Prolog as a Logic Language Negation as Failure Use of the Clause Order Cuts 220 Application of Prolog to Logic Introduction 222 Lists as Logical Expressions 222 Representing Logical Expressions as Structures 224
6 ix Sets and Relations Sets and Set Operations Introduction 230 Sets and Their Members 231 Subsets 233 Intersections 235 Unions 236 Differences and Complements 237 Expressions Involving Sets 239 Tuples, Sequences, and Powersets Introduction Tuples and Cartesian Products Sequences and Strings Powersets Types and Signatures 248 Relations Introduction 251 Relations and Their Representation Domains and Ranges 254 Some Operations on Relations 255 Composition of Relations 257 Examples 261 Properties of Relations Introduction Relations on a Set Reflective Relations Symmetric Relations Transitivity Closures Equivalence Relations Partial Orders More About Functions Representations and Manipulations Involving Functions Introduction Definitions and Notation Representations of Functions The Lambda Notation Restrictions and Overloading Composition of Functions Injections, Surjections, and Inverses Creating Inverses by Creating Types 296
7 6.2 Enumerations, Isomorphisms, and Homomorphisms Introduction Enumerations Countable and Uncountable Sets Permutations and Combinations Isomorphisms and Homomorphisms Computational Complexity Introduction Polynomials and Polynomial-time Algorithms Functions and Algorithms Related to Exponentials The Limits of Computability Asymptotic Analysis Divide and Conquer Nondeterministic Polynomial Recurrence Relations Introduction Homogeneous Recurrence Relations Nonhomogeneous Recurrence Relations Miranda Introduction Command Level Function Definitions Types, Functions, and Declarations Pattern Matching and Rewriting A Programming Problem 348 Graphs and Trees Introduction and Examples of Graph Modeling Basic Definitions of Graph Theory 362 _7.3 Paths, Reachability, and Connectedness Computing Paths from a Matrix Representation of Graphs Traversing Graphs Represented as Adjacency Lists Introduction Adjacency Lists Representation of Graphs Breadth-first Search Depth-first Search Dijkstra's Algorithm for Finding Minimum Paths Trees and Spanning Trees Introduction Free Trees Spanning Trees Minimum Spanning Trees 416
8 xi 7.7 Scheduling Networks Introduction A Project Management Model Topological Sorting Formal Requirement Specification in Z Introduction Software Life Cycle Need for Formal Specifications Introduction to Z Introduction Alphabet and Lexical Elements Types and Declarations Specifying a System with Logic and Sets Schemas Relations Functions Sequences 472 Program Correctness Proofs Preliminary Concepts Introduction Programs and Codes Assertions Correctness General Rules Involving Preconditions and Postconditions Introduction Precondition Strengthening Postcondition Weakening Conjunction and Disjunction Rules Correctness Proofs in Loopless Code Introduction Assignment Statements Concatenation of Code The If-Statement Loops and Arrays Introduction A Preliminary While Rule The General While Rule Arrays Program Termination 515
9 xii Contents 10 Grammars, Languages, and Parsing Languages and Grammars Introduction Discussion of Grammars Formal Definition of a Language Notions of Syntax Analysis Ambiguous Grammars Reduced Grammars Top-down Parsing Introduction General Top-down Parsing Strategy Deterministic Top-down Parsing with LL(1) Grammars Derivations Derivations in Propositional Calculus Introduction Basics of Natural Derivation Implementation of the Deduction Theorem Resolution Some Results from Predicate Calculus Introduction Complements Prenex Normal Forms Derivations in Predicate Calculus Introduction Canonical Derivations Quantifiers in Natural Deduction Replacing Quantifiers by Functions and Free Variables Resolution in Predicate Calculus An Overview of Relational Database Systems Basic Concepts Introduction Definitions and Concepts Introductory Example of a Relational Database Overview of a Database System Relational Data Model Introduction 600 c Overview of the Relational Structure Relations and Their Schemas 602
10 xiii Representing Relations in the Relational Model Integrity Rules Relational Algebra Introduction Basic Operations Additional Relational Operations Examples Relational Calculus Introduction Tuple Calculus Examples Structured Query Language Introduction Data Definition Data Management Data Queries Concluding Remarks 637 Bibliography 641 Solutions to Even-numbered Problems 644 Index 736
MATHEMATICAL STRUCTURES FOR COMPUTER SCIENCE
MATHEMATICAL STRUCTURES FOR COMPUTER SCIENCE A Modern Approach to Discrete Mathematics SIXTH EDITION Judith L. Gersting University of Hawaii at Hilo W. H. Freeman and Company New York Preface Note to the
More informationDISCRETE MATHEMATICS
DISCRETE MATHEMATICS WITH APPLICATIONS THIRD EDITION SUSANNA S. EPP DePaul University THOIVISON * BROOKS/COLE Australia Canada Mexico Singapore Spain United Kingdom United States CONTENTS Chapter 1 The
More informationFundamentals of Discrete Mathematical Structures
Fundamentals of Discrete Mathematical Structures THIRD EDITION K.R. Chowdhary Campus Director JIET School of Engineering and Technology for Girls Jodhpur Delhi-110092 2015 FUNDAMENTALS OF DISCRETE MATHEMATICAL
More informationIntroductory logic and sets for Computer scientists
Introductory logic and sets for Computer scientists Nimal Nissanke University of Reading ADDISON WESLEY LONGMAN Harlow, England II Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario
More informationLogic and its Applications
Logic and its Applications Edmund Burke and Eric Foxley PRENTICE HALL London New York Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Contents Preface xiii Propositional logic 1 1.1 Informal introduction
More informationEXTENSIONS OF FIRST ORDER LOGIC
EXTENSIONS OF FIRST ORDER LOGIC Maria Manzano University of Barcelona CAMBRIDGE UNIVERSITY PRESS Table of contents PREFACE xv CHAPTER I: STANDARD SECOND ORDER LOGIC. 1 1.- Introduction. 1 1.1. General
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Final exam The final exam is Saturday December 16 11:30am-2:30pm. Lecture A will take the exam in Lecture B will take the exam
More informationAbout the Author. Dependency Chart. Chapter 1: Logic and Sets 1. Chapter 2: Relations and Functions, Boolean Algebra, and Circuit Design
Preface About the Author Dependency Chart xiii xix xxi Chapter 1: Logic and Sets 1 1.1: Logical Operators: Statements and Truth Values, Negations, Conjunctions, and Disjunctions, Truth Tables, Conditional
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Final exam The final exam is Saturday March 18 8am-11am. Lecture A will take the exam in GH 242 Lecture B will take the exam
More informationCOURSE: DATA STRUCTURES USING C & C++ CODE: 05BMCAR17161 CREDITS: 05
COURSE: DATA STRUCTURES USING C & C++ CODE: 05BMCAR17161 CREDITS: 05 Unit 1 : LINEAR DATA STRUCTURES Introduction - Abstract Data Types (ADT), Arrays and its representation Structures, Stack, Queue, Circular
More informationSummary of Course Coverage
CS-227, Discrete Structures I Spring 2006 Semester Summary of Course Coverage 1) Propositional Calculus a) Negation (logical NOT) b) Conjunction (logical AND) c) Disjunction (logical inclusive-or) d) Inequalities
More informationr=1 The Binomial Theorem. 4 MA095/98G Revision
Revision Read through the whole course once Make summary sheets of important definitions and results, you can use the following pages as a start and fill in more yourself Do all assignments again Do the
More informationContents. Chapter 1 SPECIFYING SYNTAX 1
Contents Chapter 1 SPECIFYING SYNTAX 1 1.1 GRAMMARS AND BNF 2 Context-Free Grammars 4 Context-Sensitive Grammars 8 Exercises 8 1.2 THE PROGRAMMING LANGUAGE WREN 10 Ambiguity 12 Context Constraints in Wren
More informationquanüfied Statements; valid well-formed formulas; comparison of propositional and predicate wffs. Exercises
Contents Preface xix Note to the Student xxv 1 Formal Logic 1 Chapter Objectives 1 1.1 Statements, Symbolic-Representation, and Tautologies 1 Statements and logical connectives; truth tables; well-formed
More informationCIS 1.5 Course Objectives. a. Understand the concept of a program (i.e., a computer following a series of instructions)
By the end of this course, students should CIS 1.5 Course Objectives a. Understand the concept of a program (i.e., a computer following a series of instructions) b. Understand the concept of a variable
More informationIntroductory Combinatorics
Introductory Combinatorics Third Edition KENNETH P. BOGART Dartmouth College,. " A Harcourt Science and Technology Company San Diego San Francisco New York Boston London Toronto Sydney Tokyo xm CONTENTS
More informationProgramming Languages Third Edition
Programming Languages Third Edition Chapter 12 Formal Semantics Objectives Become familiar with a sample small language for the purpose of semantic specification Understand operational semantics Understand
More information1. true / false By a compiler we mean a program that translates to code that will run natively on some machine.
1. true / false By a compiler we mean a program that translates to code that will run natively on some machine. 2. true / false ML can be compiled. 3. true / false FORTRAN can reasonably be considered
More informationDiscrete Mathematics SECOND EDITION OXFORD UNIVERSITY PRESS. Norman L. Biggs. Professor of Mathematics London School of Economics University of London
Discrete Mathematics SECOND EDITION Norman L. Biggs Professor of Mathematics London School of Economics University of London OXFORD UNIVERSITY PRESS Contents PART I FOUNDATIONS Statements and proofs. 1
More information[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics
400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {
More informationMATH 139 W12 Review 1 Checklist 1. Exam Checklist. 1. Introduction to Predicates and Quantified Statements (chapters ).
MATH 139 W12 Review 1 Checklist 1 Exam Checklist 1. Introduction to Predicates and Quantified Statements (chapters 3.1-3.4). universal and existential statements truth set negations of universal and existential
More informationFoundations of AI. 9. Predicate Logic. Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution
Foundations of AI 9. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution Wolfram Burgard, Andreas Karwath, Bernhard Nebel, and Martin Riedmiller 09/1 Contents Motivation
More informationBoolean Reasoning. The Logic of Boolean Equations. Frank Markham Brown Air Force Institute of Technology
Boolean Reasoning The Logic of Boolean Equations by Frank Markham Brown Air Force Institute of Technology ff Kluwer Academic Publishers Boston/Dordrecht/London Contents Preface Two Logical Languages Boolean
More informationCSC 501 Semantics of Programming Languages
CSC 501 Semantics of Programming Languages Subtitle: An Introduction to Formal Methods. Instructor: Dr. Lutz Hamel Email: hamel@cs.uri.edu Office: Tyler, Rm 251 Books There are no required books in this
More informationPropositional Calculus: Boolean Algebra and Simplification. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus: Boolean Algebra and Simplification CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Topics Motivation: Simplifying Conditional Expressions
More informationA Survey of Mathematics with Applications 8 th Edition, 2009
A Correlation of A Survey of Mathematics with Applications 8 th Edition, 2009 South Carolina Discrete Mathematics Sample Course Outline including Alternate Topics and Related Objectives INTRODUCTION This
More informationComputation Engineering Applied Automata Theory and Logic. Ganesh Gopalakrishnan University of Utah. ^J Springer
Computation Engineering Applied Automata Theory and Logic Ganesh Gopalakrishnan University of Utah ^J Springer Foreword Preface XXV XXVII 1 Introduction 1 Computation Science and Computation Engineering
More informationFUZZY SPECIFICATION IN SOFTWARE ENGINEERING
1 FUZZY SPECIFICATION IN SOFTWARE ENGINEERING V. LOPEZ Faculty of Informatics, Complutense University Madrid, Spain E-mail: ab vlopez@fdi.ucm.es www.fdi.ucm.es J. MONTERO Faculty of Mathematics, Complutense
More informationCOMPUTATIONAL SEMANTICS WITH FUNCTIONAL PROGRAMMING JAN VAN EIJCK AND CHRISTINA UNGER. lg Cambridge UNIVERSITY PRESS
COMPUTATIONAL SEMANTICS WITH FUNCTIONAL PROGRAMMING JAN VAN EIJCK AND CHRISTINA UNGER lg Cambridge UNIVERSITY PRESS ^0 Contents Foreword page ix Preface xiii 1 Formal Study of Natural Language 1 1.1 The
More informationThe Formal Semantics of Programming Languages An Introduction. Glynn Winskel. The MIT Press Cambridge, Massachusetts London, England
The Formal Semantics of Programming Languages An Introduction Glynn Winskel The MIT Press Cambridge, Massachusetts London, England Series foreword Preface xiii xv 1 Basic set theory 1 1.1 Logical notation
More informationContents. 1 Introduction. 2 Searching and Traversal Techniques. Preface... (vii) Acknowledgements... (ix)
Contents Preface... (vii) Acknowledgements... (ix) 1 Introduction 1.1 Algorithm 1 1.2 Life Cycle of Design and Analysis of Algorithm 2 1.3 Pseudo-Code for Expressing Algorithms 5 1.4 Recursive Algorithms
More informationZ Notation. June 21, 2018
Z Notation June 21, 2018 1 Definitions There are many different ways to introduce an object in a Z specification: declarations, abbreviations, axiomatic definitions, and free types. Keep in mind that the
More informationNotation Index. Probability notation. (there exists) (such that) Fn-4 B n (Bell numbers) CL-27 s t (equivalence relation) GT-5.
Notation Index (there exists) (for all) Fn-4 Fn-4 (such that) Fn-4 B n (Bell numbers) CL-27 s t (equivalence relation) GT-5 ( n ) k (binomial coefficient) CL-15 ( n m 1,m 2,...) (multinomial coefficient)
More informationChapter 16. Logic Programming Languages
Chapter 16 Logic Programming Languages Chapter 16 Topics Introduction A Brief Introduction to Predicate Calculus Predicate Calculus and Proving Theorems An Overview of Logic Programming The Origins of
More information4.1.2 Merge Sort Sorting Lower Bound Counting Sort Sorting in Practice Solving Problems by Sorting...
Contents 1 Introduction... 1 1.1 What is Competitive Programming?... 1 1.1.1 Programming Contests.... 2 1.1.2 Tips for Practicing.... 3 1.2 About This Book... 3 1.3 CSES Problem Set... 5 1.4 Other Resources...
More informationR13 SET Discuss how producer-consumer problem and Dining philosopher s problem are solved using concurrency in ADA.
R13 SET - 1 III B. Tech I Semester Regular Examinations, November - 2015 1 a) What constitutes a programming environment? [3M] b) What mixed-mode assignments are allowed in C and Java? [4M] c) What is
More informationCopyright 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley. Chapter 6 Outline. Unary Relational Operations: SELECT and
Chapter 6 The Relational Algebra and Relational Calculus Copyright 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 6 Outline Unary Relational Operations: SELECT and PROJECT Relational
More informationCSC Discrete Math I, Spring Sets
CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its
More informationSoftware Engineering Lecture Notes
Software Engineering Lecture Notes Paul C. Attie August 30, 2013 c Paul C. Attie. All rights reserved. 2 Contents I Hoare Logic 11 1 Propositional Logic 13 1.1 Introduction and Overview..............................
More informationHANDBOOK OF LOGIC IN ARTIFICIAL INTELLIGENCE AND LOGIC PROGRAMMING
HANDBOOK OF LOGIC IN ARTIFICIAL INTELLIGENCE AND LOGIC PROGRAMMING Volume 5 Logic Programming Edited by DOV M. GABBAY and C. J. HOGGER Imperial College of Science, Technology and Medicine London and J.
More informationLogic Programming Languages
Logic Programming Languages Introduction Logic programming languages, sometimes called declarative programming languages Express programs in a form of symbolic logic Use a logical inferencing process to
More informationTHREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.
THREE LECTURES ON BASIC TOPOLOGY PHILIP FOTH 1. Basic notions. Let X be a set. To make a topological space out of X, one must specify a collection T of subsets of X, which are said to be open subsets of
More informationCSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter p. 1/27
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 2.1-2.7 p. 1/27 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
More informationSlides for Faculty Oxford University Press All rights reserved.
Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,
More informationTo prove something about all Boolean expressions, we will need the following induction principle: Axiom 7.1 (Induction over Boolean expressions):
CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 7 This lecture returns to the topic of propositional logic. Whereas in Lecture Notes 1 we studied this topic as a way of understanding
More informationAbout the Tutorial. Audience. Prerequisites. Copyright & Disclaimer. Discrete Mathematics
About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics
More informationComputational Discrete Mathematics
Computational Discrete Mathematics Combinatorics and Graph Theory with Mathematica SRIRAM PEMMARAJU The University of Iowa STEVEN SKIENA SUNY at Stony Brook CAMBRIDGE UNIVERSITY PRESS Table of Contents
More informationDefinition: A context-free grammar (CFG) is a 4- tuple. variables = nonterminals, terminals, rules = productions,,
CMPSCI 601: Recall From Last Time Lecture 5 Definition: A context-free grammar (CFG) is a 4- tuple, variables = nonterminals, terminals, rules = productions,,, are all finite. 1 ( ) $ Pumping Lemma for
More informationThe Algorithm Design Manual
Steven S. Skiena The Algorithm Design Manual With 72 Figures Includes CD-ROM THE ELECTRONIC LIBRARY OF SCIENCE Contents Preface vii I TECHNIQUES 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2 2.1 2.2 2.3
More informationExample: NFA to DFA Conversion
CPSC 121 Lecture 36 April 8, 2009 Menu April 8, 2009 Topics: Example: NFA to DFA Conversion Final Reading List Summary The End! Reminders: On-line Quiz 12 deadline 5:00pm TODAY Teaching evaluation survey
More informationRelational Databases
Relational Databases Jan Chomicki University at Buffalo Jan Chomicki () Relational databases 1 / 49 Plan of the course 1 Relational databases 2 Relational database design 3 Conceptual database design 4
More informationAn Evolution of Mathematical Tools
An Evolution of Mathematical Tools From Conceptualization to Formalization Here's what we do when we build a formal model (or do a computation): 0. Identify a collection of objects/events in the real world.
More informationNotation Index 9 (there exists) Fn-4 8 (for all) Fn-4 3 (such that) Fn-4 B n (Bell numbers) CL-25 s ο t (equivalence relation) GT-4 n k (binomial coef
Notation 9 (there exists) Fn-4 8 (for all) Fn-4 3 (such that) Fn-4 B n (Bell numbers) CL-25 s ο t (equivalence relation) GT-4 n k (binomial coefficient) CL-14 (multinomial coefficient) CL-18 n m 1 ;m 2
More informationThis is already grossly inconvenient in present formalisms. Why do we want to make this convenient? GENERAL GOALS
1 THE FORMALIZATION OF MATHEMATICS by Harvey M. Friedman Ohio State University Department of Mathematics friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ May 21, 1997 Can mathematics be
More informationLogic (or Declarative) Programming Foundations: Prolog. Overview [1]
Logic (or Declarative) Programming Foundations: Prolog In Text: Chapter 12 Formal logic Logic programming Prolog Overview [1] N. Meng, S. Arthur 2 1 Logic Programming To express programs in a form of symbolic
More informationChapter 16. Logic Programming Languages ISBN
Chapter 16 Logic Programming Languages ISBN 0-321-49362-1 Chapter 16 Topics Introduction A Brief Introduction to Predicate Calculus Predicate Calculus and Proving Theorems An Overview of Logic Programming
More informationMulti-paradigm Declarative Languages
Michael Hanus (CAU Kiel) Multi-paradigm Declarative Languages ICLP 2007 1 Multi-paradigm Declarative Languages Michael Hanus Christian-Albrechts-University of Kiel Programming Languages and Compiler Construction
More informationAn Introduction to Programming and Proving in Agda (incomplete draft)
An Introduction to Programming and Proving in Agda (incomplete draft) Peter Dybjer January 29, 2018 1 A first Agda module Your first Agda-file is called BoolModule.agda. Its contents are module BoolModule
More information2.2 Set Operations. Introduction DEFINITION 1. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, EXAMPLE 2
2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of
More informationModule 6. Knowledge Representation and Logic (First Order Logic) Version 2 CSE IIT, Kharagpur
Module 6 Knowledge Representation and Logic (First Order Logic) 6.1 Instructional Objective Students should understand the advantages of first order logic as a knowledge representation language Students
More informationIntroduction to Algorithms Third Edition
Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein Introduction to Algorithms Third Edition The MIT Press Cambridge, Massachusetts London, England Preface xiü I Foundations Introduction
More informationPart I Basic Concepts 1
Introduction xiii Part I Basic Concepts 1 Chapter 1 Integer Arithmetic 3 1.1 Example Program 3 1.2 Computer Program 4 1.3 Documentation 5 1.4 Input 6 1.5 Assignment Statement 7 1.5.1 Basics of assignment
More informationLecture 5. Logic I. Statement Logic
Ling 726: Mathematical Linguistics, Logic. Statement Logic V. Borschev and B. Partee, September 27, 2 p. Lecture 5. Logic I. Statement Logic. Statement Logic...... Goals..... Syntax of Statement Logic....2.
More informationNotes for Chapter 12 Logic Programming. The AI War Basic Concepts of Logic Programming Prolog Review questions
Notes for Chapter 12 Logic Programming The AI War Basic Concepts of Logic Programming Prolog Review questions The AI War How machines should learn: inductive or deductive? Deductive: Expert => rules =>
More informationCONTENTS Equivalence Classes Partition Intersection of Equivalence Relations Example Example Isomorphis
Contents Chapter 1. Relations 8 1. Relations and Their Properties 8 1.1. Definition of a Relation 8 1.2. Directed Graphs 9 1.3. Representing Relations with Matrices 10 1.4. Example 1.4.1 10 1.5. Inverse
More informationThomas H. Cormen Charles E. Leiserson Ronald L. Rivest. Introduction to Algorithms
Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Introduction to Algorithms Preface xiii 1 Introduction 1 1.1 Algorithms 1 1.2 Analyzing algorithms 6 1.3 Designing algorithms 1 1 1.4 Summary 1 6
More informationAppendix Set Notation and Concepts
Appendix Set Notation and Concepts In mathematics you don t understand things. You just get used to them. John von Neumann (1903 1957) This appendix is primarily a brief run-through of basic concepts from
More informationImplementação de Linguagens 2016/2017
Implementação de Linguagens Ricardo Rocha DCC-FCUP, Universidade do Porto ricroc @ dcc.fc.up.pt Ricardo Rocha DCC-FCUP 1 Logic Programming Logic programming languages, together with functional programming
More informationTyped Lambda Calculus
Department of Linguistics Ohio State University Sept. 8, 2016 The Two Sides of A typed lambda calculus (TLC) can be viewed in two complementary ways: model-theoretically, as a system of notation for functions
More informationAbout the Tutorial. Audience. Prerequisites. Copyright & Disclaimer. Discrete Mathematics
About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics
More informationIntroduction to Automata Theory. BİL405 - Automata Theory and Formal Languages 1
Introduction to Automata Theory BİL405 - Automata Theory and Formal Languages 1 Automata, Computability and Complexity Automata, Computability and Complexity are linked by the question: What are the fundamental
More informationPropositional Calculus. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus
More informationM.C.A. DEGREE EXAMINATION, MAY First Year. Paper I INFORMATION TECHNOLOGY. SECTION A (3 15 = 45 marks) Answer any THREE of the following.
Paper I INFORMATION TECHNOLOGY Answer any THREE of the following. 1. Explain Architecture of computer in detail. 2. Explain in detail about Input and Output technologies. 3. What is MODEM? What factors
More informationTo prove something about all Boolean expressions, we will need the following induction principle: Axiom 7.1 (Induction over Boolean expressions):
CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 7 This lecture returns to the topic of propositional logic. Whereas in Lecture 1 we studied this topic as a way of understanding proper reasoning
More information9/19/12. Why Study Discrete Math? What is discrete? Sets (Rosen, Chapter 2) can be described by discrete math TOPICS
What is discrete? Sets (Rosen, Chapter 2) TOPICS Discrete math Set Definition Set Operations Tuples Consisting of distinct or unconnected elements, not continuous (calculus) Helps us in Computer Science
More informationTorben./Egidius Mogensen. Introduction. to Compiler Design. ^ Springer
Torben./Egidius Mogensen Introduction to Compiler Design ^ Springer Contents 1 Lexical Analysis 1 1.1 Regular Expressions 2 1.1.1 Shorthands 4 1.1.2 Examples 5 1.2 Nondeterministic Finite Automata 6 1.3
More informationSTUDENT NUMBER: MATH Final Exam. Lakehead University. April 13, Dr. Adam Van Tuyl
Page 1 of 13 NAME: STUDENT NUMBER: MATH 1281 - Final Exam Lakehead University April 13, 2011 Dr. Adam Van Tuyl Instructions: Answer all questions in the space provided. If you need more room, answer on
More informationTheorem proving. PVS theorem prover. Hoare style verification PVS. More on embeddings. What if. Abhik Roychoudhury CS 6214
Theorem proving PVS theorem prover Abhik Roychoudhury National University of Singapore Both specification and implementation can be formalized in a suitable logic. Proof rules for proving statements in
More informationThe International Olympiad in Informatics Syllabus
The International Olympiad in Informatics Syllabus 1 Version and status information This is the official Syllabus version for IOI 2009 in Plovdiv, Bulgaria. The Syllabus is an official document related
More informationCOSC252: Programming Languages: Semantic Specification. Jeremy Bolton, PhD Adjunct Professor
COSC252: Programming Languages: Semantic Specification Jeremy Bolton, PhD Adjunct Professor Outline I. What happens after syntactic analysis (parsing)? II. Attribute Grammars: bridging the gap III. Semantic
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets and Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University Dec 4, 2014 Outline 1 2 3 4 Definition A relation R from a set A to a set
More informationAbout the Authors... iii Introduction... xvii. Chapter 1: System Software... 1
Table of Contents About the Authors... iii Introduction... xvii Chapter 1: System Software... 1 1.1 Concept of System Software... 2 Types of Software Programs... 2 Software Programs and the Computing Machine...
More informationM.Sc. (Computer Science) I Year Assignments for May Paper I DATA STRUCTURES Assignment I
Paper I DATA STRUCTURES (DMCS 01) 1. Explain in detail about the overview of Data structures. 2. Explain circular linked list and double-linked list. 3. Explain CPU scheduling in Multiprogramming Environment.
More informationSTRUCTURES AND STRATEGIES FOR STATE SPACE SEARCH
Slide 3.1 3 STRUCTURES AND STRATEGIES FOR STATE SPACE SEARCH 3.0 Introduction 3.1 Graph Theory 3.2 Strategies for State Space Search 3.3 Using the State Space to Represent Reasoning with the Predicate
More informationProving Theorems with Athena
Proving Theorems with Athena David R. Musser Aytekin Vargun August 28, 2003, revised January 26, 2005 Contents 1 Introduction 1 2 Proofs about order relations 2 3 Proofs about natural numbers 7 3.1 Term
More informationTHEORY OF COMPUTATION
THEORY OF COMPUTATION UNIT-1 INTRODUCTION Overview This chapter begins with an overview of those areas in the theory of computation that are basic foundation of learning TOC. This unit covers the introduction
More informationCLASSIC DATA STRUCTURES IN JAVA
CLASSIC DATA STRUCTURES IN JAVA Timothy Budd Oregon State University Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal CONTENTS
More informationNegations in Refinement Type Systems
Negations in Refinement Type Systems T. Tsukada (U. Tokyo) 14th March 2016 Shonan, JAPAN This Talk About refinement intersection type systems that refute judgements of other type systems. Background Refinement
More informationData Integration: Logic Query Languages
Data Integration: Logic Query Languages Jan Chomicki University at Buffalo Datalog Datalog A logic language Datalog programs consist of logical facts and rules Datalog is a subset of Prolog (no data structures)
More informationJAVA PROGRAMMING. Unit-3 :Creating Gui Using The Abstract Windowing Toolkit:
JAVA PROGRAMMING UNIT-1: Introduction To Java, Getting Started With Java, Applets And Application, Creating A Java Application, Creating A Java Applets, Object Oriented Programming In Java, Object And
More informationTHE DESIGN AND ANALYSIS OF COMPUTER ALGORITHMS
2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. THE DESIGN AND ANALYSIS OF COMPUTER ALGORITHMS Alfred V. Aho Bell
More informationOperational Semantics
15-819K: Logic Programming Lecture 4 Operational Semantics Frank Pfenning September 7, 2006 In this lecture we begin in the quest to formally capture the operational semantics in order to prove properties
More informationALGORITHMIC DECIDABILITY OF COMPUTER PROGRAM-FUNCTIONS LANGUAGE PROPERTIES. Nikolay Kosovskiy
International Journal Information Theories and Applications, Vol. 20, Number 2, 2013 131 ALGORITHMIC DECIDABILITY OF COMPUTER PROGRAM-FUNCTIONS LANGUAGE PROPERTIES Nikolay Kosovskiy Abstract: A mathematical
More informationSTABILITY AND PARADOX IN ALGORITHMIC LOGIC
STABILITY AND PARADOX IN ALGORITHMIC LOGIC WAYNE AITKEN, JEFFREY A. BARRETT Abstract. Algorithmic logic is the logic of basic statements concerning algorithms and the algorithmic rules of deduction between
More informationPart I Logic programming paradigm
Part I Logic programming paradigm 1 Logic programming and pure Prolog 1.1 Introduction 3 1.2 Syntax 4 1.3 The meaning of a program 7 1.4 Computing with equations 9 1.5 Prolog: the first steps 15 1.6 Two
More informationLOGIC SYNTHESIS AND VERIFICATION ALGORITHMS. Gary D. Hachtel University of Colorado. Fabio Somenzi University of Colorado.
LOGIC SYNTHESIS AND VERIFICATION ALGORITHMS by Gary D. Hachtel University of Colorado Fabio Somenzi University of Colorado Springer Contents I Introduction 1 1 Introduction 5 1.1 VLSI: Opportunity and
More informationI BSc(Computer Science)[ ] Semester - II Allied:DISCRETE MATHEMATICS - 207D Multiple Choice Questions.
1 of 23 1/20/2018, 2:35 PM Dr.G.R.Damodaran College of Science (Autonomous, affiliated to the Bharathiar University, recognized by the UGC)Reaccredited at the 'A' Grade Level by the NAAC and ISO 9001:2008
More informationPoint-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS
Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS Definition 1.1. Let X be a set and T a subset of the power set P(X) of X. Then T is a topology on X if and only if all of the following
More informationMadhya Pradesh Bhoj (Open) University, Bhopal
Subject- Optimization Techniques Maximum Marks: 20 Q.1 Explain the concept, scope and tools of O.R. Q.2 Explain the Graphical method for solving Linear Programming Problem. Q.3 Discuss the Two phase method
More informationA Simplified Abstract Syntax for the Dataflow Algebra. A. J. Cowling
Verification and Testing Research Group, Department of Computer Science, University of Sheffield, Regent Court, 211, Portobello Street, Sheffield, S1 4DP, United Kingdom Email: A.Cowling @ dcs.shef.ac.uk
More information