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Foundations of Computing: System Development with Set Theory and Logic / Edition 1
     

Foundations of Computing: System Development with Set Theory and Logic / Edition 1

by Thierry Scheurer
 

ISBN-10: 0201544296

ISBN-13: 9780201544299

Pub. Date: 01/28/1994

Publisher: Addison-Wesley

Set theory and logic are the twin pillars of computing science. Their mastery is an essential part of the software engineer's education. This book provides a clear introduction to the key ideas of these two subjects and shows how they can be applied successfully in formal system development.

Highlights of the book include:

  • A presentation

Overview

Set theory and logic are the twin pillars of computing science. Their mastery is an essential part of the software engineer's education. This book provides a clear introduction to the key ideas of these two subjects and shows how they can be applied successfully in formal system development.

Highlights of the book include:

  • A presentation of set theory as a modelling language of universal applicability
  • A wealth of practical examples demonstrating the remarkable simplicity and naturalness of set theory as a description tool
  • A description of logic as a formal language, and as a simple way of introducing the key concepts of formal syntax, semantics and deduction calculus
  • A practical methodology of system development based on set theory and illustrated by several substantial case studies

The book starts from first principles and requires no prior knowledge of mathematics. It will be equally valuable for students of computing science and software engineers wishing to develop the skills required to apply formal methods successfully.

Product Details

ISBN-13:
9780201544299
Publisher:
Addison-Wesley
Publication date:
01/28/1994
Series:
International Computer Science Ser.
Pages:
704
Product dimensions:
6.77(w) x 9.26(h) x 1.44(d)

Table of Contents

(Most chapters contains Exercises.)
Preface.
List of Symbols.

I. OVERVIEW.

Prologue.
1. A View of System Development.
On Systems and Computers.
Computing: a New Perspective.
General Problem Solving.
Radical Novelties.
Computers and General Problem Solving.

Universal Features of Systems.
Structure.
Taxonomy.
Modules: Components and Interfaces.
Notation.
Transformation.
Establishment: Specification and Analysis.
System Development with Set Theory and Logic.
Introduction to Set Theory.
Induction and Recursion.
Formal Notation and Symbolic Logic.
The Practice of Model Establishment.

II. SET THEORY AND INDUCTION.


2. Sets and Basic Set Operations.
Basic Definitions.
The Concept of Set.
Defining and Denoting Sets.
Identifiers.
Equivalent Definitions.
On the Caution Required in Defining Sets.
Three Fundamental Features of Sets.
Membership and Inclusion.
Special Sets and Set Cardinality.
Principle of Extensionality.
On the Method of Abstraction.
Basic Set Operations.
Pair.
Union.
Intersection.
Relative Complement.
Disjointness and Related Operations.
Powerset.
Application.
Properties of Basic Set Operations.
Commutativity and Associativity.
Distributivity and DeMorgan's Laws.
Laws of the Empty Set, Idempotence and Absorption.
Laws of Inclusion.
Properties of Operations on Disjoint Sets.
Review of Notation.
Example.
Variables, Functions and Predicates.
Logical Connective Symbols.
Quantifiers.
Punctuation.
Introducing Names.
Variants of Set Theory.
Russell's Paradox and Classes.
Set Theory as a Modelling Language.
Naive versus Axiomatic Set Theory.

3. Relations and Functions.
Couples, Cartesian Product and Disjoint Union.
Couples.
Cartesian Product.
Disjoint Union.
Relations.
The Concept of a Relation.
Examples of Relations.
Some Derived Features of a Relation.
Basic Categories of Relations.
Class Relations AT.
Functions.
The Concept of a Function.
Examples of Functions.
Basic Categories of Functions.
Fundamental Applications of Functions AT.
Characteristic Functions of Subsets and Relations.
Equivalence between Relations and Functions.
Families and Bags.
Families of Sets and Partitions.
Distributed Functions.
n-ary Relations and Functions.
n-tuples.
n-ary Cartesian Product.
n-ary Disjoint Union.
n-ary Relations.
Examples of n-ary Relations.
n-ary Functions.
Examples of n-ary Functions.
Nullary Functions and n-ary Operations.
n-ary Class Relations and Functions AT.
Operations on Relations and Functions AT.
Operations Returning Sets of Relations.
Composition of Relations and Functions.
Restrictions and Overriding.
Identity and Inclusion Functions.
Opposites and Inverses.

2.3 Induction and Recursion.
Peanos Postulates for Natural Numbers.
Deriving Peanos Postulates.
Significance of Peanos Postulates.
Inductive Sets with Respect to 0 and S.
A Concrete Definition of Natural NumbersAT.
Proof by Induction.
Proposition P(n); Example.
Proof by Induction: Theorem and Method.
Example of a Proof by Induction.
Alternative Application of the Induction Principle.
Second Induction Principle.
Simple Recursion.
Example of a Simply Recursive Function.
Simple Recursion Theorem.
Variant of the Simple Recursion Theorem.
Inductive Constructions.
Defining Arithmetic Operations in NatAT.

4. Structural Induction.
Example of a Structural Inductive Construction.
General Definition of Structural Induction.
Parse Trees.
Second Example.
Construction Sequences.
Proof by Structural Induction.

5. Free Inductive Sets; General Recursion Theorem.
Free Inductive Sets.
Structural Recursion; Example.
General Recursion Theorem.
Second Variant of the General Recursion TheoremAT.

III. SYMBOLIC LOGIC.


3.1 Introduction to Symbolic Logic.
Knowledge and Deductive Reasoning.
Two Features of Scientific Knowledge.
The Nature of Deductive Reasoning.
Logic.
Purpose of Logic.
Logic versus Symbolic Logic.
Propositional Logic versus Predicate Logic.

3.2 Propositional Logic.
Syntax of Propositional Logic.
Alphabet.
Informal Definition of Well-formed Formulas.
Formal Definition of Well-formed FormulasAT.
Semantics of Propositional Logic.
Recursive Definition of the Semantic Function M.
Example.
Well-definedness of M.
Tautological Implication.
Truth Tables.
Satisfiability.
Definition of Tautological Implication.
Tautologies.
Fundamental Tautologies and Applications.
A Catalogue.
Substitution.
Equivalent Languages.
Applications.

3.3 First-order Predicate Logic.
Introduction.
Limitations of Propositional Logic.
Requirements to Be Met by First-order Logic.
Syntax of First-order Logic.
Alphabet.
Well-formed Expressions.
Formal Definition of Well-formed ExpressionsAT.
Unique Readability of Wfes.
Free and Bound Variables.
Semantics of First-order Logic.
Interpretation.
Semantic Function of Well-formed Expressions.
Example.
A TheoremAT.
Logical Implication.
Model and Satisfiability.
Definition of Logical Implication.
Examples of Valid Formulas.

3.4 Formal Deduction in First-order Logic.
Formal Deduction.
Outline of a Formal Deduction System.
Generation of Formal Theorems.
Logical Axioms.
Fundamental Properties of FDS.
Inductive ConstructionsAT.
Fundamental Metatheorems.
Subset and Transitivity Properties of

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