Frank Markham Brown

Boolean Reasoning (eBook, ePUB)

• Format: ePub

Produktbeschreibung

A systematic treatment of Boolean reasoning, this concise, newly revised edition combines the works of early logicians with recent investigations, including previously unpublished research results.
For the benefit of readers without formal training in mathematics, the text starts with an overview of elementary mathematical concepts and outlines the theory of Boolean algebras, based on Huntington's postulate. It defines operators for elimination, division, and expansion, providing a coherent and systematic basis for subsequent discussions of syllogistic reasoning, the solution of Boolean equations, and functional deduction.
Examples and end-of-chapter problems appear throughout the book, many taken from the design for switching systems. Two concluding chapters deal with applications; one applies Boolean reasoning to diagnostic problems, and the other discusses the design of multiple-output logic-circuits.

Produktdetails

• Verlag: Dover Publications
• Erscheinungstermin: 10. Februar 2012
• Englisch
• ISBN-13: 9780486164595
• Artikelnr.: 40289648

Autorenporträt

Frank Markham Brown is Professor Emeritus of the School of Engineering at the Air Force Institute of Technology.

Inhaltsangabe

1 Fundamental Concepts.- 1.1 Formulas.- 1.2 Propositions and Predicates.- 1.3 Sets.- 1.4 Operations on Sets.- 1.5 Partitions.- 1.6 Relations.- 1.7 Functions.- 1.8 Operations and Algebraic Systems.- 2 Boolean Algebras.- 2.1 Postulates for a Boolean Algebra.- 2.2 Examples of Boolean Algebras.- 2.2.1 The Algebra of Classes (Subsets of a Set).- 2.2.2 The Algebra of Propositional Functions.- 2.2.3 Arithmetic Boolean Algebras.- 2.2.4 The Two-Element Boolean Algebra.- 2.2.5 Summary of Examples.- 2.3 The Stone Representation Theorem.- 2.4 The Inclusion-Relation.- 2.4.1 Intervals.- 2.5 Some Useful Properties.- 2.6 n-Variable Boolean Formulas.- 2.7 n-Variable Boolean Functions.- 2.8 Boole's Expansion Theorem.- 2.9 The Minterm Canonical Form.- 2.9.1 Truth-tables.- 2.9.2 Maps.- 2.10 The Löwenheim-Müller Verification Theorem.- 2.11 Switching Functions.- 2.12 Incompletely-Specified Boolean Functions.- 2.13 Boolean Algebras of Boolean Functions.- 2.13.1 Free Boolean Algebras.- 2.14 Orthonormal Expansions.- 2.14.1 Löwenheim's Expansions.- 2.15 Boolean Quotient.- 2.16 The Boolean Derivative.- 2.17 Recursive Definition of Boolean Functions.- 2.18 What Good are "Big" Boolean Algebras?.- 3 The Blake Canonical Form.- 3.1 Definitions and Terminology.- 3.2 Syllogistic & Blake Canonical Formulas.- 3.3 Generation of BCF(f).- 3.4 Exhaustion of Implicants.- 3.5 Iterated Consensus.- 3.5.1 Quine's method.- 3.5.2 Successive extraction.- 3.6 Multiplication.- 3.6.1 Recursive multiplication.- 3.6.2 Combining multiplication and iterated consensus.- 3.6.3 Unwanted syllogistic formulas.- 4 Boolean Analysis.- 4.1 Review of Elementary Properties.- 4.2 Boolean Systems.- 4.2.1 Antecedent, Consequent, and Equivalent Systems.- 4.2.2 Solutions.- 4.3 Reduction.- 4.4 The Extended Verification Theorem.- 4.5 Poretsky's Law of Forms.- 4.6 Boolean Constraints.- 4.7 Elimination.- 4.8 Eliminants.- 4.9 Rudundant Variables.- 4.10 Substitution.- 4.11 The Tautology Problem.- 4.11.1 Testing for Tautology.- 4.11.2 The Sum-to-One Theorem.- 4.11.3 Nearly-Minimal SOP Formulas.- 5 Syllogistic Reasoning.- 5.1 The Principle of Assertion.- 5.2 Deduction by Consensus.- 5.3 Syllogistic Formulas.- 5.4 Clausal Form.- 5.5 Producing and Verifying Consequents.- 5.5.1 Producing Consequents.- 5.5.2 Verifying Consequents.- 5.5.3 Comparison of Clauses.- 5.6 Class-Logic.- 5.7 Selective Deduction.- 5.8 Functional Relations.- 5.9 Dependent Sets of Functions.- 5.10 Sum-to-One Subsets.- 5.11 Irredundant Formulas.- 6 Solution of Boolean Equations.- 6.1 Particular Solutions and Consistency.- 6.2 General Solutions.- 6.3 Subsumptive General Solutions.- 6.3.1 Successive Elimination.- 6.3.2 Deriving Eliminants from Maps.- 6.3.3 Recurrent Covers and Subsumptive Solutions.- 6.3.4 Simplified Subsumptive Solutions.- 6.3.5 Simplification via Marquand Diagrams.- 6.4 Parametric General Solutions.- 6.4.1 Successive Elimination.- 6.4.2 Parametric Solutions based on Recurrent Covers.- 6.4.3 Löwenheim's Formula.- 7 Functional Deduction.- 7.1 Functionally Deducible Arguments.- 7.2 Eliminable and Determining Subsets.- 7.2.1 u-Eliminable Subsets.- 7.2.2 u-Determining Subsets.- 7.2.3 Calculation of Minimal u-Determining Subsets.- 8 Boolean Identification.- 8.1 Parametric and Diagnostic Models.- 8.1.1 Parametric Models.- 8.1.2 The Diagnostic Axiom.- 8.1.3 Diagnostic Equations and Functions.- 8.1.4 Augmentation.- 8.2 Adaptive Identification.- 8.2.1 Initial and Terminal Specifications.- 8.2.2 Updating the Model.- 8.2.3 Effective Inputs.- 8.2.4 Test-Procedure.- 9 Recursive Realizations of Combinational Circuits.- 9.1 The Design-Process.- 9.2 Specifications.- 9.2.1 Specification-Formats.- 9.2.2 Consistent Specifications.- 9.3 Tabular Specifications.- 9.4 Strongly Combinational Solutions.- 9.5 Least-Cost Recursive Solutions.- 9.6 Constructing Recursive Solutions.- 9.6.1 The Procedure.- 9.6.2 An Implementation using BORIS.- A Syllogistic Formulas.- A.1 Absorptive Formulas.- A.2 Syllogistic Formulas.- A.3 Prime Implicants.- A.4 T

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1. Introduction 2. Fundamental Concepts 3. Boolean Algebras 4. The Blake
Canonical Form 5. Boolean Analysis 6. Syllogistic Reasoning 7. Solution of
Boolean Equations 8. Functional Deduction 9. Boolean Identification 10.
Combinational Circuits: Specifications & Optimization Appendix. Syllogistic
Formulas Bibliography. Index.