In this book binary functions and their representation by implicants or implicates are described. In particular minimal representations by prime implicants or prime implicates are given. Such representations generalize the minimal representations of the usual Boolean functions. It is shown that implicants (implicates) of discrete functions may be constructed with the help of implicants (implicates) of binary functions. One substantial application is the description of the reliability structure of technical systems, another is the use of binary respectively discrete functions to classify objects which are described by the grades of certain attributes. Finally a class of Boolean algebras of practical importance (set algebras, indicator algebras, algebras of classes of propositions) are considered. The elements of such algebras have representations which are strongly connected with the representations of binary functions.
1. Introduction.- 2. Binary Functions and their Representations by Implicants.- 2.1 Cube Indicators.- 2.2 Implicants.- 2.3 Prime Implicants.- 2.4 Representations by Implicants (Prime Implicants).- 3. Representations of Binary Functions by Implicates.- 3.1 Anticube Indicators.- 3.2 Implicates.- 3.3 Prime Implicates.- 3.4 Representations by Implicates (Prime Implicates).- 4. Reduction Methods.- 4.1 Notation.- 4.2 Rule.- 4.3 Rule.- 4.4 Rule.- 4.5 Rule.- 4.6 Rule.- 4.7 Rule (Corollary).- 4.8 Rule.- 4.9 Rule.- 5. Discrete Functions.- 5.1 Representations by Binary Functions.- 5.2 Monotone Functions.- 5.3 Semimonotone Functions.- 5.4 Implicants (Prime Implicants) of Discrete Functions.- 5.5 Representations by Implicants (Prime Implicants).- 5.6 Implicates (Prime Implicates) of Discrete Functions.- 5.7 Representations by Implicates (Prime Implicates).- 6. Applications.- 6.1 Reliability Structure of Technical Systems.- 6.2 Classification (Valuation) of Objects.- 7. A Class of Finite Boolean Algebras.- 7.1 Boolean Algebras.- 7.2 Boolean Algebras Generated by Finite Partitions.- 7.3 Representations of Boolean Elements by Implicants.- 7.4 Representations of Boolean Elements by Implicates.- 7.5 Probability.- 8. Applications.- 8.1 Set Algebras (Event Algebras).- 8.2 Indicator Algebras.- 8.3 Partitions in Propositional Logic.- 8.4 Algebras of Classes of Propositions.- 8.5 Truth Function Algebras.- 8.6 Some Related Models.- 8.7 Calculation of Elements of B*.- Concluding Remark.- References.- List of Symbols.
Series: Lecture Notes in Economic and Mathematical Systems
Number Of Pages: 156
Published: 8th August 1990
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6
Weight (kg): 0.25