* Signals * Algebraic structures for signal processing * Further readings in discrete functions * Spectral techniques * Spectral transforms * Further readings in Fourier analysis on groups * Spectral interpretation of DDs * Further readings in decision diagrams * DDs and order of variables * Further readings in optimization of DD by variables ordering * Free BDDs * Further readings in free BDDs * Word-level DDs * Spectral interpretation of edge-valued DDs * Spectral interpretation of TDDs * Further readings in ternary DDs * Spectral interpretation of optimization of DDs * Group-theoretic approach to optimization of DDs * Further readings in DD on finite groups * Closing remarks * References * Index

Spectral Interpretation of Decision Diagrams

This paper relates to a recently published partly tutorial article that presents some discussion of the polynomial form of Boolean functions and its applications based on the literature published in English and German. We show that a lot of the research in this area has also been done in Eastern Europe, and this note aims to present these unknown developments. The most recent work in this area is also described.

A note on the polynomial form of Boolean functions and related topics

Preface. Acronyms. 1. SETS, RELATIONS, AND FUNCTIONS. 1. Sets. 2. Relations. 3. Functions. 4. Representations of Logic Functions. 5. Factored Expressions. 6. Exercises and Problems. 2. ALGEBRAIC STRUCTURES FOR LOGIC DESIGN. 1. Algebraic Structure. 2. Finite Groups. 3. Finite Rings. 4. Finite Fields. 5. Homomorphisms. 6. Matrices. 7. Vector spaces. 8. Algebra. 9. Boolean Algebra. 10. Graphs. 11. Exercises and Problems. 3. FUNCTIONAL EXPRESSIONS FOR SWITCHING FUNCTIONS. 1. Shannon Expansion Rule. 2. Reed-Muller Expansion Rules. 3. Fast Algorithms for Calculation of RM-expressions. 4. Negative Davio Expression. 5. Fixed Polarity Reed-Muller Expressions. 6. Algebraic Structures for Reed-Muller Expressions. 7. Interpretation of Reed-Muller Expressions. 8 Kronecker Expressions. 9. Word-Level Expressions. 10. Walsh Expressions. 11. Walsh Functions and Switching Variables. 12. Walsh Series. 13. Relationships Among Expressions. 14. Generalizations to Multiple-Valued Functions. 15. Exercises and Problems. 4. DECISION DIAGRAMS FOR REPRESENTATION OF SWITCHING FUNCTIONS. 1. Decision Diagrams. 2. Decision Diagrams over Groups. 3. Construction of Decision Diagrams. 4. Shared Decision Diagrams. 5. Multi-terminal binary decision diagrams. 6. Functional Decision Diagrams. 7. Kronecker decision diagrams. 8. Pseudo-Kronecker decision diagrams. 9. Spectral Interpretation of Decision Diagrams. 10. Reduction of Decision Diagrams. 11. Exercises and Problems. 5. CLASSIFICATION OF SWITCHING FUNCTIONS. 1. NPN-classification. 2. SD-Classification. 3. LP-classification. 4. Universal Logic Modules. 5. Exercises and Problems. 6. SYNTHESIS WITH MULTIPLEXERS. 1. Synthesis with Multiplexers. 2. Applications of Multiplexers. 3. Demultiplexers. 4. Synthesis with Demultiplexers. 5. Applications of Demultiplexers. 6. Exercises and Problems. 7. REALIZATIONS WITH ROM. 1. Realizations with ROM. 2. Two-level Addressing in ROM Realizations. 3. Characteristics of Realizations with ROM. 4. Exercises and Problems. 8. REALIZATIONS WITH PROGRAMMABLE LOGIC ARRAYS. 1. Realizations with PLA. 2. The optimization of PLA. 3. Two-level Addressing of PLA. 4. Folding of PLA. 5. Minimization of PLA by Characteristic Functions. 6. Exercises and Problems. 9. UNIVERSAL CELLULAR ARRAYS. 1. Features of Universal Cellular Arrays. 2. Realizations with Universal Cellular Arrays. 3. Synthesis with Macro Cells. 4. Exercises and Problems. 10. FIELD PROGRAMMABLE LOGIC ARRAYS. 1. Synthesis with FPGAs. 2. Synthesis with Antifuse-Based FPGAs. 3. Synthesis with LUT-FPGAs. 4. Exercises and Problems. 11. BOOLEAN DIFFERENCE AND APPLICATIONS IN TESTING LOGIC NETWORKS. 1. Boolean Difference. 2. Properties of the Boolean Difference. 3. Calculation of the Boolean Difference. 4. Boolean Difference in Testing Logic Networks. 5. Easily Testable Logic Networks. 6. Easily Testable Realizations from PPRM-expressions. 7. Easily Testable Realizations from GRM-expressions. 8. Exercises and Problems. 12. SEQUENTIAL NETWORKS. 1. Basic Sequential Machines. 2. State Tables. 3. Conversion of Sequential Machines. 4. Minimization of States. 5. Incompletely Specified Machines. 6. State Assignment. 7. Decomposition of Sequential Machines. 8. Exercises and Problems. 13. REALIZATION OF SEQUENTIAL NETWORKS. 1. Memory Elements. 2. Synthesis of Sequential Networks. 3. Realization of Binary Sequential Machines. 4. Realization of Synchronous Sequential Machines. 5. Pulse Mode Sequential Networks. 6. Asynchronous Sequential Networks. 7. Races and Hazards. 8. Exercises and Problems. References. Index

Fundamentals of Switching Theory and Logic Design - A Hands on Approach

An algorithm has been developed to calculate the arithmetic transform of Boolean functions from their Ordered Binary Decision Diagram (OBDD) representation. The method of decomposition of arithmetic spectral coefficients in terms of the cofactors of Boolean functions that resembles known Shannon decomposition of such functions has been introduced for the first time. Based on the above decomposition, a second new algorithm is presented to synthesize Ordered Binary Decision Diagrams directly from the arithmetic spectrum of Boolean functions. >

Efficient algorithms for the calculation of arithmetic spectrum from OBDD and synthesis of OBDD from arithmetic spectrum for incompletely specified Boolean functions

This chapter deals with the paradigm of handling system reliability analysis in the Boolean domain as a supplement to (rather than a replacement to) analysis in the probability domain. This paradigm is well-established within the academic circles of reliability theory and engineering, albeit virtually unknown outside these circles. The chapter lists and explains arguments in favor of this paradigm for systems described by verbal statements, fault trees, block diagrams, and network graphs. This is followed by a detailed exposition of the pertinent concept of the Real or Probability Transform of a switching (two-valued Boolean) function, and that of a Probability-Ready Expression (PRE). Some of the important rules used in generating a PRE are presented, occasionally along with succinct proofs. These include rules to achieve disjointness (orthogonality) of ORed formulas, and to preserve statistical independence, as much as possible, among ANDed formulas. Recursive relations in the Boolean domain are also discussed, with an application to the four versions of the AR algorithm for evaluating the reliability and unreliability of the k-out-of-n:G and the k-out-of-n:F systems. These four versions of the algorithm are explained in terms of signal flow graphs that are compact, regular, and acyclic, in addition to being isomorphic to the Reduced Ordered Binary Decision Diagram (ROBDD). An appendix explains some important properties of the concept of Boolean quotient, whose expectation in Boolean-based probability is the counterpart of conditional probability in event-based probability.

Switching-Algebraic Analysis of System Reliability

A method for the derivation of fault signatures for the detection of faults in single-output combinational networks is described. The approach uses the arithmetic spectrum instead of the Rademacher-Walsh spectrum. It is a form of data compression that serves to reduce the volume of the response data at test time. The price which is paid for the reduction in the storage requirements is that some of the knowledge of exact fault location is lost. The derived signatures are short and easily tested using very simple test equipment. The test circuitry could be included on the chip since the overhead involved is comparatively small. The test procedure requires a high-speed counter cycling at maximum speed through selected subsets of all input combinations. Hence, the network under test is exercised at speed, and a number of dynamic errors that are not testable by means of conventional test-set approaches will be detected. >

K.D. Heidtmann

Papers

Arithmetic spectrum applied to fault detection for combinational networks