Radomir Stankovic

Spectral Logic and Its Applications for the Design of Digital Devices: Karpovsky/Spectral Logic

PREFACE. ACKNOWLEDGMENTS. LIST OF FIGURES. LIST OF TABLES. ACRONYMS.1. LOGIC FUNCTIONS. 1.1 Discrete Functions. 1.2 Tabular Representations of Discrete Functions. 1.3 Functional Expressions. 1.4 Decision Diagrams for Discrete Functions. 1.5 Spectral Representations of Logic Functions. 1.6 Fixed-polarity Reed-Muller Expressions of Logic.Functions. 1.7 Kronecker Expressions of Logic Functions. 1.8 Circuit Implementation of Logic Functions. 2. SPECTRAL TRANSFORMS FOR LOGIC FUNCTIONS. 2.1 Algebraic Structures for Spectral Transforms. 2.2 Fourier Series. 2.3 Bases for Systems of Boolean Functions. 2.4 Walsh Related Transforms. 2.5 Bases for Systems of Multiple-Valued Functions. 2.6 Properties of DiscreteWalsh andVilenkin-Chrestenson Transforms. 2.7 Autocorrelation and Cross-Correlation Functions. 2.8 Harmonic Analysis over an Arbitrary Finite Abelian Group. 2.9 Fourier Transform on Finite Non-Abelian Groups. 3. CALCULATION OF SPECTRAL TRANSFORMS. 3.1 Calculation of Walsh Spectra. 3.2 Calculation of the Haar Spectrum. 3.3 Calculation of the Vilenkin-Chrestenson Spectrum. 3.4 Calculation of the Generalized Haar Spectrum. 3.5 Calculation of Autocorrelation Functions. 4. SPECTRAL METHODS IN OPTIMIZATION OF DECISION DIAGRAMS. 4.1 Reduction of Sizes of Decision Diagrams. 4.2 Construction of Linearly Transformed Binary Decision Diagrams. 4.3 Construction of Linearly Transformed Planar BDD. 4.4 Spectral Interpretation of Decision Diagrams. 5. ANALYSIS AND OPTIMIZATION OF LOGIC FUNCTIONS. 5.1 Spectral Analysis of Boolean Functions. 5.2 Analysis and Synthesis of Threshold Element Networks. 5.3 Complexity of Logic Functions. 5.4 Serial Decomposition of Systems of Switching Functions. 5.5 Parallel Decomposition of Systems of Switching Functions. 6. SPECTRAL METHODS IN SYNTHESIS OF LOGIC NETWORKS. 6.1 Spectral Methods of Synthesis of Combinatorial Devices. 6.2 Spectral Methods for Synthesis of Incompletely Specified Functions. 6.3 Spectral Methods of Synthesis of Multiple-Valued Functions. 6.4 Spectral Synthesis of Digital Functions and Sequences Generators. 7. SPECTRAL METHODS OF SYNTHESIS OF SEQUENTIAL MACHINES. 7.1 Realization of Finite Automata by Spectral Methods. 7.2 Assignment of States and Inputs for Completely Specified Automata. 7.3 State Assignment for Incompletely Specified Automata. 7.4 Some Special Cases of the Assignment Problem. 8. HARDWARE IMPLEMENTATION OF SPECTRAL METHODS. 8.1 Spectral Methods of Synthesis with ROM. 8.2 Serial Implementation of Spectral Methods. 8.3 Sequential Haar Networks. 8.4 Complexity of Serial Realization by Haar Series. 8.5 Parallel Realization of Spectral Methods of Synthesis. 8.6 Complexity of Parallel Realization. 8.7 Realization by Expansions over Finite Fields. 9. SPECTRAL METHODS OF ANALYSIS AND SYNTHESIS OF RELIABLE DEVICES. 9.1 Spectral Methods for Analysis of Error Correcting Capabilities. 9.2 Spectral Methods for Synthesis of Reliable Digital Devices. 9.3 Correcting Capability of Sequential Machines. 9.4 Synthesis of Fault-Tolerant Automata with Self-Error Correction. 9.5 Comparison of Spectral and Classical Methods. 10. SPECTRAL METHODS FOR TESTING OF DIGITAL SYSTEMS. 10.1 Testing and Diagnosis by Verification of Walsh Coefficients. 10.2 Functional Testing, Error Detection, and Correction by Linear Checks. 10.3 Linear Checks for Processors. 10.4 Linear Checks for Error Detection in Polynomial Computations. 10.5 Construction of Optimal Linear Checks for Polynomial Computations. 10.6 Implementations and Error-Detecting Capabilities of Linear Checks. 10.7 Testing for Numerical Computations. 10.8 Optimal Inequality Checks and Error-Correcting Codes. 10.9 Error Detection in Computer Memories by Linear Checks. 10.10 Location of Errors in ROMs by Two Orthogonal Inequality Checks. 10.11 Detection and Location of Errors in Random-Access Memories. 11. EXAMPLES OF APPLICATIONS AND GENERALIZATIONS OF SPECTRAL METHODS ON LOGIC FUNCTIONS. 11.1 Transforms Designed for Particular Applications. 11.2 Wavelet Transforms. 11.3 Fibonacci Transforms. 11.4 Two-Dimensional Spectral Transforms. 11.5 Application of the Walsh Transform in Broadband Radio. APPENDIX A. REFERENCES. INDEX.

Switching Theory in USSR

The ideas suggested by Ehrenfest about an algebra of switching circuits have been explored and elaborated by V.I. Sestakov, a student of V.I. Glivenko, and the results were reported in written form in January 1935, however, as Anovskaja [Anovskaja] states this paper has not been published at the time, and has been used as foundations for the PhD candidate Thesis by Sestakov.

Switching Theory - From Art and Skills to Scientific Methods

Majority of scholars working towards mathematical foundations for analysis and design of relay circuits focussed their efforts towards study of many particular examples attempting to disclose mathematical relationships describing such circuits. Many of them realized latter that partial mathematical results which they have developed are identical to elements of the Boolean algebra.

From Logic to Mathematical Logic

Although methods of logic and were obviously present in many cultures, which all used some intricate systems of reasoning, it is commonly accepted that explicit analysis of the principles of reasoning were developed independently in China, India, and Greece. The later being the most influential to the systems of logic in the West. In particular, the Aristotelian logic was widely accepted in the western science and mathematics. Many scholars contributed to the development, which has been continued by Islamic and medieval European scholars. The mid-fourteenth century is considered to be the period of the most respectable achievements. The period that followed including the first three or four decades of nineteenth century is viewed as a barren period or even period of declination and degradation of logic.

Convolution on Finite Groups and Fixed-Polarity Polynomial Expressions

This paper discusses relationships among convolution matrices and fixed-polarity matrices for polynomial expressions of discrete functions on finite groups. Switching and multiple-valued functions are considered as particular examples of discrete functions on finite groups. It is shown that if the negative literals for variables are defined in terms of the shift operators on domain groups, then there is a relationship between the polarity matrices and convolution matrices. Therefore, the recursive structure of polarity matrices follows from the recursive structure of convolution matrices. This structure is determined by the assumed decomposition of the domain groups for the considered functions.

Hilbert Transform on Finite Groups

The prelims comprise: Some results of Fourier analysis on finite non-Abelian groups Hilbert transform on finite non-Abelian groups Hilbert transform in finite fields References

Fourier Analysis on NonAbelian Groups

The prelims comprise: Representations of Groups Fourier Transform on Finite Groups Properties of the Fourier transform Matrix interpretation of the Fourier transform on finite non-Abelian groups Fast Fourier transform on finite non-Abelian groups References

Matrix Interpretation of the Fast Fourier Transform

The prelims comprise: Matrix interpretation of FFT on finite non-Abelian groups Illustrative examples Complexity of the FFT FFT through decision diagrams References

Gibbs Derivatives on Finite Groups

The prelims comprise: Definition and properties of Gibbs derivatives on finite non-Abelian groups Gibbs anti-derivative Partial Gibbs derivatives Gibbs differential equations Matrix interpretation of Gibbs derivatives Fast algorithms for calculation of Gibbs derivatives on finite groups Calculation of Gibbs derivatives through DDs References