Two Hierarchies of Generalized Kronecker Trees, Forms, Decision Diagrams, and Regular Layouts

| The paper presents two hierarchies of canonical AND/EXOR trees, forms and decision diagrams. The rst hierarchy generalizes the Kronecker and Generalized Kronecker representations by introducing new canonical AND/EXOR forms. We propose to call all these new forms and future AND/EXOR forms including KRO and GRM, the Zhegalkin forms 42] to honor the Russian scientist who in 1927 discovered the forms now attributed to Reed and Muller and invented by them in 1954. The new Zhegalkin representations and forms can be used for synthesis of quasi-minimum ESOP circuits and the new diagrams can represent large functions and can be used for optimal synthesis of highly testable multilevel circuits in several technologies, especially in Fine Grain Field Programmable Gate Arrays. The second hierarchy generalizes and extends the Universal Akers Array to expansions other than Shannon and neighborhoods other then 2-inputs, 2-outputs. These diagrams are called Zhegalkin Lattice Diagrams.