Efficient Algorithms for Creation of Linearly‐independent Decision Diagrams and their Mapping to Regular Layouts

Abstract

A new kind of a decision diagrams are presented: its nodes correspond to all types of nonsingular expansions for groups of input variables, in particular pairs. The diagrams are called the Linearly Independent (LI) Decision Diagrams (LI DDs). There are 840 nonsigular expansions for a pair of variables, thus 840 different types of nodes in the tree. Therefore, the number of nodes in such (exact) diagrams is usually much smaller than the number of nodes in the well-known Kronecker diagrams (which have only single-variable Shannon, Positive Davio, and Negative Davio expansions in nodes). It is usually much smaller than 1/3 of the number of nodes in Kronecker diagrams. Similarly to Kronecker diagrams, the LI Diagrams are a starting point to a synthesis of multilevel AND/OR/EXOR circuits with regular structures. Other advantages of LI diagrams include: they generalize the well-known Pseudo-Kronecker Functional Decision Diagrams, and can be used to optimize the new type of PLAs called LI PLAs. Importantly, while the known decision diagrams used AND/EXOR or AND/OR bases, the new diagrams are AND/OR/EXOR-based. Thus, because of a larger design space, multi-level structures of higher regularity can be created with them. This paper presents both new concepts and new efficient synthesis algorithms.

Family of fast transforms over GF(3) logic

“…Family of LI transformations that posses fast forward and inverse butterfly diagrams in different classes and their basic properties have been obtained [3]. LI transforms are also used as the underlying transforms to develop new types of word decision diagrams [8]. The binary Reed-Muller transform itself is only one type of LI transforms presented in [3].…”

Polynomial expansions over GF(3)based on fastest transformation

“…This paper introduces two fastest families of ternary LI transforms as the pair of transforms that has the least computational complexity among all ternary LI transforms. Moreover similarly to known LI transforms for binary case that can be used for the development of word decision diagrams [10] the new fastest ternary transforms can have the same applications as well. In this article it is also shown that the fastest transform are always advantageous over well known ternary Reed-Muller transform in terms of the necessary computation and for some ternary functions the number of non-zero spectral coefficients of the polynomial expansion based on the fastest transform can be smaller that the one for ternary Reed-Muller expansion.…”

Properties and Relations of Ternary Linearly Independent Transforms

“…Some useful properties of these LI transforms are discussed. Similar to known polynomial expansions based on binary and multiplevalued logic, the ternary LI transforms presented in this paper can have applications in spectral representations of ternary logic functions, they can also be the bases of new ternary word decision diagrams in a manner similar to the ones developed in [5,7]. …”

“…When such orthonormal basis is performed over GF (2), the concept of Linearly Independent (LI) transforms based on the basis of different two-valued Boolean functions has been developed. Linearly independent logic has proved to be not only of great theoretical value, but also of practical value, to the development of new types of decision diagrams and design of fine-grain and cellular automata types of Field Programmable Gate Arrays (FPGAs) and different Programmable Logic Devices (PLDs) with XOR gates [7]. Another important development in further consideration of different linearly independent functions as the efficient tools in the designing of modern FPGAs and PLDs was the identification of those linearly independent expansions for which there exist fast forward and inverse transformation and for efficient calculation of their spectral coefficients [4] and hence finding the final hardware implementations.…”