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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.

This paper introduces a new design approach that combines stages of logic and physical design. The logic function is synthesized and mapped to a two-dimensional array of logic cells. This array generalizes PLAs, XPLAs and cellular Maitra cascades. Each cell can be programmed to a wire, an inverter, or a two-input AND, OR or EXOR gate (with any subset of inputs negated). The gate can take any output of four neighbor cells and four neighbor buses as its inputs, and sends its result back to them. This two-dimensional geometrical model is well suited for both fine-grain FPGA realization and sea-of-gates custom ASIC layout. The comprehensive design method starts from a Boolean function, specified as SOP or ESOP, and produces a rectangularly shaped structure of (mostly) locally connected cells. Two stages: restricted factorization, and column folding, are discussed in more details to illustrate our general methodology.