Family of fast transforms over GF(3) logic

Falkowski, B.J.; Fu, Cheng

doi:10.1109/ismvl.2003.1201424

Cited by 15 publications

(33 citation statements)

References 8 publications

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“…In general, we do not claim that our circuits are always best, only that they are not worse than those obtained from Pseudo-Kronecker diagrams for single variables. B [7,24] and thus, 840 different universal modules. (All these universal modules are similar and can be realized by a single switchable universal module).…”

Section: The Nonsingular Expansionmentioning

confidence: 99%

“…Thus, in LI(2)-[n 1,i, n 2,i, n 3,i, n 4,i ] the number n j,i is a natural number corresponding to the binary vector of the j-th column of the i-th matrix M. This number is read with the bottom row as the least significant bit. In this way, the (expansion polarity) matrix is represented as a vector of four natural numbers, each corresponding to one LI function, being a column of M, starting from the left, and denoted by LI(2)- [15,3,10,7]. The name "polarity" comes from standard Reed -Muller logic, where it describes a variable or its negation consistently taken in an expansion.…”

Section: Definitionmentioning

confidence: 99%

“…where f 2 ðx 2 ; x 3 ; x 4 Þ ¼ f 0 ðx 2 ; x 3 ; x 4 Þ%f 1 ðx 2 ; x 3 ; x 4 Þ and f 0 ðx 2 ; x 3 ; x 4 Þ; f 1 ðx 2 ; x 3 ; x 4 Þ are, respectively, the negative and positive cofactors of f with respect to input variable x 1 The second level has LI(2)- [15,3,10,7] expansion for the set of LI functions on variables {x 2 ,x 4 } and the third level has Shannon expansions for variable x 3 .…”

Section: Definitionmentioning

confidence: 99%

“…[26] only outlined efficient approaches for limited types of nonsingular expansions, but no detailed synthesis algorithms were presented. Paper [7] presented a "fast transform" method to find a single expansion for certain polarities of variables, but still the problem of selecting the best polarity (and thus, the best expansion) among all polarities of two-variable nonsingular expansions was not discussed. Therefore, although there exist fast transforms (expansions), still no methods are known to select a good one among a huge number of such transforms.…”

Section: Introductionmentioning

confidence: 99%

“…However, the algorithms for expansion selection will be restricted here to pairs of variables in expansions, similarly to [7].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2000

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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.

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Section: The Nonsingular Expansionmentioning

confidence: 99%