Edinburgh Napier University

  In certain applications, AND/XOR (Reed-Muller), and ORlXNOR (Dual
form of Reed-Muller) logic have shown some attractive advantages over the
standard Sum of Products (SOP) and Product of Sums (POS). Bidirectional
conversion algorithms between SOP and AND/XOR also between POS and
ORlXNOR based on Sparse and partitioning techniques are presented for multiple
output Boolean functions. The developed programs are tested for some
benchmarks with up to 20 inputs and 40 outputs.
A new direct method is presented to calculate the coefficients of the Fixed
Polarity Dual Reed-Muller (FPDRM) from the truth vector of the POS. Any
Boolean function can be expressed by FPDRM forms. There are 211 polarities for
an n-variable function and the number of sum terms depends on these polarities.
Finding the best polarity is costly interims of CPU time, in order to search for the
best polarity which will lead to the minimum number of sums for a particular
function. Therefore, an algorithm is developed to compute all the coefficients of
the Fixed Polarity Dual Reed-Muller (FPDRM) with polarity p from any polarity q.
This technique is used to find the best polarity of FPDRM among the 211 fixed
polarities. The algorithm is based on the Dual- polarity property and the Gray code
strategy. Therefore, there is no need to start from POS form to find FPDRM
coefficients for all the polarities. The proposed methods are efficient in terms of
memory size and CPU time. A fast algorithm is developed and implemented in C
language which can convert between POSs and FPDRMs. The program was tested
for up to 23 variables. A modified version of the same program was used to find
the best polarity. For up to 13 variables the CPU time was less than 42 seconds.
To search for the optimal polarity for large number of variables and to
reduce the se arch time 0 ffinding the 0 ptimal polarity 0 fthe function, two new
algorithms are developed and presented in this thesis. The first one is used to
convert between P OS and Positive Polarity Dual Reed-Muller (PPDRM) forms.
The second algorithm will find the optimal fixed polarity for the FPDRM among
the 211 different polarities for large n-variable functions. The most popular
minimization criterion of the FPDRM form is obtained by the exhaustive search of
the entire polarity vector. A non-exhaustive method for FPDRM expansions is
presented. The new algorithms are based on separation of the truth vector (T) of
POSs around each variable Xi into two groups. Instead of generating all of the
polarity sets and searching for the best polarity, this algorithm will find the optimal
polarity using the separation and sparse techniques, which will lead to optimal
polarity. Time efficiency and computing speed are thus achieved in this technique.
The algorithms don't require a large size of memory and don't require a long CPU
time. The two algorithms are implemented in C language and tested for some
benchmark. The proposed methods are fast and efficient as shown in the
experimental results and can be used for large number of variables.