Optimization Class 1

Optimization in the number of (Complemented Edges)CEs if we follow the variable propagation either from left to right or right to left between reconvergent and nonreconvergent boolean variable.

The transformation rules in this class are as follows:

$1.\:m(u,x,m(x,y,z))\leftrightarrow m(u,\bar{y},m(\bar{y},\bar{x},z))\leftrightarrow m(u,\bar{z},m(\bar{z},y,\bar{x}))$

$2.\:m(u,x,m(x,y,\bar{z}))\leftrightarrow m(u,\bar{y},m(\bar{y},\bar{x},\bar{z}))\leftrightarrow m(u,\bar{y},M(y,x,z))$

$3.\:m(u,x,m(x,\bar{y},z))\leftrightarrow m(u,\bar{z},m(\bar{z},\bar{y},\bar{x}))\leftrightarrow m(u,\bar{z},M(y,x,z))$

$4.\:m(u,x,m(\bar{x},y,\bar{z}))\leftrightarrow m(\bar{y},x,m(\bar{x},\bar{u},\bar{z}))\leftrightarrow m(\bar{y},x,M(x,u,z))$

$5.\:m(u,x,m(\bar{x},\bar{y},z))\leftrightarrow m(y,x,m(\bar{x},\bar{u},z))\leftrightarrow m(\bar{z},x,m(\bar{x},\bar{y},\bar{u}))$

$6.\:m(u,x,m(\bar{x},\bar{y},\bar{z}))\leftrightarrow m(y,x,m(\bar{x},\bar{u},\bar{z}))\leftrightarrow m(z,x,m(\bar{x},\bar{y},\bar{u}))\\ \leftrightarrow m(y,x,M(x,u,z))\leftrightarrow m(z,x,M(x,y,u))$

$7.\:m(u,\bar{x},m(x,y,z))\leftrightarrow m(\bar{y},\bar{x},m(x,\bar{u},z))\leftrightarrow m(\bar{z},\bar{x},m(x,y,\bar{u}))$

$8.\:m(u,\bar{x},m(x,y,\bar{z}))\leftrightarrow m(\bar{y},\bar{x},m(x,\bar{u},\bar{z}))\leftrightarrow m(z,\bar{x},m(x,y,\bar{u}))$

$9.\:m(u,\bar{x},m(x,\bar{y},z))\leftrightarrow m(y,\bar{x},m(x,\bar{u},z))\leftrightarrow m(\bar{z},\bar{x},m(x,\bar{y},\bar{u}))$

$10.\:m(u,\bar{x},m(\bar{x},y,\bar{z}))\leftrightarrow m(u,z,m(z,y,x))\leftrightarrow m(u,\bar{y},m(\bar{y},x,\bar{z}))$

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Optimization Class 1

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Optimization Class 1