# Two-level Logic Minimization¶

This chapter will explain how to use PyEDA to minimize two-level “sum-of-products” forms of Boolean functions.

Logic minimization is known to be an NP-complete problem. It is equivalent to finding a minimal-cost set of subsets of a set $$S$$ that covers $$S$$. This is sometimes called the “paving problem”, because it is conceptually similar to finding the cheapest configuration of tiles that cover a floor. Due to the complexity of this operation, PyEDA uses a C extension to the famous Berkeley Espresso library [1].

All examples in this chapter assume you have interactive symbols imported:

>>> from pyeda.inter import *


## Minimize Boolean Expressions¶

Consider the three-input function $$f_{1} = a \cdot b' \cdot c' + a' \cdot b' \cdot c + a \cdot b' \cdot c + a \cdot b \cdot c + a \cdot b \cdot c'$$

>>> a, b, c = map(exprvar, 'abc')
>>> f1 = ~a & ~b & ~c | ~a & ~b & c | a & ~b & c | a & b & c | a & b & ~c


To use Espresso to perform minimization:

>>> f1m, = espresso_exprs(f1.to_dnf())
>>> f1m
Or(And(~a, ~b), And(a, b), And(~b, c))


Notice that the espresso_exprs function returns a tuple. The reason is that this function can minimize multiple input functions simultaneously. To demonstrate, let’s create a second function $$f_{2} = a' \cdot b' \cdot c + a \cdot b' \cdot c$$.

>>> f2 = ~a & ~b & c | a & ~b & c
>>> f1m, f2m = espresso_exprs(f1, f2)
>>> f1m
Or(And(~a, ~b), And(a, b), And(~b, c))
>>> f2m
And(~b, c)


It’s easy to verify that the minimal functions are equivalent to the originals:

>>> f1.equivalent(f1m)
True
>>> f2.equivalent(f2m)
True


## Minimize Truth Tables¶

An expression is a completely specified function. Sometimes, instead of minimizing an existing expression, you instead start with only a truth table that maps inputs in $${0, 1}$$ to outputs in $${0, 1, *}$$, where $$*$$ means “don’t care”. For this type of incompletely specified function, you may use the espresso_tts function to find a low-cost, equivalent Boolean expression.

Consider the following truth table with four inputs and two outputs:

Inputs Outputs
x3 x2 x1 x0 f1 f2
0 0 0 0 0 0
0 0 0 1 0 0
0 0 1 0 0 0
0 0 1 1 0 1
0 1 0 0 0 1
0 1 0 1 1 1
0 1 1 0 1 1
0 1 1 1 1 1
1 0 0 0 1 0
1 0 0 1 1 0
1 0 1 0 X X
1 0 1 1 X X
1 1 0 0 X X
1 1 0 1 X X
1 1 1 0 X X
1 1 1 1 X X

The espresso_tts function takes a sequence of input truth table functions, and returns a sequence of DNF expression instances.

>>> X = ttvars('x', 4)
>>> f1 = truthtable(X, "0000011111------")
>>> f2 = truthtable(X, "0001111100------")
>>> f1m, f2m = espresso_tts(f1, f2)
>>> f1m
Or(x[3], And(x[0], x[2]), And(x[1], x[2]))
>>> f2m
Or(x[2], And(x[0], x[1]))


You can test whether the resulting expressions are equivalent to the original truth tables by visual inspection (or some smarter method):

>>> expr2truthtable(f1m)
x[3] x[2] x[1] x[0]
0    0    0    0 : 0
0    0    0    1 : 0
0    0    1    0 : 0
0    0    1    1 : 0
0    1    0    0 : 0
0    1    0    1 : 1
0    1    1    0 : 1
0    1    1    1 : 1
1    0    0    0 : 1
1    0    0    1 : 1
1    0    1    0 : 1
1    0    1    1 : 1
1    1    0    0 : 1
1    1    0    1 : 1
1    1    1    0 : 1
1    1    1    1 : 1
>>> expr2truthtable(f2m)
x[2] x[1] x[0]
0    0    0 : 0
0    0    1 : 0
0    1    0 : 0
0    1    1 : 1
1    0    0 : 1
1    0    1 : 1
1    1    0 : 1
1    1    1 : 1


## Espresso Script¶

Starting with version 0.20, PyEDA includes a script that implements some of the functionality of the original Espresso command-line utility.

$espresso -h usage: espresso [-h] [-e {fast,ness,nirr,nunwrap,onset,strong}] [--fast] [--no-ess] [--no-irr] [--no-unwrap] [--onset] [--strong] [file] Minimize a PLA file positional arguments: file PLA file (default: stdin) optional arguments: -h, --help show this help message and exit ...  Here is an example of a simple PLA file that is part of the BOOM benchmark suite. This function has 50 input variables, 5 output variables, and 50 product terms. Also, 20% of the literals in the implicants are “don’t care”. $ cat extension/espresso/test/bb_all/bb_50x5x50_20%_0.pla
.i 50
.o 5
.p 50
.ilb x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 x41 x42 x43 x44 x45 x46 x47 x48 x49
.ob y0 y1 y2 y3 y4
.type fr
001011110--00--0100-0010-10-01010101-010--01011111 00011
0-1-1010-01000100--11011-0110001010-10001010-1-1-0 00100
0111010111110110110-100101101010010001111----1-011 10000
01-0010011-001110--000-011--11--1-0100-01101--1000 00001
001011010-1100000001101--10100-010-001100111110010 00101
011-01010-10-1101110-00-1-11001-1-0000--1-1-00-000 00011
0000000011001-0000010-000110-11011001110--100-1-10 00110
00-111111-00100-100111101000-11101100--0-1110-1-10 10001
-1-10011011000011--00-0011011101-1-1101110--1-001- 11100
-0101100-110111-010-01110-0110011-1-1---1001011111 11000
0-111011101000-11-1--10-0001001101000010-11-111101 11001
11000000-1-01--1111-10111111----0010--1-0--1--0111 01010
00-101000011-1-10101101-1101011-0101000-111111011- 11011
1-00-11111-0010-0---000--0110-0010111--000001-0001 11011
-1-1100100001--00--00001000-1-1--100-0111-00011011 11000
0-0000-010-11-1100-101-00101000111-01--11-0010-011 10000
11-1-0001100101-10-0-1-0-1100010101111-1111000-101 11101
10-01--10011111-11011-001001101100110010010-000-0- 01110
1-11010-00011101-010--101010--0111010101-11-101--1 00111
11--111-111-111-000-11000-101-1-011--1000--1111100 01111
0---0-10011101000--11001-1100-10-000011-0100011100 11110
-01--11-010-1001011-0-101000100000--10111---100-1- 11101
11-1-000010--00110-011101--11-10-1-0000110100-1101 11010
-0111110-100-11-110001001100001-100011110001001100 11110
11--00100-01--00-10---11-0001-00011101001011-01110 00000
1--010011-001-0000--0-11-001010001110-00-01-110-11 01101
100011--0101--1-1-0-101--001-0-101-1-1011101011-01 00111
0--0-01-10101-11-0100111111000-1-1011100-111-01111 10100
0-0110010--11101-0---1001-1001--001-110000---1011- 00100
0-1000-0--00000010-0--1011-1001011-01-00-011001111 10000
111-1101111-01001101-111--00-01011111000001-001001 11100
0--100111-1010001-0111-0-000--00-0111101111-101100 11000
00001101100-001001-1010010010011-1101-110-10-110-1 01011
0101-01-0100101000010111--0011-0011011110-111100-0 00100
000-1--100-00-1001-10-000000100-001100-10101010001 10000
10001001-0001011-1-1-0-00101110-10100---0010001--- 10111
01011000000100100000---1--11-0001011111101-01-1011 01111
1--01--00100110001-110-0-00001011---01001000110--- 10010
0-0001--01--11101010100000000010011001000-01100001 00011
0-0100110-00111100-001--11--00-1001-00-0-11-1-0-1- 00100
101-1-100-001001-010111-01--010-1-1011-01101001001 11000
0110-111011--1-010101-011-1-00100110-00-1111000-11 11001
011001011---010011-10-00-11-001000000101101101-0-1 00100
1001111-1-1111-1001-000111010-100--0111110011000-1 10111
1-1010-1-100111110010-101011-1001000111-0000--11-1 11000
-00110001000010000010100010010-0-0-100-1-0111011-1 00101
1110-01100111111-1-1-110-0-110--011--01-11-0000-01 00000
-01010101010-1-1-00-1111010100-1001111110110--0-00 11011
110-10000001--0-0-01001111-0011-0110110100010--111 11111
101-10111000011110000-1001-001-01111-011-0001-0100 00100
.e


After running the input file through the espresso script, it minimizes the function to 26 implicants with significantly fewer literals.

\$ espresso extension/espresso/test/bb_all/bb_50x5x50_20%_0.pla
.i 50
.o 5
.ilb x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 x41 x42 x43 x44 x45 x46 x47 x48 x49
.ob y0 y1 y2 y3 y4
.p 26
----------------------0-----1---0------0---------- 01110
------0-----------------------------1-----1--0--0- 11100
---------------------------1--------1---0------0-- 01011
-------------------------------------11-1--------1 10000
0-------------------------1---------1-1----------- 00110
--------0--------1----------0-------0------------- 00010
-------------0---------0-0-0--------------------1- 10001
--------------0-00------------------0-------1----- 00101
--------------------------0-0---0---------0------- 00011
-----------0----0-------------------------0---0--- 10000
-------------------------1---0---1---------------1 10000
-----------01-----------------------------0---1--1 11000
-------------00------------------------11--------- 11000
----------------------------0-1---------0-0---1--- 11110
--------------------------1----1--0------------1-- 00100
-----1-----------------------------1-1---1-------- 00111
------1---------------0-------1--1---0------------ 11001
------0----0------------------------------1-1-0--- 01000
-1---1--------1----------------------------------0 00001
-----------------1----------------------0-----0-1- 01010
--------------------0--------1---------1------0--- 00100
---------------------------11-------------1-0-1--- 10010
--------------------------1-----0----10----------- 00100
------0------0---------0---------0-----------0---- 00101
-------0----------0---1--1--0---0----------------- 11011
--------------------0-----------0-----------1-0--- 00100
.e


## References¶

 [1] R. Brayton, G. Hatchel, C. McMullen, and A. Sangiovanni-Vincentelli, Logic Minimization Algorithms for VLSI Synthesis, Kluwer Academic Publishers, Boston, MA, 1984.