For n variables, there exists 2^(2^n) distinct boolean functions. For example, if n=2, then there exists 16 possible boolean functions which can be written in sum of product form, or product of sum forms. The number of possible functions increases exponentially with n.
I am looking for an algorithm which can generate all these possible boolean rules for n variables. I have tried to search at various places, but have not found anything suitable till now. Most of the algorithms are related to simplifying or reducing boolean functions to standard forms.
I know even for the number of rules become too large even for n=8 or 9, but can somebody please help me out with the relevant algorithm if it exists?
Thus 16 semantically different boolean functions.
Theorem 1. There are 22n different Boolean functions on n Boolean variables.
A boolean function of n variables has 2^n possible inputs. These can be enumerated by printing out the binary representation of values in the range 0 <= x < 2^n
.
For each one of the those possible inputs, a boolean function can output 0 or 1. To enumerate all the possibilities (i.e. every possible truth table). List the binary values in range 0 <= x < 2^(2^n)
.
Here's the algorithm in Python:
from __future__ import print_function
from itertools import product # forms cartesian products
n = 3 # number of variables
print('All possible truth tables for n =', n)
inputs = list(product([0, 1], repeat=n))
for output in product([0, 1], repeat=len(inputs)):
print()
print('Truth table')
print('-----------')
for row, result in zip(inputs, output):
print(row, '-->', result)
The output looks like this:
All possible truth tables for n = 3
Truth table
-----------
(0, 0, 0) --> 0
(0, 0, 1) --> 0
(0, 1, 0) --> 0
(0, 1, 1) --> 0
(1, 0, 0) --> 0
(1, 0, 1) --> 0
(1, 1, 0) --> 0
(1, 1, 1) --> 0
Truth table
-----------
(0, 0, 0) --> 0
(0, 0, 1) --> 0
(0, 1, 0) --> 0
(0, 1, 1) --> 0
(1, 0, 0) --> 0
(1, 0, 1) --> 0
(1, 1, 0) --> 0
(1, 1, 1) --> 1
Truth table
-----------
(0, 0, 0) --> 0
(0, 0, 1) --> 0
(0, 1, 0) --> 0
(0, 1, 1) --> 0
(1, 0, 0) --> 0
(1, 0, 1) --> 0
(1, 1, 0) --> 1
(1, 1, 1) --> 0
Truth table
-----------
(0, 0, 0) --> 0
(0, 0, 1) --> 0
(0, 1, 0) --> 0
(0, 1, 1) --> 0
(1, 0, 0) --> 0
(1, 0, 1) --> 0
(1, 1, 0) --> 1
(1, 1, 1) --> 1
... and so on
If you want the output in algebraic form rather than truth tables, the algorithm is the same:
from __future__ import print_function
from itertools import product # forms cartesian products
n = 3 # number of variables
variables = 'abcdefghijklmnopqrstuvwxyz'[:n]
pairs = [('~'+var, var) for var in variables]
print('All possible algebraic expressions for n =', n)
inputs = list(product(*pairs))
for i, outputs in enumerate(product([0, 1], repeat=len(inputs))):
terms = [''.join(row) for row, output in zip(inputs, outputs) if output]
if not terms:
terms = ['False']
print('Function %d:' % i, ' or '.join(terms))
The output looks like this:
All possible algebraic expressions for n = 3
Function 0: False
Function 1: abc
Function 2: ab~c
Function 3: ab~c or abc
Function 4: a~bc
Function 5: a~bc or abc
Function 6: a~bc or ab~c
Function 7: a~bc or ab~c or abc
Function 8: a~b~c
Function 9: a~b~c or abc
Function 10: a~b~c or ab~c
Function 11: a~b~c or ab~c or abc
Function 12: a~b~c or a~bc
Function 13: a~b~c or a~bc or abc
Function 14: a~b~c or a~bc or ab~c
Function 15: a~b~c or a~bc or ab~c or abc
Function 16: ~abc
Function 17: ~abc or abc
Function 18: ~abc or ab~c
Function 19: ~abc or ab~c or abc
Function 20: ~abc or a~bc
Function 21: ~abc or a~bc or abc
Function 22: ~abc or a~bc or ab~c
Function 23: ~abc or a~bc or ab~c or abc
Function 24: ~abc or a~b~c
Function 25: ~abc or a~b~c or abc
Function 26: ~abc or a~b~c or ab~c
Function 27: ~abc or a~b~c or ab~c or abc
Function 28: ~abc or a~b~c or a~bc
Function 29: ~abc or a~b~c or a~bc or abc
Function 30: ~abc or a~b~c or a~bc or ab~c
Function 31: ~abc or a~b~c or a~bc or ab~c or abc
Function 32: ~ab~c
Function 33: ~ab~c or abc
... and so on
As mentioned in comments, there's a one-to-one relation between numbers and truth tables. For example, we can represent the truth table
0 0 0 | 1
0 0 1 | 1
0 1 0 | 0
0 1 1 | 0
1 0 0 | 1
1 0 1 | 0
1 1 0 | 1
1 1 1 | 0
by the binary number 01010011
(the topmost row is represented by the least-significant bit).
It is obviously just a matter of looping over numbers to generate these representations:
for f := 0 to 2^(2^n) - 1:
# do something with f
What can we do with f
? We can evaluate it, for example. Say we want to know f(0,1,0)
. It's as simple as interpreting the argument as the binary number x = 010
and doing some bit-magic:
def evaluate(f, x):
return (f & (1<<x)) != 0
We can also find its disjunctive normal form by just checking which bits are 0:
def dnf(f):
for x := 0 to 2^n - 1:
if f & (1<<x) != 0:
print binary(x) + " OR "
Giving a result like 000 OR 001 OR 100 OR 110 (OR)
for the function above.
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