# Compressing boolean formulae

## Syntax

~ not
/\ and
\/ or
t true
f false
P, Q, FISH, etc: variables

(Operators are given in order of precedence)

## Introduction

Some boolean formulae can be changed to different forms to make them shorter. For example, the formula

~(~P /\ ~Q)


can be changed to the shorter form

P\/Q


while the formula

P \/ ~P


can be changed to the shorter form

t


## Challenge

In this challenge, you are required to write a program that, given any boolean formula using only /\, \/, ~, t, f, parentheses, boolean variables (in uppercase), and whitespace, outputs a shortest form (since there may be more than one shortest form) in characters of that expression which is equivalent for all assignments of the variables. Shortest code (in any language) wins. I/O can be done in any reasonable manner.

Also, since answers are difficult to verify, it would be helpful (but isn't required) to include a brief explanation of how the code works.

• In your "Challenge" section you do not mention any whitespace, but your examples have them. Should I handle them? Mar 10, 2014 at 1:53
• I think Florent's point is that it is a difficult problem to solve in general, let alone to golf. Among other things, the parser will need to be able to determine if two arbitrary formulae have the same truth conditions. Reducing p ^ ~p is easy enough if p is atomic, but how about ((A^B)v(A^C)) ^ ~(A^(BvC)) ? I think it is a cool problem and I am curious to see some responses. But if you want short solutions, the problem could be made more conducive to golfing by A. using prefix notation and B. providing a list of required reductions. Mar 12, 2014 at 2:34
• I have a valid (golfed) solution in more than 5000 characters. This is ridiculous... It is composed of a tokenizer, an AST-parser, an AST-rewriter and an expression generator. Mar 12, 2014 at 13:27
• Mathematica can do it in one function call (BooleanMinimize) Mar 12, 2014 at 15:07
• For the record, I have a 498-character solution now, whose sha256sum is b9c98d088b78c30bb2108008a064a7b95722a4694d90ddad94a025c2eb4ed30a. I'll post the actual code at a later date, as I don't want to stifle creativity. Mar 13, 2014 at 2:40

# Python 616

Not particularly efficient, but works in reasonable time for inputs whose results are around 5 or 6 characters. To check a string to see if it matches, it loops through every possible combination of truth/false values for all the variables and makes sure each one agrees. Using this it checks every possible string comprised of the relevant characters (not even necessarily a syntactically correct one).

It actually will print every equivalent expression (of every size) and does not actually ever terminate.

Code:

c=['t','f'];o=['1 ','0 ']
def e(s,v):
for k in v:s=s.replace(k,v[k])
return eval(s)
def z(t,p='~/\\() '):
w=[]
if p=='':return[t]*(t not in['']+c)
for s in t.split(p):w.extend(z(s,p[1:]))
w.sort(key=lambda v:-len(v));return w
def m(v):
l=list('~\\/()')+v
for s in l:yield s
for r in m(v):
for s in l:yield s+r
def n(x):
if x<1:yield []
else:
for l in n(x-1):
for b in o:yield[b]+l
t=raw_input();v=z(t)+c;l=len(v)
for s in m(v):
g=1
for y in n(l):
y[-2:]=o;d=dict(zip(v+['/\\','\\/','~'],y+['and ','or ','not ']))
try:
if e(s,d)!=e(t,d):g=0
except:g=0
if g:print s


Input/Ouput:

> ~(~P /\ ~Q)
Q\/P
P\/Q
...

> P /\ ~P
f
~t
...

> (P \/ Q) /\ P
P
(P)
...


Oh right, I forgot to ever actually post my answer. It uses essentially the exact same approach which KSab's answer uses, but prints only the shortest valid expression.

# Python3, 493 484

e=lambda x:eval(x.replace('\\/','+').replace('/\\','%'),None,w)
class V(int):
def __mod__(s,o):return V(s*o)
def __invert__(s):return V(1-s)
import re;from itertools import product as P;t=V(1);f=V(0);i=input();v=re.findall('[A-Z]+',i)
for k in range(1,len(i)):
for m in P(i+'~/\\tf()',repeat=k):
m=''.join(m)
try:
for d in P((V(0),V(1)),repeat=len(v)):
w=dict(zip(v,d))
if e(m)!=e(i):raise
except:continue
print(m);exit()
print(i)


Edit: fixed a bug where parentheses might not be generated, and golfed the computation of m a bit. Also replaced -s+1 with 1-s.