## [Z3][SMT][Python] (Mostly for CS geeks) constructing minimal DFA accepting specific binary numbers

A problem: construct a regular expression accepting binary number divisible by 3.

"1111011" (123) is, "101010101011" (2731) is not.

Some discussion and the correct expressions:

I couldn't generate RE, but I can generate minimal DFA (Deterministic finite automaton):

from z3 import *
import sys, random, os

DIVISOR=3
#DIVISOR=5
#DIVISOR=7
#DIVISOR=9
#DIVISOR=10
#DIVISOR=11 # no luck

state=[[Int('state_%d_%d' % (s, b)) for b in range(2)] for s in range(100)]

INITIAL_STATE=0
INVALID_STATE=999

# construct FA for z3
def transition (STATES, s, i):
# this is like switch()

rt=IntVal(INVALID_STATE)
for j in range(STATES):
rt=If(And(s==IntVal(j), i==0), state[j][0], rt)
rt=If(And(s==IntVal(j), i==1), state[j][1], rt)
return rt

# construct FA for z3
def FA(STATES, input_string):
s=IntVal(INITIAL_STATE)
for i in input_string:
s=transition(STATES, s, int(i))
return s

# simulate FA for testing purpose:
def simulate_FA(input_, FA):
s=INITIAL_STATE
for i in input_:
if i=='0':
s=FA[s][0]
else:
s=FA[s][1]
return s

def test_FA (STATES, FA):
ACCEPTING_STATE=STATES-1
for i in range(10000):
rnd=random.randint(1,100000000000000)
b=bin(rnd)[2:]
final_state=simulate_FA(b, FA)
if (rnd % DIVISOR)==0:
if final_state!=ACCEPTING_STATE:
raise AssertionError
else:
if final_state==ACCEPTING_STATE:
raise AssertionError
print "test OK"

def print_model(STATES, m):
print "[state, input, new state]"
for i in range(STATES):
print "[%d, \"0\", %d]," % (i, m[state[i][0]].as_long())
print "[%d, \"1\", %d]," % (i, m[state[i][1]].as_long())

f=open("1.gv", "wt")
f.write("digraph finite_state_machine {\n")
f.write("\trankdir=LR;\n")
f.write("\tsize=\"8,5\"\n")
f.write("\tnode [shape = doublecircle]; S_0 S_"+str(STATES-1)+";\n");
f.write("\tnode [shape = circle];\n");

FA={}
for s in range(STATES):
f.write("\tS_%d -> S_%d [ label = \"0\" ];\n" % (s, m[state[s][0]].as_long()))
f.write("\tS_%d -> S_%d [ label = \"1\" ];\n" % (s, m[state[s][1]].as_long()))
FA[s]=(m[state[s][0]].as_long(), m[state[s][1]].as_long())
f.write("}\n")
f.close()
os.system("dot -Tpng 1.gv > 1.png") # run GraphViz
test_FA (STATES, FA)

def attempt(STATES):
print "STATES=", STATES
sl=Solver()
# Z3 multithreading, starting at 4.8.x:
set_param("parallel.enable", True)

for s in range(STATES):
for b in range(2):

ACCEPTING_STATE=STATES-1

# may be lower for low DIVISOR's like 3 or 5.
# but 256 is safe choice for DIVISOR's up to 9
for i in range(256):
b=bin(i)[2:]
if (i % DIVISOR)==0:
else:

result=[]

if sl.check() == unsat:
return
m = sl.model()
print_model(STATES, m)
exit(0)

for i in range(2, 100):
attempt(i)



As you can see, it has testing procedure, which is, in turn, can be used instead of RE matcher, if you really need to match numbers divisible by 3.

DFA for numbers divisible by 3. States in double circles - initial (S_0) and accepting:

... is almost like the one someone posted here, but my solution has two separate states as initial and accepting.

Divisible by 5:

Divisible by 7:

Divisible by 9. Probably not an easy thing to find it manually:

Divisible by 10. Not a prime, so the DFA is smaller:

These DFAs are guaranteed to be minimal. Further work: convert them to RE...