Question

How would you write a regular expression to define all strings of 0’s and 1’s that, as a binary number, represent an integer that is multiple of 3.

Some valid binary numbers would be:

11
110
1001
1100
1111

Answer 1

Using the DFA here we can make a regular expression the following way, where A, B, C represent the states of the DFA.

A = 1B + 0A
B = 1A + 0C
C = 1C + 0B

C = 1*0B // Eliminate recursion

B = 1A + 0(1*0B)
B = 01*0B + 1A
B = (01*0)*1A // Eliminate recursion

A = 1(01*0)*1A + 0A
A = (1(01*0)*1 + 0)A
A = (1(01*0)*1 + 0)* // Eliminate recursion

Resulting in a PCRE regex like:

/^(1(01*0)*1|0)+$/

Perl test/example:

use strict;

for(qw(
11
110
1001
1100
1111
0
1
10
111
)){
    print "$_ (", eval "0b$_", ") ";
    print /^(1(01*0)*1|0)+$/? "matched": "didnt match";
    print "\n";
}

Outputs:

11 (3) matched
110 (6) matched
1001 (9) matched
1100 (12) matched
1111 (15) matched
0 (0) matched
1 (1) didnt match
10 (2) didnt match
111 (7) didnt match