How to do a goedel numbering for bit strings? - binary-data

Question

I'm looking for a concept for doing a Gödel numbering for bit strings, i.e. for arbitrary binary data.
Approach 1 (failing): Simply interpret the binary data as data of an unsigned integer.
This fails, because e.g. the two different strings "01" and "001" both represent the same integer 1.
Is there a standard way of doing this? Is 0 usually included or excluded from the Gödel numbering?

Answer 1

The original Gödel numbering used prime numbers and unique encoding of symbols. If you want to do it for strings consisting of "0" and "1", you need positive codes for "0" (say 1) and "1" (say 2). Then numbering of "01" is
21 * 32
while numbering of "001" is
21 * 31 * 52
For longer strings use next prime numbers. However, note that Gödel numbering goals did not include any practical considerations, he simply needed numbering as a tool in the proof of his theorem. In practice for fairly short strings you will exceed range of integers in your language, so you need to use either a language with arbitrary large integers built-in (like Scheme) or a library supporting bignums in language without them built-in.

Answer 2

A super simple solution is to prepend a 1 to the binary data and then interpret the result as an unsigned integer value. This way, no 0-digits get lost at the left side of the bit string.
Illustration how well this works:
One obvious way to order bit strings is to order them first by length and then lexicographically:
+------------+
| bit string |
+------------+
| ε |
| 0 |
| 1 |
| 00 |
| 01 |
| 10 |
| 11 |
| 000 |
| 001 |
| 010 |
| 011 |
| 100 |
| 101 |
| 110 |
| ... |
+------------+
(ε denotes the empty string with no digits.)
Now we add an index number n to this table, starting with 1, and then look at the binary representation of the index number n. We will make a nice discovery there:
+------------+--------------+-------------+
| bit string | n in decimal | n in binary |
+------------+--------------+-------------+
| ε | 1 | 1 |
| 0 | 2 | 10 |
| 1 | 3 | 11 |
| 00 | 4 | 100 |
| 01 | 5 | 101 |
| 10 | 6 | 110 |
| 11 | 7 | 111 |
| 000 | 8 | 1000 |
| 001 | 9 | 1001 |
| 010 | 10 | 1010 |
| 011 | 11 | 1011 |
| 100 | 12 | 1100 |
| 101 | 13 | 1101 |
| 110 | 14 | 1110 |
| ... | ... | ... |
+------------+--------------+-------------+
This works out surprisingly well, because the binary representation of n (the index of each bit string when ordering in a very obvious way) is nothing else than a 1 prepended to the original bit string and then the whole thing interpreted as an unsigned integral value.
If you prefer a 0-based Goedel numbering, then subtract 1 from the resulting integer value.
Conversion formulas in pseudo code:
// for starting with 1
n_base1 = integer(prepend1(s))
s = removeFirstDigit(bitString(n_base1))
// for starting with 0
n_base0 = integer(prepend1(s)) - 1
s = removeFirstDigit(bitString(n_base0 + 1))