An interesting thing I noticed was that I couldn’t perfectly emulate the input instruction. I’m feeding the brainfuck code to the interpreter from STDIN and so that might be the problem. I’ve noticed that brainfuck interpreters written in brainfuck have the same problem. You have to specify program input before hand. This is what I’ve decided to do. You write brainfuck code, and then mark the end of program code by an exclamation mark. After the exclamation mark, you provide any input, and then mark the end of input by another exclamation mark. Programs that do not have any input end with two exclamation marks. After I finished writing the interpreter, I commented it. While I was doing this, I noticed a lack of labels in bAdkOde. So, I decided to update the interpreter to include them. Speaking of which, I really ought to rewrite the interpreter sometime…

Anyway, here is the code to the interpreter. I’m providing a link to it, because the commented version is rather large. But here’s the expanded (without labels or macros), unformatted version:

>3a>1b{!b?b>b[a)b-91b{=b+1[b>1b}+91b-93b{=b-1[b>1b}+93b(b+1a-33b}>0[a+1a>2b>a[b>1b{!b?b>b[a+1a-33b}>1b>a[b>0a>[aa>ab{=a>1a>[ab>3a{![a)a>[aa-62a{=a+1b-30000b)a>ba{+b>1b>[bb)b>0b-1b>0a}{-a+30000b)b>0a}(b(a+62a}+62a-60a{=a-1b)a>1a>[aa-ab>ba{-b>1a>[aa+30000a>ab)b>0b>0a-1a}{+a>1a>[aa+ab)b>0a-1a}(b(a+60a}+60a-43a{=a+1[b+43a}+43a-45a{=a-1[b+45a}+45a-46a{=a"[b+46a}+46a-44a{=a)a>2a>[aa>[a[b>2a+1[a(a+44a}+44a-91a{=a(a)b)a>[bb{=b>0b>1[b{![b(a+1a)a>[aa)a-91a{=a+1[a>1a}+91a-93a{=a-1[a>1a}+93a(a}(a-1a)a>1b}(a(b)a+91a}+91a-93a{=a(a)b)a>[bb{!b>0b>0[b-1[b{![b(a-1a)a>[aa)a-91a{=a+1[a>1a}+91a-93a{=a-1[a>1a}+93a(a}>0b}(a(b)a+93a}+93a(a+1a}>1a>0b}{!b"85"110"109"97"116"99"104"101"100"32"98"114"97"99"107"101"116"115>0b}

Also, this is additional proof for Turing Completeness. Yes, I’m a nerd!

1. The a register is our memory pointer.
2. We initialize the a register to 0.

Proof

We translate Brainfuck to bAdkOde using the following mappings or rules:

1. > maps to +1a
2. < maps to -1a
3. + maps to +1[a
4. - maps to -1[a
5. . maps to "[a
6. , maps to ?[a
7. [ maps to {![a
8. ] maps to }

Conclusion

Since we can completely translate or map Brainfuck, an already Turing-Complete language into bAdkOde, we can say that bAdkOde is also Turing Complete.

Translated example

helloworld.bf:

+++ +++ +++ +
[
    > +++ +++ +
    > +++ +++ +++ +
    > +++
    > +
    <<< < -
]
>++ .
>+.
+++ +++ +.
.
+++ .
>++ .
<<+ +++ +++ +++ +++ ++.
>.
+++ .
--- --- .
--- --- --.
>+.

helloworld.bad

>0a+1[a+1[a+1[a +1[a+1[a+1[a +1[a+1[a+1[a +1[a
{![a
    +1a +1[a+1[a+1[a +1[a+1[a+1[a +1[a
    +1a +1[a+1[a+1[a +1[a+1[a+1[a +1[a+1[a+1[a +1[a
    +1a +1[a+1[a+1[a
    +1a +1[a
    -1a-1a-1a -1a -1[a
}
+1a+1[a+1[a "[a
+1a+1[a"[a
+1[a+1[a+1[a +1[a+1[a+1[a +1[a"[a
"[a
+1[a+1[a+1[a "[a
+1a+1[a+1[a "[a
-1a-1a+1[a +1[a+1[a+1[a +1[a+1[a+1[a +1[a+1[a+1[a +1[a+1[a+1[a +1[a+1[a"[a
+1a"[a
+1[a+1[a+1[a "[a
-1[a-1[a-1[a -1[a-1[a-1[a "[a
-1[a-1[a-1[a -1[a-1[a-1[a -1[a-1[a"[a
+1a+1[a"[a

Here's a quick perl script that I whipped up, to translate Brainfuck to bAdkOde:

#!/usr/bin/perl

 use strict;

 my $terminator = $/;
 undef $/;
 my $bf = <STDIN>;
 $bf =~ s/[^<>\[\]\.,\+\-\s\n]//g;
 $/ = $terminator;

 my $i = 0;

 print ">0a";

 while($i < length($bf)) {

    my $bf_instruction = substr($bf, $i, 1);

    if($bf_instruction eq '>') {
       print "+1a";
    }

    elsif($bf_instruction eq '<') {
       print "-1a";
    }

    elsif($bf_instruction eq '+') {
       print "+1[a";
    }

    elsif($bf_instruction eq '-') {
       print "-1[a";
    }

    elsif($bf_instruction eq '.') {
       print "\"[a";
    }

    elsif($bf_instruction eq ',') {
       print "?[a";
    }

    elsif($bf_instruction eq '[') {
       print "{![a";
    }

    elsif($bf_instruction eq ']') {
       print "}";
    }

    else {
       print $bf_instruction;
    }

    $i++;
 }

Additional proof

Since there exist self-interpreters for Brainfuck (i.e., Brainfuck interpreters written in Brainfuck), and since it has been shown that there exists a valid translation from Brainfuck to bAdkOde, it follows that it is possible to write a Brainfuck interpreter in bAdkOde. This provides additional proof for the Turing Completeness of bAdkOde.

 print " \@body = (\n";
 map { printf "0x%x, ", $_ } @body;
 print " );\n";
 map { printf "%c", $_ } @body;

Here, you can see that I’ve decided to use an array called @body to store the ASCII values (in hexadecimal). The first three lines print out the definition for the array that will hold the ASCII values. The last line reads the array, translates the code into actual characters, and prints them out. Now all we have to do is get the hexadecimal codes and create the array. To do that, we use a utility called xxd:

vivin@dauntless ~/Projects/code/perl/quine
$ xxd -i quinegen.pl >qhex.pl

The output is an array definition in C:

unsigned char quinegen_pl[] = {
  0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x5c, 0x40, 0x62,
  0x6f, 0x64, 0x79, 0x20, 0x3d, 0x20, 0x28, 0x5c, 0x6e, 0x22, 0x3b, 0x0a,
  0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70, 0x72, 0x69, 0x6e, 0x74,
  0x66, 0x20, 0x22, 0x30, 0x78, 0x25, 0x78, 0x2c, 0x20, 0x22, 0x2c, 0x20,
  0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0x0a,
  0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x29, 0x3b, 0x5c,
  0x6e, 0x22, 0x3b, 0x0a, 0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70,
  0x72, 0x69, 0x6e, 0x74, 0x66, 0x20, 0x22, 0x25, 0x63, 0x22, 0x2c, 0x20,
  0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0x0a
};
unsigned int quinegen_pl_len = 108;

We need to change this to valid Perl, and that’s quite simple:

 @body = (
   0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x5c, 0x40, 0x62,
   0x6f, 0x64, 0x79, 0x20, 0x3d, 0x20, 0x28, 0x5c, 0x6e, 0x22, 0x3b, 0x0a,
   0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70, 0x72, 0x69, 0x6e, 0x74,
   0x66, 0x20, 0x22, 0x30, 0x78, 0x25, 0x78, 0x2c, 0x20, 0x22, 0x2c, 0x20,
   0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0x0a,
   0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x29, 0x3b, 0x5c,
   0x6e, 0x22, 0x3b, 0x0a, 0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70,
   0x72, 0x69, 0x6e, 0x74, 0x66, 0x20, 0x22, 0x25, 0x63, 0x22, 0x2c, 0x20,
   0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0x0a
 );

We take this and paste it on top of our core:

 @body = (
   0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x5c, 0x40, 0x62,
   0x6f, 0x64, 0x79, 0x20, 0x3d, 0x20, 0x28, 0x5c, 0x6e, 0x22, 0x3b, 0x0a,
   0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70, 0x72, 0x69, 0x6e, 0x74,
   0x66, 0x20, 0x22, 0x30, 0x78, 0x25, 0x78, 0x2c, 0x20, 0x22, 0x2c, 0x20,
   0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0x0a,
   0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x29, 0x3b, 0x5c,
   0x6e, 0x22, 0x3b, 0x0a, 0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70,
   0x72, 0x69, 0x6e, 0x74, 0x66, 0x20, 0x22, 0x25, 0x63, 0x22, 0x2c, 0x20,
   0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0x0a
 );
 print " \@body = (\n";
 map { printf "0x%x, ", $_ } @body;
 print " );\n";
 map { printf "%c", $_ } @body;

Now the interesting thing here is that this piece of code is not the actual quine. Rather, it generates the quine. But the generated code isn’t that different from the generator:

 @body = (
0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x5c, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x20, 0x3d, 0x20, 0x28, 0x5c, 0x6e, 0x22, 0x3b, 0xa, 0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x66, 0x20, 0x22, 0x30, 0x78, 0x25, 0x78, 0x2c, 0x20, 0x22, 0x2c, 0x20, 0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0xa, 0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x20, 0x22, 0x20, 0x29, 0x3b, 0x5c, 0x6e, 0x22, 0x3b, 0xa, 0x20, 0x6d, 0x61, 0x70, 0x20, 0x7b, 0x20, 0x70, 0x72, 0x69, 0x6e, 0x74, 0x66, 0x20, 0x22, 0x25, 0x63, 0x22, 0x2c, 0x20, 0x24, 0x5f, 0x20, 0x7d, 0x20, 0x40, 0x62, 0x6f, 0x64, 0x79, 0x3b, 0xa,  );
 print " \@body = (\n";
 map { printf "0x%x, ", $_ } @body;
 print " );\n";
 map { printf "%c", $_ } @body;

The only difference is the extra comma in the array definition, and the fact that all the ASCII codes are on one line. If you run this perl script, you will get an exact copy of its source code. Now how do we write a quine in bAdkOde? We use a similar approach. First we write the core:

>0a
"62"48"97"10
{![a
 "62'[a"91"97"43"49"97
 +1a
}
"10
>0a
{![a
 "[a
 +1a
}

To implement a quine in bAdkOde, I decided to store the representation of the core (in decimal ASCII values) in memory starting at location zero. The first line initializes the a register to zero. The second line prints out the string >0a with a newline. This line is analogous to the print " \@body = (\n"; line in the perl quine. Within our first loop, we're printing out the actual code that will store the representation of the core in memory. So the first line within the loop prints >ASCII[a+1a where ASCII is the ASCII value of a character from the core. The line is analogous to the printf "0x%x, ", $_ line in the perl quine. There is another print statement right outside the first loop, which simply prints a newline. This line is analogous to the print " );\n"; line in the perl quine. The second loop performs the actual translation of the ASCII code into the core. Now, just like the perl quine, we use xxd:

vivin@dauntless ~/Projects/code/perl/bAdkOde
$ xxd -i quine.bad >qhex.txt

After doing this, we get C code that looks like this:

unsigned char quine_bad[] = {
  0x3e, 0x30, 0x61, 0x0a, 0x22, 0x36, 0x32, 0x22, 0x34, 0x38, 0x22, 0x39,
  0x37, 0x22, 0x31, 0x30, 0x0a, 0x7b, 0x21, 0x5b, 0x61, 0x0a, 0x20, 0x22,
  0x36, 0x32, 0x27, 0x5b, 0x61, 0x22, 0x39, 0x31, 0x22, 0x39, 0x37, 0x22,
  0x34, 0x33, 0x22, 0x34, 0x39, 0x22, 0x39, 0x37, 0x0a, 0x20, 0x2b, 0x31,
  0x61, 0x0a, 0x7d, 0x0a, 0x22, 0x31, 0x30, 0x0a, 0x3e, 0x30, 0x61, 0x0a,
  0x7b, 0x21, 0x5b, 0x61, 0x0a, 0x20, 0x22, 0x5b, 0x61, 0x0a, 0x20, 0x2b,
  0x31, 0x61, 0x0a, 0x7d, 0x0a
};
unsigned int quine_bad_len = 77;

Translating this into bAdkOde is not as straightforward, but it’s not that difficult either. I removed the first line, and the last two lines from qhex.txt and then used a oneliner:

vivin@dauntless ~/Projects/code/perl/bAdkOde
$ cat qhex.txt | perl -e 'chomp($a=<STDIN>); map { print ">" . hex($_) . "[a+1a"; } \
   split(",", $a);' > qdec.txt

You end up with this snippet of bAdkOde:

>62[a+1a>48[a+1a>97[a+1a>10[a+1a>34[a+1a>54[a+1a>50[a+1a>34[a+1a>52[a+1a>56[a+1a>34[a+1a>57[a+1a>55[a+1a>34[a+1a>49[a+1a>48[a+1a>10[a+1a>123[a+1a>33[a+1a>91[a+1a>97[a+1a>10[a+1a>32[a+1a>34[a+1a>54[a+1a>50[a+
1a>39[a+1a>91[a+1a>97[a+1a>34[a+1a>57[a+1a>49[a+1a>34[a+1a>57[a+1a>55[a+1a>34[a+1a>52[a+1a>51[a+1a>34[a+1a>52[a+1a>57[a+1a>34[a+1a>57[a+1a>55[a+1a>10[a+1a>32[a+1a>43[a+1a>49[a+1a>97[a+1a>10[a+1a>125[a+1a>10[
a+1a>34[a+1a>49[a+1a>48[a+1a>10[a+1a>62[a+1a>48[a+1a>97[a+1a>10[a+1a>123[a+1a>33[a+1a>91[a+1a>97[a+1a>10[a+1a>32[a+1a>34[a+1a>91[a+1a>97[a+1a>10[a+1a>32[a+1a>43[a+1a>49[a+1a>97[a+1a>10[a+1a>125[a+1a>10[a+1a

Now all you need to do is copy this snippet into quine.bad (with one minor change) to get a bAdkOde quine:

>0a
>62[a+1a>48[a+1a>97[a+1a>10[a+1a>34[a+1a>54[a+1a>50[a+1a>34[a+1a>52[a+1a>56[a+1a>34[a+1a>57[a+1a>55[a+1a>34[a+1a>49[a+1a>48[a+1a>10[a+1a>123[a+1a>33[a+1a>91[a+1a>97[a+1a>10[a+1a>32[a+1a>34[a+1a>54[a+1a>50[a+
1a>39[a+1a>91[a+1a>97[a+1a>34[a+1a>57[a+1a>49[a+1a>34[a+1a>57[a+1a>55[a+1a>34[a+1a>52[a+1a>51[a+1a>34[a+1a>52[a+1a>57[a+1a>34[a+1a>57[a+1a>55[a+1a>10[a+1a>32[a+1a>43[a+1a>49[a+1a>97[a+1a>10[a+1a>125[a+1a>10[
a+1a>34[a+1a>49[a+1a>48[a+1a>10[a+1a>62[a+1a>48[a+1a>97[a+1a>10[a+1a>123[a+1a>33[a+1a>91[a+1a>97[a+1a>10[a+1a>32[a+1a>34[a+1a>91[a+1a>97[a+1a>10[a+1a>32[a+1a>43[a+1a>49[a+1a>97[a+1a>10[a+1a>125[a+1a>10[a+1a
>0a
"62"48"97"10
{![a
 "62'[a"91"97"43"49"97
 +1a
}
"10
>0a
{![a
 "[a
 +1a
}

The small change I had to make, was to add >0a at the beginning. If you run this bAdkOde program, it will print its source code as output. So there you have it. A bAdkOde quine, which proves that bAdkOde is Turing Complete. I have to thank Dave Cope for his quine tutorial, without which it would have been extremely difficult for me to write a bAdkOde quine!

Update (12/26/2009)

Dave mentioned that writing a quine only suggests that a language is Turing Complete, but not that it is. He suggested that I write an interpreter for Brainfuck which would then prove that bAdkOde is Turing Complete. That was actually my next project!

Update (12/26/2009)

I’ve been able to provide a direct translation of Brainfuck into bAdkOde, which proves the Turing Completeness of bAdkOde.

If you know assembly, then you know that a while is simply a set of statements wrapped with a conditional branch at the top and a backwards branch at the bottom (or in other words, an if with a goto at the end. A do-while is simply a set of statements with a conditional branch (at the bottom) that branches to the top of the loop. In fact, in assembly programming there really aren’t any for loops or while loops. These keywords are simply abstractions and syntactic sugar. In bAdkOde, you can implement an if with the existing while statement if you explicitly make it break out of the loop. For example, let’s say that we want to check whether the user entered the character “0″:

?a
-48a
# print a line break
"10
# if a is zero
{=a
 # print the string "zero"
 "122"101"114"111"10
 # set the a register to a non-zero value so that we can break out of the loop
 >1a
}

I knew there was a way to implement an if-else with just while statements but I didn’t remember exactly how. Then my friend and co-worker Juan reminded me that I needed two variables. In our case, we need to use two registers:

?a
-48a
# copy the value of a into b
>ab
"10
# if a is zero
{=a
 # print the string "zero"
 "122"101"114"111"10
 # set the a register to a non-zero value so that we can break out of the loop
 >1a
}
# if the top loop failed, it means that b (which holds the same value as a) is
# non-zero and so we can enter this block.
# if the top loop was successful, it means that b (which holds the same value
# as a) is zero and so we won't enter this block. Hence, the second block acts
# as an 'else'
{!b
 # print the string "non-zero"
 "110"111"110"45"122"101"114"111"10
 # zero out the b register
 >0b
}

Yes, quite ugly. But that’s what I’m going for!