Jolverine Turing-completeness proof

From Esolang
Jump to: navigation, search

Jolverine can be shown Turing complete by a reduction from ternary Reversible Brainfuck. We put the construction on a separate page because of the size of one of its parts.

A ternary Reversible Brainfuck program is converted to Jolverine as a series of diagonal elements concatenated with each other. The first element is the initialization; then follows the elements for the parts of the RB program in order. Loops are considerably more complicated to represent than any of the other RB instructions, and contain their inner instructions embedded within them.

Note that I/O is not needed for theoretical Turing-completeness, and may not have completely matching semantics between all esolangs involved. For best results only use input on a cell containing a zero value.

The following properties are kept for all elements other than initialization (whose job is to set them up in the first place):

  • Each element takes a multiple of 7 ticks to execute. Apart from loop elements, this means each of them is a multiple of 7 long.
  • Each element starts with the instruction wheel in the state
left <--
adddx
input
output
adddy
rot
right
with the top instruction (left) the current one, and with the next executed instruction to be re-inserted at the top.
  • Instructions which may turn the IP will only be executed as the sixth instruction on the wheel. This ensures that the wheel sixth instructions are always at 7x7 grid positions in the larger program, thus ensuring there can be no position mismatch not fixable simply by adding a multiple of 7 non-* symbols.
  • At the start of an element, the current cell of the Jolverine tape corresponds to the current RB cell. RB cells are spaced two cells apart on the Jolverine tape, with the intermediate cell being zero.

Since editing this stuff has been rather awkward, the reduction has only been tested on one program:

[>,.]

Reduction tables[edit]

The following table lists the elements other than loops.

initialization
==============
..*.*
     .        +      -      >      <      .      ,
      .       =      =      =      =      =      =
       *      .      .      *      *      .      .
        .      .      .      .      .      .      .
         .      .      .      .      .      .      *
          .      .      .      .      .      .      .
           .      .      .      .      .      .      .
            .      *      *      .      .      *      .
             .      .      .      *      .      .      *
              *      *      *      .      *      *      .
               .      .      .      .      .      .      .
                .      .      .      .      .      .      *
                 .      .      .      .      .      .      .
                  .      .      .      .      .      .      .
                   .      .      .      .      .      .      .
                    .      *      .      *      *      *      .
                     *      *      *      *      .      .      .
                      *      .      .      .      .      .      .
                       .      .      .      .      .      .      *
                        .      .      .      .      .      .      .
                         .      .      .      .      .      *      .
                          .      .      *      .      .      *      .
                           .      .      .      .      *      .      *
                                   *                           .      .
                                    .                           .      .
                                     .                           .      .
                                      .                           .      .
                                       .                           .      .
                                        *                           *      .
                                         .                           .      *
                                                                      .
                                                                       .
                                                                        *
                                                                         .
                                                                          .
                                                                           .
                                                                            .
                                                                             .
                                                                              .
                                                                               *
                                                                                .
                                                                                 .
                                                                                  .
                                                                                   *
                                                                                    *
                                                                                     .
                                                                                      .
                                                                                       .
                                                                                        .
                                                                                         .
                                                                                          .
                                                                                           *
                                                                                            *
                                                                                             .
                                                                                              .
                                                                                               .
                                                                                                .
                                                                                                 .

The following shows the general structure of the large loop elements. The instructions for the loop contents are embedded at the spot marked "INSIDE LOOP". The surrounding paths must be stretched as necessary by inserting multiples of 7 non-* elements in order to fit the inside content. (The version below has space enough for the loop [>,.] as that was what it was tested with.)

                                                                                                       *                                                                                                 *
                                                                                                      . .                                                                                               . .
                                                                                                     .   *                                                                                             .   *
                                                                                                    .     .                                                                                           .     .
                                                                                                   .       .                                                                                         .       .
                                                                                                  .         .                                                                                       .         .
                                                                                                 *           .                                                                                     *           .
                                                                                                *             .                                                                                   *             .
                                                                                               .               *                                                                                 .               *
                                                                                              .                 .                                                                               .                 .
                                                                                             .                   .                                                                             .                   .
                                                                                            .                     .                                                                           .                     .
                                                                                           *                       .                                                                         *                       .
                                                                                          .                         .                                                                       .                         .
                                                                                         .                           *                                                                     .                           *
                                                                                        *                             .                                                                   *                             .
                     B                                                                 .                               .                                                                 .                               .
                      E                                                               .                                 *                                                               .                                 *
                       F                                                             .                                   .                                                             .                                   .
                        O                                                           *                                     .                                                           *                                     .
                         R                                                         .                                       .                                                         .                                       .
                          E                                                       .                                         .                                                       .                                         .
                                                                                 .                                           *                                                     .                                           *
                            L                                                   .                                             *                                                   .                                             *
                             O                                                 .                                               *                                                 .                                               *
                              O                                               .                                                 .                                               .                                                 .
                               P                                             *                                                   .                                             *                                                   .
                                ,                                           *                                                     .                                           *                                                     .
                                 ,                                         .                                                       .                                         .                                                       .
                                  ,                                       .                                                         .                                       .                                                         .
                                   .                                     .                                                           .                                     .                                                           .
                                    .                                   .                                                             *                                   .                                                             *
                                     .                                 .                                                               .                                 .                                                               .
                                      .                               *                                                                 .                               *                                                                 .
                                       .                             *                                                                   .                             *                                                                   .
                                        *                           .                                                                     *                           .                                                                     .
                                         .                         .                                                                       .                         .                                                                       .
                                          *                       .                                                                         *                       .                                                                         ,
                                           .                     .                                                                           .                     .                                                                           ,
                                            .                   .                                                                             .                   .                                                                             ,
                                             .                 *                                                                               .                 *                                                                               ,
                                              .               .                                                                                 .               .                                                                                 ,
                                               .             .                                                                                   .             .                                                                                   ,
                                                *           .                                                                                     *           .                                                                                     ,
                                                 .         .                                                                                       .         .                                                                                       ,
                                                  .       .                                                                                         .       .                                                                                         ,
                                                   .     .                                                                                           .     .                                                                                           ,
                                                    .   *                                                                                             .   *                                                                                             ,
                                                     . .                                                                                               . .                                                                                               ,
                                                      *.*......*.....**.....**......*...*..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....*.*.....*.....*                                                                                                 ,
                                                       .                                                                                               .                                                                                                   ,
                                                        *                                                                                             .                                                                                                     ,
                                                         .                                                                                           .                                                                                                       ,
                                                          .                                                                                         .                                                                                                         ,
                                                           .                                                                                       .                                                                                                           ,
                                                            .                                                                                     *                                                                                                             ,
                                                             .                                                                                   .                                                                                                               ,
                                                              .                                                                                 .                                                                                                                 ,
                                                               *                                                                               .                                                                                                                   ,
                                                                .                                                                             .                                                                                                                     ,
                                                                 .                                                                           .                                                                                                                       ,
                                                                  .                                                                         *                                                                                                                         ,
                                                                   .                                                                       .                                                                                                                           ,
                                                                    .                                                                     *                                                                                                                             ,
                                                                     *                                                                   .                                                                                                                               ,
                                                                      *                                                                 .                                                                                                                                 ,
                                                                       .                                                               .                                                                                                                                   ,
                                                                        .                                                             .                                                                                                                                     ,
                                                                         .                                                           .                                                                                                                                       ,
                                                                          .                                                         ,                                                                                                                                         ,
                                                                           .                                                       ,                                                                                                                                           ,
                                                                            *                                                     ,                                                                                                                                             ,
                                                                             *                                                   ,                                                                                                                                               ,
                                                                              .                                                 ,                                                                                                                                                 ,
                                                                               .                                               ,                                                                                                                                                   ,
                                                                                .                                             ,                                                                                                                                                     ,
                                                                                 .                                           ,                                                                                                                                                       ,
                                                                                  .                                         ,                                                                                                                                                         ,
                                                                                   .                                       ,                                                                                                                                                           ,
                                                                                    *                                     ,                                                                                                                                                             ,
                                                                                     .                                   ,                                                                                                                                                               ,
                                                                                      .                                 ,                                                                                                                                                                 ,
                                                                                       .                               ,                                                                                                                                                                   ,
                                                                                        *                             ,                                                                                                                                                                     ,
                                                                                         .                           ,                                                                                                                                                                       ,
                                                                                          .                         ,                                                                                                                                                                         ,
                                                                                           ,                       ,                                                                                                                                                                           ,
                                                                                            ,                     ,                                                                                                                                                                             ,
                                                                                             ,                   ,                                                                                                                                                                               ,
                                                                                              ,                 ,                                                                                                                                                                                 ,
                                                                                               ,               ,                                                                                                                                                                                   ,
                                                                                                ,             ,                                                                                                                                                                                     ,
                                                                                                 ,           ,                                                                                                                                                                                       ,
                                                                                                  ,         ,                                                                                                                                                                                         ,
                                                                                                   ,       ,                                                                                                                                                                                           ,
                                                                                                    ,     ,                                                                                                                                                                                             ,
                                                                                                     ,   ,                                                                                                                                                                                               ,
                                                                                                      , ,                                                                                                                                                                                                 ,
                                                                                                       ,                                                                                                                                                                                                   ,
                                                                                                      , ,                                                                                                                                                                                                   ,
                                                                                                     ,   ,                                                                                                                                                                                                   ,
                                                                                                    ,     ,                                                                                                                                                                                                   ,
                                                                                                   ,       ,                                                                                                                                                                                                   ,
                                                                                                  ,         ,                                                                                                                                                                                                   ,
                                                                                                 ,           ,                                                                                                                                                                                                   ,
                                                                                                ,             ,                                                                                                                                                                                                   ,
                                                                                               ,               ,                                                                                                                                                                                                   ,
                                                                                              ,                 ,                                                                                                                                                                                                   ,
                                                                                             ,                   ,                                                                                                                                                                                                   ,
                                                                                            ,                     ,                                                                                                                                                                                                   ,
                                                                                           ,                       ,                                                                                                                                                                                                   ,
                                                                                          .                         ,                                                                                                                                                                                                   ,
                                                                                         .                           ,                                                                                                                                                                                                   ,
                                                                                        *                             ,                                                                                                                                                                                                   ,
                                                                                       .                               ,                                                                                                                                                                                                   ,
                                                                                      .                                 ,                                                                                                                                                                                                   ,
                                                                                     .                                   ,                                                                                                                                                                                                   ,
                                                                                    *                                     ,                                                                                                                                                                                                   ,
                                                                                   .                                       ,                                                                                                                                                                                                   ,
                                                                                  .                                         ,                                                                                                                                                                                                   ,
                                                                                 .                                           ,                                                                                                                                                                                                   ,
                                                                                .                                             ,                                                                                                                                                                                                   ,
                                                                               .                                               ,                                                                                                                                                                                                   ,
                                                                              .                                                 ,                                                                                                                                                                                                   ,
                                                                             *                                                   ,                                                                                                                                                                                                   ,
                                                                            *                                                     ,                                                                                                                                                                                                   ,
                                                                           .                                                       ,                                                                                                                                                                                                   ,
                                                                          .                                                         ,                                                                                                                                                                                                   ,
                                                                         .                                                           .                                                                                                                                                                                                   ,
                                                                        .                                                             .                                                                                                                                                                                                   ,
                                                                       .                                                               .                                                                                                                                                                                                   ,
                                                                      *                                                                 .                                                                                                                                                                                                   ,
                                                                     *                                                                   .                                                                                                                                                                                                   ,
                                                                    .                                                                     *                                                                                                                                                                                                   ,
                                                                   .                                                                       .                                                                                                                                                                                                   ,
                                                                  .                                                                         *                                                                                                                                                                                                   ,
                                                                 .                                                                           .                                                                                                                                                                                                   ,
                                                                .                                                                             .                                                                                                                                                                                                   ,
                                                               *                                                                               .                                                                                                                                                                                                   ,
                                                              .                                                                                 .                                                                                                                                                                                                   ,
                                                             .                                                                                   .                                                                                                                                                                                                   ,
                                                            .                                                                                     *                                                                                                                                                                                                   ,
                                                           .                                                                                       .                                                                                                                                                                                                   ,
                                                          .                                                                                         .                                                                                                                                                                                                   ,
                                                         .                                                                                           .                                                                                                                                                                                                   ,
                                                        *                                                                                             .                                                                                                                                                                                                   ,
                                                       .                                                                                               .                                                                                                                                                                                                   ,
                                                      *.*......*.....**.....**......*...*..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....*.*.....*.....*                                                                                                                                                                                                   ,
                                                     . .                                                                                               . .                                                                                                                                                                                                   ,
                                                    .   *                                                                                             .   *                                                                                                                                                                                                   ,
                                                   .     .                                                                                           .     .                                                                                                                                                                                                   ,
                                                  .       .                                                                                         .       .                                                                                                                                                                                                   ,
                                                 .         .                                                                                       .         .                                                                                                                                                                                                   ,
                                                *           .                                                                                     *           .                                                                                                                                                                                                   ,
                                               .             .                                                                                   .             .                                                                                                                                                                                                   ,
                                              .               .                                                                                 .               .                                                                                                                                                                                                   ,
                                             .                 *                                                                               .                 *                                                                                                                                                                                                   ,
                                            .                   .                                                                             .                   .                                                                                                                                                                                                   ,
                                           .                     .                                                                           .                     .                                                                                                                                                                                                   ,
                                          *                       .                                                                         *                       .                                                                                                                                                                                                   ,
                                         .                         .                                                                       .                         .                                                                                                                                                                                                   ,
                                        *                           .                                                                     *                           .                                                                                                                                                                                                   ,
                                       .                             *                                                                   .                             *                                                                                                                                                                                                   ,
                                      .                               *                                                                 .                               *                                                                                                                                                                                                   ,
                                     *                                 .                                                               .                                 .                                                                                                                                                                                                   ,
                                    .                                   .                                                             *                                   .                                                                                                                                                                                                   ,
                                   *                                     .                                                           *                                     .                                                                                                                                                                                                   ,
                                  .                                       .                                                         .                                       .                                                                                                                                                                                                   ,
                                 .                                         .                                                       .                                         .                                                                                                                                                                                                   ,
                                .                                           *                                                     .                                           *                                                                                                                                                                                                   ,
                               .                                             *                                                   .                                             *                                                                                                                                                                                                   ,
                              .                                               .                                                 .                                               .                                                                                                                                                                                                   ,
                             .                                                 .                                               .                                                 .                                                                                                                                                                                                   ,
                            *                                                   .                                             *                                                   .                                                                                                                                                                                                   ,
                           *                                                     .                                           *                                                     .                                                                                                                                                                                                   ,
                          .                                                       .                                         .                                                       .                                                                                                                                                                                                   ,
                         .                                                         .                                       .                                                         .                                                                                                                                                                                                   ,
                        .                                                           *                                     .                                                           *                                                                                                                                                                                                   ,
                       .                                                             .                                   .                                                             .                                                                                                                                                                                                   ,
                      .                                                               .                                 *                                                               .                                                                                                                                                                                                   ,
                     .                                                                 .                               .                                                                 .                                                                                                                                                                                                   ,
                    *                                                                   *                             .                                                                   *                                                                                                                                                                                                   ,
                   .                                                                     .                           .                                                                     .                                                                                                                                                                                                   ,
                  .                                                                       *                         .                                                                       .                                                                                                                                                                                                   ,
                 .                                                                         .                       .                                                                         I                                                                                                                                                                                                   ,
                *                                                                           .                     *                                                                           N                                                                                                                                                                                                   ,
               .                                                                             .                   *                                                                             S                                                                                                                                                                                                   ,
              .                                                                               .                 .                                                                               I                                                                                                                                                                                                   ,
             *                                                                                 .               *                                                                                 D                                                                                                                                                                                                   ,
            *                                                                                   .             .                                                                                   E                                                                                                                                                                                                   ,
             .                                                                                   *           .                                                                                                                                                                                                                                                                                         ,
              .                                                                                   .         .                                                                                       L                                                                                                                                                                                                   ,
               .                                                                                   .       .                                                                                         O                                                                                                                                                                                                   ,
                .                                                                                   .     .                                                                                           O                                                                                                                                                                                                   ,
                 *                                                                                   .   .                                                                                             P                                                                                                                                                                                                   ,
                  .                                                                                   . *                                                                                               ,                                                                                                                                                                                                   ,
                   *                                                                                   *                                                                                                 ,                                                                                                                                                                                                   ,
                    .                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                     .                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                      *                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                       .                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                        .                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                         *                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                          .                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                           .                                                                                                                                                                                     ,                                                                                                                                                                                                   ,
                            .                                                                                                                                                                                     ,                                                                                                                                                                                                   *
                             .                                                                                                                                                                                     ,                                                                                                                                                                                                   *
                              .                                                                                                                                                                                     ,                                                                                                                                                                                                   .
                               .                                                                                                                                                                                     ,                                                                                                                                                                                                   *
                                ,                                                                                                                                                                                     ,                                                                                                                                                                                                   *
                                 ,                                                                                                                                                                                     ,                                                                                                                                                                                                   *
                                  ,                                                                                                                                                                                     ,                                                                                                                                                                                                 *
                                   ,                                                                                                                                                                                     ,                                                                                                                                                                                               .
                                    ,                                                                                                                                                                                     ,                                                                                                                                                                                             .
                                     ,                                                                                                                                                                                     ,                                                                                                                                                                                           .
                                      ,                                                                                                                                                                                     ,                                                                                                                                                                                         .
                                       ,                                                                                                                                                                                     ,                                                                                                                                                                                       .
                                        ,                                                                                                                                                                                     ,                                                                                                                                                                                     .
                                         ,                                                                                                                                                                                     ,                                                                                                                                                                                   *
                                          ,                                                                                                                                                                                     ,                                                                                                                                                                                 .
                                           ,                                                                                                                                                                                     ,                                                                                                                                                                               .
                                            ,                                                                                                                                                                                     ,                                                                                                                                                                             *
                                             ,                                                                                                                                                                                     ,                                                                                                                                                                           .
                                              ,                                                                                                                                                                                     ,                                                                                                                                                                         .
                                               ,                                                                                                                                                                                     ,                                                                                                                                                                       *
                                                ,                                                                                                                                                                                     ,                                                                                                                                                                     .
                                                 ,                                                                                                                                                                                     ,                                                                                                                                                                   .
                                                  ,                                                                                                                                                                                     ,                                                                                                                                                                 *
                                                   ,                                                                                                                                                                                     ,                                                                                                                                                               .
                                                    ,                                                                                                                                                                                     ,                                                                                                                                                             .
                                                     ,                                                                                                                                                                                     ,                                                                                                                                                           .
                                                      ,                                                                                                                                                                                     ,                                                                                                                                                         .
                                                       ,                                                                                                                                                                                     ,                                                                                                                                                       *
                                                        ,                                                                                                                                                                                     ,                                                                                                                                                     *
                                                         ,                                                                                                                                                                                     ,                                                                                                                                                   .
                                                          ,                                                                                                                                                                                     ,                                                                                                                                                 *
                                                           ,                                                                                                                                                                                     ,                                                                                                                                               .
                                                            ,                                                                                                                                                                                     ,                                                                                                                                             .
                                                             ,                                                                                                                                                                                     ,                                                                                                                                           .
                                                              ,                                                                                                                                                                                     ,                                                                                                                                         .
                                                               ,                                                                                                                                                                                     ,                                                                                                                                       .
                                                                ,                                                                                                                                                                                     ,                                                                                                                                     *
                                                                 ,                                                                                                                                                                                     ,                                                                                                                                   .
                                                                  ,                                                                                                                                                                                     ,                                                                                                                                 .
                                                                   ,                                                                                                                                                                                     ,                                                                                                                               .
                                                                    ,                                                                                                                                                                                     ,                                                                                                                             .
                                                                     ,                                                                                                                                                                                     ,                                                                                                                           .
                                                                      ,                                                                                                                                                                                     ,                                                                                                                         ,
                                                                       ,                                                                                                                                                                                     ,                                                                                                                       ,
                                                                        ,                                                                                                                                                                                     ,                                                                                                                     ,
                                                                         ,                                                                                                                                                                                     ,                                                                                                                   ,
                                                                          ,                                                                                                                                                                                     ,                                                                                                                 ,
                                                                           ,                                                                                                                                                                                     ,                                                                                                               ,
                                                                            ,                                                                                                                                                                                     ,                                                                                                             ,
                                                                             ,                                                                                                                                                                                     ,                                                                                                           ,
                                                                              ,                                                                                                                                                                                     ,                                                                                                         ,
                                                                               ,                                                                                                                                                                                     ,                                                                                                       ,
                                                                                ,                                                                                                                                                                                     ,                                                                                                     ,
                                                                                 ,                                                                                                                                                                                     ,                                                                                                   ,
                                                                                  ,                                                                                                                                                                                     ,                                                                                                 ,
                                                                                   ,                                                                                                                                                                                     ,                                                                                               ,
                                                                                    ,                                                                                                                                                                                     ,                                                                                             ,
                                                                                     ,                                                                                                                                                                                     ,                                                                                           ,
                                                                                      ,                                                                                                                                                                                     ,                                                                                         ,
                                                                                       ,                                                                                                                                                                                     ,                                                                                       ,
                                                                                        ,                                                                                                                                                                                     ,                                                                                     ,
                                                                                         ,                                                                                                                                                                                     ,                                                                                   ,
                                                                                          ,                                                                                                                                                                                     ,                                                                                 ,
                                                                                           ,                                                                                                                                                                                     ,                                                                               ,
                                                                                            ,                                                                                                                                                                                     ,                                                                             ,
                                                                                             ,                                                                                                                                                                                     ,                                                                           ,
                                                                                              ,                                                                                                                                                                                     ,                                                                         ,
                                                                                               ,                                                                                                                                                                                     ,                                                                       ,
                                                                                                ,                                                                                                                                                                                     ,                                                                     ,
                                                                                                 ,                                                                                                                                                                                     ,                                                                   ,
                                                                                                  ,                                                                                                                                                                                     ,                                                                 ,
                                                                                                   ,                                                                                                                                                                                     ,                                                               ,
                                                                                                    ,                                                                                                                                                                                     ,                                                             ,
                                                                                                     ,                                                                                                                                                                                     ,                                                           ,
                                                                                                      ,                                                                                                                                                                                     ,                                                         ,
                                                                                                       ,                                                                                                                                                                                     ,                                                       ,
                                                                                                        ,                                                                                                                                                                                     ,                                                     ,
                                                                                                         ,                                                                                                                                                                                     ,                                                   ,
                                                                                                          ,                                                                                                                                                                                     ,                                                 ,
                                                                                                           ,                                                                                                                                                                                     ,                                               ,
                                                                                                            ,                                                                                                                                                                                     ,                                             ,
                                                                                                             ,                                                                                                                                                                                     ,                                           ,
                                                                                                              ,                                                                                                                                                                                     ,                                         ,
                                                                                                               ,                                                                                                                                                                                     ,                                       ,
                                                                                                                ,                                                                                                                                                                                     ,                                     .
                                                                                                                 ,                                                                                                                                                                                     ,                                   .
                                                                                                                  ,                                                                                                                                                                                     ,                                 .
                                                                                                                   ,                                                                                                                                                                                     ,                               .
                                                                                                                    ,                                                                                                                                                                                     ,                             .
                                                                                                                     ,                                                                                                                                                                                     ,                           .
                                                                                                                      ,                                                                                                                                                                                     ,                         *
                                                                                                                       ,                                                                                                                                                                                     *                       .
                                                                                                                        ,                                                                                                                                                                                     *                     .
                                                                                                                         ,                                                                                                                                                                                     .                   *
                                                                                                                          ,                                                                                                                                                                                     *                 .
                                                                                                                           ,                                                                                                                                                                                     *               .
                                                                                                                            ,                                                                                                                                                                                     *             *
                                                                                                                             ,                                                                                                                                                                                   *             .
                                                                                                                              ,                                                                                                                                                                                 .             *
                                                                                                                               ,                                                                                                                                                                               .             .
                                                                                                                                ,                                                                                                                                                                             .             .
                                                                                                                                 ,                                                                                                                                                                           .             .
                                                                                                                                  ,                                                                                                                                                                         .             .
                                                                                                                                   ,                                                                                                                                                                       .             *
                                                                                                                                    ,                                                                                                                                                                     *               *
                                                                                                                                     ,                                                                                                                                                                   .                 .
                                                                                                                                      ,                                                                                                                                                                 .                   .
                                                                                                                                       ,                                                                                                                                                               *                     *
                                                                                                                                        ,                                                                                                                                                             .                       .
                                                                                                                                         .                                                                                                                                                           .                         .
                                                                                                                                          .                                                                                                                                                         *                           .
                                                                                                                                           .                                                                                                                                                       .                             *
                                                                                                                                            .                                                                                                                                                     .                               .
                                                                                                                                             .                                                                                                                                                   *                                 .
                                                                                                                                              *                                                                                                                                                 .                                   .
                                                                                                                                               .                                                                                                                                               .                                     .
                                                                                                                                                .                                                                                                                                             .                                       .
                                                                                                                                                 .                                                                                                                                           .                                         .
                                                                                                                                                  .                                                                                                                                         *                                           *
                                                                                                                                                   .                                                                                                                                       *                                             *
                                                                                                                                                    .                                                                                                                                     .                                               .
                                                                                                                                                     *                                                                                                                                   *                                                 .
                                                                                                                                                      *                                                                                                                                 .                                                   .
                                                                                                                                                       *                                                                                                                               .                                                     .
                                                                                                                                                        .                                                                                                                             .                                                       .
                                                                                                                                                         .                                                                                                                           .                                                         .
                                                                                                                                                          .                                                                                                                         .                                                           *
                                                                                                                                                           .                                                                                                                       *                                                             .
                                                                                                                                                            *                                                                                                                     .                                                               *
                                                                                                                                                             .                                                                                                                   .                                                                 .
                                                                                                                                                              .                                                                                                                 .                                                                   .
                                                                                                                                                               *                                                                                                               .                                                                     .
                                                                                                                                                                .                                                                                                             .                                                                       .
                                                                                                                                                                 .                                                                                                           *                                                                         A
                                                                                                                                                                  .                                                                                                         .                                                                           F
                                                                                                                                                                   .                                                                                                       .                                                                             T
                                                                                                                                                                    .                                                                                                     .                                                                               E
                                                                                                                                                                     *                                                                                                   .                                                                                 R
                                                                                                                                                                      .                                                                                                 *                                                                                    
                                                                                                                                                                       .                                                                                               *                                                                                     L
                                                                                                                                                                        .                                                                                             .                                                                                       O
                                                                                                                                                                         .                                                                                           .                                                                                         O
                                                                                                                                                                          .                                                                                         .                                                                                           P
                                                                                                                                                                           *                                                                                       .                                                                                             ,
                                                                                                                                                                            .                                                                                     .                                                                                               ,
                                                                                                                                                                             *.....**....*,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....*......***....*..*.....*.....*.*                                                                                                 ,
                                                                                                                                                                                                                                                                                                                                                                    ,