apoelstra/RamseyScript

GETTING STARTED
======================


RamseyScript is a high-level scripting language for recursively
generating long pattern-free objects. It is case-insensitive, so
it ignores capitalization, and also does not distinguish between
'-' and '_'.


Essentially, every script does the same thing:

  1. Start with some seed object (e.g. the empty sequence)

  2a. Attempt to extend the object.

  2b. Check that the object is still pattern-free (as defined by
      a set of filters given by the script). If it's still good,
      repeat (2a) on the extended object.

  2c. Otherwise, try a different extension and repeat (2b). If
      there are no more possible extensions, quit.

  3. Output the longest pattern-free extension found.


As a concrete example, consider the calculation of the van der Waerden
number w(3; 2), the longest 3-AP-free 2-coloring of the interval [1, n].
The script would be

  set n-colors 2
  filter no-3-aps
  search colorings

The effect of the final line is to:

  1. Start with the empty coloring.

  2a. Attempt to add the next number, colored red. (On the first
      iteration, the "next number" is 1.)

  2b. If the coloring is monochromatic-3-AP-free, repeat (2a).

  2c. Otherwise, try making the next number blue and repeat (2b).

  3. Output the longest monochromatic-3-AP-free extension found (which
     will be some 2-coloring of [1, 8]). Conclude that every 2-coloring
     of [1, 9] has a monochromatic 3-AP, and therefore that w(2;3) is 9.


Advanced features are:

  1. Change the final output. This is done by the "target" command, as

       target any-length

     This will output every object found; the default target, max-length,
     only outputs the longest object.

     To remove all targets, use "target clear". To see a list of all
     available targets, just type "target".


  2. Output auxiliary data. This is done by the "dump" command, as

       dump iters-per-length

     This will also output a list which counts the number of objects
     found at each recursion depth, to allow for data analysis/plotting.


  3. Start with a non-trivial seed. For example, typing

       search sequences [1,3,7,9]

     would search through the space of sequences beginning in 1,3,7,9.
     The special seed "random" will create a random seed:

       set random-length 5
       search colorings random


  4. Run in quiet mode. By default RamseyScript outputs a lot of human
     readable information which is inappropriate for automated scripts.
     To prevent any output, add the command "quiet" to a script before
     running it. To see the status of your targets, run "state".


  5. Split up a problem for parallel processing. RamseyScript uses only
     one thread, one processor, which is inefficient if parallelizable
     hardware is available. The special target "fork" will output lines
     of the form "search [space] [seed]" suitable for creating extra
     scripts (which can then be run in separate RamseyScript instances).
     It is used as such:

       set max-depth 11
       set fork-depth 10
       target clear    # remove the default max-depth target
       target fork
       filter no-double-3-aps
       search colorings


   6. Manually iterate. If you have a split-up problem as in the previous
      step, you will wind up with candidate solutions from every instance
      of RamseyScript. To combine these (e.g., to find the -real- maximum
      length object), use the command "process", like so:

        quiet
        process [[1, 2, 4, 7, 8] [3, 5, 6, 9, 10]]
        process [[1, 3, 4, 7, 9, 10, 12] [2, 5, 6, 8, 11, 13]]
        process [[1, 3, 6, 8, 9, 11, 14, 16] [2, 4, 5, 7, 10, 12, 13, 15]]
        state

      (If you do not use "quiet" and "state" as explained in item (4) above,
       you will get a bunch of extraneous output.)



DETAILED LANGUAGE SPECIFICATION
=======================


===============
Variables
===============

  set <variable> <value>

Sets a variable. Not all variables are effective for each search space.
``variable'' should be one of:

      alphabet: The alphabet to use when searching words. Specified in the
                format [c1 c2 c3 ... cn].
                Default value: (none)

     ap_length: The length of AP's filtered out by the no-double-n-ap filter
                Default value: (none)

 base_sequence: A sequence that will be used instead of 1, 2, 3, ... when
                creating colorings. If unset, the positive integers will be
                used.

     dump-file: The file to output dump output (see 'dump'), or "-" to use
                stdout.
                Default value: -

       gap-set: The set of allowable gap sizes when searching sequences or
                colorings. For sequences, the value must be a 1D sequence
                of the form [x y z] containing the allowable gap sizes.

                For colorings, if the gap-set is a sequence, this sequence
                defines the allowable gap sizes for each color. Alternately,
                if the gap-set is a coloring (i.e., a sequence of sequences)
                then each color of the gap-set defines the allowable gap
                sizes for the respective colors in the search space.
                Default value: (none)

     max-depth: The maximum depth to search the space.
                Default value: (none)

  max-run-time: The maximum time (in seconds) to search before stopping.
                Default value: (none)

max-iterations: The number of iterations to run before stopping.
                Default value: (none)

      n-colors: The number of colors to use when searching colorings.
                Default value: 3

    prune-tree: Whether or not the search tree should be pruned; i.e., if some
                element is filtered out, should new elements be built from it?

                For example, if a sequence S contains a double-3-AP, then every
                sequence starting with S will too, so we would set prune-tree
                to nonzero.

                But if a permutation P contains a double-3-AP, it still might
                be that permutations built from P could be double-3-AP-free,
                so we would set prune-tree to 0.
           
                Default value: 1

 random-length: If the seed is set to RANDOM on a supported space, this
                sets the length of the generated seed.
                Default value: 10

   stall-after: Like max-iterations, but resets its counter every time a target
                (e.g., new object of maximum length) is reached.
                Default value: (none)



  get <variable>

Prints the value of <variable>.



  unset <variable>

Unsets a previously-set variable. If the variable does not exist, does nothing.



===============
Filters
===============

  filter <filter>

Adds a filter on the space to be searched. Multiple filters may be
used. To delete filters, use "filter clear" and re-add the ones you
want kept. ``filter'' should be one of:

             no-3-aps: Only recurse on objects with no 3-AP's

             no-n-aps: Only recurse on objects with no n-AP's, with
                       n set by "set ap-length"

      no-double-3-aps: Only recurse on objects with no double-3-AP's

      no-double-n-aps: Only recurse on objects with no double-AP's of
                       length n, with n set by "set ap-length".

       no-rainbow-aps: Only recurse on colorings with no rainbow-AP's
                       (i.e., arithmetic progressions with one entry
                       of each color). Note that this filter must be
                       used with a different filter, since by itself,
                       it simply tells the program to color every
                       number the same.

   no-odd-lattice-aps: No arithmetic progressions on lattices, with
                       length equal to the number of columns on the
                       lattice. (So named because if the lattice has
                       4 columns, all 4-AP's with odd gap size will
                       appear as straight lines through grid points.)

  no-additive-squares: Only recurse on words with no additive squares

no-pythagorean-triples: Only recurse on objects with no solutions to
                        X^2 + Y^2 = Z^2.

   no-schur-solutions: Only recurse on objects with no solutions to
                       X + Y = Z.



  filter clear

Removes all set filters.


===============
Search Spaces
===============

  search <space> [seed]

Selects a search space and recursively explores it, looking for longer
and longer elements which satisfy the given filter. There is no default
value. If you do not specify a 'search' line, the program will do
nothing. ``space'' should be one of:


   colorings: The space of r-colorings of integers, for some given r.
              For colorings, the seed may be RANDOM, in which case a
              seed of length random-length is generated.
              Default seed: [[1] [] []]

    lattices: The space of r-colorings of integers, organized as a
              grid of m columns (with r and m given). This allows a
              geometric interpretation of some Ramsey-type problems.
              Default seed: []

  partitions: synonym of 'colorings'

permutations: The space of permutations of [1,n]. You cannot change
              the seed for this space.
              Default seed: [1]

   sequences: The space of strictly increasing sequences
              Default seed: [1]

       words: The space of words on some given alphabet
              Default seed: []



==============
Targets
==============

  target <target>

Chooses a goal to achieve when searching through Ramsey objects. Whenever
a target is reached (e.g., a new object of maximal length is found), the
current object is output. Multiple targets may be specified.

By default, the "max-length" target is set. ``target'' should be one of:

  max-length: Output the current object every time one is found one maximal
              length is found.

  any-length: Output every object found while recursing.



  target clear

Removes all set targets.



===============
Additional Output
===============

  dump <iterations-per-length>

Output auxiliary data about the search space or program operation.

iterations-per-length: dump the number of iterations spent at each
                       search-space depth




  dump clear

Removes all data dumps.




===============
Manual Recursion
===============

It is possible to operate RamseyScript manually (i.e., when the Ramsey
objects are obtained externally, and no recursion needs to be done, just
filtering). This is done using the following three commands:


  reset 

Resets all output dumps and targets.



  process <space> <object> 

Runs filters, outputs and targets on ``object'', which is a Ramsey object
of type ``space''. ``space'' is any valid search space (see the ``search''
command).



  state 

Output all targets and run all output dumps.



EXAMPLES
===================

## Look for 2-colorings of [1, N] with no double-3-ap's
## This will find that there is no such coloring of [1, 17],
## though there is one of [1, 16].

set n-colors 2
filter no-double-3-aps
search colorings

# (end file)


## Explore the space of 3-colorings of [1, N] with no double-3-ap's.
## One does exist of [1, 390], but this is all that is known so far.

set n-colors 3
filter no-double-3-aps
set max-iterations 10000000

search colorings RANDOM
search colorings RANDOM
search colorings RANDOM
search colorings RANDOM
search colorings RANDOM

# (end file)



## Explore the space of permutations with no 3-AP's. Output the iterations
## at each depth (i.e., the number of valid permutations of each length)
## to output.dump.

# The resultant output will match the values computed by G.J. Simmons.
# See American Mathematical Monthly 82 (1975) pp. 76-77.

filter no-3-aps

set dump-depth 25
set dump-file output.dump
dump iterations-per-length

set max-depth 20
search permutations

# (end file)



## Check some 3-colorings for double-3-aps

target clear
target any-length

filter no-double-3-aps
process colorings [[2, 5, 6, 8, 9] [1, 3, 7] [4, 10]]
process colorings [[2, 5, 6, 8, 9] [1, 3, 7, 10] [4]]
process colorings [[2, 5, 6] [1, 3, 7, 8, 9, 10] [4]]
state