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Explore mathematical spaces related to Ramsey-type problems.
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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: [  ] 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:  sequences: The space of strictly increasing sequences Default seed:  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] ] process colorings [[2, 5, 6] [1, 3, 7, 8, 9, 10] ] state