fragmentary ideas ䷿ intellectual what-nots ䷷ and haskell programming ䷴

Like the Lambda Calculus (on which Lisp is based), Combinatory Logic is a formal system that can be used to model and study computation. What makes it fascinating is that it is as powerful as the (already simple) lambda calculus, but has no need for the variables that LC requires to perform substitution!

Here I show how we can use Haskell’s type classes and other language features to model the Combinator Calculus. The goals of this module were:

1. don’t force any one evaluation strategy
2. allow users to define new combinators without exposing the implementation
3. allow new combinators to be defined in terms of other already-defined combinators.
4. discourage creation of non-sensical combinator expressions, or possible mis-use of the module
5. hide existential types and anything else exotic

If you want to play with it, you can do a:

$> darcs get http://coder.bsimmons.name/code/CombinatorCalculus

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Note: this is part of a series of posts is related to the “17×17 Challenge” posted by Bill Gasarch. The goal is to color cells of a 17 by 17 grid, using only four colors, such that no rectangle is formed from four cells of the same color.

In my last post, I gave a symmetrical rotation of the known 74-cell single-coloring. I want to use a slight variation of that grid to show why I think treating the grids as symmetrical will help us in solving this problem.

Here is a new spreadsheet, with several panes you can click through on the bottom. Panes ONE through FIVE represent the first few rows of a single-coloring in which we avoid making any real choices, thanks to a few assumptions we make (I’ll come back to that).

Orange represents the row we’re coloring, gray cells are rectangle forming cells (in which marks are not allowed), and blue cells represent what I consider to be the real search-space:

NOTE: We’re simply using another automorphism of the original 74-coloring, i.e. a series of rotations that preserve all the significant attributes of the graph.

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I’ve been puzzling over some of the problems presented by the 17 x 17 Challenge, and wanted to share some of what I’ve learned and have been wondering about. The problem and ultimate goal is to color a 17 x 17 grid with four colors such that every cell is colored and no rectangle is formed by any four cells of the same color.

Single-Color Subsets

I’ve started by trying to find an algorithm to find optimal single-color subsets, i.e. a way of coloring a grid with one color such that no rectangle is formed by the colored cells, and we have the greatest number of cells colored as possible. I started thinking of coloring cells one at a time, following a path based on the notion of an “extended rectangle” where the fourth side was longer or shorter than the first, to avoid forming rectangles.

I also noticed that we could traverse all the cells in the largest known 17×17 subset by following a simple spiraling algorithm. It didn’t seem to matter which cell or direction we started in either:

go until you hit a colored cell, then rotate 90 degrees