unfinished/path: some jottings towards a solver.

I still don't actually have a solver design, but a recent conversation
sent my brain in some more useful directions than I've come up with
before, so I might as well at least note it all down.

unfinished/path.c

@@ -68,6 +68,103 @@
  *    has precisely the set of link changes to cause that effect).
  */
 
+/*
+ * 2020-05-11: some thoughts on a solver.
+ *
+ * Consider this example puzzle, from Wikipedia:
+ *
+ *     ---4---
+ *     -3--25-
+ *     ---31--
+ *     ---5---
+ *     -------
+ *     --1----
+ *     2---4--
+ *
+ * The kind of deduction that a human wants to make here is: which way
+ * does the path between the 4s go? In particular, does it go round
+ * the left of the W-shaped cluster of endpoints, or round the right
+ * of it? It's clear at a glance that it must go to the right, because
+ * _any_ path between the 4s that goes to the left of that cluster, no
+ * matter what detailed direction it takes, will disconnect the
+ * remaining grid squares into two components, with the two 2s not in
+ * the same component. So we immediately know that the path between
+ * the 4s _must_ go round the right-hand side of the grid.
+ *
+ * How do you model that global and topological reasoning in a
+ * computer?
+ *
+ * The most plausible idea I've seen so far is to use fundamental
+ * groups. The fundamental group of loops based at a given point in a
+ * space is a free group, under loop concatenation and up to homotopy,
+ * generated by the loops that go in each direction around each hole
+ * in the space. In this case, the 'holes' are clues, or connected
+ * groups of clues.
+ *
+ * So you might be able to enumerate all the homotopy classes of paths
+ * between (say) the two 4s as follows. Start with any old path
+ * between them (say, find the first one that breadth-first search
+ * will give you). Choose one of the 4s to regard as the base point
+ * (arbitrarily). Then breadth-first search among the space of _paths_
+ * by the following procedure. Given a candidate path, append to it
+ * each of the possible loops that starts from the base point,
+ * circumnavigates one clue cluster, and returns to the base point.
+ * The result will typically be a path that retraces its steps and
+ * self-intersects. Now adjust it homotopically so that it doesn't. If
+ * that can't be done, then we haven't generated a fresh candidate
+ * path; if it can, then we've got a new path that is not homotopic to
+ * any path we already had, so add it to our list and queue it up to
+ * become the starting point of this search later.
+ *
+ * The idea is that this should exhaustively enumerate, up to
+ * homotopy, the different ways in which the two 4s can connect to
+ * each other within the constraint that you have to actually fit the
+ * path non-self-intersectingly into this grid. Then you can keep a
+ * list of those homotopy classes in mind, and start ruling them out
+ * by techniques like the connectivity approach described above.
+ * Hopefully you end up narrowing down to few enough homotopy classes
+ * that you can deduce something concrete about actual squares of the
+ * grid - for example, here, that if the path between 4s has to go
+ * round the right, then we know some specific squares it must go
+ * through, so we can fill those in. And then, having filled in a
+ * piece of the middle of a path, you can now regard connecting the
+ * ultimate endpoints to that mid-section as two separate subproblems,
+ * so you've reduced to a simpler instance of the same puzzle.
+ *
+ * But I don't know whether all of this actually works. I more or less
+ * believe the process for enumerating elements of the free group; but
+ * I'm not as confident that when you find a group element that won't
+ * fit in the grid, you'll never have to consider its descendants in
+ * the BFS either. And I'm assuming that 'unwind the self-intersection
+ * homotopically' is a thing that can actually be turned into a
+ * sensible algorithm.
+ *
+ * --------
+ *
+ * Another thing that might be needed is to characterise _which_
+ * homotopy class a given path is in.
+ *
+ * For this I think it's sufficient to choose a collection of paths
+ * along the _edges_ of the square grid, each of which connects two of
+ * the holes in the grid (including the grid exterior, which counts as
+ * a huge hole), such that they form a spanning tree between the
+ * holes. Then assign each of those paths an orientation, so that
+ * crossing it in one direction counts as 'positive' and the other
+ * 'negative'. Now analyse a candidate path from one square to another
+ * by following it and noting down which of those paths it crosses in
+ * which direction, then simplifying the result like a free group word
+ * (i.e. adjacent + and - crossings of the same path cancel out).
+ *
+ * --------
+ *
+ * If we choose those paths to be of minimal length, then we can get
+ * an upper bound on the number of homotopy classes by observing that
+ * you can't traverse any of those barriers more times than will fit
+ * non-self-intersectingly in the grid. That might be an alternative
+ * method of bounding the search through the fundamental group to only
+ * finitely many possibilities.
+ */
+
 /*
  * Standard notation for directions.
  */