Sunday, May 25, 2014

Gödel Numbering for LYNE

What is the Gödel numbering for LYNE puzzles?

I propose, there exists a number, L, such that it describes each LYNE puzzle, such that Lx → puzzlex and puzzlex →  Lx uniquely.

That one is simple enough, for each shape node and terminus we introduce a prime, for each board size we introduce a prime (such that a 3x4 board shape is a different prime than a 4x3 board shape) ... or we could describe the board as numbers: 3 x 4 is sssz x ssssz and we have an unique prime for s, z, and x ('by'), for each empty space on the grid we introduce a prime (2?), for each n-pass-through unit we introduce a prime (I've seen up to 4, is there a 5?) or we could go type-theory here and have the different shapes be in a type-family, the different pass-throughs be in an other type-family (inhabited by singleton type instances). And we have from that a Gödel number that we can wrap a puzzle into and unwrap a puzzle from.

Now, that's the Gödel-numbering, or part I of the two-parter of LYNE-as-Gödel numbers. Simple enough. It's just simply a matter of programming.

Now I further propose the following. There exists some Gödel numbering, L', such that the encoding of the board includes the solution, that is, 'for squares, start at x, N, W, NW, S, S, S, terminated. For triangles, start at y, E, E, S, S, NW, terminate, ... etc.'  That is to say, the construction of the board renders a Gödel number, and the destruction of the board, that is to say, that each move is a Gödel number, and that each move, when dividing the board L' number yields a new, smaller Gödel number such that the final move, completing the puzzle yields a 'finished' Gödel number for that board, that is 'no more moves, all nodes and pass-throughs touched and filled.'

Now, to construct such a number may entail a foray into Graph theory, I don't know, for it is demonstrable that each LYNE puzzle is a DAG (how?), or it may require using the Gödel numbers for each board as proofs (and the solving of the puzzle the co-construction of the proof).

That's part II of the two-parter of LYNE-as-Gödel numbers.