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Re: 5x5 Arithmetic solver
From: |
Vincent Belaïche |
Subject: |
Re: 5x5 Arithmetic solver |
Date: |
Sun, 22 May 2011 22:35:43 +0200 |
Cher Stéfan & tous,
> Beware: I'm talking about make-variable-buffer-local, which is
> different (but I think it's what you want here).
Yes, this is what I meant, but I use this function so seldom that I
mispelled it, and as I am my own spell-checker...
Ok, I removed the function list and used a macro 5x5-defvar-local
instead according to your kind recommendations.
I also added a final message "5x5 Solution computation done." ---
hopefully this is correct English --- to mask the obscure Calc verbiage.
I also realized that there was a bug in 5x5, when making random grids,
it was sometimes impossible for the solver to find a solution, and that
happened while I was demonstrating the solver to my 6 years old son ---
then I had to explained the reason for that kind of failure, which was
incredibly more difficult than understand it and find a correction for
it...
I realized that the random grid generator was indeed making a grid whose
each position is random, rather than making a set of random 5x5
moves. The former does not ensure the existence of a solution, while the
latter does. I don't know if that was the initial intention to have this
behaviour, but as « à l'impossible nul n'est tenu », I considered it a
bug, and I corrected it along with providing the solver. So my
contribution is 3 fold:
I Add an arithmetic solver
II Make 5x5 multisession
III Ensure that random grids always have a solution --- this is a quick
patch that makes it only when the grid size is 5, but I am not sure
that that would work for other size, as I have not proved it
mathematically.
Here is the updated Changelog, I commonalized comment when possible.
-----------------------------------------------------------------------
2011-05-22 Vincent Belaïche <address@hidden>
* play/5x5.el: I/ Add an arithmetic solver to suggest positions to
click on. II/ Make 5x5 multisession. III/ Ensure that random grids
always have a solution in grid size = 5 cases
(5x5-mode-map): Add keybinding to function `5x5-solve-suggest'.
(5x5-solver-output): New defvar, output of function
`5x5-solve-suggest'.
(5x5-grid, 5x5-x-pos, 5x5-y-pos, 5x5-moves, 5x5-cracking): Make
these variables buffer local to achieve 5x5 multi-session-ness.
(5x5): Set 5x5-grid-size only if SIZE is non-negative.
(5x5-grid-to-vec): New defun. Convert a 5x5 game grid into a Calc
matrix in Z/2Z.
(5x5-vec-to-grid): New defun. Convert a Calc matrix in Z/2Z into a
5x5 game grid.
(5x5-log-buffer): New defvar. Not defined, provisionned for
debugging Arithmetric solver.
(5x5-log-init): New defun. Defined to dummy, provisionned for
debugging Arithmetric solver.
(5x5-log): New defun. Defined to dummy, provisionned for debugging
Arithmetric solver.
(5x5-solver): New defun, make the actual algorithm of arithmetic
solver, to be usable that function needs some wrapper to the 5x5
user interace, this wapper is function `5x5-solve-suggest' for
invocation, and function `5x5-draw-grid' for the diplay.
(5x5-solve-suggest): New defun, invoke the arithmetic solver, and
display solution.
(5x5-randomize): Use 5x5-make-move instead of 5x5-flip-cell to
randomize a grid so that we ensure that there is always a
solution.
(5x5-make-random-grid) allow other movement than flipping.
-----------------------------------------------------------------------
VBR,
Vincent
=== modified file 'lisp/play/5x5.el'
--- lisp/play/5x5.el 2011-04-21 12:24:46 +0000
+++ lisp/play/5x5.el 2011-05-22 20:20:45 +0000
@@ -24,15 +24,15 @@
;;; Commentary:
-;; The aim of 5x5 is to fill in all the squares. If you need any more of an
+;; The aim of 5x5 is to fill in all the squares. If you need any more of an
;; explanation you probably shouldn't play the game.
;;; TODO:
-;; o The code for updating the grid needs to be re-done. At the moment it
+;; o The code for updating the grid needs to be re-done. At the moment it
;; simply re-draws the grid every time a move is made.
;;
-;; o Look into tarting up the display with color. gamegrid.el looks
+;; o Look into tarting up the display with color. gamegrid.el looks
;; interesting, perhaps that is the way to go?
;;; Thanks:
@@ -41,7 +41,10 @@
;; emacs mode.
;;
;; Pascal Q. Porcupine <address@hidden> for inspiring the animated
-;; solver.
+;; cracker.
+;;
+;; Vincent Belaïche <address@hidden> & Jay P. Belanger
+;; <address@hidden> for the math solver.
;;; Code:
@@ -89,19 +92,25 @@
;; Non-customize variables.
-(defvar 5x5-grid nil
+(defmacro 5x5-defvar-local (var value doc)
+ "Define VAR to VALUE with documentation DOC and make it buffer local."
+ `(progn
+ (defvar ,var ,value ,doc)
+ (make-variable-buffer-local (quote ,var))))
+
+(5x5-defvar-local 5x5-grid nil
"5x5 grid contents.")
-(defvar 5x5-x-pos 2
+(5x5-defvar-local 5x5-x-pos 2
"X position of cursor.")
-(defvar 5x5-y-pos 2
+(5x5-defvar-local 5x5-y-pos 2
"Y position of cursor.")
-(defvar 5x5-moves 0
+(5x5-defvar-local 5x5-moves 0
"Moves made.")
-(defvar 5x5-cracking nil
+(5x5-defvar-local 5x5-cracking nil
"Are we in cracking mode?")
(defvar 5x5-buffer-name "*5x5*"
@@ -134,10 +143,28 @@
(define-key map [(control c) (control b)] #'5x5-crack-mutating-best)
(define-key map [(control c) (control x)] #'5x5-crack-xor-mutate)
(define-key map "n" #'5x5-new-game)
+ (define-key map "s" #'5x5-solve-suggest)
(define-key map "q" #'5x5-quit-game)
map)
"Local keymap for the 5x5 game.")
+(5x5-defvar-local 5x5-solver-output nil
+ "List that is is the output of artihmetic solver.
+
+This list L is such that
+
+L = (M S_1 S_2 ... S_N)
+
+M is the move count when the solve output was stored.
+
+S_1 ... S_N are all the solutions ordered from least to greatest
+number of strokes. S_1 is the solution to be displayed.
+
+Each solution S_1, ..., S_N is a a list (STROKE-COUNT GRID) where
+STROKE-COUNT is to number of strokes to achieve the solution and
+GRID is the grid of positions to click.")
+
+
;; Menu definition.
(easy-menu-define 5x5-mode-menu 5x5-mode-map "5x5 menu."
@@ -146,6 +173,7 @@
["Random game" 5x5-randomize t]
["Quit game" 5x5-quit-game t]
"---"
+ ["Use Calc solver" 5x5-solve-suggest t]
["Crack randomly" 5x5-crack-randomly t]
["Crack mutating current" 5x5-crack-mutating-current t]
["Crack mutating best" 5x5-crack-mutating-best t]
@@ -158,7 +186,7 @@
(defun 5x5-mode ()
"A mode for playing `5x5'.
-The key bindings for 5x5-mode are:
+The key bindings for `5x5-mode' are:
\\{5x5-mode-map}"
(kill-all-local-variables)
@@ -194,14 +222,14 @@
(interactive "P")
(setq 5x5-cracking nil)
- (when size
- (setq 5x5-grid-size size))
(switch-to-buffer 5x5-buffer-name)
+ (5x5-mode)
+ (when (natnump size)
+ (setq 5x5-grid-size size))
(if (or (not 5x5-grid) (not (= 5x5-grid-size (length (aref 5x5-grid 0)))))
(5x5-new-game))
(5x5-draw-grid (list 5x5-grid))
- (5x5-position-cursor)
- (5x5-mode))
+ (5x5-position-cursor))
(defun 5x5-new-game ()
"Start a new game of `5x5'."
@@ -277,10 +305,11 @@
(defun 5x5-draw-grid (grids)
"Draw the grids GRIDS into the current buffer."
- (let ((buffer-read-only nil))
+ (let ((buffer-read-only nil) grid-org)
(erase-buffer)
(loop for grid in grids do (5x5-draw-grid-end))
(insert "\n")
+ (setq grid-org (point))
(loop for y from 0 to (1- 5x5-grid-size) do
(loop for lines from 0 to (1- 5x5-y-scale) do
(loop for grid in grids do
@@ -290,6 +319,23 @@
(if (5x5-cell grid y x) ?#
?.))))
(insert " | "))
(insert "\n")))
+ (when 5x5-solver-output
+ (if (= (car 5x5-solver-output) 5x5-moves)
+ (save-excursion
+ (goto-char grid-org)
+ (beginning-of-line (+ 1 (/ 5x5-y-scale 2)))
+ (let ((solution-grid (cdadr 5x5-solver-output)))
+ (dotimes (y 5x5-grid-size)
+ (save-excursion
+ (forward-char (+ 1 (/ (1+ 5x5-x-scale) 2)))
+ (dotimes (x 5x5-grid-size)
+ (when (5x5-cell solution-grid y x)
+ (insert-char ?O 1)
+ (delete-char 1)
+ (backward-char))
+ (forward-char (1+ 5x5-x-scale))))
+ (forward-line 5x5-y-scale))))
+ (setq 5x5-solver-output nil)))
(loop for grid in grids do (5x5-draw-grid-end))
(insert "\n")
(insert (format "On: %d Moves: %d" (5x5-grid-value (car grids))
5x5-moves))))
@@ -304,13 +350,14 @@
"Keep track of how many moves have been made."
(incf 5x5-moves))
-(defun 5x5-make-random-grid ()
+(defun 5x5-make-random-grid (&optional move)
"Make a random grid."
+ (setq move (or move (symbol-function '5x5-flip-cell)))
(let ((grid (5x5-make-new-grid)))
(loop for y from 0 to (1- 5x5-grid-size) do
(loop for x from 0 to (1- 5x5-grid-size) do
(if (zerop (random 2))
- (5x5-flip-cell grid y x))))
+ (funcall move grid y x))))
grid))
;; Cracker functions.
@@ -415,6 +462,312 @@
(sit-for 5x5-animate-delay))))
5x5-grid)
+;; Arithmetic solver
+;;===========================================================================
+(defun 5x5-grid-to-vec (grid)
+ "Convert GRID to an equivalent Calc matrix of (mod X 2) forms
+where X is 1 for setting a position, and 0 for unsetting a
+position."
+ (cons 'vec
+ (mapcar (lambda (y)
+ (cons 'vec
+ (mapcar (lambda (x)
+ (if x '(mod 1 2) '(mod 0 2)))
+ y)))
+ grid)))
+
+(defun 5x5-vec-to-grid (grid-matrix)
+ "Convert a grid matrix GRID-MATRIX in Calc format to a grid in
+5x5 format. See function `5x5-grid-to-vec'."
+ (apply
+ 'vector
+ (mapcar
+ (lambda (x)
+ (apply
+ 'vector
+ (mapcar
+ (lambda (y) (/= (cadr y) 0))
+ (cdr x))))
+ (cdr grid-matrix))))
+
+(if nil; set to t to enable solver logging
+ (progn
+ (defvar 5x5-log-buffer nil)
+ (defun 5x5-log-init ()
+ (if (buffer-live-p 5x5-log-buffer)
+ (with-current-buffer 5x5-log-buffer (erase-buffer))
+ (setq 5x5-log-buffer (get-buffer-create "*5x5 LOG*"))))
+
+ (defun 5x5-log (name value)
+ "Debug purpuse only.
+
+Log a matrix VALUE of (mod B 2) forms, only B is output and
+Scilab matrix notation is used. VALUE is returned so that it is
+easy to log a value with minimal rewrite of code."
+ (when (buffer-live-p 5x5-log-buffer)
+ (let* ((unpacked-value
+ (math-map-vec
+ (lambda (row) (math-map-vec 'cadr row))
+ value))
+ (calc-vector-commas "")
+ (calc-matrix-brackets '(C O))
+ (value-to-log (math-format-value unpacked-value)))
+ (with-current-buffer 5x5-log-buffer
+ (insert name ?= value-to-log ?\n))))
+ value))
+ (defmacro 5x5-log-init ())
+ (defmacro 5x5-log (name value) value))
+
+(defun 5x5-solver (grid)
+ "Return a list of solutions for GRID.
+
+Given some grid GRID, the returned a list of solution LIST is
+sorted from least Hamming weight to geatest one.
+
+ LIST = (SOLUTION-1 ... SOLUTION-N)
+
+Each solution SOLUTION-I is a cons cell (HW . G) where HW is the
+Hamming weight of the solution --- ie the number of strokes to
+achieves it --- and G is the grid of positions to click in order
+to complete the 5x5.
+
+Solutions are sorted from least to greatest Hamming weight."
+ (require 'calc-ext)
+ (flet ((5x5-mat-mode-2
+ (a)
+ (math-map-vec
+ (lambda (y)
+ (math-map-vec
+ (lambda (x) `(mod ,x 2))
+ y))
+ a)))
+ (let* (calc-command-flags
+ (grid-size-squared (* 5x5-grid-size 5x5-grid-size))
+
+ ;; targetv is the vector the origine of which is org="current
+ ;; grid" and the end of which is dest="all ones".
+ (targetv
+ (5x5-log
+ "b"
+ (let (
+ ;; org point is the current grid
+ (org (calcFunc-arrange (5x5-grid-to-vec grid)
+ 1))
+
+ ;; end point of game is the all ones matrix
+ (dest (calcFunc-cvec '(mod 1 2) grid-size-squared 1)))
+ (math-sub dest org))))
+
+ ;; transferm is the transfer matrix, ie it is the 25x25
+ ;; matrix applied everytime a flip is carried out where a
+ ;; flip is defined by a 25x1 Dirac vector --- ie all zeros
+ ;; but 1 in the position that is flipped.
+ (transferm
+ (5x5-log
+ "a"
+ ;; transfer-grid is not a play grid, but this is the
+ ;; transfer matrix in the format of a vector of vectors, we
+ ;; do it this way because random access in vectors is
+ ;; faster. The motivation is just speed as we build it
+ ;; element by element, but that could have been created
+ ;; using only Calc primitives. Probably that would be a
+ ;; better idea to use Calc with some vector manipulation
+ ;; rather than going this way...
+ (5x5-grid-to-vec (let ((transfer-grid
+ (let ((5x5-grid-size grid-size-squared))
+ (5x5-make-new-grid))))
+ (dotimes (i 5x5-grid-size)
+ (dotimes (j 5x5-grid-size)
+ ;; k0 = flattened flip position
corresponding
+ ;; to (i, j) on the grid.
+ (let* ((k0 (+ (* 5 i) j)))
+ ;; cross center
+ (5x5-set-cell transfer-grid k0 k0 t)
+ ;; Cross top.
+ (and
+ (> i 0)
+ (5x5-set-cell transfer-grid
+ (- k0 5x5-grid-size) k0 t))
+ ;; Cross bottom.
+ (and
+ (< (1+ i) 5x5-grid-size)
+ (5x5-set-cell transfer-grid
+ (+ k0 5x5-grid-size) k0 t))
+ ;; Cross left.
+ (and
+ (> j 0)
+ (5x5-set-cell transfer-grid (1- k0) k0
t))
+ ;; Cross right.
+ (and
+ (< (1+ j) 5x5-grid-size)
+ (5x5-set-cell transfer-grid
+ (1+ k0) k0 t)))))
+ transfer-grid))))
+ ;; TODO: this is hard-coded for grid-size = 5, make it generic.
+ (transferm-kernel-size
+ (if (= 5x5-grid-size 5) 2
+ (error "Transfer matrix rank not known for grid-size != 5")))
+
+ ;; TODO: this is hard-coded for grid-size = 5, make it generic.
+ ;;
+ ;; base-change is a 25x25 matrix, where topleft submatrix
+ ;; 23x25 is a diagonal of 1, and the two last columns are a
+ ;; base of kernel of transferm.
+ ;;
+ ;; base-change must be by construction inversible.
+ (base-change
+ (5x5-log
+ "p"
+ (let ((id (5x5-mat-mode-2 (calcFunc-diag 1 grid-size-squared))))
+ (setcdr (last id (1+ transferm-kernel-size))
+ (cdr (5x5-mat-mode-2
+ '(vec (vec 0 1 1 1 0 1 0 1 0 1 1 1 0 1
+ 1 1 0 1 0 1 0 1 1 1 0)
+ (vec 1 1 0 1 1 0 0 0 0 0 1 1 0 1
+ 1 0 0 0 0 0 1 1 0 1 1)))))
+ (calcFunc-trn id))))
+
+ (inv-base-change
+ (5x5-log "invp"
+ (calcFunc-inv base-change)))
+
+ ;; B:= targetv
+ ;; A:= transferm
+ ;; P:= base-change
+ ;; P^-1 := inv-base-change
+ ;; X := solution
+
+ ;; B = A * X
+ ;; P^-1 * B = P^-1 * A * P * P^-1 * X
+ ;; CX = P^-1 * X
+ ;; CA = P^-1 * A * P
+ ;; CB = P^-1 * B
+ ;; CB = CA * CX
+ ;; CX = CA^-1 * CB
+ ;; X = P * CX
+ (ctransferm
+ (5x5-log
+ "ca"
+ (math-mul
+ inv-base-change
+ (math-mul transferm base-change)))); CA
+ (ctarget
+ (5x5-log
+ "cb"
+ (math-mul inv-base-change targetv))); CB
+ (row-1 (math-make-intv 3 1 transferm-kernel-size)) ; 1..2
+ (row-2 (math-make-intv 1 transferm-kernel-size
+ grid-size-squared)); 3..25
+ (col-1 (math-make-intv 3 1 (- grid-size-squared
+ transferm-kernel-size))); 1..23
+ (col-2 (math-make-intv 1 (- grid-size-squared
+ transferm-kernel-size)
+ grid-size-squared)); 24..25
+ (ctransferm-1-: (calcFunc-mrow ctransferm row-1))
+ (ctransferm-1-1 (calcFunc-mcol ctransferm-1-: col-1))
+
+ ;; By construction ctransferm-:-2 = 0, so ctransferm-1-2 = 0
+ ;; and ctransferm-2-2 = 0.
+
+ ;;(ctransferm-1-2 (calcFunc-mcol ctransferm-1-: col-2))
+ (ctransferm-2-: (calcFunc-mrow ctransferm row-2))
+ (ctransferm-2-1
+ (5x5-log
+ "ca_2_1"
+ (calcFunc-mcol ctransferm-2-: col-1)))
+
+ ;; By construction ctransferm-2-2 = 0.
+ ;;
+ ;;(ctransferm-2-2 (calcFunc-mcol ctransferm-2-: col-2))
+
+ (ctarget-1 (calcFunc-mrow ctarget row-1))
+ (ctarget-2 (calcFunc-mrow ctarget row-2))
+
+ ;; ctarget-1(2x1) = ctransferm-1-1(2x23) *cx-1(23x1)
+ ;; + ctransferm-1-2(2x2) *cx-2(2x1);
+ ;; ctarget-2(23x1) = ctransferm-2-1(23x23)*cx-1(23x1)
+ ;; + ctransferm-2-2(23x2)*cx-2(2x1);
+ ;; By construction:
+ ;;
+ ;; ctransferm-1-2 == zeros(2,2) and ctransferm-2-2 == zeros(23,2)
+ ;;
+ ;; So:
+ ;;
+ ;; ctarget-2 = ctransferm-2-1*cx-1
+ ;;
+ ;; So:
+ ;;
+ ;; cx-1 = inv-ctransferm-2-1 * ctarget-2
+ (cx-1 (math-mul (calcFunc-inv ctransferm-2-1) ctarget-2))
+
+ ;; Any cx-2 can do, so there are 2^{transferm-kernel-size} solutions.
+ (solution-list
+ ;; Within solution-list each element is a cons cell:
+ ;;
+ ;; (HW . SOL)
+ ;;
+ ;; where HW is the Hamming weight of solution, and SOL is
+ ;; the solution in the form of a grid.
+ (sort
+ (cdr
+ (math-map-vec
+ (lambda (cx-2)
+ ;; Compute `solution' in the form of a 25x1 matrix of
+ ;; (mod B 2) forms --- with B = 0 or 1 --- and
+ ;; return (HW . SOL) where HW is the Hamming weight
+ ;; of solution and SOL a grid.
+ (let ((solution (math-mul
+ base-change
+ (calcFunc-vconcat cx-1 cx-2)))); X = P * CX
+ (cons
+ ;; The Hamming Weight is computed by matrix reduction
+ ;; with an ad-hoc operator.
+ (math-reduce-vec
+ ;; (cadadr '(vec (mod x 2))) => x
+ (lambda (r x) (+ (if (integerp r) r (cadadr r))
+ (cadadr x)))
+ solution); car
+ (5x5-vec-to-grid
+ (calcFunc-arrange solution 5x5-grid-size));cdr
+ )))
+ ;; A (2^K) x K matrix, where K is the dimension of kernel
+ ;; of transfer matrix --- i.e. K=2 in if the grid is 5x5
+ ;; --- for I from 0 to K-1, each row rI correspond to the
+ ;; binary representation of number I, that is to say row
+ ;; rI is a 1xK vector:
+ ;; [ n{I,0} n{I,1} ... n{I,K-1} ]
+ ;; such that:
+ ;; I = sum for J=0..K-1 of 2^(n{I,J})
+ (let ((calc-number-radix 2)
+ (calc-leading-zeros t)
+ (calc-word-size transferm-kernel-size))
+ (math-map-vec
+ (lambda (x)
+ (cons 'vec
+ (mapcar (lambda (x) `(vec (mod ,(logand x 1) 2)))
+ (substring (math-format-number x)
+ (- transferm-kernel-size)))))
+ (calcFunc-index (math-pow 2 transferm-kernel-size) 0))) ))
+ ;; Sort solutions according to respective Hamming weight.
+ (lambda (x y) (< (car x) (car y)))
+ )))
+ (message "5x5 Solution computation done.")
+ solution-list)))
+
+(defun 5x5-solve-suggest (&optional n)
+ "Suggest to the user where to click.
+
+Argument N is ignored."
+ ;; For the time being n is ignored, the idea was to use some numeric
+ ;; argument to show a limited amount of positions.
+ (interactive "P")
+ (5x5-log-init)
+ (let ((solutions (5x5-solver 5x5-grid)))
+ (setq 5x5-solver-output
+ (cons 5x5-moves solutions)))
+ (5x5-draw-grid (list 5x5-grid))
+ (5x5-position-cursor))
+
;; Keyboard response functions.
(defun 5x5-flip-current ()
@@ -490,7 +843,7 @@
(setq 5x5-x-pos (/ 5x5-grid-size 2)
5x5-y-pos (/ 5x5-grid-size 2)
5x5-moves 0
- 5x5-grid (5x5-make-random-grid))
+ 5x5-grid (5x5-make-random-grid (symbol-function '5x5-make-move)))
(unless 5x5-cracking
(5x5-draw-grid (list 5x5-grid)))
(5x5-position-cursor)))
- Re: 5x5 Arithmetic solver, (continued)