source: branches/f4grobner/division.lisp@ 2013

;;; -*-  Mode: Lisp -*- 
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;                                                                              
;;;  Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>          
;;;                                                                              
;;;  This program is free software; you can redistribute it and/or modify        
;;;  it under the terms of the GNU General Public License as published by        
;;;  the Free Software Foundation; either version 2 of the License, or           
;;;  (at your option) any later version.                                         
;;;                                                                              
;;;  This program is distributed in the hope that it will be useful,             
;;;  but WITHOUT ANY WARRANTY; without even the implied warranty of              
;;;  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the               
;;;  GNU General Public License for more details.                                
;;;                                                                              
;;;  You should have received a copy of the GNU General Public License           
;;;  along with this program; if not, write to the Free Software                 
;;;  Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.  
;;;                                                                              
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defpackage "DIVISION"
  (:use :cl :utils :ring :monom :polynomial :grobner-debug :term :ring-and-order)
  (:export "$POLY_TOP_REDUCTION_ONLY"
           "POLY-PSEUDO-DIVIDE"
           "POLY-EXACT-DIVIDE"
           "NORMAL-FORM-STEP"
           "NORMAL-FORM"
           "POLY-NORMALIZE"
           "POLY-NORMALIZE-LIST"
           "BUCHBERGER-CRITERION"
           "GROBNER-TEST"
           ))

(in-package :division)

(defvar $poly_top_reduction_only nil
  "If not FALSE, use top reduction only whenever possible.
Top reduction means that division algorithm stops after the first reduction.")

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; An implementation of the division algorithm
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defun grobner-op (ring-and-order c1 c2 m f g
                   &aux 
                     (ring (ro-ring ring-and-order)))
  "Returns C2*F-C1*M*G, where F and G are polynomials M is a monomial.
Assume that the leading terms will cancel."
  (declare (type ring-and-order ring-and-order)
           (type monom m)
           (type poly f g))
  #+grobner-check(funcall (ring-zerop ring)
                          (funcall (ring-sub ring)
                                   (funcall (ring-mul ring) c2 (poly-lc f))
                                   (funcall (ring-mul ring) c1 (poly-lc g))))
  #+grobner-check(monom-equal-p (poly-lm f) (monom-mul m (poly-lm g)))
  ;; Note that below we can drop the leading terms of f ang g for the
  ;; purpose of polynomial arithmetic.  
  ;;
  ;; TODO: Make sure that the sugar calculation is correct if leading
  ;; terms are dropped.
  (poly-sub ring-and-order
            (scalar-times-poly-1 ring c2 f)
            (scalar-times-poly-1 ring c1 (monom-times-poly m g))))

(defun check-loop-invariant (ring-and-order c f a fl r p
                             &aux 
                               (ring (ro-ring ring-and-order))
                               (p-zero (make-poly-zero))
                               (a (mapcar #'poly-reverse a))
                               (r (poly-reverse r)))
  "Check loop invariant of division algorithms, when we divide a
polynomial F by the list of polynomials FL. The invariant is the
identity C*F=SUM AI*FI+R+P, where F0 is the initial value of F, A is
the list of partial quotients, R is the intermediate value of the
remainder, and P is the intermediate value which eventually becomes
0. A thing to remember is that the terms of polynomials in A and
the polynomial R have their terms in reversed order. Hence, before
the arithmetic is performed, we need to fix the order of terms"
  #|
  (format t "~&----------------------------------------------------------------~%")
  (format t "#### Loop invariant check ####:~%C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%" 
          c f a fl r p)
  |#
  (flet ((p-add (x y) (poly-add ring-and-order x y))
         (p-sub (x y) (poly-sub ring-and-order x y))
         (p-mul (x y) (poly-mul ring-and-order x y)))
    (let* ((prod (inner-product a fl p-add p-mul p-zero))
           (succeeded-p 
            (poly-zerop
             (p-sub
              (scalar-times-poly ring c f)
              (reduce #'p-add (list prod r p))))))
      (unless succeeded-p 
          (error "#### Polynomial division Loop invariant failed ####:~%C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%" 
                 c f a fl r p))
      succeeded-p)))
  

(defun poly-pseudo-divide (ring-and-order f fl
                           &aux
                             (ring (ro-ring ring-and-order)))
  "Pseudo-divide a polynomial F by the list of polynomials FL. Return
multiple values. The first value is a list of quotients A.  The second
value is the remainder R. The third argument is a scalar coefficient
C, such that C*F can be divided by FL within the ring of coefficients,
which is not necessarily a field. Finally, the fourth value is an
integer count of the number of reductions performed.  The resulting
objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of 
the quotients is initialized to default."
  (declare (type poly f) (list fl))
  ;; Loop invariant: c*f=sum ai*fi+r+p, where p must eventually become 0
  (do ((r (make-poly-zero))
       (c (funcall (ring-unit ring)))
       (a (make-list (length fl) :initial-element (make-poly-zero)))
       (division-count 0)
       (p f))
      ((poly-zerop p)
       #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
       (debug-cgb "~&~3T~d reduction~:p" division-count)
       (when (poly-zerop r) (debug-cgb " ---> 0"))
       ;; We obtained the terms in reverse order, so must fix that
       (setf a (mapcar #'poly-nreverse a)
             r (poly-nreverse r))
       ;; Initialize the sugar of the quotients
       (mapc #'poly-reset-sugar a)
       (values a r c division-count))
    (declare (fixnum division-count))
    ;; Check the loop invariant here
    #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
    (do ((fl fl (rest fl))              ;scan list of divisors
         (b a (rest b)))
        ((cond
           ((endp fl)                           ;no division occurred
            (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
            (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
            (pop (poly-termlist p))     ;remove lt(p) from p
            t)
           ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
            (incf division-count)
            (multiple-value-bind (gcd c1 c2)
                (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
              (declare (ignore gcd))
              (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
                ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
                (mapl #'(lambda (x)
                          (setf (car x) (scalar-times-poly ring c1 (car x))))
                      a)
                (setf r (scalar-times-poly ring c1 r)
                      c (funcall (ring-mul ring) c c1)
                      p (grobner-op ring-and-order c2 c1 m p (car fl)))
                (push (make-term :monom m :coeff c2) (poly-termlist (car b))))
              t))))
      )))

(defun poly-exact-divide (ring-and-order f g)
  "Divide a polynomial F by another polynomial G. Assume that exact division
with no remainder is possible. Returns the quotient."
  (declare (type poly f g) (type ring-and-order ring-and-order))
  (multiple-value-bind (quot rem coeff division-count)
      (poly-pseudo-divide ring-and-order f (list g))
    (declare (ignore division-count coeff)
             (list quot)
             (type poly rem)
             (type fixnum division-count))
    (unless (poly-zerop rem) (error "Exact division failed."))
    (car quot)))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; An implementation of the normal form
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defun normal-form-step (ring-and-order fl p r c division-count
                         &aux 
                           (ring (ro-ring ring-and-order))
                           (g (find (poly-lm p) fl
                                    :test #'monom-divisible-by-p
                                    :key #'poly-lm)))
  (cond
   (g                                   ;division possible
    (incf division-count)
    (multiple-value-bind (gcd cg cp)
        (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
      (declare (ignore gcd))
      (let ((m (monom-div (poly-lm p) (poly-lm g))))
        ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
        (setf r (scalar-times-poly ring cg r)
              c (funcall (ring-mul ring) c cg)
              ;; p := cg*p-cp*m*g
              p (grobner-op ring-and-order cp cg m p g))))
    (debug-cgb "/"))
   (t                                                   ;no division possible
    (push (poly-lt p) (poly-termlist r))                ;move lt(p) to remainder
    (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
    (pop (poly-termlist p))                             ;remove lt(p) from p
    (debug-cgb "+")))
  (values p r c division-count))

;;
;; Merge NORMAL-FORM someday with POLY-PSEUDO-DIVIDE.
;;
;; TODO: It is hard to test normal form as there is no loop invariant,
;; like for POLY-PSEUDO-DIVIDE.  Is there a testing strategy? One
;; method would be to test NORMAL-FORM using POLY-PSEUDO-DIVIDE.
;;
(defun normal-form (ring-and-order f fl
                    &optional 
                      (top-reduction-only $poly_top_reduction_only)
                      (ring (ro-ring ring-and-order)))
  #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
  (do ((r (make-poly-zero))
       (c (funcall (ring-unit ring)))
       (division-count 0))
      ((or (poly-zerop f)
           ;;(endp fl)
           (and top-reduction-only (not (poly-zerop r))))
       (progn
         (debug-cgb "~&~3T~D reduction~:P" division-count)
         (when (poly-zerop r)
           (debug-cgb " ---> 0")))
       (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
       (values f c division-count))
    (declare (fixnum division-count)
             (type poly r))
    (multiple-value-setq (f r c division-count)
      (normal-form-step ring-and-order fl f r c division-count))))

(defun buchberger-criterion (ring-and-order g)
  "Returns T if G is a Grobner basis, by using the Buchberger
criterion: for every two polynomials h1 and h2 in G the S-polynomial
S(h1,h2) reduces to 0 modulo G."
  (every #'poly-zerop
         (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
                   (i 0 (- (length g) 2))
                   (j (1+ i) (1- (length g))))))


(defun poly-normalize (ring p &aux (c (poly-lc p)))
  "Divide a polynomial by its leading coefficient. It assumes
that the division is possible, which may not always be the
case in rings which are not fields. The exact division operator
is assumed to be provided by the RING structure."
  (mapc #'(lambda (term)
            (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
        (poly-termlist p))
  p)

(defun poly-normalize-list (ring plist)
  "Divide every polynomial in a list PLIST by its leading coefficient. "
  (mapcar #'(lambda (x) (poly-normalize ring x)) plist))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; The function GROBNER-CHECK is provided primarily for debugging purposes. To
;; enable verification of grobner bases with BUCHBERGER-CRITERION, do
;; (pushnew :grobner-check *features*) and compile/load this file.
;; With this feature, the calculations will slow down CONSIDERABLY.
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defun grobner-test (ring-and-order g f)
  "Test whether G is a Grobner basis and F is contained in G. Return T
upon success and NIL otherwise."
  (debug-cgb "~&GROBNER CHECK: ")
  (let (($poly_grobner_debug nil)
        (stat1 (buchberger-criterion ring-and-order g))
        (stat2
          (every #'poly-zerop
                 (makelist (normal-form ring-and-order (copy-tree (elt f i)) g nil)
                           (i 0 (1- (length f)))))))
    (unless stat1 (error "~&Buchberger criterion failed, not a grobner basis: ~A" g))
    (unless stat2
      (error "~&Original polynomials not in ideal spanned by Grobner basis: ~A" f)))
  (debug-cgb "~&GROBNER CHECK END")
  t)