767 lines
30 KiB
Scheme
767 lines
30 KiB
Scheme
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;;; MATRIX -- Obtained from Andrew Wright.
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; Chez-Scheme compatibility stuff:
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(define (chez-box x) (cons x '()))
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(define (chez-unbox x) (car x))
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(define (chez-set-box! x y) (set-car! x y))
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; Test that a matrix with entries in {+1, -1} is maximal among the matricies
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; obtainable by
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; re-ordering the rows
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; re-ordering the columns
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; negating any subset of the columns
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; negating any subset of the rows
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; Where we compare two matricies by lexicographically comparing the first row,
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; then the next to last, etc., and we compare a row by lexicographically
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; comparing the first entry, the second entry, etc., and we compare two
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; entries by +1 > -1.
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; Note, this scheme obeys the useful fact that if (append mat1 mat2) is
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; maximal, then so is mat1. Thus, we can build up maximal matricies
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; row by row.
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;
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; Once you have chosen the row re-ordering so that you know which row goes
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; last, the set of columns to negate is fixed (since the last row must be
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; all +1's).
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;
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; Note, the column ordering is really totally determined as follows:
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; all columns for which the second row is +1 must come before all
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; columns for which the second row is -1.
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; among columns for which the second row is +1, all columns for which
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; the third row is +1 come before those for which the third is
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; -1, and similarly for columns in which the second row is -1.
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; etc
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; Thus, each succeeding row sorts columns withing refinings equivalence
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; classes.
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;
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; Maximal? assumes that mat has atleast one row, and that the first row
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; is all +1's.
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(define maximal?
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(lambda (mat)
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(let pick-first-row
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((first-row-perm
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(gen-perms mat)))
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(if first-row-perm
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(and (zunda first-row-perm mat)
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(pick-first-row (first-row-perm 'brother)))
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#t))))
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(define zunda
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(lambda (first-row-perm mat)
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(let* ((first-row
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(first-row-perm 'now))
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(number-of-cols
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(length first-row))
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(make-row->func
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(lambda (if-equal if-different)
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(lambda (row)
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(let ((vec
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(make-vector number-of-cols)))
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(do ((i 0 (+ i 1))
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(first first-row
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(cdr first))
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(row row
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(cdr row)))
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((= i number-of-cols))
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(vector-set! vec
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i
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(if (= (car first) (car row))
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if-equal
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if-different)))
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(lambda (i)
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(vector-ref vec i))))))
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(mat
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(cdr mat)))
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(zebra (first-row-perm 'child)
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(make-row->func 1 -1)
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(make-row->func -1 1)
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mat
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number-of-cols))))
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(define zebra
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(lambda (row-perm row->func+ row->func- mat number-of-cols)
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(let _-*-
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((row-perm
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row-perm)
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(mat
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mat)
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(partitions
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(list (miota number-of-cols))))
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(or (not row-perm)
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(and
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(zulu (car mat)
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(row->func+ (row-perm 'now))
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partitions
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(lambda (new-partitions)
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(_-*- (row-perm 'child)
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(cdr mat)
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new-partitions)))
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(zulu (car mat)
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(row->func- (row-perm 'now))
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partitions
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(lambda (new-partitions)
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(_-*- (row-perm 'child)
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(cdr mat)
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new-partitions)))
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(let ((new-row-perm
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(row-perm 'brother)))
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(or (not new-row-perm)
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(_-*- new-row-perm
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mat
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partitions))))))))
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(define zulu
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(let ((cons-if-not-null
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(lambda (lhs rhs)
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(if (null? lhs)
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rhs
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(cons lhs rhs)))))
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(lambda (old-row new-row-func partitions equal-cont)
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(let _-*-
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((p-in
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partitions)
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(old-row
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old-row)
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(rev-p-out
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'()))
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(let _-split-
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((partition
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(car p-in))
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(old-row
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old-row)
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(plus
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'())
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(minus
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'()))
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(if (null? partition)
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(let _-minus-
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((old-row
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old-row)
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(m
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minus))
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(if (null? m)
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(let ((rev-p-out
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(cons-if-not-null
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minus
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(cons-if-not-null
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plus
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rev-p-out)))
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(p-in
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(cdr p-in)))
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(if (null? p-in)
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(equal-cont (reverse rev-p-out))
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(_-*- p-in old-row rev-p-out)))
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(or (= 1 (car old-row))
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(_-minus- (cdr old-row)
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(cdr m)))))
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(let ((next
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(car partition)))
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(case (new-row-func next)
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((1)
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(and (= 1 (car old-row))
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(_-split- (cdr partition)
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(cdr old-row)
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(cons next plus)
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minus)))
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((-1)
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(_-split- (cdr partition)
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old-row
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plus
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(cons next minus)))))))))))
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(define all?
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(lambda (ok? lst)
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(let _-*-
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((lst
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lst))
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(or (null? lst)
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(and (ok? (car lst))
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(_-*- (cdr lst)))))))
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(define gen-perms
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(lambda (objects)
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(let _-*-
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((zulu-future
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objects)
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(past
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'()))
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(if (null? zulu-future)
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#f
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(lambda (msg)
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(case msg
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((now)
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(car zulu-future))
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((brother)
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(_-*- (cdr zulu-future)
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(cons (car zulu-future)
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past)))
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((child)
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(gen-perms
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(fold past cons (cdr zulu-future))))
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((puke)
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(cons (car zulu-future)
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(fold past cons (cdr zulu-future))))
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(else
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(error gen-perms "Bad msg: ~a" msg))))))))
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(define fold
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(lambda (lst folder state)
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(let _-*-
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((lst
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lst)
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(state
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state))
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(if (null? lst)
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state
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(_-*- (cdr lst)
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(folder (car lst)
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state))))))
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(define miota
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(lambda (len)
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(let _-*-
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((i 0))
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(if (= i len)
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'()
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(cons i
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(_-*- (+ i 1)))))))
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(define proc->vector
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(lambda (size proc)
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(let ((res
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(make-vector size)))
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(do ((i 0
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(+ i 1)))
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((= i size))
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(vector-set! res
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i
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(proc i)))
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res)))
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; Given a prime number P, return a procedure which, given a `maker' procedure,
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; calls it on the operations for the field Z/PZ.
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(define make-modular
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(lambda (modulus)
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(let* ((reduce
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(lambda (x)
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(modulo x modulus)))
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(coef-zero?
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(lambda (x)
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(zero? (reduce x))))
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(coef-+
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(lambda (x y)
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(reduce (+ x y))))
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(coef-negate
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(lambda (x)
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(reduce (- x))))
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(coef-*
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(lambda (x y)
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(reduce (* x y))))
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(coef-recip
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(let ((inverses
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(proc->vector (- modulus 1)
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(lambda (i)
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(extended-gcd (+ i 1)
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modulus
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(lambda (gcd inverse ignore)
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inverse))))))
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; Coef-recip.
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(lambda (x)
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(let ((x
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(reduce x)))
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(vector-ref inverses (- x 1)))))))
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(lambda (maker)
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(maker 0 ; coef-zero
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1 ; coef-one
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coef-zero?
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coef-+
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coef-negate
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coef-*
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coef-recip)))))
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; Extended Euclidean algorithm.
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; (extended-gcd a b cont) computes the gcd of a and b, and expresses it
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; as a linear combination of a and b. It returns calling cont via
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; (cont gcd a-coef b-coef)
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; where gcd is the GCD and is equal to a-coef * a + b-coef * b.
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(define extended-gcd
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(let ((n->sgn/abs
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(lambda (x cont)
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(if (>= x 0)
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(cont 1 x)
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(cons -1 (- x))))))
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(lambda (a b cont)
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(n->sgn/abs a
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(lambda (p-a p)
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(n->sgn/abs b
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(lambda (q-b q)
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(let _-*-
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((p
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p)
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(p-a
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p-a)
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(p-b
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0)
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(q
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q)
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(q-a
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0)
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(q-b
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q-b))
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(if (zero? q)
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(cont p p-a p-b)
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(let ((mult
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(quotient p q)))
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(_-*- q
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q-a
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q-b
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(- p (* mult q))
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(- p-a (* mult q-a))
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(- p-b (* mult q-b)))))))))))))
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; Given elements and operations on the base field, return a procedure which
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; computes the row-reduced version of a matrix over that field. The result
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; is a list of rows where the first non-zero entry in each row is a 1 (in
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; the coefficient field) and occurs to the right of all the leading non-zero
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; entries of previous rows. In particular, the number of rows is the rank
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; of the original matrix, and they have the same row-space.
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; The items related to the base field which are needed are:
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; coef-zero additive identity
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; coef-one multiplicative identity
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; coef-zero? test for additive identity
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; coef-+ addition (two args)
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; coef-negate additive inverse
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; coef-* multiplication (two args)
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; coef-recip multiplicative inverse
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; Note, matricies are stored as lists of rows (i.e., lists of lists).
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(define make-row-reduce
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(lambda (coef-zero coef-one coef-zero? coef-+ coef-negate coef-* coef-recip)
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(lambda (mat)
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(let _-*-
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((mat
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mat))
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(if (or (null? mat)
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(null? (car mat)))
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'()
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(let _-**-
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((in
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mat)
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(out
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'()))
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(if (null? in)
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(map
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(lambda (x)
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(cons coef-zero x))
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(_-*- out))
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(let* ((prow
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(car in))
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(pivot
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(car prow))
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(prest
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(cdr prow))
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(in
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(cdr in)))
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(if (coef-zero? pivot)
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(_-**- in
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(cons prest out))
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(let ((zap-row
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(map
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(let ((mult
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(coef-recip pivot)))
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(lambda (x)
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(coef-* mult x)))
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prest)))
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(cons (cons coef-one zap-row)
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(map
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(lambda (x)
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(cons coef-zero x))
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(_-*-
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(fold in
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(lambda (row mat)
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(cons
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(let ((first-col
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(car row))
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(rest-row
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(cdr row)))
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(if (coef-zero? first-col)
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rest-row
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(map
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(let ((mult
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(coef-negate first-col)))
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(lambda (f z)
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(coef-+ f
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(coef-* mult z))))
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rest-row
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zap-row)))
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mat))
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out))))))))))))))
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; Given elements and operations on the base field, return a procedure which
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|
|
; when given a matrix and a vector tests to see if the vector is in the
|
||
|
|
; row-space of the matrix. This returned function is curried.
|
||
|
|
; The items related to the base field which are needed are:
|
||
|
|
; coef-zero additive identity
|
||
|
|
; coef-one multiplicative identity
|
||
|
|
; coef-zero? test for additive identity
|
||
|
|
; coef-+ addition (two args)
|
||
|
|
; coef-negate additive inverse
|
||
|
|
; coef-* multiplication (two args)
|
||
|
|
; coef-recip multiplicative inverse
|
||
|
|
; Note, matricies are stored as lists of rows (i.e., lists of lists).
|
||
|
|
(define make-in-row-space?
|
||
|
|
(lambda (coef-zero coef-one coef-zero? coef-+ coef-negate coef-* coef-recip)
|
||
|
|
(let ((row-reduce
|
||
|
|
(make-row-reduce coef-zero
|
||
|
|
coef-one
|
||
|
|
coef-zero?
|
||
|
|
coef-+
|
||
|
|
coef-negate
|
||
|
|
coef-*
|
||
|
|
coef-recip)))
|
||
|
|
(lambda (mat)
|
||
|
|
(let ((mat
|
||
|
|
(row-reduce mat)))
|
||
|
|
(lambda (row)
|
||
|
|
(let _-*-
|
||
|
|
((row
|
||
|
|
row)
|
||
|
|
(mat
|
||
|
|
mat))
|
||
|
|
(if (null? row)
|
||
|
|
#t
|
||
|
|
(let ((r-first
|
||
|
|
(car row))
|
||
|
|
(r-rest
|
||
|
|
(cdr row)))
|
||
|
|
(cond ((coef-zero? r-first)
|
||
|
|
(_-*- r-rest
|
||
|
|
(map cdr
|
||
|
|
(if (or (null? mat)
|
||
|
|
(coef-zero? (caar mat)))
|
||
|
|
mat
|
||
|
|
(cdr mat)))))
|
||
|
|
((null? mat)
|
||
|
|
#f)
|
||
|
|
(else
|
||
|
|
(let* ((zap-row
|
||
|
|
(car mat))
|
||
|
|
(z-first
|
||
|
|
(car zap-row))
|
||
|
|
(z-rest
|
||
|
|
(cdr zap-row))
|
||
|
|
(mat
|
||
|
|
(cdr mat)))
|
||
|
|
(if (coef-zero? z-first)
|
||
|
|
#f
|
||
|
|
(_-*-
|
||
|
|
(map
|
||
|
|
(let ((mult
|
||
|
|
(coef-negate r-first)))
|
||
|
|
(lambda (r z)
|
||
|
|
(coef-+ r
|
||
|
|
(coef-* mult z))))
|
||
|
|
r-rest
|
||
|
|
z-rest)
|
||
|
|
(map cdr mat)))))))))))))))
|
||
|
|
|
||
|
|
|
||
|
|
; Given a prime number, return a procedure which takes integer matricies
|
||
|
|
; and returns their row-reduced form, modulo the prime.
|
||
|
|
(define make-modular-row-reduce
|
||
|
|
(lambda (modulus)
|
||
|
|
((make-modular modulus)
|
||
|
|
make-row-reduce)))
|
||
|
|
|
||
|
|
|
||
|
|
(define make-modular-in-row-space?
|
||
|
|
(lambda (modulus)
|
||
|
|
((make-modular modulus)
|
||
|
|
make-in-row-space?)))
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
; Usual utilities.
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
; Given a bound, find a prime greater than the bound.
|
||
|
|
(define find-prime
|
||
|
|
(lambda (bound)
|
||
|
|
(let* ((primes
|
||
|
|
(list 2))
|
||
|
|
(last
|
||
|
|
(chez-box primes))
|
||
|
|
(is-next-prime?
|
||
|
|
(lambda (trial)
|
||
|
|
(let _-*-
|
||
|
|
((primes
|
||
|
|
primes))
|
||
|
|
(or (null? primes)
|
||
|
|
(let ((p
|
||
|
|
(car primes)))
|
||
|
|
(or (< trial (* p p))
|
||
|
|
(and (not (zero? (modulo trial p)))
|
||
|
|
(_-*- (cdr primes))))))))))
|
||
|
|
(if (> 2 bound)
|
||
|
|
2
|
||
|
|
(let _-*-
|
||
|
|
((trial
|
||
|
|
3))
|
||
|
|
(if (is-next-prime? trial)
|
||
|
|
(let ((entry
|
||
|
|
(list trial)))
|
||
|
|
(set-cdr! (chez-unbox last) entry)
|
||
|
|
(chez-set-box! last entry)
|
||
|
|
(if (> trial bound)
|
||
|
|
trial
|
||
|
|
(_-*- (+ trial 2))))
|
||
|
|
(_-*- (+ trial 2))))))))
|
||
|
|
|
||
|
|
; Given the size of a square matrix consisting only of +1's and -1's,
|
||
|
|
; return an upper bound on the determinant.
|
||
|
|
(define det-upper-bound
|
||
|
|
(lambda (size)
|
||
|
|
(let ((main-part
|
||
|
|
(expt size
|
||
|
|
(quotient size 2))))
|
||
|
|
(if (even? size)
|
||
|
|
main-part
|
||
|
|
(* main-part
|
||
|
|
(do ((i 0 (+ i 1)))
|
||
|
|
((>= (* i i) size)
|
||
|
|
i)))))))
|
||
|
|
|
||
|
|
; Fold over all maximal matrices.
|
||
|
|
(define go
|
||
|
|
(lambda (number-of-cols inv-size folder state)
|
||
|
|
(let* ((in-row-space?
|
||
|
|
(make-modular-in-row-space?
|
||
|
|
(find-prime
|
||
|
|
(det-upper-bound inv-size))))
|
||
|
|
(make-tester
|
||
|
|
(lambda (mat)
|
||
|
|
(let ((tests
|
||
|
|
(let ((old-mat
|
||
|
|
(cdr mat))
|
||
|
|
(new-row
|
||
|
|
(car mat)))
|
||
|
|
(fold-over-subs-of-size old-mat
|
||
|
|
(- inv-size 2)
|
||
|
|
(lambda (sub tests)
|
||
|
|
(cons
|
||
|
|
(in-row-space?
|
||
|
|
(cons new-row sub))
|
||
|
|
tests))
|
||
|
|
'()))))
|
||
|
|
(lambda (row)
|
||
|
|
(let _-*-
|
||
|
|
((tests
|
||
|
|
tests))
|
||
|
|
(and (not (null? tests))
|
||
|
|
(or ((car tests) row)
|
||
|
|
(_-*- (cdr tests)))))))))
|
||
|
|
(all-rows ; all rows starting with +1 in decreasing order
|
||
|
|
(fold
|
||
|
|
(fold-over-rows (- number-of-cols 1)
|
||
|
|
cons
|
||
|
|
'())
|
||
|
|
(lambda (row rows)
|
||
|
|
(cons (cons 1 row)
|
||
|
|
rows))
|
||
|
|
'())))
|
||
|
|
(let _-*-
|
||
|
|
((number-of-rows
|
||
|
|
1)
|
||
|
|
(rev-mat
|
||
|
|
(list
|
||
|
|
(car all-rows)))
|
||
|
|
(possible-future
|
||
|
|
(cdr all-rows))
|
||
|
|
(state
|
||
|
|
state))
|
||
|
|
(let ((zulu-future
|
||
|
|
(remove-in-order
|
||
|
|
(if (< number-of-rows inv-size)
|
||
|
|
(in-row-space? rev-mat)
|
||
|
|
(make-tester rev-mat))
|
||
|
|
possible-future)))
|
||
|
|
(if (null? zulu-future)
|
||
|
|
(folder (reverse rev-mat)
|
||
|
|
state)
|
||
|
|
(let _-**-
|
||
|
|
((zulu-future
|
||
|
|
zulu-future)
|
||
|
|
(state
|
||
|
|
state))
|
||
|
|
(if (null? zulu-future)
|
||
|
|
state
|
||
|
|
(let ((rest-of-future
|
||
|
|
(cdr zulu-future)))
|
||
|
|
(_-**- rest-of-future
|
||
|
|
(let* ((first
|
||
|
|
(car zulu-future))
|
||
|
|
(new-rev-mat
|
||
|
|
(cons first rev-mat)))
|
||
|
|
(if (maximal? (reverse new-rev-mat))
|
||
|
|
(_-*- (+ number-of-rows 1)
|
||
|
|
new-rev-mat
|
||
|
|
rest-of-future
|
||
|
|
state)
|
||
|
|
state))))))))))))
|
||
|
|
|
||
|
|
(define go-folder
|
||
|
|
(lambda (mat bsize.blen.blist)
|
||
|
|
(let ((bsize
|
||
|
|
(car bsize.blen.blist))
|
||
|
|
(size
|
||
|
|
(length mat)))
|
||
|
|
(if (< size bsize)
|
||
|
|
bsize.blen.blist
|
||
|
|
(let ((blen
|
||
|
|
(cadr bsize.blen.blist))
|
||
|
|
(blist
|
||
|
|
(cddr bsize.blen.blist)))
|
||
|
|
(if (= size bsize)
|
||
|
|
(let ((blen
|
||
|
|
(+ blen 1)))
|
||
|
|
; (if
|
||
|
|
; (let _-*-
|
||
|
|
; ((blen
|
||
|
|
; blen))
|
||
|
|
; (or (< blen 10)
|
||
|
|
; (and (zero? (remainder blen 10))
|
||
|
|
; (_-*- (quotient blen 10)))))
|
||
|
|
;
|
||
|
|
; (begin
|
||
|
|
; (display blen)
|
||
|
|
; (display " of size ")
|
||
|
|
; (display bsize)
|
||
|
|
; (newline)))
|
||
|
|
|
||
|
|
(cons bsize
|
||
|
|
(cons blen
|
||
|
|
(cond ((< blen 3000)
|
||
|
|
(cons mat blist))
|
||
|
|
((= blen 3000)
|
||
|
|
(cons "..." blist))
|
||
|
|
(else
|
||
|
|
blist)))))
|
||
|
|
; (begin
|
||
|
|
; (newline)
|
||
|
|
; (display "First of size ")
|
||
|
|
; (display size)
|
||
|
|
; (display ":")
|
||
|
|
; (newline)
|
||
|
|
; (for-each
|
||
|
|
; (lambda (row)
|
||
|
|
; (display " ")
|
||
|
|
; (for-each
|
||
|
|
; (lambda (e)
|
||
|
|
; (case e
|
||
|
|
; ((1)
|
||
|
|
; (display " 1"))
|
||
|
|
; ((-1)
|
||
|
|
; (display " -1"))))
|
||
|
|
; row)
|
||
|
|
; (newline))
|
||
|
|
; mat)
|
||
|
|
|
||
|
|
(list size 1 mat)))))))
|
||
|
|
|
||
|
|
(define really-go
|
||
|
|
(lambda (number-of-cols inv-size)
|
||
|
|
(cddr
|
||
|
|
(go number-of-cols
|
||
|
|
inv-size
|
||
|
|
go-folder
|
||
|
|
(list -1 -1)))))
|
||
|
|
|
||
|
|
(define remove-in-order
|
||
|
|
(lambda (remove? lst)
|
||
|
|
(reverse
|
||
|
|
(fold lst
|
||
|
|
(lambda (e lst)
|
||
|
|
(if (remove? e)
|
||
|
|
lst
|
||
|
|
(cons e lst)))
|
||
|
|
'()))))
|
||
|
|
|
||
|
|
; The first fold-over-rows is slower than the second one, but folds
|
||
|
|
; over rows in lexical order (large to small).
|
||
|
|
(define fold-over-rows
|
||
|
|
(lambda (number-of-cols folder state)
|
||
|
|
(if (zero? number-of-cols)
|
||
|
|
(folder '()
|
||
|
|
state)
|
||
|
|
(fold-over-rows (- number-of-cols 1)
|
||
|
|
(lambda (tail state)
|
||
|
|
(folder (cons -1 tail)
|
||
|
|
state))
|
||
|
|
(fold-over-rows (- number-of-cols 1)
|
||
|
|
(lambda (tail state)
|
||
|
|
(folder (cons 1 tail)
|
||
|
|
state))
|
||
|
|
state)))))
|
||
|
|
|
||
|
|
; Fold over subsets of a given size.
|
||
|
|
(define fold-over-subs-of-size
|
||
|
|
(lambda (universe size folder state)
|
||
|
|
(let ((usize
|
||
|
|
(length universe)))
|
||
|
|
(if (< usize size)
|
||
|
|
state
|
||
|
|
(let _-*-
|
||
|
|
((size
|
||
|
|
size)
|
||
|
|
(universe
|
||
|
|
universe)
|
||
|
|
(folder
|
||
|
|
folder)
|
||
|
|
(csize
|
||
|
|
(- usize size))
|
||
|
|
(state
|
||
|
|
state))
|
||
|
|
(cond ((zero? csize)
|
||
|
|
(folder universe state))
|
||
|
|
((zero? size)
|
||
|
|
(folder '() state))
|
||
|
|
(else
|
||
|
|
(let ((first-u
|
||
|
|
(car universe))
|
||
|
|
(rest-u
|
||
|
|
(cdr universe)))
|
||
|
|
(_-*- size
|
||
|
|
rest-u
|
||
|
|
folder
|
||
|
|
(- csize 1)
|
||
|
|
(_-*- (- size 1)
|
||
|
|
rest-u
|
||
|
|
(lambda (tail state)
|
||
|
|
(folder (cons first-u tail)
|
||
|
|
state))
|
||
|
|
csize
|
||
|
|
state))))))))))
|
||
|
|
|
||
|
|
(define (main)
|
||
|
|
(run-benchmark
|
||
|
|
"matrix"
|
||
|
|
matrix-iters
|
||
|
|
(lambda (result)
|
||
|
|
(equal? result
|
||
|
|
'(((1 1 1 1 1) (1 1 1 1 -1) (1 1 1 -1 1)
|
||
|
|
(1 1 -1 -1 -1) (1 -1 1 -1 -1) (1 -1 -1 1 1))
|
||
|
|
((1 1 1 1 1) (1 1 1 1 -1) (1 1 1 -1 1)
|
||
|
|
(1 1 -1 1 -1) (1 -1 1 -1 -1) (1 -1 -1 1 1))
|
||
|
|
((1 1 1 1 1) (1 1 1 1 -1) (1 1 1 -1 1)
|
||
|
|
(1 1 -1 1 -1) (1 -1 1 -1 1) (1 -1 -1 1 1))
|
||
|
|
((1 1 1 1 1) (1 1 1 1 -1) (1 1 1 -1 1)
|
||
|
|
(1 1 -1 1 1) (1 -1 1 1 -1) (1 -1 -1 -1 1))
|
||
|
|
((1 1 1 1 1) (1 1 1 1 -1) (1 1 1 -1 1)
|
||
|
|
(1 1 -1 1 1) (1 -1 1 1 1) (1 -1 -1 -1 -1)))))
|
||
|
|
(lambda (number-of-cols inv-size) (lambda () (really-go number-of-cols inv-size)))
|
||
|
|
5
|
||
|
|
5))
|