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VerifyThis 2017: Maximum-sum submatrix

solution to VerifyThis 2017 challenge 2


Authors: Jean-Christophe Filliâtre

Topics: Matrices

Tools: Why3

References: VerifyThis @ ETAPS 2017

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VerifyThis @ ETAPS 2017 competition Challenge 2: Maximum-sum submatrix

See https://formal.iti.kit.edu/ulbrich/verifythis2017/

Author: Jean-Christophe FilliĆ¢tre (CNRS)

(* note: this is a 2D-version of maximum-sum subarray, for which several
   verified implementations can be found in maximum_subarray.mlw,
   including Kadane's linear algorithm *)

module Kadane2D

  use int.Int
  use ref.Refint
  use int.Sum
  use array.Array
  use matrix.Matrix

  (* maximum-sum subarray problem is assumed *)

  function array_sum (a: array int) (l h: int) : int
    = sum (fun i -> a[i]) l h

  val maximum_subarray (a: array int) : (s: int, lo: int, hi: int)
    ensures { 0 <= lo <= hi <= length a /\ s = array_sum a lo hi /\
              forall l h. 0 <= l <= h <= length a -> s >= array_sum a l h }

  (* sum of a submatrix *)

  function col (m: matrix int) (j i: int) : int = m.elts i j

  function matrix_sum (m: matrix int) (rl rh cl ch: int) : int
    = sum (fun j -> sum (col m j) rl rh) cl ch

  let maximum_submatrix (m: matrix int) :
      (s: int, rlo: int, rhi: int, clo: int, chi: int)
    ensures { (* this is a legal submatrix *)
              0 <= rlo <= rhi <= rows m /\
              0 <= clo <= chi <= columns m /\
              (* s is its sum *)
              s = matrix_sum m rlo rhi clo chi /\
              (* and it is maximal *)
              forall rl rh. 0 <= rl <= rh <= rows m ->
              forall cl ch. 0 <= cl <= ch <= columns m ->
              s >= matrix_sum m rl rh cl ch }
  = let a = Array.make m.columns 0 in
    let maxsum = ref 0 in
    let rlo = ref 0 in
    let rhi = ref 0 in
    let clo = ref 0 in
    let chi = ref 0 in
    for rl = 0 to rows m - 1 do
      invariant { 0 <= !rlo <= !rhi <= rows m }
      invariant { 0 <= !clo <= !chi <= columns m }
      invariant { !maxsum = matrix_sum m !rlo !rhi !clo !chi >= 0 }
      invariant { forall rl' rh. 0 <= rl' < rl -> rl' <= rh <= rows m ->
                  forall cl ch. 0 <= cl <= ch <= columns m ->
                  !maxsum >= matrix_sum m rl' rh cl ch }
      fill a 0 (columns m) 0;
      for rh = rl + 1 to rows m do
        invariant { 0 <= !rlo <= !rhi <= rows m }
        invariant { 0 <= !clo <= !chi <= columns m }
        invariant { !maxsum = matrix_sum m !rlo !rhi !clo !chi >= 0 }
        invariant { forall rl' rh'. 0 <= rl' <= rh' <= rows m ->
                    (rl' < rl \/ rl' = rl /\ rh' < rh) ->
                    forall cl ch. 0 <= cl <= ch <= columns m ->
                    !maxsum >= matrix_sum m rl' rh' cl ch }
        invariant { forall j. 0 <= j < columns m ->
                    a[j] = sum (col m j) rl (rh - 1) }
        (* update array a *)
        for c = 0 to columns m -1 do
          invariant { forall j. 0 <= j < c ->
                      a[j] = sum (col m j) rl rh }
          invariant { forall j. c <= j < columns m ->
                      a[j] = sum (col m j) rl (rh - 1) }
          a[c] <- a[c] + get m (rh - 1) c
        done;
        (* then use Kadane algorithme on array a *)
        let sum, lo, hi = maximum_subarray a in
        assert { sum = matrix_sum m rl rh lo hi };
        assert { forall cl ch. 0 <= cl <= ch <= columns m ->
                 sum >= matrix_sum m rl rh cl ch
                 by array_sum a cl ch = matrix_sum m rl rh cl ch };
        (* update the maximum if needed *)
        if sum > !maxsum then begin
          maxsum := sum;
          rlo := rl; rhi := rh;
          clo := lo; chi := hi
        end
      done;
    done;
    !maxsum, !rlo, !rhi, !clo, !chi

end

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Why3 Proof Results for Project "verifythis_2017_maximum_sum_submatrix"

Theory "verifythis_2017_maximum_sum_submatrix.Kadane2D": fully verified

ObligationsAlt-Ergo 2.0.0CVC4 1.5Z3 4.6.0
VC for maximum_submatrix---------
split_goal_right
array creation size0.00------
loop invariant init0.01------
loop invariant init0.00------
loop invariant init0.01------
loop invariant init0.00------
precondition0.01------
loop invariant init0.01------
loop invariant init0.01------
loop invariant init0.00------
loop invariant init------0.01
loop invariant init0.01------
loop invariant init0.01------
loop invariant init0.01------
index in matrix bounds0.01------
index in array bounds0.01------
index in array bounds0.00------
loop invariant preservation0.04------
loop invariant preservation0.01------
assertion1.76------
assertion---------
split_goal_right
assertion0.34------
VC for maximum_submatrix0.01------
loop invariant preservation0.01------
loop invariant preservation0.01------
loop invariant preservation0.02------
loop invariant preservation---0.05---
loop invariant preservation0.01------
loop invariant preservation0.01------
loop invariant preservation0.00------
loop invariant preservation0.01------
loop invariant preservation---0.05---
loop invariant preservation0.01------
out of loop bounds0.01------
loop invariant preservation0.01------
loop invariant preservation0.01------
loop invariant preservation0.01------
loop invariant preservation---0.030.01
out of loop bounds0.00------
postcondition---------
split_goal_right
VC for maximum_submatrix0.00------
VC for maximum_submatrix0.01------
VC for maximum_submatrix0.00------
VC for maximum_submatrix0.01------
VC for maximum_submatrix0.00------
VC for maximum_submatrix0.01------
VC for maximum_submatrix0.00------
VC for maximum_submatrix------0.02
out of loop bounds0.01------