ada - 使用 SPARK 证明选择排序算法

Question

我试图证明我在 Ada 中实现的选择排序是正确的。我已经尝试了一些循环不变量，但使用 gnatprove 只能证明内部循环的不变量:

package body Selection with SPARK_Mode is

procedure Sort (A : in out Arr) is
    I: Integer := A'First;
    J: Integer;
    Min_Idx: Integer;
    Tmp: Integer;
begin
    while I < A'Last loop

        pragma Loop_Invariant
            (Sorted( A (A'First .. I) ));

        Min_Idx := I;
        J := I + 1;

        while J <= A'Last loop

            if A (J) < A (Min_Idx) then
                Min_Idx := J;
            end if;

            pragma Loop_Invariant
                (for all Index in I .. J => (A (Min_Idx) <= A (Index)));

            J := J + 1;
        end loop;

        Tmp := A (Min_Idx);
        A (Min_Idx) := A (I);
        A (I) := Tmp;

        I := I + 1;

    end loop;
end Sort;
end Selection;

package Selection with SPARK_Mode is
    type Arr is array (Integer range <>) of Integer;

    function Sorted (A : Arr) return Boolean
    is (for all I in A'First .. A'Last - 1 => A(I) <= A(I + 1))
    with
        Ghost,
        Pre => A'Last > Integer'First;

    procedure Sort (A : in out Arr)
    with
        Pre => A'First in Integer'First + 1 .. Integer'Last - 1 and
            A'Last in Integer'First + 1 .. Integer'Last - 1,
        Post => Sorted (A);

end Selection;

Gnatprove 告诉我selection.adb:15:14: medium: 循环不变量可能不会被任意迭代保留，无法证明 Sorted( A (A'First..I)) (例如，当 A = (-1 => 0, 0 => 0，其他 => 1) 和 A'First = -1) 您有解决此问题的想法吗？

Answer 1

我稍微修改了例程，向外层循环添加了两个循环不变量，并将它们全部移动到循环的末尾。两个额外的循环不变量声明正在处理的元素总是大于或等于那些已经处理过的元素，小于或等于那些尚未处理的元素。

我还更改了 Sorted幽灵函数/谓词仅将量化表达式应用于长度大于 1 的数组。这是为了防止溢出问题。对于长度为 0 或 1 的数组，函数返回 True根据定义为 (if False then <bool_expr>)是True (或 vacuously true，如果我没记错的话)。

所有 VC 都可以用 gnatprove 来解除/证明随 GNAT/SPARK CE 2020 一起提供的 1 级:

$ gnatprove -Pdefault.gpr -j0 --report=all --level=1

selection.ads

package Selection with SPARK_Mode is
   
   type Arr is array (Integer range <>) of Integer;

   function Sorted (A : Arr) return Boolean is
     (if A'Length > 1 then
         (for all I in A'First + 1 .. A'Last => A (I - 1) <= A (I)))
       with Ghost;
         
   procedure Sort (A : in out Arr)
     with Post => Sorted (A);

end Selection;

selection.adb

package body Selection with SPARK_Mode is
   
   ----------
   -- Sort --
   ----------
   
   procedure Sort (A : in out Arr) is
      M : Integer;      
   begin
      if A'Length > 1 then
         for I in A'First .. A'Last - 1 loop
         
            M := I;
         
            for J in I + 1 .. A'Last loop            
               if A (J) <= A (M) then
                  M := J;
               end if;
            
               pragma Loop_Invariant (M in I .. J);            
               pragma Loop_Invariant (for all K in I .. J => A (M) <= A (K));
            
            end loop;
            
            declare
               T : constant Integer := A (I);
            begin
               A (I) := A (M);
               A (M) := T;
            end;
                  
            --  Linear incremental sorting in ascending order.
            pragma Loop_Invariant (for all K in A'First .. I => A (K) <= A (I));
            pragma Loop_Invariant (for all K in I .. A'Last  => A (I) <= A (K));
         
            pragma Loop_Invariant (Sorted (A (A'First .. I)));         

         end loop;
      end if;
   end Sort;
   
end Selection;