Question

我正在研究用验证创建的几种语言的语言比较（Whiley，Dafny和Frama-C等）我给出了一个函数示例，该函数将一个数组的区域复制到另一个具有不同位置的数组在目标数组中。我提出的规范在Dafny中看起来像这样：

method copy( src: array<int>, sStart: nat, dest: array<int>, dStart: nat, len: nat)
    returns (r: array<int>)
  // both arrays cannot be null
   requires dest != null && src != null
  // Source array must contain enough elements to be copied
   requires src.Length >= sStart + len
  // Destination array must have enough space for copied elements
   requires dest.Length >= dStart + len
  // Result is same size as dest
  ensures r != null
  ensures r.Length == dest.Length
  // All elements before copied region are same
   ensures r[..dStart] == dest[..dStart]
  // All elements above copied region are same
   ensures r[dStart + len..] == dest[dStart + len..]
  // All elements in copied region match src
   ensures forall k: nat :: k < len ==> r[dStart + k] == src[sStart + k]

{
    if len == 0 { return dest; }
    assert len > 0;
    var i: nat := 0;
    r := new int[dest.Length];
    while (i < r.Length)
      invariant i <= r.Length
      decreases r.Length - i
      invariant r.Length == dest.Length
      invariant forall k: nat :: k < i ==> r[k] == dest[k]
    {
        r[i] := dest[i];
        i := i + 1;
    }
    assume r[..] == dest[..];
    i := 0;
    while (i < len)
      invariant i <= len
      decreases len - i
      invariant r.Length == dest.Length
      invariant r.Length >= dStart + i
      invariant src.Length >= sStart + i
      invariant r[..dStart] == dest[..dStart]
      invariant r[(dStart + len)..] == dest[(dStart + len)..]
      invariant forall k: nat :: k < i ==> r[dStart + k] == src[sStart + k]
    {
        r[dStart + i] := src[sStart + i];
        i := i + 1;
    }
}

在上面的第二个while循环中，可能有一些不需要的不变量，因为我试图涵盖我能想到的一切。但是，是的，这并没有验证，我很困惑为什么......

Dafny/copy.dfy(35,4): Error BP5003: A postcondition might not hold on this return path.
Dafny/copy.dfy(17,11): Related location: This is the postcondition that might not hold.
Execution trace:
    (0,0): anon0
    (0,0): anon19_Else
    Dafny/copy.dfy(24,5): anon20_LoopHead
    (0,0): anon20_LoopBody
    Dafny/copy.dfy(24,5): anon21_Else
    (0,0): anon23_Then
    Dafny/copy.dfy(35,5): anon24_LoopHead
    (0,0): anon24_LoopBody
    Dafny/copy.dfy(35,5): anon25_Else
    (0,0): anon27_Then
Dafny/copy.dfy(43,16): Error BP5005: This loop invariant might not be maintained by the loop.
Execution trace:
    (0,0): anon0
    (0,0): anon19_Else
    Dafny/copy.dfy(24,5): anon20_LoopHead
    (0,0): anon20_LoopBody
    Dafny/copy.dfy(24,5): anon21_Else
    (0,0): anon23_Then
    Dafny/copy.dfy(35,5): anon24_LoopHead
    (0,0): anon24_LoopBody
    Dafny/copy.dfy(35,5): anon25_Else
    Dafny/copy.dfy(35,5): anon27_Else

Dafny program verifier finished with 1 verified, 2 errors

Answer 1

您可以通过添加一个不变的

来验证它

invariant r[dStart .. dStart + i] == src[sStart .. sStart + i]

如：

while (i < len)
  invariant i <= len
  decreases len - i
  invariant r.Length == dest.Length
  invariant r.Length >= dStart + i
  invariant src.Length >= sStart + i
  invariant r[..dStart] == dest[..dStart]
  invariant r[(dStart + len)..] == dest[(dStart + len)..]
  invariant r[dStart .. dStart + i] == src[sStart .. sStart + i]
  invariant forall k: nat :: k < i ==> r[dStart + k] == src[sStart + k]

顺便说一下，如果你想

，我认为你可以删除许多不变量

method copy( src: array<int>, sStart: nat, dest: array<int>, dStart: nat, len: nat)
    returns (r: array<int>)
  // both arrays cannot be null
   requires dest != null && src != null
  // Source array must contain enough elements to be copied
   requires src.Length >= sStart + len
  // Destination array must have enough space for copied elements
   requires dest.Length >= dStart + len
  // Result is same size as dest
  ensures r != null
  ensures r.Length == dest.Length
  // All elements before copied region are same
   ensures r[..dStart] == dest[..dStart]
  // All elements above copied region are same
   ensures r[dStart + len..] == dest[dStart + len..]
  // All elements in copied region match src
   ensures forall k: nat :: k < len ==> r[dStart + k] == src[sStart + k]

{
    if len == 0 { return dest; }
    var i: nat := 0;
    r := new int[dest.Length];
    while (i < r.Length)
      invariant i <= r.Length
      invariant r[..i] == dest[..i]
    {
        r[i] := dest[i];
        i := i + 1;
    }

    i := 0;
    while (i < len)
      invariant i <= len
      invariant r[..dStart] == dest[..dStart]
      invariant r[(dStart + len)..] == dest[(dStart + len)..]
      invariant r[dStart .. dStart + i] == src[sStart .. sStart + i]
      {
        r[dStart + i] := src[sStart + i];
        i := i + 1;
    }
}

Dafny：复制数组区域方法验证

1 个答案: