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/*
Rustan Leino, 5 Oct 2011
COST Verification Competition, Challenge 2: Maximum in a tree
http://foveoos2011.cost-ic0701.org/verification-competition
Given: A non-empty binary tree, where every node carries an integer.
Implement and verify a program that computes the maximum of the values
in the tree.
Please base your program on the following data structure signature:
public class Tree {
int value;
Tree left;
Tree right;
}
You may represent empty trees as null references or as you consider
appropriate.
*/
// Remarks:
// The specification of this program uses the common dynamic-frames idiom in Dafny: the
// ghost field 'Contents' stores the abstract value of an object, the ghost field 'Repr'
// stores the set of (references to) objects that make up the representation of the object
// (which in this case is the Tree itself plus the 'Repr' sets of the left and right
// subtrees), and a function 'Valid()' that returns 'true' when an object is in a
// consistent state (that is, when an object satisfies the "class invariant").
// The design I used was to represent an empty tree as a Tree object whose left and
// right pointers point to the object iself. This is convenient, because it lets
// clients of Tree and the implementation of Tree always use non-null pointers to
// Tree objects.
// What needs to be human-trusted about this program is that the 'requires' and
// 'ensures' clauses (that is, the pre- and postconditions, respectively) of
// 'ComputeMax' are correct. And, since the specification talks about the ghost
// variable 'Contents', one also needs to trust that the 'Valid()' function
// constrains 'Contents' in a way that a human thinks matches the intuitive
// definition of what the contents of a tree is.
// To give a taste of that the 'Valid()' function does not over-constrain the
// object, I have included two instance constructors, 'Empty()' and 'Node(...)'.
// To take this a step further, one could also write a 'Main' method that
// builds somme tree and then calls 'ComputeMax', but I didn't do that here.
// About Dafny:
// As always (when it is successful), Dafny verifies that the program does not
// cause any run-time errors (like array index bounds errors), that the program
// terminates, that expressions and functions are well defined, and that all
// specifications are satisfied. The language prevents type errors by being type
// safe, prevents dangling pointers by not having an "address-of" or "deallocate"
// operation (which is accommodated at run time by a garbage collector), and
// prevents arithmetic overflow errors by using mathematical integers (which
// is accommodated at run time by using BigNum's). By proving that programs
// terminate, Dafny proves that a program's time usage is finite, which implies
// that the program's space usage is finite too. However, executing the
// program may fall short of your hopes if you don't have enough time or
// space; that is, the program may run out of space or may fail to terminate in
// your lifetime, because Dafny does not prove that the time or space needed by
// the program matches your execution environment. The only input fed to
// the Dafny verifier/compiler is the program text below; Dafny then automatically
// verifies and compiles the program (for this program in less than 2.5 seconds)
// without further human intervention.
class Tree {
// an empty tree is represented by a Tree object with left==this==right
var value: int;
var left: Tree;
var right: Tree;
ghost var Contents: seq<int>;
ghost var Repr: set<object>;
function Valid(): bool
reads this, Repr;
{
this in Repr &&
left != null && right != null &&
((left == this == right && Contents == []) ||
(left in Repr && left.Repr <= Repr && this !in left.Repr &&
right in Repr && right.Repr <= Repr && this !in right.Repr &&
left.Valid() && right.Valid() &&
Contents == left.Contents + [value] + right.Contents))
}
function method IsEmpty(): bool
requires Valid();
reads Repr;
ensures IsEmpty() <==> Contents == [];
{
left == this
}
constructor Empty()
modifies this;
ensures Valid() && Contents == [];
{
left, right := this, this;
Contents := [];
Repr := {this};
}
constructor Node(lft: Tree, val: int, rgt: Tree)
requires lft != null && rgt != null && lft.Valid() && rgt.Valid();
requires this !in lft.Repr && this !in rgt.Repr;
modifies this;
ensures Valid() && Contents == lft.Contents + [val] + rgt.Contents;
{
left, value, right := lft, val, rgt;
Contents := lft.Contents + [val] + rgt.Contents;
Repr := lft.Repr + {this} + rgt.Repr;
}
method ComputeMax() returns (mx: int)
requires Valid() && !IsEmpty();
ensures forall x :: x in Contents ==> x <= mx;
ensures exists x :: x in Contents && x == mx;
decreases Repr;
{
mx := value;
if (!left.IsEmpty()) {
var m := left.ComputeMax();
mx := if mx < m then m else mx;
}
if (!right.IsEmpty()) {
var m := right.ComputeMax();
mx := if mx < m then m else mx;
}
}
}
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