diff options
Diffstat (limited to 'Test/textbook')
| -rw-r--r-- | Test/textbook/Answer | 4 | ||||
| -rw-r--r-- | Test/textbook/BQueue.bpl | 430 | ||||
| -rw-r--r-- | Test/textbook/runtest.bat | 2 |
3 files changed, 435 insertions, 1 deletions
diff --git a/Test/textbook/Answer b/Test/textbook/Answer index dace3eb3..42e41c5a 100644 --- a/Test/textbook/Answer +++ b/Test/textbook/Answer @@ -22,3 +22,7 @@ Boogie program verifier finished with 1 verified, 0 errors ------------------------------ TuringFactorial.bpl ---------------------
Boogie program verifier finished with 1 verified, 0 errors
+
+------------------------------ BQueue.bpl ---------------------
+
+Boogie program verifier finished with 8 verified, 0 errors
diff --git a/Test/textbook/BQueue.bpl b/Test/textbook/BQueue.bpl new file mode 100644 index 00000000..77f1efb3 --- /dev/null +++ b/Test/textbook/BQueue.bpl @@ -0,0 +1,430 @@ +// BQueue.bpl
+// A queue program specified in the style of dynamic frames.
+// Rustan Leino, Michal Moskal, and Wolfram Schulte, 2007.
+
+// ---------------------------------------------------------------
+
+type ref;
+const null: ref;
+
+type Field x;
+
+// this variable represents the heap; read its type as \forall \alpha. ref * Field \alpha --> \alpha
+type HeapType = <x>[ref, Field x]x;
+var H: HeapType;
+
+// every object has an 'alloc' field, which says whether or not the object has been allocated
+const unique alloc: Field bool;
+
+// for simplicity, we say that every object has one field representing its abstract value and one
+// field representing its footprint (aka frame aka data group).
+
+const unique abstractValue: Field Seq;
+const unique footprint: Field [ref]bool;
+
+// ---------------------------------------------------------------
+
+type T; // the type of the elements of the queue
+const NullT: T; // some value of type T
+
+// ---------------------------------------------------------------
+
+// Queue:
+const unique head: Field ref;
+const unique tail: Field ref;
+const unique mynodes: Field [ref]bool;
+// Node:
+const unique data: Field T;
+const unique next: Field ref;
+
+function ValidQueue(HeapType, ref) returns (bool);
+axiom (forall h: HeapType, q: ref ::
+ { ValidQueue(h, q) }
+ q != null && h[q,alloc] ==>
+ (ValidQueue(h, q) <==>
+ h[q,head] != null && h[h[q,head],alloc] &&
+ h[q,tail] != null && h[h[q,tail],alloc] &&
+ h[h[q,tail], next] == null &&
+ // The following line can be suppressed now that we have a ValidFootprint invariant
+ (forall o: ref :: { h[q,footprint][o] } o != null && h[q,footprint][o] ==> h[o,alloc]) &&
+ h[q,footprint][q] &&
+ h[q,mynodes][h[q,head]] && h[q,mynodes][h[q,tail]] &&
+ (forall n: ref :: { h[q,mynodes][n] }
+ h[q,mynodes][n] ==>
+ n != null && h[n,alloc] && ValidNode(h, n) &&
+ SubSet(h[n,footprint], h[q,footprint]) &&
+ !h[n,footprint][q] &&
+ (h[n,next] == null ==> n == h[q,tail])
+ ) &&
+ (forall n: ref :: { h[n,next] }
+ h[q,mynodes][n] ==>
+ (h[n,next] != null ==> h[q,mynodes][h[n,next]])
+ ) &&
+ h[q,abstractValue] == h[h[q,head],abstractValue]
+ ));
+
+// frame axiom for ValidQueue
+axiom (forall h0: HeapType, h1: HeapType, n: ref ::
+ { ValidQueue(h0,n), ValidQueue(h1,n) }
+ (forall<alpha> o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o]
+ ==> h0[o,f] == h1[o,f])
+ &&
+ (forall<alpha> o: ref, f: Field alpha :: o != null && h1[o,alloc] && h1[n,footprint][o]
+ ==> h0[o,f] == h1[o,f])
+ ==>
+ ValidQueue(h0,n) == ValidQueue(h1,n));
+
+function ValidNode(HeapType, ref) returns (bool);
+axiom (forall h: HeapType, n: ref ::
+ { ValidNode(h, n) }
+ n != null && h[n,alloc] ==>
+ (ValidNode(h, n) <==>
+ // The following line can be suppressed now that we have a ValidFootprint invariant
+ (forall o: ref :: { h[n,footprint][o] } o != null && h[n,footprint][o] ==> h[o,alloc]) &&
+ h[n,footprint][n] &&
+ (h[n,next] != null ==>
+ h[h[n,next],alloc] &&
+ SubSet(h[h[n,next], footprint], h[n,footprint]) &&
+ !h[h[n,next], footprint][n]) &&
+ (h[n,next] == null ==> EqualSeq(h[n,abstractValue], EmptySeq)) &&
+ (h[n,next] != null ==> EqualSeq(h[n,abstractValue],
+ Append(Singleton(h[h[n,next],data]), h[h[n,next],abstractValue])))
+ ));
+
+// frame axiom for ValidNode
+axiom (forall h0: HeapType, h1: HeapType, n: ref ::
+ { ValidNode(h0,n), ValidNode(h1,n) }
+ (forall<alpha> o: ref, f: Field alpha :: o != null && h0[o,alloc] && h0[n,footprint][o]
+ ==> h0[o,f] == h1[o,f])
+ &&
+ (forall<alpha> o: ref, f: Field alpha :: o != null && h1[o,alloc] && h1[n,footprint][o]
+ ==> h0[o,f] == h1[o,f])
+ ==>
+ ValidNode(h0,n) == ValidNode(h1,n));
+
+// ---------------------------------------------------------------
+
+procedure MakeQueue() returns (q: ref)
+ requires ValidFootprints(H);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySet(old(H), H, EmptySet);
+ ensures q != null && H[q,alloc];
+ ensures AllNewSet(old(H), H[q,footprint]);
+ ensures ValidQueue(H, q);
+ ensures Length(H[q,abstractValue]) == 0;
+{
+ var n: ref;
+
+ assume Fresh(H,q);
+ H[q,alloc] := true;
+
+ call n := MakeNode(NullT);
+ H[q,head] := n;
+ H[q,tail] := n;
+ H[q,mynodes] := SingletonSet(n);
+ H[q,footprint] := UnionSet(SingletonSet(q), H[n,footprint]);
+ H[q,abstractValue] := H[n,abstractValue];
+}
+
+procedure IsEmpty(q: ref) returns (isEmpty: bool)
+ requires ValidFootprints(H);
+ requires q != null && H[q,alloc] && ValidQueue(H, q);
+ ensures isEmpty <==> Length(H[q,abstractValue]) == 0;
+{
+ isEmpty := H[q,head] == H[q,tail];
+}
+
+procedure Enqueue(q: ref, t: T)
+ requires ValidFootprints(H);
+ requires q != null && H[q,alloc] && ValidQueue(H, q);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySet(old(H), H, old(H)[q,footprint]);
+ ensures DifferenceIsNew(old(H), old(H)[q,footprint], H[q,footprint]);
+ ensures ValidQueue(H, q);
+ ensures EqualSeq(H[q,abstractValue], Append(old(H)[q,abstractValue], Singleton(t)));
+{
+ var n: ref;
+
+ call n := MakeNode(t);
+
+ // foreach m in q.mynodes { m.footprint := m.footprint U n.footprint }
+ call BulkUpdateFootprint(H[q,mynodes], H[n,footprint]);
+ H[q,footprint] := UnionSet(H[q,footprint], H[n,footprint]);
+
+ // foreach m in q.mynodes { m.abstractValue := Append(m.abstractValue, Singleton(t)) }
+ call BulkUpdateAbstractValue(H[q,mynodes], t);
+ H[q,abstractValue] := H[H[q,head],abstractValue];
+
+ H[q,mynodes] := UnionSet(H[q,mynodes], SingletonSet(n));
+
+ H[H[q,tail], next] := n;
+ H[q,tail] := n;
+}
+
+procedure BulkUpdateFootprint(targetSet: [ref]bool, delta: [ref]bool);
+ requires ValidFootprints(H);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySetField(old(H), H, targetSet, footprint);
+ ensures (forall o: ref ::
+ o != null && old(H)[o,alloc] && targetSet[o]
+ ==> H[o,footprint] == UnionSet(old(H)[o,footprint], delta));
+
+procedure BulkUpdateAbstractValue(targetSet: [ref]bool, t: T);
+ requires ValidFootprints(H);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySetField(old(H), H, targetSet, abstractValue);
+ ensures (forall o: ref ::
+ o != null && old(H)[o,alloc] && targetSet[o]
+ ==> EqualSeq(H[o,abstractValue], Append(old(H)[o,abstractValue], Singleton(t))));
+
+procedure Front(q: ref) returns (t: T)
+ requires ValidFootprints(H);
+ requires q != null && H[q,alloc] && ValidQueue(H, q);
+ requires 0 < Length(H[q,abstractValue]);
+ ensures t == Index(H[q,abstractValue], 0);
+{
+ t := H[H[H[q,head], next], data];
+}
+
+procedure Dequeue(q: ref)
+ requires ValidFootprints(H);
+ requires q != null && H[q,alloc] && ValidQueue(H, q);
+ requires 0 < Length(H[q,abstractValue]);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySet(old(H), H, old(H)[q,footprint]);
+ ensures DifferenceIsNew(old(H), old(H)[q,footprint], H[q,footprint]);
+ ensures ValidQueue(H, q);
+ ensures EqualSeq(H[q,abstractValue], Drop(old(H)[q,abstractValue], 1));
+{
+ var n: ref;
+
+ n := H[H[q,head], next];
+ H[q,head] := n;
+ // we could also remove old(H)[q,head] from H[q,mynodes], and similar for the footprints
+ H[q,abstractValue] := H[n,abstractValue];
+}
+
+// --------------------------------------------------------------------------------
+
+procedure MakeNode(t: T) returns (n: ref)
+ requires ValidFootprints(H);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySet(old(H), H, EmptySet);
+ ensures n != null && H[n,alloc];
+ ensures AllNewSet(old(H), H[n,footprint]);
+ ensures ValidNode(H, n);
+ ensures H[n,data] == t && H[n,next] == null;
+{
+ assume Fresh(H,n);
+ H[n,alloc] := true;
+
+ H[n,next] := null;
+ H[n,data] := t;
+ H[n,footprint] := SingletonSet(n);
+ H[n,abstractValue] := EmptySeq;
+}
+
+// --------------------------------------------------------------------------------
+
+procedure Main(t: T, u: T, v: T)
+ requires ValidFootprints(H);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySet(old(H), H, EmptySet);
+{
+ var q0, q1: ref;
+ var w: T;
+
+ call q0 := MakeQueue();
+ call q1 := MakeQueue();
+
+ call Enqueue(q0, t);
+ call Enqueue(q0, u);
+
+ call Enqueue(q1, v);
+
+ assert Length(H[q0,abstractValue]) == 2;
+
+ call w := Front(q0);
+ assert w == t;
+ call Dequeue(q0);
+
+ call w := Front(q0);
+ assert w == u;
+
+ assert Length(H[q0,abstractValue]) == 1;
+ assert Length(H[q1,abstractValue]) == 1;
+}
+
+// --------------------------------------------------------------------------------
+
+procedure Main2(t: T, u: T, v: T, q0: ref, q1: ref)
+ requires q0 != null && H[q0,alloc] && ValidQueue(H, q0);
+ requires q1 != null && H[q1,alloc] && ValidQueue(H, q1);
+ requires DisjointSet(H[q0,footprint], H[q1,footprint]);
+ requires Length(H[q0,abstractValue]) == 0;
+
+ requires ValidFootprints(H);
+ modifies H;
+ ensures ValidFootprints(H);
+ ensures ModifiesOnlySet(old(H), H, UnionSet(old(H)[q0,footprint], old(H)[q1,footprint]));
+{
+ var w: T;
+
+ call Enqueue(q0, t);
+ call Enqueue(q0, u);
+
+ call Enqueue(q1, v);
+
+ assert Length(H[q0,abstractValue]) == 2;
+
+ call w := Front(q0);
+ assert w == t;
+ call Dequeue(q0);
+
+ call w := Front(q0);
+ assert w == u;
+
+ assert Length(H[q0,abstractValue]) == 1;
+ assert Length(H[q1,abstractValue]) == old(Length(H[q1,abstractValue])) + 1;
+}
+
+// ---------------------------------------------------------------
+
+// Helpful predicates used in specs
+
+function ModifiesOnlySet(oldHeap: HeapType, newHeap: HeapType, set: [ref]bool) returns (bool);
+axiom (forall oldHeap: HeapType, newHeap: HeapType, set: [ref]bool ::
+ { ModifiesOnlySet(oldHeap, newHeap, set) }
+ ModifiesOnlySet(oldHeap, newHeap, set) <==>
+ NoDeallocs(oldHeap, newHeap) &&
+ (forall<alpha> o: ref, f: Field alpha :: { newHeap[o,f] }
+ o != null && oldHeap[o,alloc] ==>
+ oldHeap[o,f] == newHeap[o,f] || set[o]));
+
+function ModifiesOnlySetField<alpha>(oldHeap: HeapType, newHeap: HeapType,
+ set: [ref]bool, field: Field alpha) returns (bool);
+axiom (forall<alpha> oldHeap: HeapType, newHeap: HeapType, set: [ref]bool, field: Field alpha ::
+ { ModifiesOnlySetField(oldHeap, newHeap, set, field) }
+ ModifiesOnlySetField(oldHeap, newHeap, set, field) <==>
+ NoDeallocs(oldHeap, newHeap) &&
+ (forall<beta> o: ref, f: Field beta :: { newHeap[o,f] }
+ o != null && oldHeap[o,alloc] ==>
+ oldHeap[o,f] == newHeap[o,f] || (set[o] && f == field)));
+
+function NoDeallocs(oldHeap: HeapType, newHeap: HeapType) returns (bool);
+axiom (forall oldHeap: HeapType, newHeap: HeapType ::
+ { NoDeallocs(oldHeap, newHeap) }
+ NoDeallocs(oldHeap, newHeap) <==>
+ (forall o: ref :: { newHeap[o,alloc] }
+ o != null && oldHeap[o,alloc] ==> newHeap[o,alloc]));
+
+function AllNewSet(oldHeap: HeapType, set: [ref]bool) returns (bool);
+axiom (forall oldHeap: HeapType, set: [ref]bool ::
+ { AllNewSet(oldHeap, set) }
+ AllNewSet(oldHeap, set) <==>
+ (forall o: ref :: { oldHeap[o,alloc] }
+ o != null && set[o] ==> !oldHeap[o,alloc]));
+
+function DifferenceIsNew(oldHeap: HeapType, oldSet: [ref]bool, newSet: [ref]bool) returns (bool);
+axiom (forall oldHeap: HeapType, oldSet: [ref]bool, newSet: [ref]bool ::
+ { DifferenceIsNew(oldHeap, oldSet, newSet) }
+ DifferenceIsNew(oldHeap, oldSet, newSet) <==>
+ (forall o: ref :: { oldHeap[o,alloc] }
+ o != null && !oldSet[o] && newSet[o] ==> !oldHeap[o,alloc]));
+
+function ValidFootprints(h: HeapType) returns (bool);
+axiom (forall h: HeapType ::
+ { ValidFootprints(h) }
+ ValidFootprints(h) <==>
+ (forall o: ref, r: ref :: { h[o,footprint][r] }
+ o != null && h[o,alloc] && r != null && h[o,footprint][r] ==> h[r,alloc]));
+
+function Fresh(h: HeapType, o: ref) returns (bool);
+axiom (forall h: HeapType, o: ref ::
+ { Fresh(h,o) }
+ Fresh(h,o) <==>
+ o != null && !h[o,alloc] && h[o,footprint] == SingletonSet(o));
+
+// ---------------------------------------------------------------
+
+const EmptySet: [ref]bool;
+axiom (forall o: ref :: { EmptySet[o] } !EmptySet[o]);
+
+function SingletonSet(ref) returns ([ref]bool);
+axiom (forall r: ref :: { SingletonSet(r) } SingletonSet(r)[r]);
+axiom (forall r: ref, o: ref :: { SingletonSet(r)[o] } SingletonSet(r)[o] <==> r == o);
+
+function UnionSet([ref]bool, [ref]bool) returns ([ref]bool);
+axiom (forall a: [ref]bool, b: [ref]bool, o: ref :: { UnionSet(a,b)[o] }
+ UnionSet(a,b)[o] <==> a[o] || b[o]);
+
+function SubSet([ref]bool, [ref]bool) returns (bool);
+axiom(forall a: [ref]bool, b: [ref]bool :: { SubSet(a,b) }
+ SubSet(a,b) <==> (forall o: ref :: {a[o]} {b[o]} a[o] ==> b[o]));
+
+function EqualSet([ref]bool, [ref]bool) returns (bool);
+axiom(forall a: [ref]bool, b: [ref]bool :: { EqualSet(a,b) }
+ EqualSet(a,b) <==> (forall o: ref :: {a[o]} {b[o]} a[o] <==> b[o]));
+
+function DisjointSet([ref]bool, [ref]bool) returns (bool);
+axiom (forall a: [ref]bool, b: [ref]bool :: { DisjointSet(a,b) }
+ DisjointSet(a,b) <==> (forall o: ref :: {a[o]} {b[o]} !a[o] || !b[o]));
+
+// ---------------------------------------------------------------
+
+// Sequence of T
+type Seq;
+
+function Length(Seq) returns (int);
+axiom (forall s: Seq :: { Length(s) } 0 <= Length(s));
+
+const EmptySeq: Seq;
+axiom Length(EmptySeq) == 0;
+axiom (forall s: Seq :: { Length(s) } Length(s) == 0 ==> s == EmptySeq);
+
+function Singleton(T) returns (Seq);
+axiom (forall t: T :: { Length(Singleton(t)) } Length(Singleton(t)) == 1);
+
+function Append(Seq, Seq) returns (Seq);
+axiom (forall s0: Seq, s1: Seq :: { Length(Append(s0,s1)) }
+ Length(Append(s0,s1)) == Length(s0) + Length(s1));
+
+function Index(Seq, int) returns (T);
+axiom (forall t: T :: { Index(Singleton(t), 0) } Index(Singleton(t), 0) == t);
+axiom (forall s0: Seq, s1: Seq, n: int :: { Index(Append(s0,s1), n) }
+ (n < Length(s0) ==> Index(Append(s0,s1), n) == Index(s0, n)) &&
+ (Length(s0) <= n ==> Index(Append(s0,s1), n) == Index(s1, n - Length(s0))));
+
+function EqualSeq(Seq, Seq) returns (bool);
+axiom (forall s0: Seq, s1: Seq :: { EqualSeq(s0,s1) }
+ EqualSeq(s0,s1) <==>
+ Length(s0) == Length(s1) &&
+ (forall j: int :: { Index(s0,j) } { Index(s1,j) }
+ 0 <= j && j < Length(s0) ==> Index(s0,j) == Index(s1,j)));
+
+function Take(s: Seq, howMany: int) returns (Seq);
+axiom (forall s: Seq, n: int :: { Length(Take(s,n)) }
+ 0 <= n ==>
+ (n <= Length(s) ==> Length(Take(s,n)) == n) &&
+ (Length(s) < n ==> Length(Take(s,n)) == Length(s)));
+axiom (forall s: Seq, n: int, j: int :: { Index(Take(s,n), j) }
+ 0 <= j && j < n && j < Length(s) ==>
+ Index(Take(s,n), j) == Index(s, j));
+
+function Drop(s: Seq, howMany: int) returns (Seq);
+axiom (forall s: Seq, n: int :: { Length(Drop(s,n)) }
+ 0 <= n ==>
+ (n <= Length(s) ==> Length(Drop(s,n)) == Length(s) - n) &&
+ (Length(s) < n ==> Length(Drop(s,n)) == 0));
+axiom (forall s: Seq, n: int, j: int :: { Index(Drop(s,n), j) }
+ 0 <= n && 0 <= j && j < Length(s)-n ==>
+ Index(Drop(s,n), j) == Index(s, j+n));
+
+// ---------------------------------------------------------------
diff --git a/Test/textbook/runtest.bat b/Test/textbook/runtest.bat index b747312d..f43e54d9 100644 --- a/Test/textbook/runtest.bat +++ b/Test/textbook/runtest.bat @@ -7,7 +7,7 @@ REM ====================== REM ====================== Examples written in Boogie
REM ======================
for %%f in (Find.bpl DutchFlag.bpl Bubble.bpl DivMod.bpl McCarthy-91.bpl
- TuringFactorial.bpl) do (
+ TuringFactorial.bpl BQueue.bpl) do (
echo.
echo ------------------------------ %%f ---------------------
%BPLEXE% %* %%f
|