WiSE.implem.bugfinder
From Coq Require Import List Lia String ZArith.
From WiSE Require Import streams imp symbolic.symex.
Import ListNotations.
From WiSE Require Import streams imp symbolic.symex.
Import ListNotations.
Implementation of a naive bug finder for IMP
A status corresponds to the current state of the bugfinding loop
Inductive status :=
| BugFound (s : sym_state)
| Pending
| Finished.
Definition then_do q '(((path, env), p) : sym_state) :=
((path, env), Seq p q).
| BugFound (s : sym_state)
| Pending
| Finished.
Definition then_do q '(((path, env), p) : sym_state) :=
((path, env), Seq p q).
Executing a task according to sym_step
Fixpoint expand path env prog : list sym_state :=
match prog with
| Skip => []
| Aff x e =>
[(path, sym_update env x (sym_aeval env e), Skip)]
| Err => []
| Ite b p1 p2 =>
[
(Band path (sym_beval env b), env, p1);
(Band path (Bnot (sym_beval env b)), env, p2)
]
| Loop b p =>
[
(Band path (sym_beval env b), env, Seq p (Loop b p));
(Band path (Bnot (sym_beval env b)), env, Skip)
]
| Seq Skip p2 => [(path, env, p2)]
| Seq p1 p2 =>
List.map (then_do p2) (expand path env p1)
end.
match prog with
| Skip => []
| Aff x e =>
[(path, sym_update env x (sym_aeval env e), Skip)]
| Err => []
| Ite b p1 p2 =>
[
(Band path (sym_beval env b), env, p1);
(Band path (Bnot (sym_beval env b)), env, p2)
]
| Loop b p =>
[
(Band path (sym_beval env b), env, Seq p (Loop b p));
(Band path (Bnot (sym_beval env b)), env, Skip)
]
| Seq Skip p2 => [(path, env, p2)]
| Seq p1 p2 =>
List.map (then_do p2) (expand path env p1)
end.
Bugfinding loop: at every iteration, a task is choosen and then executed.
If the execution results in an error, a BugFound message is
emitted. If executing the task terminates in a state s without error,
a message Clear s is emitted. If executing the task generates a list l of subtasks,
then the loop signals that further computations are pending and add l to the worklist.
Finally, if the task list is empty, the loop emits the token Finished continuously.
CoFixpoint run (l : tasks) : stream (option sym_state) :=
match l with
| [] => scons None (run [])
| ((path, env), prog)::next =>
let next_next := expand path env prog in
scons (Some (path, env, prog)) (run (next ++ next_next))
end.
match l with
| [] => scons None (run [])
| ((path, env), prog)::next =>
let next_next := expand path env prog in
scons (Some (path, env, prog)) (run (next ++ next_next))
end.
run_n predicts the tasks list after n iteration
Fixpoint run_n (n : nat) (tasks : list sym_state) : list sym_state :=
match n, tasks with
| 0, _ => tasks
| S n, [] => []
| S n, (path, env, prog)::next =>
let next_next := expand path env prog in
run_n n (next ++ next_next)
end.
Definition display (st : option sym_state) : status :=
match st with
| None => Finished
| Some (path, env, prog) =>
if is_error prog then BugFound (path, env, prog)
else Pending
(* else if is_skip prog then Clear (path, env, prog)
else Pending (path, env, prog) *)
end.
Definition init (p : IMP) : list sym_state :=
[((Bcst true, id), p)].
Definition find_bugs (p : IMP) : stream status :=
map display (run (init p)).
Definition init_assuming (p : IMP) (cond : bexpr) :=
[(cond, id, p)].
Definition find_bugs_assuming (p : IMP) (cond : bexpr) : stream status :=
map display (run (init_assuming p cond)).
match n, tasks with
| 0, _ => tasks
| S n, [] => []
| S n, (path, env, prog)::next =>
let next_next := expand path env prog in
run_n n (next ++ next_next)
end.
Definition display (st : option sym_state) : status :=
match st with
| None => Finished
| Some (path, env, prog) =>
if is_error prog then BugFound (path, env, prog)
else Pending
(* else if is_skip prog then Clear (path, env, prog)
else Pending (path, env, prog) *)
end.
Definition init (p : IMP) : list sym_state :=
[((Bcst true, id), p)].
Definition find_bugs (p : IMP) : stream status :=
map display (run (init p)).
Definition init_assuming (p : IMP) (cond : bexpr) :=
[(cond, id, p)].
Definition find_bugs_assuming (p : IMP) (cond : bexpr) : stream status :=
map display (run (init_assuming p cond)).
Examples
Section Examples.
Section Simple.
Example my_prog :=
Ite
(Beq (Var "x"%string) (Cst 0%Z))
Err
Skip.
Compute (peek 10 (find_bugs my_prog)).
Example a := (Var "a"%string).
Example b := (Var "b"%string).
Example olda := (Var "olda"%string).
Example oldb := (Var "oldb"%string).
Example trivial_assertion :=
Seq
(Aff "a"%string b)
(Assert (Bnot (Beq a b))).
Compute (peek 10 (find_bugs trivial_assertion)).
End Simple.
Section Gcd.
(*
function gcd(a, b)
while a ≠ b
if a <= b
b := b − a
else
a := a − b
return a
*)
Example gcd :=
Loop
(Bnot (Beq a b))
(Ite
(Ble a b)
(Aff "b"%string (Sub b a))
(Aff "a"%string (Sub a b))).
Example init_store (x : ident) : Z :=
match x with
| "a"%string => 9
| "b"%string => 6
| _ => 0
end.
(* gcd(9, 6) --> (3, 3) *)
Example execute_gcd:
exec
init_store
gcd
(update (update init_store "a"%string 3%Z) "b"%string 3%Z).
Proof.
unfold gcd.
(* First Iteration *)
apply exec_loop_true with
(update init_store "a"%string 3%Z);
auto.
1: (* correctness of provided store *)
apply exec_Ite_false; auto.
apply exec_Aff.
auto.
(* Second Iteration *)
apply exec_loop_true with
(update (update init_store "a"%string 3%Z) "b"%string 3%Z); auto.
1: (* correctness of provided store *)
apply exec_Ite_true; auto.
apply exec_Aff.
auto.
(* Termination *)
apply exec_loop_false.
auto.
Qed.
Example a0 := (Var "a0"%string).
Example b0 := (Var "b0"%string).
(* Now, a correct version of `gcd` with assertions. *)
(*
function gcd(a, b)
while a ≠ b
if a <= b
b := b − a
else
a := a − b
assert a + b + 1 <= \old(a) + \old(b)
return a
*)
Example gcd_assertions :=
Loop
(Bnot (Beq a b))
(Seq
(Aff "a0"%string a)
(Seq
(Aff "b0"%string b)
(Seq
(Ite
(Ble a b)
(Aff "b"%string (Sub b a))
(Aff "a"%string (Sub a b)))
(Assert (Ble (Add (Add a b) (Cst 1%Z)) (Add a0 b0)))))).
Compute (peek 15 (find_bugs gcd_assertions)).
Example init_store' (x : ident) : Z :=
match x with
| "a"%string => 4
| "b"%string => 2
| _ => 0
end.
(* gcd(4, 2) --> (2, 2) *)
Example execute_gcd_noerr:
exec
init_store'
gcd_assertions
(update
(update
(update init_store' "a0"%string 4)
"b0"%string 2)
"a"%string 2).
unfold gcd_assertions.
(* First Iteration *)
apply exec_loop_true with
(update
(update
(update init_store' "a0"%string 4)
"b0"%string 2)
"a"%string 2);
auto.
1: (* correctness of provided store *)
(* `a0 = a` *)
apply exec_Seq with (update init_store' "a0"%string 4%Z);
try constructor; auto.
(* `b0 = b` *)
apply exec_Seq with
(update (update init_store' "a0"%string 4%Z) "b0"%string 2%Z);
try constructor; auto.
(* if... ; ... *)
apply exec_Seq with
(update
(update
(update init_store' "a0"%string 4%Z)
"b0"%string 2%Z)
"a"%string 2%Z).
1: (* if ... *)
apply exec_Ite_false; auto.
apply exec_Aff.
auto.
(* assert ... *)
unfold Assert.
apply exec_Ite_true; auto.
apply exec_Skip.
(* Termination *)
apply exec_loop_false.
auto.
Qed.
(* Now, a *buggy* version of `gcd` with assertions. *)
(*
function gcd_buggy(a, b)
while a ≠ b
if a <= b
b := b − a
else
a := a + b // Should be `a - b`
assert a + b + 1 <= \old(a) + \old(b)
return a
*)
Example gcd_buggy :=
Loop
(Bnot (Beq a b))
(Seq
(Aff "a0"%string a)
(Seq
(Aff "b0"%string b)
(Seq
(Ite
(Ble a b)
(Aff "b"%string (Sub b a))
(Aff "a"%string (Add a b)))
(Assert (Ble (Add (Add a b) (Cst 1%Z)) (Add a0 b0)))))).
(* gcd_buggy(4, 2) --> Err *)
Example execute_gcd_err:
forall s, ~ exec init_store' gcd_buggy s.
intros s.
unfold not; intros.
inversion H; try (inversion H4; fail).
inversion H4; clear H4 H2; subst; simpl in *.
inversion H12; clear H12; subst; simpl in *.
inversion H5; clear H5; subst; simpl in *.
inversion H4; clear H4; subst; simpl in *.
- (* then case *)
inversion H3; clear H3; subst; simpl in *.
inversion H10; clear H10; subst; simpl in *.
inversion H9.
- (* else case *)
inversion H11; clear H11; subst; simpl in *.
inversion H3; clear H3; subst; simpl in *.
inversion H10; clear H10; subst; simpl in *.
inversion H8; clear H8; subst; simpl in *; inversion H5.
inversion H7.
Qed.
Notation "'Bug' π" := (BugFound (π, _, _)) (at level 80).
Eval compute in (peek 20 (find_bugs gcd_buggy)).
End Gcd.
End Examples.
Section Simple.
Example my_prog :=
Ite
(Beq (Var "x"%string) (Cst 0%Z))
Err
Skip.
Compute (peek 10 (find_bugs my_prog)).
Example a := (Var "a"%string).
Example b := (Var "b"%string).
Example olda := (Var "olda"%string).
Example oldb := (Var "oldb"%string).
Example trivial_assertion :=
Seq
(Aff "a"%string b)
(Assert (Bnot (Beq a b))).
Compute (peek 10 (find_bugs trivial_assertion)).
End Simple.
Section Gcd.
(*
function gcd(a, b)
while a ≠ b
if a <= b
b := b − a
else
a := a − b
return a
*)
Example gcd :=
Loop
(Bnot (Beq a b))
(Ite
(Ble a b)
(Aff "b"%string (Sub b a))
(Aff "a"%string (Sub a b))).
Example init_store (x : ident) : Z :=
match x with
| "a"%string => 9
| "b"%string => 6
| _ => 0
end.
(* gcd(9, 6) --> (3, 3) *)
Example execute_gcd:
exec
init_store
gcd
(update (update init_store "a"%string 3%Z) "b"%string 3%Z).
Proof.
unfold gcd.
(* First Iteration *)
apply exec_loop_true with
(update init_store "a"%string 3%Z);
auto.
1: (* correctness of provided store *)
apply exec_Ite_false; auto.
apply exec_Aff.
auto.
(* Second Iteration *)
apply exec_loop_true with
(update (update init_store "a"%string 3%Z) "b"%string 3%Z); auto.
1: (* correctness of provided store *)
apply exec_Ite_true; auto.
apply exec_Aff.
auto.
(* Termination *)
apply exec_loop_false.
auto.
Qed.
Example a0 := (Var "a0"%string).
Example b0 := (Var "b0"%string).
(* Now, a correct version of `gcd` with assertions. *)
(*
function gcd(a, b)
while a ≠ b
if a <= b
b := b − a
else
a := a − b
assert a + b + 1 <= \old(a) + \old(b)
return a
*)
Example gcd_assertions :=
Loop
(Bnot (Beq a b))
(Seq
(Aff "a0"%string a)
(Seq
(Aff "b0"%string b)
(Seq
(Ite
(Ble a b)
(Aff "b"%string (Sub b a))
(Aff "a"%string (Sub a b)))
(Assert (Ble (Add (Add a b) (Cst 1%Z)) (Add a0 b0)))))).
Compute (peek 15 (find_bugs gcd_assertions)).
Example init_store' (x : ident) : Z :=
match x with
| "a"%string => 4
| "b"%string => 2
| _ => 0
end.
(* gcd(4, 2) --> (2, 2) *)
Example execute_gcd_noerr:
exec
init_store'
gcd_assertions
(update
(update
(update init_store' "a0"%string 4)
"b0"%string 2)
"a"%string 2).
unfold gcd_assertions.
(* First Iteration *)
apply exec_loop_true with
(update
(update
(update init_store' "a0"%string 4)
"b0"%string 2)
"a"%string 2);
auto.
1: (* correctness of provided store *)
(* `a0 = a` *)
apply exec_Seq with (update init_store' "a0"%string 4%Z);
try constructor; auto.
(* `b0 = b` *)
apply exec_Seq with
(update (update init_store' "a0"%string 4%Z) "b0"%string 2%Z);
try constructor; auto.
(* if... ; ... *)
apply exec_Seq with
(update
(update
(update init_store' "a0"%string 4%Z)
"b0"%string 2%Z)
"a"%string 2%Z).
1: (* if ... *)
apply exec_Ite_false; auto.
apply exec_Aff.
auto.
(* assert ... *)
unfold Assert.
apply exec_Ite_true; auto.
apply exec_Skip.
(* Termination *)
apply exec_loop_false.
auto.
Qed.
(* Now, a *buggy* version of `gcd` with assertions. *)
(*
function gcd_buggy(a, b)
while a ≠ b
if a <= b
b := b − a
else
a := a + b // Should be `a - b`
assert a + b + 1 <= \old(a) + \old(b)
return a
*)
Example gcd_buggy :=
Loop
(Bnot (Beq a b))
(Seq
(Aff "a0"%string a)
(Seq
(Aff "b0"%string b)
(Seq
(Ite
(Ble a b)
(Aff "b"%string (Sub b a))
(Aff "a"%string (Add a b)))
(Assert (Ble (Add (Add a b) (Cst 1%Z)) (Add a0 b0)))))).
(* gcd_buggy(4, 2) --> Err *)
Example execute_gcd_err:
forall s, ~ exec init_store' gcd_buggy s.
intros s.
unfold not; intros.
inversion H; try (inversion H4; fail).
inversion H4; clear H4 H2; subst; simpl in *.
inversion H12; clear H12; subst; simpl in *.
inversion H5; clear H5; subst; simpl in *.
inversion H4; clear H4; subst; simpl in *.
- (* then case *)
inversion H3; clear H3; subst; simpl in *.
inversion H10; clear H10; subst; simpl in *.
inversion H9.
- (* else case *)
inversion H11; clear H11; subst; simpl in *.
inversion H3; clear H3; subst; simpl in *.
inversion H10; clear H10; subst; simpl in *.
inversion H8; clear H8; subst; simpl in *; inversion H5.
inversion H7.
Qed.
Notation "'Bug' π" := (BugFound (π, _, _)) (at level 80).
Eval compute in (peek 20 (find_bugs gcd_buggy)).
End Gcd.
End Examples.