## Find the shortest path in a directed graph using BFS

Problem 5 of the second verified software competition.

The ZIP file below contains both the source code, the Why3 proof session file and the Coq scripts of the proofs made in Coq. The Why3 source code is then displayed, followed by a summary of the proofs.

Authors: Jean-Christophe Filliâtre

Topics: Graph Algorithms

Tools: Why3

References: The 2nd Verified Software Competition

see also the index (by topic, by tool, by reference, by year)

```(* The 2nd Verified Software Competition (VSTTE 2012)

Problem 5:
Shortest path in a graph using breadth-first search *)

theory Graph

use int.Int
use export set.Fset

type vertex

val predicate eq (x y: vertex)
ensures { result <-> x = y }

function succ vertex : fset vertex

(* there is a path of length n from v1 to v2 *)
inductive path (v1 v2: vertex) (n: int) =
| path_empty:
forall v: vertex. path v v 0
| path_succ:
forall v1 v2 v3: vertex, n: int.
path v1 v2 n -> mem v3 (succ v2) -> path v1 v3 (n+1)

(* path length is non-negative *)
lemma path_nonneg:
forall v1 v2: vertex, n: int. path v1 v2 n -> n >= 0

(* a non-empty path has a last but one node *)
lemma path_inversion:
forall v1 v3: vertex, n: int. n >= 0 ->
path v1 v3 (n+1) ->
exists v2: vertex. path v1 v2 n /\ mem v3 (succ v2)

(* a path starting in a set S that is closed under succ ends up in S *)
lemma path_closure:
forall s: fset vertex.
(forall x: vertex. mem x s ->
forall y: vertex. mem y (succ x) -> mem y s) ->
forall v1 v2: vertex, n: int. path v1 v2 n ->
mem v1 s -> mem v2 s

predicate shortest_path (v1 v2: vertex) (n: int) =
path v1 v2 n /\
forall m: int. m < n -> not (path v1 v2 m)

end

module BFS

use int.Int
use Graph
clone set.SetImp as B with type elt = vertex, val eq = eq
use ref.Refint

exception Found int

(* global invariant *)
predicate inv (s t: vertex) (visited current next: fset vertex) (d: int) =
(* current *)
subset current visited /\
(forall x: vertex. mem x current -> shortest_path s x d) /\
(* next *)
subset next visited /\
(forall x: vertex. mem x next -> shortest_path s x (d + 1)) /\
(* visited *)
(forall x: vertex, m: int. path s x m -> m <= d -> mem x visited) /\
(forall x: vertex. mem x visited ->
exists m: int. path s x m /\ m <= d+1) /\
(* nodes at distance d+1 are either in next or not yet visited *)
(forall x: vertex. shortest_path s x (d + 1) ->
mem x next \/ not (mem x visited)) /\
(* target t *)
(mem t visited -> mem t current \/ mem t next)

(* visited\current\next is closed under succ *)
predicate closure (visited current next: fset vertex) (x: vertex) =
mem x visited -> not (mem x current) -> not (mem x next) ->
forall y: vertex. mem y (succ x) -> mem y visited

function (!!) (s: B.set) : fset vertex = B.to_fset s

val pick_succ (v: vertex) : B.set
ensures { !!result = succ v }

(* function fill_next fills set next with the unvisited successors of v *)
let fill_next (s t v: vertex) (visited current next: B.set) (d:ref int)
requires { inv s t !!visited !!current !!next !d /\
shortest_path s v !d /\
(forall x: vertex. x<> v -> closure !!visited !!current !!next x) }
ensures { inv s t !!visited !!current !!next !d /\
subset (succ v) !!visited  /\
(forall x: vertex. closure !!visited !!current !!next x) }
= let acc = pick_succ v in
while not (B.is_empty acc) do
invariant {
inv s t !!visited !!current !!next !d /\
subset !!acc (succ v) /\
subset (diff (succ v) !!acc) !!visited /\
(forall x: vertex. x <> v -> closure !!visited !!current !!next x)
}
variant { cardinal !!acc }
let w = B.choose_and_remove acc in
if not (B.mem w visited) then begin
end
done

let bfs (s: vertex) (t: vertex)
ensures { forall d: int. not (path s t d) }
raises { Found r -> shortest_path s t r }
diverges (* the set of reachable nodes may be infinite *)
= let visited = B.empty () in
let current = ref (B.empty ()) in
let next    = ref (B.empty ()) in
let d = ref 0 in
while not (B.is_empty !current) do
invariant {
inv s t !!visited !! !current !! !next !d /\
(is_empty !! !current -> is_empty !! !next) /\
(forall x: vertex. closure !!visited !! !current !! !next x) /\
0 <= !d
}
let v = B.choose_and_remove !current in
if eq v t then raise (Found !d);
fill_next s t v visited !current !next d;
if B.is_empty !current then begin
(current.contents, next.contents) <- (!next, B.empty ());
incr d
end
done;
assert { not (mem t !!visited) }

end
```

# Why3 Proof Results for Project "vstte12_bfs"

## Theory "vstte12_bfs.Graph": fully verified

 Obligations Alt-Ergo 2.0.0 CVC4 1.5 path_nonneg --- --- induction_pr path_nonneg.0 0.00 --- path_nonneg.1 --- 0.00 path_inversion --- --- inversion_pr path_inversion.0 0.00 --- path_inversion.1 0.00 --- path_closure --- --- induction_pr path_closure.0 0.00 --- path_closure.1 --- 0.01

## Theory "vstte12_bfs.BFS": fully verified

 Obligations Alt-Ergo 2.0.0 CVC4 1.6 Coq 8.11.2 Eprover 2.0 VC for eq'refn --- 0.02 --- --- VC for fill_next --- --- --- --- split_goal_right loop invariant init 0.02 --- --- --- precondition 0.01 --- --- --- loop variant decrease 0.02 --- --- --- loop invariant preservation --- --- --- --- introduce_premises loop invariant preservation --- --- --- --- inline_goal loop invariant preservation --- --- --- --- split_goal_right VC for fill_next 0.03 --- --- --- VC for fill_next 0.01 --- --- --- VC for fill_next 0.03 --- --- --- VC for fill_next 0.04 --- --- --- VC for fill_next 0.01 --- --- --- VC for fill_next 0.37 --- --- --- VC for fill_next 0.01 --- --- --- VC for fill_next 0.02 --- --- --- VC for fill_next 0.11 --- --- --- VC for fill_next 0.01 --- --- --- VC for fill_next --- --- --- 0.42 loop variant decrease 0.01 --- --- --- loop invariant preservation 0.15 --- --- --- postcondition 0.09 --- --- --- VC for bfs --- --- --- --- split_goal_right loop invariant init 0.04 --- --- --- precondition 0.00 --- --- --- exceptional postcondition 0.01 --- --- --- precondition --- --- --- --- split_goal_right VC for bfs 0.02 --- --- --- VC for bfs 0.01 --- --- --- VC for bfs --- --- --- --- introduce_premises VC for bfs --- --- --- --- inline_goal VC for bfs --- --- --- 0.34 loop invariant preservation --- --- --- --- split_goal_right VC for bfs --- --- --- --- introduce_premises VC for bfs --- --- --- --- inline_goal VC for bfs --- --- --- --- split_goal_right VC for bfs 0.01 --- --- --- VC for bfs 0.01 --- --- --- VC for bfs 0.02 --- --- --- VC for bfs 0.02 --- --- --- VC for bfs --- --- --- --- inline_all VC for bfs --- 0.21 --- --- VC for bfs 0.02 --- --- --- VC for bfs 0.00 --- --- --- VC for bfs 0.02 --- --- --- VC for bfs 0.00 --- --- --- VC for bfs --- --- --- --- introduce_premises VC for bfs --- --- --- --- inline_goal VC for bfs --- --- --- 0.10 VC for bfs 0.01 --- --- --- loop invariant preservation 0.01 --- --- --- assertion 0.02 --- --- --- postcondition --- --- 0.34 ---