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# Random search in graph

The graph is represented by a pair (`vertices`, `successor`)

• `vertices` : this constant is the set of graph vertices
• `successor` : this function gives for each vertex the set of vertices directly joinable from it
The algorithm is a ministep of a random search in the graph. It is generic to any search strategy.
This completeness proof is adapted from Dowek and Munoz and is fully automatic.

```
module RandomSearch
use import int.Int
use import list.List
use import list.Append
use import list.Mem as L
use import init_graph.GraphSetSucc
use import list.NumOcc

lemma path_lst_occ:
forall x y z l. L.mem x l -> path y l z ->
exists l1 l2. l = l1 ++ Cons x l2 /\ path x (Cons x l2) z /\ not L.mem x l2

lemma path_inv:
forall x z l. path x (Cons x l) z -> exists x'. edge x x' /\ path x' l z

predicate white_vertex (x: vertex) (v: set vertex) =
not (mem x v)

predicate whitepath (x: vertex) (l: list vertex) (z: vertex) (v: set vertex) =
path x l z /\ (forall y. L.mem y l -> white_vertex y v) /\ white_vertex z v

lemma whitepath_suffix:
forall x y z v l1 l2. whitepath y (l1 ++ Cons x l2) z v -> whitepath x (Cons x l2) z v

lemma whitepath_inv:
forall x z l v. whitepath x (Cons x l) z v -> not L.mem x l -> z <> x -> exists x'. edge x x' /\ whitepath x' l z (add x v)

lemma whitepath_split_lst_occ:
forall x y l z v. L.mem x l -> whitepath y l z v -> z <> x ->
exists x' l'. edge x x' /\ whitepath x' l' z (add x v)

```

### program

```
let rec random_search r v =
requires {subset r vertices }
requires {subset v vertices }
ensures {subset v result}
ensures {forall z y l. mem y r -> whitepath y l z v -> mem z (diff result v)}

if is_empty r then v else
let x = choose r in
let r' = remove x r in
if mem x v then random_search r' v else
let v' = random_search (union r' (successors x)) (add x v) in
assert {forall z y l. mem y r -> whitepath y l z v -> z <> x ->
whitepath y l z (add x v) \/
exists x' l'. edge x x' /\ whitepath x' l' z (add x v) } ;
v'

let random_search_main (roots: set vertex) =
requires {subset roots vertices}
ensures {forall z y l. mem y roots -> path y l z -> mem z result}
random_search roots empty

end

```

#### [session]

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