## Schorr-Waite algorithm, proof via recursion

Schorr-Waite's graph marking algorithm, an alternative proof using recursion

Authors: Martin Clochard

Tools: Why3

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Schorr-Waite algorithm for general-size records.

Author: Martin Clochard (Université Paris Sud)

Here the proof is carried out by proving an equivalent recursive program. The recursive program can be justified to be equivalent to the looping one: all side-effects/exception throwing are done by running the loop body (which features a minor modification to exit by exception), so the recursive programs amounts to execute an arbitrary number of time the loop body. It is immediately followed by an absurd statement to enforce the equivalence with the loop which runs its body an infinite number of times. Although the added recursive structure can be seen to be computationally irrelevant, it allows to discharge details about Schorr-Waite stack management through a recursive procedure proof. The method basically make explicit the derecursification used to obtain Schorr-Waite algorithm.

See example verifythis_2016_tree_traversal for details about the technique, applied to a similar algorithm for trees.

```module Memory
```

Component-as-array memory model, with null pointers and arbitrary-sized memory blocks.

```  use int.Int
use option.Option
use map.Map
```

Memory locations

```  type loc
```

Null pointer

```  constant null : loc
```

Marks used by Schorr-Waite

```  type color =
| White
| Black (option int)
type pmem = map loc (map int loc)
```

Memory blocks have two parts: a marking part containing in particular Schorr-Waite internal management data, and a sequence of pointers to other memory blocks.

```  type memory = abstract {
```

Associate block size to location.

```    block_size : map loc int;
```

Pointers to other memory blocks.

```    mutable accessor : pmem;
```

Marks.

```    mutable colors : map loc color;
}
```

Global instance for memory

```  val memo : memory
```

null creation

```  val null () : loc ensures { result = null }
```

null test

```  val is_null (l:loc) : bool ensures { result <-> l = null }
```

Get block size associated to a given location

```  val get_block_size (l:loc) : int
requires { l <> null }
reads { memo }
ensures { result = memo.block_size[l] /\ result >= 0 }
```

```  val get_color (l:loc) : color
requires { l <> null }
reads { memo }
ensures { result = memo.colors[l] }
```

Set a mark. We also impose the restriction that when a block is marked black, the given index must be coherent with the block size. This impose special treatment for 0-sized memory blocks.

```  val set_color (l:loc) (c:color) : unit
requires { l <> null }
requires { match c with
| White -> true
| Black None -> memo.block_size[l] = 0
| Black (Some ind) -> 0 <= ind < memo.block_size[l]
end }
writes { memo.colors }
ensures { memo.colors = old (memo.colors[l <- c]) }
```

Getter/Setter for pointer buffer

```  val get_acc (l:loc) (k:int) : loc
requires { l <> null /\ 0 <= k < memo.block_size[l] }
reads { memo }
ensures { result = memo.accessor[l][k] }
val set_acc (l:loc) (k:int) (d:loc) : unit
requires { l <> null /\ 0 <= k < memo.block_size[l] }
writes { memo.accessor }
ensures { memo.accessor =
old (memo.accessor[l <- memo.accessor[l][k <- d]]) }
```

Take ghost snapshots of memory.

```  val ghost snapshot_acc () : pmem
reads { memo }
ensures { result = memo.accessor }
val ghost snapshot_colors () : map loc color
reads { memo }
ensures { result = memo.colors }

end

module GraphShape
```

Define notions about the memory graph

```  use int.Int
use set.Fset
use map.Map
use Memory

predicate black (c:color) = c <> White

predicate edge (m:memory) (x y:loc) =
x <> null /\ (exists n. 0 <= n < m.block_size[x] /\ m.accessor[x][n] = y)
```

Edges

Paths

```  inductive path memory (x y:loc) =
| path_nil : forall m x. path m x x
| path_cons : forall m x y z. edge m x y /\ path m y z -> path m x z
```

DFS invariant. For technical reason, it must refer to different parts of the memory at different time. The graph structure is given via the initial memory, but the coloring is given via the current one.

```  predicate well_colored_on (graph gray:set loc) (m:memory) (cl:map loc color) =
subset gray graph /\
(forall x y. mem x graph /\ edge m x y /\ y <> null /\ black cl[x] ->
mem x gray \/ black cl[y]) /\
(forall x. mem x gray -> black cl[x])
```

Unchanged only concerns the graph shape, not the marks

```  predicate unchanged (m1 m2:memory) =
forall x n. x <> null /\ 0 <= n < m1.block_size[x] ->
m2.accessor[x][n] = m1.accessor[x][n]

end

module SchorrWaite
```

Proof of Schorr-Waite algorithm

```  use int.Int
use option.Option
use set.Fset
use map.Map
use map.Const
use ref.Ref
use Memory
use GraphShape

let black (l: loc) : bool
requires { l <> null }
reads { memo }
ensures { result <-> black memo.colors[l] }
= match get_color l with White -> false | _ -> true end

exception Stop

let schorr_waite (root: loc) (ghost graph: set loc) : unit
```

Root belong to graph (note: for simplicity, the graph set may (and likely does) contain the null pointer.

```    requires { mem root graph }
```

Graph is closed by following edges

```    requires { forall l n.
mem l graph /\ l <> null /\ 0 <= n < memo.block_size[l] ->
mem memo.accessor[l][n] graph }
writes { memo.accessor, memo.colors }
```

The graph starts fully unmarked.

```    requires { forall x. mem x graph -> not black memo.colors[x] }
```

The graph structure is left unchanged.

```    ensures { unchanged (old memo) memo }
```

Every non-null location reachable from the root is marked black.

```    ensures { forall x. path (old memo) root x /\ x <> null ->
black memo.colors[x] }
```

Every other location is left with its previous color.

```    ensures { forall x. not path (old memo) root x /\ x <> null ->
memo.colors[x] = (old memo).colors[x] }
=
label I in
let t = ref root in
let p = ref (null ()) in
```

Schorr-Waite loop body.

```    let body () : unit
```

Loop body specification: mindlessly repeat the underlying code.

```      requires { !p <> null /\ (!t = null \/ black memo.colors[!t]) ->
match memo.colors[!p] with
| Black (Some m) -> 0 <= m < memo.block_size[!p]
| _ -> false
end }
ensures { old (!p <> null \/ (!t <> null /\ not black memo.colors[!t])) }
ensures { old (!t <> null /\ not black memo.colors[!t] /\
memo.block_size[!t] = 0) ->
memo.colors = old memo.colors[!t <- Black None] /\
!t = old !t /\ !p = old (!p) /\ memo.accessor = old memo.accessor
}
ensures { old (!t <> null /\ not black memo.colors[!t] /\
memo.block_size[!t] > 0) ->
memo.colors = old memo.colors[!t <- Black (Some 0)] /\
!t = old memo.accessor[!t][0] /\ !p = old (!t) /\
memo.accessor = old memo.accessor[!t <- memo.accessor[!t][0 <- !p]] }
ensures { old (!t = null \/ black memo.colors[!t]) ->
match old (memo.colors[!p]) with
| Black (Some m) ->
let n = m + 1 in
if n = old (memo.block_size[!p])
then
!t = old (!p) /\ !p = (old memo.accessor[!p])[m] /\
memo.colors = old (memo.colors) /\
memo.accessor =
(old memo.accessor)[old !p <-
(old memo.accessor[!p])[m <- old !t]]
else !p = old !p /\ !t = (old memo.accessor[!p])[n] /\
memo.colors = (old memo.colors)[old !p <- Black (Some n)] /\
let pi = (old memo.accessor[!p])[m] in
memo.accessor =
(old memo.accessor)[old !p <-
(old memo.accessor[!p])[n <- pi][m <- old !t]]
| _ -> false
end }
raises { Stop -> old(!p = null /\ (!t = null \/ black memo.colors[!t])) /\
memo.colors = old memo.colors /\ memo.accessor = old memo.accessor }
=```

Minor modification to the loop: it exits by exception.

```      if is_null !p && (is_null !t || black !t) then raise Stop;
if is_null !t || black !t then begin
match get_color !p with
| Black (Some m) ->
let s = get_block_size !p in
let n = m + 1 in
if n = s then begin (* Pop *)
let q = !t in
t := !p;
p := get_acc !p m;
set_acc !t m q
end else begin (* Swing *)
let q = !t in
t := get_acc !p n;
set_acc !p n (get_acc !p m);
set_acc !p m q;
set_color !p (Black (Some n))
end
| _ -> absurd
end
end else
let s = get_block_size !t in
if s = 0 then (* Mark & continue *) set_color !t (Black None)
else begin (* Mark & Push *)
let q = !p in
p := !t;
t := get_acc !t 0;
set_acc !p 0 q;
set_color !p (Black (Some 0))
end
in
let rec aux (ghost gray:set loc) : unit
(* DFS invariant *)
requires { well_colored_on graph gray (memo at I) memo.colors }
requires { mem !t graph }
(* Non-marked nodes have unchanged structure with
respect to the initial one. *)
requires { forall x n.
x <> null /\ not black memo.colors[x] /\ 0 <= n < memo.block_size[x] ->
memo.accessor[x][n] = (memo.accessor at I)[x][n] }
(* 'stack frames' are maintained correctly *)
ensures { !t = old !t /\ !p = old !p }
(* Pointer buffer is left unchanged *)
ensures { unchanged (old memo) memo }
(* Maintain DFS invariant *)
ensures { well_colored_on graph gray (memo at I) memo.colors }
(* The top node get marked *)
ensures { black memo.colors[!t] \/ !t = null }
(* May not mark unreachable node, neither change marked node. *)
ensures { forall x.
x <> null -> not path (memo at I) !t x \/ black (old memo.colors)[x] ->
memo.colors[x] = (old memo.colors)[x] }
(* Never 'exit' the loop during the recursive calls *)
raises { Stop -> false }
(* Terminates because there is a limited number of 'grayable' nodes. *)
variant { cardinal graph - cardinal gray }
= label J in
if is_null !t || black !t then () else begin
let s = get_block_size !t in
let ghost g2 = add !t gray in
assert { path (memo at I) !t !t };
body (); (* Either push or mark & continue. *)
if s <> 0 then begin
for i = 0 to s - 2 do (* Over all sub-blocs... *)
(* DFS invariant. *)
invariant { well_colored_on graph g2 (memo at I) memo.colors }
(* Current stack frame invariants *)
invariant { !p = !t at J }
invariant { !t = (memo.accessor at I)[!p][i] }
invariant { memo.colors[!p] = Black (Some i) }
invariant { forall j. 0 <= j < s /\ j <> i ->
memo.accessor[!p][j] = (memo.accessor at J)[!p][j] }
invariant { memo.accessor[!p][i] = !p at J }
(* Outside structure is unchanged. *)
invariant { forall l n.
l <> null /\ l <> !p /\ 0 <= n < memo.block_size[l] ->
memo.accessor[l][n] = (memo.accessor at J)[l][n] }
(* All nodes under !p & before i are either null or marked black. *)
invariant { forall j. 0 <= j < i ->
let l = memo.accessor[!p][j] in l = null \/ black memo.colors[l] }
(* Unreachable/pre-marked blocks do not change. *)
invariant { forall l. l <> null ->
not path (memo at I) !p l \/ black (memo.colors at J)[l] ->
memo.colors[l] = (memo.colors at J)[l] }
label K in
aux g2; (* Explore sub-bloc. *)
body (); (* Swing to next sub-bloc. *)
assert { !t = (memo.accessor at K)[!p][i+1]
= (memo.accessor at J)[!p][i+1] }
done;
aux g2; (* Explore last sub-bloc. *)
body (); (* Pop. *)
end
end in
try aux (ghost empty); (* Explore main bloc *)
body (); (* Exit *)
absurd; (* Done. *)
with Stop ->
(* Need induction to recover path-based specification. *)
assert { forall x y m. m = memo at I /\ x <> null /\ y <> null /\
mem x graph /\ black memo.colors[x] ->
([@induction] path m x y) -> black memo.colors[y] }
end

end
```