## Schorr-Waite algorithm

Schorr-Waite's graph marking algorithm

**Authors:** Mário Pereira

**Topics:** Ghost code / Graph Algorithms / Pointer Programs

**Tools:** Why3

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

Schorr-Waite algorithm

The Schorr-Waite algorithm is the first mountain that any formalism for pointer aliasing should climb. -- Richard Bornat, 2000

Author: Mário Pereira (UBI, then Université Paris Sud)

The code, specification, and invariants below follow those of the following two proofs:

- Thierry Hubert and Claude Marché, using Caduceus and Coq

A case study of C source code verification: the Schorr-Waite algorithm. SEFM 2005. http://www.lri.fr/~marche/hubert05sefm.ps

- Rustan Leino, using Dafny

Dafny: An Automatic Program Verifier for Functional Correctness. LPAR-16. http://research.microsoft.com/en-us/um/people/leino/papers/krml203.pdf

module SchorrWaite use seq.Seq use map.Map use ref.Ref use int.Int use list.List use list.Length use list.Append use set.Fset as S

Why3 has no support for arbitrary pointers, so we introduce a small component-as-array memory model

type loc val constant null: loc

the type of pointers and the null pointer

val (=) (a b: loc) : bool ensures { result = (a = b) } val m: ref (map loc bool) val c: ref (map loc bool) val left: ref (map loc loc) val right: ref (map loc loc)

each (non-null) location holds four fields: two Boolean marks m and c and two pointers left and right

val get_left (p: loc) : loc requires { p <> null } ensures { result = !left[p] } val set_left (p: loc) (v: loc) : unit requires { p <> null } writes { left } ensures { !left = set (old !left) p v } val get_right (p: loc) : loc requires { p <> null } ensures { result = !right[p] } val set_right (p: loc) (v: loc) : unit requires { p <> null } writes { right } ensures { !right = set (old !right) p v } val get_m (p: loc) : bool requires { p <> null } ensures { result = !m[p] } val set_m (p: loc) (v: bool) : unit requires { p <> null } writes { m } ensures { !m = set (old !m) p v } val get_c (p: loc) : bool requires { p <> null } ensures { result = !c[p] } val set_c (p: loc) (v: bool) : unit requires { p <> null } writes { c } ensures { !c = set (old !c) p v } val ghost path_from_root : ref (map loc (list loc))

for the purpose of the proof, we add a fifth, ghost, field, which records the path from the root (when relevant)

val ghost get_path_from_root (p : loc) : list loc ensures { result = !path_from_root[p] } val ghost set_path_from_root (p: loc) (l : list loc) : unit requires { p <> null } writes { path_from_root } ensures { !path_from_root = set (old !path_from_root) p l }

Stack of nodes, from the root to the current location, in reverse order (i.e. the current location is the first element in the stack)

type stacknodes = Seq.seq loc predicate not_in_stack (n: loc) (s: stacknodes) = forall i: int. 0 <= i < Seq.length s -> n <> Seq.get s i let ghost tl_stackNodes (stack: stacknodes) : stacknodes requires { Seq.length stack > 0 } ensures { result = stack[1..] } ensures { forall n. not_in_stack n stack -> not_in_stack n result } = stack[1..]

two lemmas about stacks

let lemma cons_not_in (s: stacknodes) (n t: loc) requires { not_in_stack n (cons t s) } ensures { not_in_stack n s } = assert { forall i: int. 0 <= i < Seq.length s -> Seq.get s i = Seq.get (cons t s) (i+1) } let lemma tl_cons (s1 s2: stacknodes) (p: loc) requires { Seq.length s2 > 0 } requires { s1 = s2[1..] } requires { p = Seq.get s2 0 } ensures { s2 = cons p s1 } = assert { Seq.(==) s2 (cons p s1) } function last (s: stacknodes) : loc = Seq.get s (Seq.length s - 1) predicate distinct (s: stacknodes) = forall i j. 0 <= i < Seq.length s -> 0 <= j < Seq.length s -> i <> j -> Seq.get s i <> Seq.get s j

Paths

predicate edge (x y : loc) (left right : map loc loc) = x <> null /\ (left[x] = y \/ right[x] = y) inductive path (left right : map loc loc) (x y : loc) (p : list loc) = | path_nil : forall x : loc, l r : map loc loc. path l r x x Nil | path_cons : forall x y z : loc, l r : (map loc loc), p : list loc. edge x z l r -> path l r z y p -> path l r x y (Cons x p) let rec lemma trans_path (x y z : loc) (l r : map loc loc) (p1 p2 : list loc) variant { length p1 } requires { path l r x y p1 } requires { path l r y z p2 } ensures { path l r x z (p1 ++ p2) } = match p1 with | Cons _ (Cons b _ as p') -> trans_path b y z l r p' p2 | _ -> () end lemma path_edge : forall x y : loc, left right : map loc loc. edge x y left right -> path left right x y (Cons x Nil) lemma path_edge_cons: forall n x y : loc, left right : map loc loc, pth : list loc. path left right n x pth -> edge x y left right -> path left right n y (pth ++ (Cons x Nil)) predicate reachable (left right: map loc loc) (x y : loc) = exists p : list loc. path left right x y p

Schorr-Waite algorithm

let schorr_waite (root: loc) (ghost graph : S.fset loc) : unit requires { root <> null /\ S.mem root graph } (* graph is closed under left and right *) requires { forall n : loc. S.mem n graph -> n <> null /\ (!left[n] = null \/ S.mem !left[n] graph) /\ (!right[n] = null \/ S.mem !right[n] graph) } (* graph starts with nothing marked *) requires { forall x : loc. S.mem x graph -> not !m[x] } (* the structure of the graph is not changed *) ensures { forall n : loc. S.mem n graph -> (old !left)[n] = !left[n] /\ (old !right)[n] = !right[n] } (* all the non-null vertices reachable from root are marked at the end of the algorithm, and only these *) ensures { forall n : loc. S.mem n graph -> !m[n] -> reachable (old !left) (old !right) root n } ensures { !m[root] } ensures { forall n : loc. S.mem n graph -> !m[n] -> (forall ch : loc. edge n ch !left !right -> ch <> null -> !m[ch]) } = let t = ref root in let p = ref null in let ghost stackNodes = ref Seq.empty in let ghost pth = ref Nil in ghost set_path_from_root !t !pth; let ghost unmarked_nodes = ref graph in let ghost c_false_nodes = ref graph in while !p <> null || (!t <> null && not get_m !t) do invariant { forall n. S.mem n graph -> not_in_stack n !stackNodes \/ exists i : int. Seq.get !stackNodes i = n } invariant { not_in_stack null !stackNodes } invariant { Seq.length !stackNodes = 0 <-> !p = null } invariant { !p <> null -> S.mem !p graph } invariant { Seq.length !stackNodes <> 0 -> Seq.get !stackNodes 0 = !p } invariant { forall n : loc. S.mem n graph -> not !m[n] -> S.mem n !unmarked_nodes } invariant { forall n : loc. S.mem n graph -> not !c[n] -> S.mem n !c_false_nodes } invariant { forall i. 0 <= i < Seq.length !stackNodes -> S.mem (Seq.get !stackNodes i) graph } invariant { forall i. 0 <= i < Seq.length !stackNodes - 1 -> let p1 = Seq.get !stackNodes i in let p2 = Seq.get !stackNodes (i+1) in (!c[p2] -> old !left[p2] = !left[p2] /\ old !right[p2] = p1) /\ (not !c[p2] -> old !left[p2] = p1 /\ old !right[p2] = !right[p2]) } invariant { !path_from_root[root] = Nil } invariant { forall n : loc. S.mem n graph -> not_in_stack n !stackNodes -> !left[n] = old !left[n] /\ !right[n] = old !right[n] } (* something like Leino's line 74; this is useful to prove that * the stack is empty iff p = null *) invariant { Seq.length !stackNodes <> 0 -> let first = last !stackNodes in if !c[first] then !right[first] = null else !left[first] = null } invariant { Seq.length !stackNodes <> 0 -> last !stackNodes = root } (* something like lines 75-76 from Leino's paper *) invariant { forall k : int. 0 <= k < Seq.length !stackNodes - 1 -> if !c[Seq.get !stackNodes k] then !right[Seq.get !stackNodes k] = Seq.get !stackNodes (k+1) else !left [Seq.get !stackNodes k] = Seq.get !stackNodes (k+1) } (* all nodes in the stack are marked * (I4a in Hubert and Marché and something alike line 57 in Leino) *) invariant { forall i. 0 <= i < Seq.length !stackNodes -> !m[Seq.get !stackNodes i] } (* stack has no duplicates ---> line 55 from Leino's paper *) invariant { distinct !stackNodes } (* something like Leino's line 68 *) invariant { forall i. 0 <= i < Seq.length !stackNodes -> let n = Seq.get !stackNodes i in if !c[n] then !left[n] = old !left[n] else !right[n] = old !right[n] } (* lines 80-81 from Leino's paper *) invariant { Seq.length !stackNodes <> 0 -> if !c[!p] then old !right[!p] = !t else old !left[!p] = !t } (* lines 78-79 from Leino's paper *) invariant { forall k : int. 0 < k < Seq.length !stackNodes -> let n = Seq.get !stackNodes k in if !c[n] then Seq.get !stackNodes (k - 1) = old !right[n] else Seq.get !stackNodes (k - 1) = old !left[n] } (* line 70 from Leino's paper *) invariant { !p <> null -> path (old !left) (old !right) root !p !pth } (* line 72 from Leino's paper *) invariant { forall n : loc. S.mem n graph -> !m[n] -> reachable (old !left) (old !right) root n } invariant { !p = null -> !t = root } (* line 70 from Leino's paper *) invariant { forall n : loc, pth : list loc. S.mem n graph -> !m[n] -> pth = !path_from_root[n] -> path (old !left) (old !right) root n pth } (* lines 61-62 from Leinos' paper *) invariant { forall n : loc. S.mem n graph -> n <> null -> !m[n] -> not_in_stack n !stackNodes -> (!left[n] <> null -> !m[!left[n]]) /\ (!right[n] <> null -> !m[!right[n]]) } (* something like Leino's lines 57-59 *) invariant { forall i. 0 <= i < Seq.length !stackNodes -> let n = Seq.get !stackNodes i in !c[n] -> (!left[n] <> null -> !m[!left[n]]) /\ (!right[n] <> null -> !m[!right[n]]) } (* termination proved using lexicographic order over a triple *) variant { S.cardinal !unmarked_nodes, S.cardinal !c_false_nodes, Seq.length !stackNodes } if !t = null || get_m !t then begin if get_c !p then begin (* pop *) let q = !t in t := !p; ghost stackNodes := tl_stackNodes !stackNodes; p := get_right !p; set_right !t q; ghost pth := get_path_from_root !p; end else begin (* swing *) let q = !t in t := get_right !p; set_right !p (get_left !p); set_left !p q; ghost c_false_nodes := S.remove !p !c_false_nodes; set_c !p true; end end else begin (* push *) let q = !p in p := !t; ghost stackNodes := Seq.cons !p !stackNodes; t := get_left !t; set_left !p q; set_m !p true; ghost if !p <> root then pth := !pth ++ (Cons q Nil) else pth := Nil; ghost set_path_from_root !p !pth; assert { !p = Seq.get !stackNodes 0 }; assert { path (old !left) (old !right) root !p !pth }; set_c !p false; ghost c_false_nodes := S.add !p !c_false_nodes; ghost unmarked_nodes := S.remove !p !unmarked_nodes end done end

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# Why3 Proof Results for Project "schorr_waite"

## Theory "schorr_waite.SchorrWaite": fully verified

Obligations | Alt-Ergo 2.0.0 | CVC4 1.4 | CVC4 1.5 | |||

VC for tl_stackNodes | 0.01 | --- | --- | |||

VC for cons_not_in | --- | --- | --- | |||

split_goal_right | ||||||

assertion | 0.01 | --- | --- | |||

postcondition | 0.01 | --- | --- | |||

VC for tl_cons | 0.01 | --- | --- | |||

VC for trans_path | 0.09 | --- | --- | |||

path_edge | --- | 0.16 | --- | |||

path_edge_cons | 0.01 | --- | --- | |||

VC for schorr_waite | --- | --- | --- | |||

split_goal_right | ||||||

precondition | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.00 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.00 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.00 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.00 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.00 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

loop invariant init | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.02 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

loop variant decrease | 0.12 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.03 | --- | --- | |||

loop invariant preservation | 0.22 | --- | --- | |||

loop invariant preservation | 0.10 | --- | --- | |||

loop invariant preservation | 1.10 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.92 | --- | --- | |||

loop invariant preservation | --- | 0.44 | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | --- | 0.57 | --- | |||

loop invariant preservation | --- | --- | --- | |||

split_goal_right | ||||||

loop invariant preservation | --- | --- | --- | |||

split_goal_right | ||||||

loop invariant preservation | 0.53 | 2.68 | --- | |||

loop invariant preservation | 1.33 | --- | --- | |||

loop invariant preservation | --- | --- | --- | |||

split_goal_right | ||||||

loop invariant preservation | 0.43 | --- | --- | |||

loop invariant preservation | 1.34 | 3.15 | --- | |||

loop invariant preservation | 1.80 | --- | --- | |||

loop invariant preservation | --- | 0.65 | --- | |||

loop invariant preservation | 0.12 | --- | --- | |||

loop invariant preservation | --- | --- | 0.33 | |||

loop invariant preservation | 1.30 | --- | --- | |||

loop invariant preservation | 0.61 | --- | --- | |||

loop invariant preservation | 3.79 | --- | --- | |||

loop invariant preservation | 0.11 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.06 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | --- | 1.05 | --- | |||

loop invariant preservation | 2.60 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.02 | --- | --- | |||

loop variant decrease | 0.03 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.07 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.05 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.06 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.04 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.03 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.04 | --- | --- | |||

loop invariant preservation | 0.27 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

precondition | 0.01 | --- | --- | |||

assertion | 0.02 | --- | --- | |||

assertion | 0.02 | --- | --- | |||

precondition | 0.02 | --- | --- | |||

loop variant decrease | 0.04 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.03 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.06 | --- | --- | |||

loop invariant preservation | 0.33 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.05 | --- | --- | |||

loop invariant preservation | 0.11 | --- | --- | |||

loop invariant preservation | 0.03 | --- | --- | |||

loop invariant preservation | 0.09 | --- | --- | |||

loop invariant preservation | 0.07 | --- | --- | |||

loop invariant preservation | 0.06 | --- | --- | |||

loop invariant preservation | 0.30 | --- | --- | |||

loop invariant preservation | 0.03 | --- | --- | |||

loop invariant preservation | 0.18 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.32 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 3.52 | --- | --- | |||

precondition | 0.02 | --- | --- | |||

assertion | 0.02 | --- | --- | |||

assertion | 0.01 | --- | --- | |||

precondition | 0.02 | --- | --- | |||

loop variant decrease | 0.03 | --- | --- | |||

loop invariant preservation | 0.00 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.03 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.04 | --- | --- | |||

loop invariant preservation | 0.21 | --- | --- | |||

loop invariant preservation | 0.01 | --- | --- | |||

loop invariant preservation | 0.05 | --- | --- | |||

loop invariant preservation | 0.14 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.08 | --- | --- | |||

loop invariant preservation | 0.04 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.04 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.14 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.04 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.02 | --- | --- | |||

loop invariant preservation | 0.04 | --- | --- | |||

loop invariant preservation | 0.03 | --- | --- | |||

postcondition | 0.02 | --- | --- | |||

postcondition | 0.02 | --- | --- | |||

postcondition | 0.01 | --- | --- | |||

postcondition | 0.02 | --- | --- |