## Warshall algorithm

**Authors:** Jean-Christophe Filliâtre / Martin Clochard

**Topics:** Graph Algorithms

**Tools:** Why3

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Warshall algorithm

Computes the transitive closure of a graph implemented as a Boolean matrix.

Authors: Martin Clochard (École Normale Supérieure) Jean-Christophe Filliâtre (CNRS)

module WarshallAlgorithm use int.Int use matrix.Matrix (* path m i j k = there is a path from i to j, using only vertices smaller than k *) inductive path (matrix bool) int int int = | Path_empty: forall m: matrix bool, i j k: int. m.elts i j -> path m i j k | Path_cons: forall m: matrix bool, i x j k: int. 0 <= x < k -> path m i x k -> path m x j k -> path m i j k lemma weakening: forall m i j k1 k2. 0 <= k1 <= k2 -> path m i j k1 -> path m i j k2 lemma decomposition: forall m k. 0 <= k -> forall i j. path m i j (k+1) -> (path m i j k \/ (path m i k k /\ path m k j k)) let transitive_closure (m: matrix bool) : matrix bool requires { m.rows = m.columns } ensures { let n = m.rows in forall x y: int. 0 <= x < n -> 0 <= y < n -> result.elts x y <-> path m x y n } = let t = copy m in let n = m.rows in for k = 0 to n - 1 do invariant { forall x y. 0 <= x < n -> 0 <= y < n -> t.elts x y <-> path m x y k } for i = 0 to n - 1 do invariant { forall x y. 0 <= x < n -> 0 <= y < n -> t.elts x y <-> if x < i then path m x y (k+1) else path m x y k } for j = 0 to n - 1 do invariant { forall x y. 0 <= x < n -> 0 <= y < n -> t.elts x y <-> if x < i \/ (x = i /\ y < j) then path m x y (k+1) else path m x y k } set t i j (get t i j || get t i k && get t k j) done done done; t end

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# Why3 Proof Results for Project "warshall_algorithm"

## Theory "warshall_algorithm.WarshallAlgorithm": fully verified

Obligations | Alt-Ergo 2.0.0 | CVC4 1.5 | Coq 8.11.2 | Z3 4.6.0 | ||

weakening | --- | --- | --- | --- | ||

induction_pr | ||||||

weakening.0 | 0.00 | --- | --- | --- | ||

weakening.1 | --- | 0.01 | --- | --- | ||

decomposition | --- | --- | 0.96 | --- | ||

VC for transitive_closure | --- | --- | --- | --- | ||

split_goal_right | ||||||

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

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

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

index in matrix bounds | 0.00 | --- | --- | --- | ||

index in matrix bounds | 0.00 | --- | --- | --- | ||

index in matrix bounds | 0.00 | --- | --- | --- | ||

index in matrix bounds | 0.00 | --- | --- | --- | ||

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

split_goal_right | ||||||

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

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

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

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

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

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

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

postcondition | 0.00 | --- | --- | --- | ||

out of loop bounds | 0.00 | --- | --- | --- |