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VerifyThis 2015: solution to problem 3


Authors: Jean-Christophe Filliâtre / Guillaume Melquiond

Topics: Data Structures

Tools: Why3

References: VerifyThis @ ETAPS 2015

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VerifyThis @ ETAPS 2015 competition, Challenge 3: Dancing Links

The following is the original description of the verification task, reproduced verbatim from the competition web site.

DANCING LINKS (90 minutes)
==========================

Dancing links is a technique introduced in 1979 by Hitotumatu and
Noshita and later popularized by Knuth. The technique can be used to
efficiently implement a search for all solutions of the exact cover
problem, which in its turn can be used to solve Tiling, Sudoku,
N-Queens, and other problems.

The technique
-------------

Suppose x points to a node of a doubly linked list; let L[x] and R[x]
point to the predecessor and successor of that node. Then the operations

L[R[x]] := L[x];
R[L[x]] := R[x];

remove x from the list. The subsequent operations

L[R[x]] := x;
R[L[x]] := x;

will put x back into the list again.

A graphical illustration of the process is available at
http://formal.iti.kit.edu/~klebanov/DLX.png


Verification task
-----------------

Implement the data structure with these operations, and specify and
verify that they behave in the way described above.

The following is the solution by Jean-Christophe FilliĆ¢tre (CNRS) and Guillaume Melquiond (Inria) who entered the competition as "team Why3".

module DancingLinks

  use int.Int
  use ref.Ref
  use array.Array

  type dll = { prev: array int; next: array int; ghost n: int }
    invariant { length prev = length next = n }
    by { prev = make 0 0; next = make 0 0; n = 0 }

we model the data structure with two arrays, nodes being represented by array indices

  predicate valid_in (l: dll) (i: int) =
    0 <= i < l.n /\ 0 <= l.prev[i] < l.n /\ 0 <= l.next[i] < l.n /\
    l.next[l.prev[i]] = i /\
    l.prev[l.next[i]] = i

node i is a valid node i.e. it has consistent neighbors

  predicate valid_out (l: dll) (i: int) =
    0 <= i < l.n /\ 0 <= l.prev[i] < l.n /\ 0 <= l.next[i] < l.n /\
    l.next[l.prev[i]] = l.next[i] /\
    l.prev[l.next[i]] = l.prev[i]

node i is ready to be put back in a list

  use seq.Seq as S
  function nth (s: S.seq 'a) (i: int) : 'a = S.([]) s i

  predicate is_list (l: dll) (s: S.seq int) =
    forall k: int. 0 <= k < S.length s ->
      0 <= nth s k < l.n /\
      l.prev[nth s k] = nth s (if k = 0 then S.length s - 1 else k - 1) /\
      l.next[nth s k] = nth s (if k = S.length s - 1 then 0 else k + 1) /\
      (forall k': int. 0 <= k' < S.length s -> k <> k' -> nth s k <> nth s k')

Representation predicate: Sequence s is the list of indices of a valid circular list in l. We choose to model circular lists, since this is the way the data structure is used in Knuth's dancing links algorithm.

  let remove (l: dll) (i: int) (ghost s: S.seq int)
    requires { valid_in l i }
    requires { is_list l (S.cons i s) }
    ensures  { valid_out l i }
    ensures  { is_list l s }
  =
    l.prev[l.next[i]] <- l.prev[i];
    l.next[l.prev[i]] <- l.next[i];
    assert { forall k: int. 0 <= k < S.length s ->
       nth (S.cons i s) (k + 1) = nth s k } (* to help SMT with triggers *)

Note: the code below works fine even when the list has one element (necessarily i in that case).

  let put_back (l: dll) (i: int) (ghost s: S.seq int)
    requires { valid_out l i } (* `i` is ready to be reinserted *)
    requires { is_list l s }
    requires { 0 < S.length s } (* `s` must contain at least one element *)
    requires { l.next[i] = nth s 0 <> i } (* do not link `i` to itself *)
    ensures  { valid_in l i }
    ensures  { is_list l (S.cons i s) }
  =
    l.prev[l.next[i]] <- i;
    l.next[l.prev[i]] <- i

end

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Why3 Proof Results for Project "verifythis_2015_dancing_links"

Theory "verifythis_2015_dancing_links.DancingLinks": fully verified

ObligationsAlt-Ergo 2.0.0
VC dll0.00
VC remove2.24
split_goal_right
VC remove.00.01
VC remove.10.01
VC remove.20.01
VC remove.30.00
VC remove.40.01
VC remove.50.02
VC remove.60.01
VC remove.70.02
VC remove.80.06
VC remove.91.71
VC put_back1.78