Octagonal Domains for Continuous Constraints

Transcription

1 Octagonal Domains for Continuous Constraints Marie Pelleau Charlotte Truchet Frédéric Benhamou TASC, University of Nantes CP 2011 September 13, 2011 September 13, / 22

2 Outline Table of contents 1 Context 2 Computer Representation Octagonal Consistency Octagonal Solving 3 Results 4 Conclusion September 13, / 22

3 Context Context Continuous Constraint Satisfaction Problem: CSP < V, D, C > V = (v 1... v n ) real variables D = (D 1... D n ) interval domains C = (C 1... C p ) continuous constraints Solving process in CP relies on Cartesian product of intervals September 13, / 22

4 Context Context Continuous Constraint Satisfaction Problem: CSP < V, D, C > V = (v 1... v n ) real variables D = (D 1... D n ) interval domains C = (C 1... C p ) continuous constraints Solving process in CP relies on Cartesian product of intervals September 13, / 22

5 Context Context Continuous Constraint Satisfaction Problem: CSP < V, D, C > V = (v 1... v n ) real variables D = (D 1... D n ) interval domains C = (C 1... C p ) continuous constraints Solving process in CP relies on Cartesian product of intervals September 13, / 22

6 Context Context Continuous Constraint Satisfaction Problem: CSP < V, D, C > V = (v 1... v n ) real variables D = (D 1... D n ) interval domains C = (C 1... C p ) continuous constraints Solving process in CP relies on Cartesian product of intervals September 13, / 22

7 Context Context Continuous Constraint Satisfaction Problem: CSP < V, D, C > V = (v 1... v n ) real variables D = (D 1... D n ) interval domains C = (C 1... C p ) continuous constraints Solving process in CP relies on Cartesian product of intervals September 13, / 22

8 Context Context Our goal Continuous Constraint Solving We don t live in a Cartesian world! Can we use other domain representations? There exist several domain representations in other fields (e.g. ellipsoids, zonotopes in Abstract Interpretation) Our Contribution Show that the basic tools of CP can be defined for non-cartesian domain representations September 13, / 22

9 Computer Representation The Octagon Abstract Domain Definition (Octagon [Miné, 2006]) Set of points satisfying a conjunction of constraints of the form ±v i ± v j c, called octagonal constraints v 2 v 2 v 1 2 v 1 + v 2 3 v 1 1 v 2 5 In dimension n, an octagon has at most 2n 2 faces v 2 1 v 1 5 An octagon can be unbounded v 1 v 1 v September 13, / 22

10 Computer Representation The Octagon Abstract Domain A Difference Bound Matrix (DBM) to represent an octagon v i v j. c v j v i c September 13, / 22

11 Computer Representation The Octagon Abstract Domain A Difference Bound Matrix (DBM) to represent an octagon v i v j. c v j v i c v 1 + v 2 c v 1 ( v 2 ) c and v 2 ( v 1 ) c September 13, / 22

12 Computer Representation The Octagon Abstract Domain A Difference Bound Matrix (DBM) to represent an octagon v i v j. c v j v i c v 1 + v 2 c v 1 ( v 2 ) c and v 2 ( v 1 ) c Need of two rows and columns for each variable (v i, v i ) September 13, / 22

13 Computer Representation The Octagon Abstract Domain v 2 v 2 v 1 2 v 1 + v 2 3 v 2 1 v 1 1 v 1 5 v 2 5 v 1 v 1 v 1 v 2 v 2 v 1 v 1 v 2 v v 1 v September 13, / 22

14 Computer Representation Canonical Representation Different DBMs can correspond to the same octagon need for a canonical form It has been proved that a modified version of Floyd-Warshall shortest path algorithm computes in O(n 3 ) the smallest DBM representing an octagon of dimension n [Miné, 2006] This canonical form corresponds to the consistency of the difference constraints [Dechter et al., 1991] September 13, / 22

15 Computer Representation Representation for CP v 2 v 2 v 1 2 v 1 1 v 2 5 v 1 + v 2 3 v 2 1 v 1 v v 1 5 v 1 September 13, / 22

16 Computer Representation Representation for CP v 2 v 1 1 v 2 v 1 2 v 2 5 v 1 = v 1 cos ( ) π 4 + v 2 sin ( ) π 4 v 2 = v 2 cos ( ) π 4 v 1 sin ( ) π 4 v 2 v 1 v 1 + v 2 3 v 2 v 2 1 v 1 5 v 1 π 4 v 1 v 1 v September 13, / 22

17 Computer Representation Representation for CP Rotation v 1 = v 1 cos ( ) π 4 + v 2 sin ( ) π 4 v 2 = v 2 cos ( ) π 4 v 1 sin ( ) π 4 Constraint Translation 2v 1 v 2 4 September 13, / 22

18 Computer Representation Representation for CP Rotation v 1 = v 1 cos ( ) π 4 + v 2 sin ( ) π 4 v 2 = v 2 cos ( ) π 4 v 1 sin ( ) π 4 Constraint Translation 2v 1 v 2 4 2(v 1 cos ( π 4 ) + v 2 sin ( π 4 ) ) v 2 cos ( ) π 4 + v 1 sin ( ) π 4 4 September 13, / 22

19 Computer Representation Representation for CP Rotation v 1 = v 1 cos ( ) π 4 + v 2 sin ( ) π 4 v 2 = v 2 cos ( ) π 4 v 1 sin ( ) π 4 Constraint Translation 2v 1 v 2 4 2(v 1 cos ( π 4 ) + v 2 sin ( π v v 2 4 ) ) v 2 cos ( ) π 4 + v 1 sin ( ) π 4 4 September 13, / 22

20 Computer Representation Representation for CP Representation in O(n 2 ) for a CSP with n variables and p constraints n 2 variables p(n(n 1) + 2)/2 constraints Back to the boxes Direct definition of the needed tools for the resolution September 13, / 22

21 Octagonal Consistency Octagonal HC4 Octagonal Consistency v 2 v 1 September 13, / 22

22 Octagonal Consistency Octagonal HC4 Octagonal Consistency v 2 v 1 HC4 September 13, / 22

23 Octagonal Consistency Octagonal HC4 Octagonal Consistency v 2 v 1 v 2 HC4 v 1 September 13, / 22

24 Octagonal Consistency Octagonal HC4 Octagonal Consistency v 2 v 1 v 2 v 1 September 13, / 22

25 Octagonal Consistency Octagonal HC4 Octagonal Consistency September 13, / 22

26 Octagonal Consistency Octagonal HC4 Octagonal Consistency September 13, / 22

27 Octagonal Consistency Octagonal HC4 Octagonal Consistency Use the DBM and apply the consistency September 13, / 22

28 Octagonal Consistency Octagonal HC4 Octagonal Consistency Use the DBM and apply the consistency The canonical DBM corresponds to the smallest octagon September 13, / 22

29 Octagonal Exploration Splitting Process Octagonal Solving A splitting operator, splits a variable domain v 2 v 2 v 2 v 1 v 1 v 1 September 13, / 22

30 Octagonal Exploration Splitting Process Octagonal Solving A splitting operator, splits a variable domain v 2 v 1 v 2 v 1 v 2 v 1 v 2 v 2 v 2 v 1 v 1 v 1 September 13, / 22

31 Octagonal Exploration Choice Heuristic Octagonal Solving September 13, / 22

32 Octagonal Exploration Choice Heuristic Octagonal Solving September 13, / 22

33 Octagonal Exploration Choice Heuristic Octagonal Solving Take the "best" basis, the box with the minimum of the maximum width Split the largest domain in this basis, the domain with the maximum width September 13, / 22

34 Octagonal Solving Octagonal Solving What we got We defined: an octagonal consistency Correctness Assume that, for all constraint C there exists a propagator ρ C, such that ρ C reaches Hull consistency, that is, ρ C (D 1... D n ) is the Hull consistent box for C. Then the propagation scheme as defined computes the Oct-consistent octagon for the constraints September 13, / 22

35 Octagonal Solving What we got Octagonal Solving We defined: an octagonal consistency a splitting operator Completeness The union of the two octagonal subdomains is the original octagon, thus the split does not lose solutions September 13, / 22

36 Octagonal Solving Octagonal Solving What we got We defined: an octagonal consistency a splitting operator a choice heuristic a precision (see paper) We obtain an Octagonal Solver which is correct and complete September 13, / 22

37 Results Octagonal Exploration Experiments Protocol Prototype in Ibex [Chabert and Jaulin, 2009] Different type of problems from the COCONUT benchmark Same configuration Timeout of 3 hours September 13, / 22

38 Results Octagonal Exploration Results Octagonal Exploration First solution All the solutions name nbvar ctrs I n Oct I n Oct h hs h KinematicPair pramanik 3 = trigo1 10 = brent = h74 5 = fredtest 6 = CPU time in seconds September 13, / 22

39 Results Octagonal Exploration Results Octagonal Exploration First solution All the solutions name nbvar ctrs I n Oct I n Oct h hs h KinematicPair pramanik 3 = trigo1 10 = brent = h74 5 = fredtest 6 = Number of created nodes during the search September 13, / 22

40 Conclusion and Perspectives Conclusion and Perspectives Conclusion No need to be Cartesian Paradox: New representation in O(n 2 ) but efficient Efficient Octagonal-consistency thanks to the modified version of Floyd-Warshall algorithm Good choice heuristic Use existent consistencies to define the Octagonal-consistency Future work Improvement of the implementation Constraints rewriting Improvement of the octagonal constraints Other numerical abstract domains September 13, / 22

41 CPAIOR 2012 C for CPAIOR 2012 Remember, Remember the 5th of December 2011 (Deadline) M. Pelleau, C. Truchet, F. Benhamou (TASC, University TASC, of Nantes) Nantes September 13, / 22

42 Questions Questions? September 13, / 22