Octagonal Domains for Continuous Constraints Marie Pelleau - PowerPoint PPT Presentation

Octagonal Domains for Continuous Constraints Marie Pelleau Charlotte Truchet Frédéric Benhamou TASC, University of Nantes CP 2011 September 13, 2011 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 1 / 22

Table of contents Outline Context 1 Octagons 2 Computer Representation Octagonal Consistency Octagonal Solving Results 3 Conclusion 4 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 2 / 22

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 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 3 / 22

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 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 3 / 22

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 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 3 / 22

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 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 3 / 22

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 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 3 / 22

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 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 4 / 22

Octagons Computer Representation Octagons 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 2 ≤ 5 v 1 ≥ 1 In dimension n , an octagon v 1 + v 2 ≥ 3 has at most 2 n 2 faces An octagon can be unbounded v 2 ≥ 1 v 1 ≤ 5 v 1 v 1 − v 2 ≤ 2 . 5 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 5 / 22

Octagons Computer Representation Octagons The Octagon Abstract Domain A Difference Bound Matrix (DBM) to represent an octagon v j .   . . v i v j − v i ≤ c   · · · c   M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 6 / 22

Octagons Computer Representation Octagons The Octagon Abstract Domain A Difference Bound Matrix (DBM) to represent an octagon v j .   . . v i v j − v i ≤ c   · · · c   v 1 + v 2 ≤ c v 1 − ( − v 2 ) ≤ c and v 2 − ( − v 1 ) ≤ c → M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 6 / 22

Octagons Computer Representation Octagons The Octagon Abstract Domain A Difference Bound Matrix (DBM) to represent an octagon v j .   . . v i v j − v i ≤ c   · · · 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 ) M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 6 / 22

Octagons Computer Representation Octagons The Octagon Abstract Domain v 2 v 2 − v 1 ≤ 2 v 2 ≤ 5 v 1 ≥ 1 v 1 − v 1 v 2 − v 2   v 1 0 − 2 2 − 3 v 1 + v 2 ≥ 3 − v 1 10 0 + ∞ 2 . 5     v 2 2 . 5 − 3 0 − 2   v 2 ≥ 1 + ∞ 2 10 0 − v 2 v 1 ≤ 5 v 1 v 1 − v 2 ≤ 2 . 5 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 7 / 22

Octagons Computer Representation Octagons 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] M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 8 / 22

Octagons Computer Representation Octagons Representation for CP v 2 v 2 − v 1 ≤ 2 v 2 ≤ 5 v 1 ≥ 1 v 1 + v 2 ≥ 3 v 2 ≥ 1 v 1 ≤ 5 v 1 v 1 − v 2 ≤ 2 . 5 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 9 / 22

Octagons Computer Representation Octagons Representation for CP � π � π v 1 = v ′ � + v ′ � 1 cos 2 sin 4 4 � π � π v 2 = v ′ � − v ′ � 2 cos 1 sin v 2 v 2 4 4 v ′ v 2 − v 1 ≤ 2 1 v 2 ≤ 5 v 1 ≥ 1 v 1 + v 2 ≥ 3 v ′ 2 v 2 ≥ 1 v 1 ≤ 5 π 4 v 1 v 1 v 1 − v 2 ≤ 2 . 5 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 9 / 22

Octagons Computer Representation Octagons Representation for CP Rotation � π � π v 1 = v ′ � + v ′ � 1 cos 2 sin 4 4 � π � π v 2 = v ′ � − v ′ � 2 cos 1 sin 4 4 Constraint Translation 2 v 1 − v 2 ≤ 4 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 10 / 22

Octagons Computer Representation Octagons Representation for CP Rotation � π � π v 1 = v ′ � + v ′ � 1 cos 2 sin 4 4 � π � π v 2 = v ′ � − v ′ � 2 cos 1 sin 4 4 Constraint Translation 2 v 1 − v 2 ≤ 4 � π � π � π � π 2 ( v ′ � + v ′ � ) − v ′ � + v ′ � 1 cos 2 sin 2 cos 1 sin ≤ 4 ⇔ 4 4 4 4 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 10 / 22

Octagons Computer Representation Octagons Representation for CP Rotation � π � π v 1 = v ′ � + v ′ � 1 cos 2 sin 4 4 � π � π v 2 = v ′ � − v ′ � 2 cos 1 sin 4 4 Constraint Translation 2 v 1 − v 2 ≤ 4 � π � π � π � π 2 ( v ′ � + v ′ � ) − v ′ � + v ′ � 1 cos 2 sin 2 cos 1 sin ≤ 4 ⇔ 4 4 4 4 √ √ 2 2 2 v ′ 2 v ′ ⇔ 3 1 + 2 ≤ 4 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 10 / 22

Octagons Computer Representation Octagons 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 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 11 / 22

Octagons Octagonal Consistency Octagonal Consistency Octagonal HC4 v 2 v 1 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 12 / 22

Octagons Octagonal Consistency Octagonal Consistency Octagonal HC4 v 2 v 1 HC4 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 12 / 22

Octagons Octagonal Consistency Octagonal Consistency Octagonal HC4 v 2 v ′ 1 v ′ 2 4 v 1 C H M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 12 / 22

Octagons Octagonal Consistency Octagonal Consistency Octagonal HC4 v 2 v ′ 1 v ′ 2 v 1 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 12 / 22

Octagons Octagonal Consistency Octagonal Consistency Octagonal HC4 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 13 / 22

Octagons Octagonal Consistency Octagonal Consistency Octagonal HC4 M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 13 / 22

Octagons Octagonal Consistency Octagonal Consistency Octagonal HC4 Use the DBM and apply the consistency M. Pelleau, C. Truchet, F. Benhamou (TASC, University of Nantes) TASC, Nantes September 13, 2011 13 / 22