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asprin : Answer Set Programming with Preferences Javier Romero University of Potsdam Javier Romero (KRR@UP) asprin : ASP with Preferences 1 / 55 Outline 1 Introduction 2 Declarative Problem Solving 3 An Example 4 Preliminaries 5 Language 6

  1. An Example Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city { travel(X,Y) } :- road(X,Y,D). visited(Y) :- start(X), travel(X,Y). visited(Y) :- visited(X), travel(X,Y). :- city(X), not visited(X). :- city(X), 2 { travel(X,Y) }. Javier Romero (KRR@UP) asprin : ASP with Preferences 14 / 55

  2. An Example Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city { travel(X,Y) } :- road(X,Y,D). visited(Y) :- start(X), travel(X,Y). visited(Y) :- visited(X), travel(X,Y). :- city(X), not visited(X). :- city(X), 2 { travel(X,Y) }. Javier Romero (KRR@UP) asprin : ASP with Preferences 14 / 55

  3. An Example Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city { travel(X,Y) } :- road(X,Y,D). visited(Y) :- start(X), travel(X,Y). visited(Y) :- visited(X), travel(X,Y). :- city(X), not visited(X). :- city(X), 2 { travel(X,Y) }. Javier Romero (KRR@UP) asprin : ASP with Preferences 14 / 55

  4. An Example Travelling Salesperson Problem Solution ✻ Modeling Interpreting ❄ ✲ Logic Program Stable Models Solving Javier Romero (KRR@UP) asprin : ASP with Preferences 15 / 55

  5. An Example Travelling Salesperson $ clingo tsp.lp 0 clingo version 5.1.0 Reading from tsp.lp Solving... Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... Answer: 2 travel(a,b) travel(b,d) travel(d,c) travel(c,a) ... Answer: 3 travel(a,c) travel(c,b) travel(b,d) travel(d,a) ... SATISFIABLE Models : 3 Javier Romero (KRR@UP) asprin : ASP with Preferences 16 / 55

  6. An Example Travelling Salesperson $ clingo tsp.lp 0 clingo version 5.1.0 Reading from tsp.lp Solving... Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... Answer: 2 travel(a,b) travel(b,d) travel(d,c) travel(c,a) ... Answer: 3 travel(a,c) travel(c,b) travel(b,d) travel(d,a) ... SATISFIABLE Models : 3 Javier Romero (KRR@UP) asprin : ASP with Preferences 16 / 55

  7. An Example Travelling Salesperson Problem Solution ✻ Modeling Interpreting ❄ ✲ Logic Program Stable Models Solving Javier Romero (KRR@UP) asprin : ASP with Preferences 17 / 55

  8. An Example A route Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... Javier Romero (KRR@UP) asprin : ASP with Preferences 18 / 55

  9. An Example A route Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... 4 a b 3 3 4 c d Javier Romero (KRR@UP) asprin : ASP with Preferences 18 / 55

  10. An Example A route Answer: 2 travel(a,b) travel(b,d) travel(d,c) travel(c,a) ... 4 a b 5 5 4 c d Javier Romero (KRR@UP) asprin : ASP with Preferences 18 / 55

  11. An Example A route Answer: 3 travel(a,c) travel(c,b) travel(b,d) travel(d,a) ... a b 5 3 3 5 c d Javier Romero (KRR@UP) asprin : ASP with Preferences 18 / 55

  12. An Example asprin for Declarative Problem Solving Problem Solution ✻ Modeling Interpreting ❄ Logic Program ✲ Optimal with Preferences Stable Models Solving Javier Romero (KRR@UP) asprin : ASP with Preferences 19 / 55

  13. An Example asprin for Declarative Problem Solving Problem Solution ✻ Modeling Interpreting ❄ Logic Program Optimal ✲ with Preferences Stable Models Solving Javier Romero (KRR@UP) asprin : ASP with Preferences 19 / 55

  14. An Example Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer routes of minimum distance. 4 a b 5 3 3 5 4 c d Javier Romero (KRR@UP) asprin : ASP with Preferences 20 / 55

  15. An Example Travelling Salesperson Problem Solution ✻ Modeling Interpreting ❄ Logic Program Optimal ✲ Stable Models with Preferences Solving Javier Romero (KRR@UP) asprin : ASP with Preferences 21 / 55

  16. An Example Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer routes of minimum distance. #preference(distance,less(weight)){ D,(X,Y) :: travel(X,Y), road(X,Y,D) }. #optimize(distance). Javier Romero (KRR@UP) asprin : ASP with Preferences 22 / 55

  17. An Example Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer routes of minimum distance. #preference(distance,less(weight)){ D,(X,Y) :: travel(X,Y), road(X,Y,D) }. #optimize(distance). Javier Romero (KRR@UP) asprin : ASP with Preferences 22 / 55

  18. An Example Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer routes of minimum distance. #preference(distance,less(weight)){ D,(X,Y) :: travel(X,Y), road(X,Y,D) }. #optimize(distance). Javier Romero (KRR@UP) asprin : ASP with Preferences 22 / 55

  19. An Example Travelling Salesperson Problem Solution ✻ Modeling Interpreting ❄ Logic Program Optimal ✲ with Preferences Stable Models Solving Javier Romero (KRR@UP) asprin : ASP with Preferences 23 / 55

  20. An Example Travelling Salesperson $ asprin tsp.lp preference.lp 0 asprin version 3.0.0 Reading from tsp.lp ... Solving... Answer: 1 travel(a,c) travel(c,b) travel(b,d) travel(d,a) ... Answer: 2 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... OPTIMUM FOUND Models : 2 Optimum : yes Optimal : 1 Javier Romero (KRR@UP) asprin : ASP with Preferences 24 / 55

  21. An Example Travelling Salesperson $ asprin tsp.lp preference.lp 0 asprin version 3.0.0 Reading from tsp.lp ... Solving... Answer: 1 travel(a,c) travel(c,b) travel(b,d) travel(d,a) ... Answer: 2 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... OPTIMUM FOUND Models : 2 Optimum : yes Optimal : 1 Javier Romero (KRR@UP) asprin : ASP with Preferences 24 / 55

  22. An Example Travelling Salesperson Problem Solution ✻ Modeling Interpreting ❄ Logic Program Optimal ✲ with Preferences Stable Models Solving Javier Romero (KRR@UP) asprin : ASP with Preferences 25 / 55

  23. An Example A route Answer: 1 travel(a,c) travel(c,b) travel(b,d) travel(d,a) ... Answer: 2 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... OPTIMUM FOUND a b 5 3 3 5 c d Javier Romero (KRR@UP) asprin : ASP with Preferences 26 / 55

  24. An Example An optimal route Answer: 1 travel(a,c) travel(c,b) travel(b,d) travel(d,a) ... Answer: 2 travel(a,b) travel(b,c) travel(c,d) travel(d,a) ... OPTIMUM FOUND 4 a b 3 3 4 c d Javier Romero (KRR@UP) asprin : ASP with Preferences 26 / 55

  25. An Example Extending Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer going along the outside coast if it is cloudy, else prefer going through the inside mountains. type(X,Y,mountain) :- road(X,Y,D), D == 5. type(X,Y,coast) :- road(X,Y,D), D != 5. #preference(weather,aso){ travel(X,Y), type(X,Y,coast) >> travel(X’,Y), type(X’,Y,mountain) || cloudy; travel(X,Y), type(X,Y,mountain) >> travel(X’,Y), type(X’,Y,coast) || not cloudy }. #optimize(weather). Javier Romero (KRR@UP) asprin : ASP with Preferences 27 / 55

  26. An Example Extending Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer going along the outside coast if it is cloudy, else prefer going through the inside mountains. type(X,Y,mountain) :- road(X,Y,D), D == 5. type(X,Y,coast) :- road(X,Y,D), D != 5. #preference(weather,aso){ travel(X,Y), type(X,Y,coast) >> travel(X’,Y), type(X’,Y,mountain) || cloudy; travel(X,Y), type(X,Y,mountain) >> travel(X’,Y), type(X’,Y,coast) || not cloudy }. #optimize(weather). Javier Romero (KRR@UP) asprin : ASP with Preferences 27 / 55

  27. An Example Extending Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer going along the outside coast if it is cloudy, else prefer going through the inside mountains. type(X,Y,mountain) :- road(X,Y,D), D == 5. type(X,Y,coast) :- road(X,Y,D), D != 5. #preference(weather,aso){ travel(X,Y), type(X,Y,coast) >> travel(X’,Y), type(X’,Y,mountain) || cloudy; travel(X,Y), type(X,Y,mountain) >> travel(X’,Y), type(X’,Y,coast) || not cloudy }. #optimize(weather). Javier Romero (KRR@UP) asprin : ASP with Preferences 27 / 55

  28. An Example Extending Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer going along the outside coast if it is cloudy, else prefer going through the inside mountains. type(X,Y,mountain) :- road(X,Y,D), D == 5. type(X,Y,coast) :- road(X,Y,D), D != 5. #preference(weather,aso){ travel(X,Y), type(X,Y,coast) >> travel(X’,Y), type(X’,Y,mountain) || cloudy; travel(X,Y), type(X,Y,mountain) >> travel(X’,Y), type(X’,Y,coast) || not cloudy }. #optimize(weather). Javier Romero (KRR@UP) asprin : ASP with Preferences 27 / 55

  29. An Example Extending Travelling Salesperson $ asprin tsp.lp preference.lp 0 asprin version 3.0.0 Reading from tsp.lp ... Solving... Solving... Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) Answer: 2 travel(a,c) travel(b,d) travel(c,b) travel(d,a) OPTIMUM FOUND Answer: 3 travel(a,b) travel(b,d) travel(c,a) travel(d,c) OPTIMUM FOUND Models : 3 Optimum : yes Optimal : 2 Javier Romero (KRR@UP) asprin : ASP with Preferences 28 / 55

  30. An Example Extending Travelling Salesperson $ asprin tsp.lp preference.lp 0 asprin version 3.0.0 Reading from tsp.lp ... Solving... Solving... Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) Answer: 2 travel(a,c) travel(b,d) travel(c,b) travel(d,a) OPTIMUM FOUND Answer: 3 travel(a,b) travel(b,d) travel(c,a) travel(d,c) OPTIMUM FOUND Models : 3 Optimum : yes Optimal : 2 Javier Romero (KRR@UP) asprin : ASP with Preferences 28 / 55

  31. An Example Extending Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer routes of minimum distance. Prefer going through the outside coast if it is cloudy, else prefer going through the inside mountains. Combine both preferences with Pareto. #preference(all,pareto){ **distance; **weather }. #optimize(all). Javier Romero (KRR@UP) asprin : ASP with Preferences 29 / 55

  32. An Example Extending Travelling Salesperson Problem class Find a route visiting every city once, and returning to the starting city. Prefer routes of minimum distance. Prefer going through the outside coast if it is cloudy, else prefer going through the inside mountains. Combine both preferences with Pareto. #preference(all,pareto){ **distance; **weather }. #optimize(all). Javier Romero (KRR@UP) asprin : ASP with Preferences 29 / 55

  33. An Example Extending Travelling Salesperson $ asprin tsp.lp preference.lp 0 asprin version 3.0.0 Reading from tsp.lp ... Solving... Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) OPTIMUM FOUND Answer: 2 travel(a,c) travel(b,d) travel(c,b) travel(d,a) OPTIMUM FOUND Answer: 3 travel(a,b) travel(b,d) travel(c,a) travel(d,c) OPTIMUM FOUND Models : 3 Optimum : yes Optimal : 3 Javier Romero (KRR@UP) asprin : ASP with Preferences 30 / 55

  34. An Example Extending Travelling Salesperson $ asprin tsp.lp preference.lp 0 asprin version 3.0.0 Reading from tsp.lp ... Solving... Answer: 1 travel(a,b) travel(b,c) travel(c,d) travel(d,a) OPTIMUM FOUND Answer: 2 travel(a,c) travel(b,d) travel(c,b) travel(d,a) OPTIMUM FOUND Answer: 3 travel(a,b) travel(b,d) travel(c,a) travel(d,c) OPTIMUM FOUND Models : 3 Optimum : yes Optimal : 3 Javier Romero (KRR@UP) asprin : ASP with Preferences 30 / 55

  35. Preliminaries Outline 1 Introduction 2 Declarative Problem Solving 3 An Example 4 Preliminaries 5 Language 6 Implementation 7 Experimental analysis Boosting optimization via heuristics Dedicated systems versus asprin 8 Summary Javier Romero (KRR@UP) asprin : ASP with Preferences 31 / 55

  36. Preliminaries Preference A strict partial order ≻ on the stable models of a logic program That is, X ≻ Y means that X is preferred to Y A stable model X is ≻ -preferred, if there is no other stable model Y such that Y ≻ X A preference type is a (parametric) class of preference relations Example subset , pareto , etc. Javier Romero (KRR@UP) asprin : ASP with Preferences 32 / 55

  37. Preliminaries Preference A strict partial order ≻ on the stable models of a logic program That is, X ≻ Y means that X is preferred to Y A stable model X is ≻ -preferred, if there is no other stable model Y such that Y ≻ X A preference type is a (parametric) class of preference relations Example subset , pareto , etc. Javier Romero (KRR@UP) asprin : ASP with Preferences 32 / 55

  38. Preliminaries Preference A strict partial order ≻ on the stable models of a logic program That is, X ≻ Y means that X is preferred to Y A stable model X is ≻ -preferred, if there is no other stable model Y such that Y ≻ X A preference type is a (parametric) class of preference relations Example subset , pareto , etc. Javier Romero (KRR@UP) asprin : ASP with Preferences 32 / 55

  39. Language Outline 1 Introduction 2 Declarative Problem Solving 3 An Example 4 Preliminaries 5 Language 6 Implementation 7 Experimental analysis Boosting optimization via heuristics Dedicated systems versus asprin 8 Summary Javier Romero (KRR@UP) asprin : ASP with Preferences 33 / 55

  40. Language Language naming atom ∗∗ s where s is the name of a preference weighted formula w 1 , . . . , w l : φ where each w i is a term and φ is a Boolean formula preference element Φ 1 ≫ · · · ≫ Φ m � Φ m +1 where each Φ r is a set of weighted formulas preference statement # preference ( s , t ) { e 1 , . . . , e n } where s and t represent the preference statement and its type and each e j is a preference element optimization directive # optimize ( s ) where s is the name of a preference preference specification is a set S of preference statements and a directive # optimize ( s ) such that S is acyclic, closed, and s ∈ S Javier Romero (KRR@UP) asprin : ASP with Preferences 34 / 55

  41. Language Language naming atom ∗∗ s where s is the name of a preference weighted formula w 1 , . . . , w l : φ where each w i is a term and φ is a Boolean formula preference element Φ 1 ≫ · · · ≫ Φ m � Φ m +1 where each Φ r is a set of weighted formulas preference statement # preference ( s , t ) { e 1 , . . . , e n } where s and t represent the preference statement and its type and each e j is a preference element optimization directive # optimize ( s ) where s is the name of a preference preference specification is a set S of preference statements and a directive # optimize ( s ) such that S is acyclic, closed, and s ∈ S Javier Romero (KRR@UP) asprin : ASP with Preferences 34 / 55

  42. Language Language naming atom ∗∗ s where s is the name of a preference weighted formula w 1 , . . . , w l : φ where each w i is a term and φ is a Boolean formula preference element Φ 1 ≫ · · · ≫ Φ m � Φ m +1 where each Φ r is a set of weighted formulas preference statement # preference ( s , t ) { e 1 , . . . , e n } where s and t represent the preference statement and its type and each e j is a preference element optimization directive # optimize ( s ) where s is the name of a preference preference specification is a set S of preference statements and a directive # optimize ( s ) such that S is acyclic, closed, and s ∈ S Javier Romero (KRR@UP) asprin : ASP with Preferences 34 / 55

  43. Language Language naming atom ∗∗ s where s is the name of a preference weighted formula w 1 , . . . , w l : φ where each w i is a term and φ is a Boolean formula preference element Φ 1 ≫ · · · ≫ Φ m � Φ m +1 where each Φ r is a set of weighted formulas preference statement # preference ( s , t ) { e 1 , . . . , e n } where s and t represent the preference statement and its type and each e j is a preference element optimization directive # optimize ( s ) where s is the name of a preference preference specification is a set S of preference statements and a directive # optimize ( s ) such that S is acyclic, closed, and s ∈ S Javier Romero (KRR@UP) asprin : ASP with Preferences 34 / 55

  44. Language Language naming atom ∗∗ s where s is the name of a preference weighted formula w 1 , . . . , w l : φ where each w i is a term and φ is a Boolean formula preference element Φ 1 ≫ · · · ≫ Φ m � Φ m +1 where each Φ r is a set of weighted formulas preference statement # preference ( s , t ) { e 1 , . . . , e n } where s and t represent the preference statement and its type and each e j is a preference element optimization directive # optimize ( s ) where s is the name of a preference preference specification is a set S of preference statements and a directive # optimize ( s ) such that S is acyclic, closed, and s ∈ S Javier Romero (KRR@UP) asprin : ASP with Preferences 34 / 55

  45. Language Language naming atom ∗∗ s where s is the name of a preference weighted formula w 1 , . . . , w l : φ where each w i is a term and φ is a Boolean formula preference element Φ 1 ≫ · · · ≫ Φ m � Φ m +1 where each Φ r is a set of weighted formulas preference statement # preference ( s , t ) { e 1 , . . . , e n } where s and t represent the preference statement and its type and each e j is a preference element optimization directive # optimize ( s ) where s is the name of a preference preference specification is a set S of preference statements and a directive # optimize ( s ) such that S is acyclic, closed, and s ∈ S Javier Romero (KRR@UP) asprin : ASP with Preferences 34 / 55

  46. Language Language naming atom ∗∗ s where s is the name of a preference weighted formula w 1 , . . . , w l : φ where each w i is a term and φ is a Boolean formula preference element Φ 1 ≫ · · · ≫ Φ m � Φ m +1 where each Φ r is a set of weighted formulas preference statement # preference ( s , t ) { e 1 , . . . , e n } where s and t represent the preference statement and its type and each e j is a preference element optimization directive # optimize ( s ) where s is the name of a preference preference specification is a set S of preference statements and a directive # optimize ( s ) such that S is acyclic, closed, and s ∈ S Javier Romero (KRR@UP) asprin : ASP with Preferences 34 / 55

  47. Language Preference type A preference type t is a function mapping a set of preference elements E to a preference relation t ( E ) ⊆ A × A Examples ( X , Y ) ∈ subset ( E ) iff { ℓ ∈ E | X | = ℓ } ⊂ { ℓ ∈ E | Y | = ℓ } ( X , Y ) ∈ pareto ( E ) iff � ∗∗ s ∈ E ( X � s Y ) ∧ � ∗∗ s ∈ E ( X ≻ s Y ) Javier Romero (KRR@UP) asprin : ASP with Preferences 35 / 55

  48. Language Preference type A preference type t is a function mapping a set of preference elements E to a preference relation t ( E ) ⊆ A × A Examples ( X , Y ) ∈ subset ( E ) iff { ℓ ∈ E | X | = ℓ } ⊂ { ℓ ∈ E | Y | = ℓ } ( X , Y ) ∈ pareto ( E ) iff � ∗∗ s ∈ E ( X � s Y ) ∧ � ∗∗ s ∈ E ( X ≻ s Y ) Javier Romero (KRR@UP) asprin : ASP with Preferences 35 / 55

  49. Language Preference relation A preference relation is obtained by applying a preference type to a set of preference elements # preference ( s , t ) E declares preference relation t ( E ), denoted by ≻ s Example # preference (1 , subset ) { a , b , c } declares X ≻ 1 Y iff { ℓ ∈ { a , b , c } | X | = ℓ } ⊂ { ℓ ∈ { a , b , c } | Y | = ℓ } Javier Romero (KRR@UP) asprin : ASP with Preferences 36 / 55

  50. Language Preference relation A preference relation is obtained by applying a preference type to a set of preference elements # preference ( s , t ) E declares preference relation t ( E ), denoted by ≻ s Example # preference (1 , subset ) { a , b , c } declares X ≻ 1 Y iff { ℓ ∈ { a , b , c } | X | = ℓ } ⊂ { ℓ ∈ { a , b , c } | Y | = ℓ } Javier Romero (KRR@UP) asprin : ASP with Preferences 36 / 55

  51. Language Preference relation A preference relation is obtained by applying a preference type to a set of preference elements # preference ( s , t ) E declares preference relation t ( E ), denoted by ≻ s Example # preference (1 , subset ) { a , b , c } declares X ≻ 1 Y iff { ℓ ∈ { a , b , c } | X | = ℓ } ⊂ { ℓ ∈ { a , b , c } | Y | = ℓ } Javier Romero (KRR@UP) asprin : ASP with Preferences 36 / 55

  52. Implementation Outline 1 Introduction 2 Declarative Problem Solving 3 An Example 4 Preliminaries 5 Language 6 Implementation 7 Experimental analysis Boosting optimization via heuristics Dedicated systems versus asprin 8 Summary Javier Romero (KRR@UP) asprin : ASP with Preferences 37 / 55

  53. Implementation Preference program X = { holds ′ ( a ) | a ∈ X } Reification H X = { holds ( a ) | a ∈ X } and H ′ Preference program Let s be a preference statement declaring ≻ s . We define Q s as a preference program for s , if for all sets X , Y ⊆ A , we have iff Q s ∪ H X ∪ H ′ X ≻ s Y Y is satisfiable Note Q s is implemented as F s ∪ E t s ∪ C Note asprin ’s expressiveness is delineated by the decision problem Javier Romero (KRR@UP) asprin : ASP with Preferences 38 / 55

  54. Implementation Preference program X = { holds ′ ( a ) | a ∈ X } Reification H X = { holds ( a ) | a ∈ X } and H ′ Preference program Let s be a preference statement declaring ≻ s . We define Q s as a preference program for s , if for all sets X , Y ⊆ A , we have iff Q s ∪ H X ∪ H ′ X ≻ s Y Y is satisfiable Note Q s is implemented as F s ∪ E t s ∪ C Note asprin ’s expressiveness is delineated by the decision problem Javier Romero (KRR@UP) asprin : ASP with Preferences 38 / 55

  55. Implementation Preference program X = { holds ′ ( a ) | a ∈ X } Reification H X = { holds ( a ) | a ∈ X } and H ′ Preference program Let s be a preference statement declaring ≻ s . We define Q s as a preference program for s , if for all sets X , Y ⊆ A , we have iff Q s ∪ H X ∪ H ′ X ≻ s Y Y is satisfiable Note Q s is implemented as F s ∪ E t s ∪ C Note asprin ’s expressiveness is delineated by the decision problem Javier Romero (KRR@UP) asprin : ASP with Preferences 38 / 55

  56. Implementation Preference program X = { holds ′ ( a ) | a ∈ X } Reification H X = { holds ( a ) | a ∈ X } and H ′ Preference program Let s be a preference statement declaring ≻ s . We define Q s as a preference program for s , if for all sets X , Y ⊆ A , we have iff Q s ∪ H X ∪ H ′ X ≻ s Y Y is satisfiable Note Q s is implemented as F s ∪ E t s ∪ C Note asprin ’s expressiveness is delineated by the decision problem Javier Romero (KRR@UP) asprin : ASP with Preferences 38 / 55

  57. Implementation # preference (1 , subset ) { a , b , c } # optimize (1) � � H { a } = holds(a). H ′ � � { a , b } = holds ’(a). holds ’(b). � � preference (1, subset ). preference (1,1,1,for(a) ,()). = F 1 preference (1,2,1,for(b) ,()). optimize (1). preference (1,3,1,for(c) ,()). � � better(P) :- preference (P,subset), E subset = not holds(X), holds ’(X), preference(P,_,_,for(X),_), not holds(Y) : not holds ’(Y), preference(P,_,_,for(Y),_). � � C = :- optimize(P), not better(P). There is a stable model, indicating that { a } ≻ 1 { a , b } Javier Romero (KRR@UP) asprin : ASP with Preferences 39 / 55

  58. Implementation # preference (1 , subset ) { a , b , c } # optimize (1) � � H { a } = holds(a). H ′ � � { a , b } = holds ’(a). holds ’(b). � � preference (1, subset ). preference (1,1,1,for(a) ,()). = F 1 preference (1,2,1,for(b) ,()). optimize (1). preference (1,3,1,for(c) ,()). � � better(P) :- preference (P,subset), E subset = not holds(X), holds ’(X), preference(P,_,_,for(X),_), not holds(Y) : not holds ’(Y), preference(P,_,_,for(Y),_). � � C = :- optimize(P), not better(P). There is a stable model, indicating that { a } ≻ 1 { a , b } Javier Romero (KRR@UP) asprin : ASP with Preferences 39 / 55

  59. Implementation # preference (1 , subset ) { a , b , c } # optimize (1) � � H { a } = holds(a). H ′ � � { a , b } = holds ’(a). holds ’(b). � � preference (1, subset ). preference (1,1,1,for(a) ,()). = F 1 preference (1,2,1,for(b) ,()). optimize (1). preference (1,3,1,for(c) ,()). � � better(P) :- preference (P,subset), E subset = not holds(X), holds ’(X), preference(P,_,_,for(X),_), not holds(Y) : not holds ’(Y), preference(P,_,_,for(Y),_). � � C = :- optimize(P), not better(P). There is a stable model, indicating that { a } ≻ 1 { a , b } Javier Romero (KRR@UP) asprin : ASP with Preferences 39 / 55

  60. Implementation # preference (1 , subset ) { a , b , c } # optimize (1) � � H { a , b } = holds(a). holds(b). H ′ � � = holds ’(a). { a } � � preference (1, subset ). preference (1,1,1,for(a) ,()). = F 1 preference (1,2,1,for(b) ,()). optimize (1). preference (1,3,1,for(c) ,()). � � better(P) :- preference (P,subset), E subset = not holds(X), holds ’(X), preference(P,_,_,for(X),_), not holds(Y) : not holds ’(Y), preference(P,_,_,for(Y),_). � � C = :- optimize(P), not better(P). There is no stable model, indicating that { a , b } ⊁ 1 { a } Javier Romero (KRR@UP) asprin : ASP with Preferences 40 / 55

  61. Implementation Basic algorithm solveOpt ( P , s ) : A program P over A and preference statement s Input Output : A ≻ s -preferred stable model of P , if P is satisfiable, and ⊥ otherwise Y ← solve ( P ) if Y = ⊥ then return ⊥ repeat X ← Y P ∪ Q s ∪ R ∪ H ′ � � Y ← solve X until Y = ⊥ return X where R = { holds ( a ) ← a | a ∈ A} Javier Romero (KRR@UP) asprin : ASP with Preferences 41 / 55

  62. Implementation asprin ’s library Basic preference types subset and superset less(cardinality) and more(cardinality) less(weight) and more(weight) maxmin and minmax aso (Answer Set Optimization) poset (Qualitative Preferences) Composite preference types neg and pareto lexico And more to come. . . cp (restricted CP nets) Javier Romero (KRR@UP) asprin : ASP with Preferences 42 / 55

  63. Implementation asprin ’s library Basic preference types subset and superset less(cardinality) and more(cardinality) less(weight) and more(weight) maxmin and minmax aso (Answer Set Optimization) poset (Qualitative Preferences) Composite preference types neg and pareto lexico And more to come. . . cp (restricted CP nets) Javier Romero (KRR@UP) asprin : ASP with Preferences 42 / 55

  64. Experimental analysis Outline 1 Introduction 2 Declarative Problem Solving 3 An Example 4 Preliminaries 5 Language 6 Implementation 7 Experimental analysis Boosting optimization via heuristics Dedicated systems versus asprin 8 Summary Javier Romero (KRR@UP) asprin : ASP with Preferences 43 / 55

  65. Experimental analysis Boosting optimization via heuristics Outline 1 Introduction 2 Declarative Problem Solving 3 An Example 4 Preliminaries 5 Language 6 Implementation 7 Experimental analysis Boosting optimization via heuristics Dedicated systems versus asprin 8 Summary Javier Romero (KRR@UP) asprin : ASP with Preferences 44 / 55

  66. Experimental analysis Boosting optimization via heuristics Heuristic framework clingo allows for incorporating domain-specific heuristics into ASP solving input language for expressing domain-specific heuristics solving capacities for integrating domain-specific heuristics Heuristic directive #heuristic Heuristic modifiers (atom, a , and integer, v ) init for initializing the heuristic value of a with v factor for amplifying the heuristic value of a by factor v level for ranking all atoms; the rank of a is v sign for attributing the sign of v as truth value to a Heuristic statements #heuristic occurs(A,T) : action(A), time(T). [T,factor] Javier Romero (KRR@UP) asprin : ASP with Preferences 45 / 55

  67. Experimental analysis Boosting optimization via heuristics Heuristic framework clingo allows for incorporating domain-specific heuristics into ASP solving input language for expressing domain-specific heuristics solving capacities for integrating domain-specific heuristics Heuristic directive #heuristic Heuristic modifiers (atom, a , and integer, v ) init for initializing the heuristic value of a with v factor for amplifying the heuristic value of a by factor v level for ranking all atoms; the rank of a is v sign for attributing the sign of v as truth value to a Heuristic statements #heuristic occurs(A,T) : action(A), time(T). [T,factor] Javier Romero (KRR@UP) asprin : ASP with Preferences 45 / 55

  68. Experimental analysis Boosting optimization via heuristics Heuristic framework clingo allows for incorporating domain-specific heuristics into ASP solving input language for expressing domain-specific heuristics solving capacities for integrating domain-specific heuristics Heuristic directive #heuristic Heuristic modifiers (atom, a , and integer, v ) init for initializing the heuristic value of a with v factor for amplifying the heuristic value of a by factor v level for ranking all atoms; the rank of a is v sign for attributing the sign of v as truth value to a Heuristic statements #heuristic occurs(A,T) : action(A), time(T). [T,factor] Javier Romero (KRR@UP) asprin : ASP with Preferences 45 / 55

  69. Experimental analysis Boosting optimization via heuristics Heuristic framework clingo allows for incorporating domain-specific heuristics into ASP solving input language for expressing domain-specific heuristics solving capacities for integrating domain-specific heuristics Heuristic directive #heuristic Heuristic modifiers (atom, a , and integer, v ) init for initializing the heuristic value of a with v factor for amplifying the heuristic value of a by factor v level for ranking all atoms; the rank of a is v sign for attributing the sign of v as truth value to a Heuristic statements #heuristic occurs(A,T) : action(A), time(T). [T,factor] Javier Romero (KRR@UP) asprin : ASP with Preferences 45 / 55

  70. Experimental analysis Boosting optimization via heuristics Heuristic framework clingo allows for incorporating domain-specific heuristics into ASP solving input language for expressing domain-specific heuristics solving capacities for integrating domain-specific heuristics Heuristic directive #heuristic Heuristic modifiers (atom, a , and integer, v ) init for initializing the heuristic value of a with v factor for amplifying the heuristic value of a by factor v level for ranking all atoms; the rank of a is v sign for attributing the sign of v as truth value to a Heuristic statements #heuristic occurs(mv,5) : action(mv), time(5). [5,factor] Javier Romero (KRR@UP) asprin : ASP with Preferences 45 / 55

  71. Experimental analysis Boosting optimization via heuristics asprin with different heuristic settings Benchmark \ System asprin asprin w + s asprin w + l asprin w + f asprin asprin s + s asprin s + l asprin s + f w s Ricochet (30) 20.00 432 ( 8, 4) 407 ( 7, 4) 68 ( 1, 0) 71 (1, 0) 365 ( 8, 3) 461 ( 7, 10) 69 (1, 0) 71 (1, 0) Timetabling (12) 23687.75 345 (285, 3) 255 (202, 2) 900 ( 4, 12) 6 (1, 0) 217 (144, 2) 21 (18, 0) 900 (2, 12) 5 (1, 0) Puzzle (7) 580.57 82 ( 2, 0) 112 ( 2, 0) 136 ( 2, 0) 416 (2, 1) 31 ( 1, 0) 32 ( 1, 0) 21 (1, 0) 51 (1, 0) Crossing (24) 211.92 104 ( 42, 1) 98 ( 35, 0) 805 (19, 20) 387 (6, 6) 0 ( 6, 0) 1 ( 6, 0) 7 (9, 0) 3 (1, 0) Valves (30) 56.63 69 ( 7, 0) 65 ( 6, 0) 460 ( 8, 11) 715 (0, 22) 38 ( 4, 0) 39 ( 4, 0) 339 (4, 6) 673 (0, 21) Expansion (30) 7501.87 216 (299, 0) 10 ( 15, 0) 38 ( 7, 0) 12 (3, 0) 64 (295, 0) 14 (54, 0) 4 (4, 0) 3 (1, 0) Repair (30) 6750.73 76 ( 48, 0) 15 ( 47, 0) 71 ( 3, 2) 8 (2, 0) 8 ( 43, 0) 3 (11, 0) 1 (1, 0) 1 (1, 0) Diagnosis (30) 1669.00 196 (341, 3) 76 ( 66, 0) 43 ( 4, 0) 118 (3, 2) 19 (338, 0) 2 (39, 0) 0 (1, 0) 0 (1, 0) ∅ ( ∅ , Σ) 190 (129, 11) 130 ( 48, 6) 315 ( 6, 45) 217 (2, 31) 93 (105, 5) 72 (18, 10) 168 (3, 18) 101 (1, 21) asprin w asprin s asprin w + s asprin s + s asprin w + l asprin s + l asprin w + f asprin s + f w — weight-based s — subset-based s — sign heuristic s — sign heuristic l — level-based heuristic l — level-based heuristic f — factor-based heuristic f — factor-based heuristic Javier Romero (KRR@UP) asprin : ASP with Preferences 46 / 55

  72. Experimental analysis Boosting optimization via heuristics asprin with different heuristic settings Benchmark \ System asprin asprin w + s asprin w + l asprin w + f asprin asprin s + s asprin s + l asprin s + f w s Ricochet (30) 20.00 432 ( 8, 4) 407 ( 7, 4) 68 ( 1, 0) 71 (1, 0) 365 ( 8, 3) 461 ( 7, 10) 69 (1, 0) 71 (1, 0) Timetabling (12) 23687.75 345 (285, 3) 255 (202, 2) 900 ( 4, 12) 6 (1, 0) 217 (144, 2) 21 (18, 0) 900 (2, 12) 5 (1, 0) Puzzle (7) 580.57 82 ( 2, 0) 112 ( 2, 0) 136 ( 2, 0) 416 (2, 1) 31 ( 1, 0) 32 ( 1, 0) 21 (1, 0) 51 (1, 0) Crossing (24) 211.92 104 ( 42, 1) 98 ( 35, 0) 805 (19, 20) 387 (6, 6) 0 ( 6, 0) 1 ( 6, 0) 7 (9, 0) 3 (1, 0) Valves (30) 56.63 69 ( 7, 0) 65 ( 6, 0) 460 ( 8, 11) 715 (0, 22) 38 ( 4, 0) 39 ( 4, 0) 339 (4, 6) 673 (0, 21) Expansion (30) 7501.87 216 (299, 0) 10 ( 15, 0) 38 ( 7, 0) 12 (3, 0) 64 (295, 0) 14 (54, 0) 4 (4, 0) 3 (1, 0) Repair (30) 6750.73 76 ( 48, 0) 15 ( 47, 0) 71 ( 3, 2) 8 (2, 0) 8 ( 43, 0) 3 (11, 0) 1 (1, 0) 1 (1, 0) Diagnosis (30) 1669.00 196 (341, 3) 76 ( 66, 0) 43 ( 4, 0) 118 (3, 2) 19 (338, 0) 2 (39, 0) 0 (1, 0) 0 (1, 0) ∅ ( ∅ , Σ) 190 (129, 11) 130 ( 48, 6) 315 ( 6, 45) 217 (2, 31) 93 (105, 5) 72 (18, 10) 168 (3, 18) 101 (1, 21) asprin w asprin s asprin w + s asprin s + s asprin w + l asprin s + l asprin w + f asprin s + f w — weight-based s — subset-based s — sign heuristic s — sign heuristic l — level-based heuristic l — level-based heuristic f — factor-based heuristic f — factor-based heuristic Javier Romero (KRR@UP) asprin : ASP with Preferences 47 / 55

  73. Experimental analysis Boosting optimization via heuristics asprin with different heuristic settings Benchmark \ System asprin asprin w + s asprin w + l asprin w + f asprin asprin s + s asprin s + l asprin s + f w s Ricochet (30) 20.00 432 ( 8, 4) 407 ( 7, 4) 68 ( 1, 0) 71 (1, 0) 365 ( 8, 3) 461 ( 7, 10) 69 (1, 0) 71 (1, 0) Timetabling (12) 23687.75 345 (285, 3) 255 (202, 2) 900 ( 4, 12) 6 (1, 0) 217 (144, 2) 21 (18, 0) 900 (2, 12) 5 (1, 0) Puzzle (7) 580.57 82 ( 2, 0) 112 ( 2, 0) 136 ( 2, 0) 416 (2, 1) 31 ( 1, 0) 32 ( 1, 0) 21 (1, 0) 51 (1, 0) Crossing (24) 211.92 104 ( 42, 1) 98 ( 35, 0) 805 (19, 20) 387 (6, 6) 0 ( 6, 0) 1 ( 6, 0) 7 (9, 0) 3 (1, 0) Valves (30) 56.63 69 ( 7, 0) 65 ( 6, 0) 460 ( 8, 11) 715 (0, 22) 38 ( 4, 0) 39 ( 4, 0) 339 (4, 6) 673 (0, 21) Expansion (30) 7501.87 216 (299, 0) 10 ( 15, 0) 38 ( 7, 0) 12 (3, 0) 64 (295, 0) 14 (54, 0) 4 (4, 0) 3 (1, 0) Repair (30) 6750.73 76 ( 48, 0) 15 ( 47, 0) 71 ( 3, 2) 8 (2, 0) 8 ( 43, 0) 3 (11, 0) 1 (1, 0) 1 (1, 0) Diagnosis (30) 1669.00 196 (341, 3) 76 ( 66, 0) 43 ( 4, 0) 118 (3, 2) 19 (338, 0) 2 (39, 0) 0 (1, 0) 0 (1, 0) ∅ ( ∅ , Σ) 190 (129, 11) 130 ( 48, 6) 315 ( 6, 45) 217 (2, 31) 93 (105, 5) 72 (18, 10) 168 (3, 18) 101 (1, 21) asprin w asprin s asprin w + s asprin s + s asprin w + l asprin s + l asprin w + f asprin s + f w — weight-based s — subset-based s — sign heuristic s — sign heuristic l — level-based heuristic l — level-based heuristic f — factor-based heuristic f — factor-based heuristic Javier Romero (KRR@UP) asprin : ASP with Preferences 48 / 55

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