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Answer Set Solving in Practice Torsten Schaub University of Potsdam Torsten Schaub (KRR@UP) ASP in Practice 1 / 69 Outline 1 Motivation 2 Introduction 3 Modeling by Example Graph Coloring Queens Traveling Salesperson 4 Meta Programming 5

  1. Introduction Basic idea Consider the logical formula Φ and its three Φ q ∧ ( q ∧ ¬ r → p ) (classical) models: { p , q } , { q , r } , and { p , q , r } Formula Φ has one stable model, ← P Φ q often called answer set: ← q , ∼ r p { p , q } Informally, a set X of atoms is an stable model of a logic program P if X is a (classical) model of P and if all atoms in X are justified by some rule in P (rooted in intuitionistic logics HT (Heyting, 1930) and G3 (G¨ odel, 1932)) Torsten Schaub (KRR@UP) ASP in Practice 8 / 69

  2. Introduction Basic idea Consider the logical formula Φ and its three Φ q ∧ ( q ∧ ¬ r → p ) (classical) models: { p , q } , { q , r } , and { p , q , r } Formula Φ has one stable model, ← P Φ q often called answer set: ← q , ∼ r p { p , q } Informally, a set X of atoms is an stable model of a logic program P if X is a (classical) model of P and if all atoms in X are justified by some rule in P (rooted in intuitionistic logics HT (Heyting, 1930) and G3 (G¨ odel, 1932)) Torsten Schaub (KRR@UP) ASP in Practice 8 / 69

  3. Introduction ASP versus SAT ASP SAT Model generation Bottom-up Constructive Logic Classical Logic Closed (and open) Open world reasoning world reasoning Modeling language — Complex reasoning modes Satisfiability testing Satisfiability Satisfiability Enumeration/Projection — Optimization — Intersection/Union — (Turing +) NP ( NP ) NP Torsten Schaub (KRR@UP) ASP in Practice 9 / 69

  4. Introduction ASP versus SAT ASP SAT Model generation Bottom-up Constructive Logic Classical Logic Closed (and open) Open world reasoning world reasoning Modeling language — Complex reasoning modes Satisfiability testing Satisfiability Satisfiability Enumeration/Projection — Optimization — Intersection/Union — (Turing +) NP ( NP ) NP Torsten Schaub (KRR@UP) ASP in Practice 9 / 69

  5. Introduction Formal Definition Syntax A rule, r , is an expression of the form a ← b 1 , . . . , b m , ∼ c 1 , . . . , ∼ c n , where 0 ≤ m , n and each a , b i , c j is an atom A logic program is a finite set of rules Semantics The reduct, P X , of a program P relative to a set X of atoms is defined by P X = { a ← b 1 , . . . , b m | r ∈ P and { c 1 , . . . , c n } ∩ X = ∅} The ⊆ –smallest model of P X is denoted by Cn ( P X ) A set X of atoms is an stable model of a program P , if X = Cn ( P X ) Torsten Schaub (KRR@UP) ASP in Practice 10 / 69

  6. Introduction Formal Definition Syntax A rule, r , is an expression of the form a ← b 1 , . . . , b m , ∼ c 1 , . . . , ∼ c n , where 0 ≤ m , n and each a , b i , c j is an atom A logic program is a finite set of rules Semantics The reduct, P X , of a program P relative to a set X of atoms is defined by P X = { a ← b 1 , . . . , b m | r ∈ P and { c 1 , . . . , c n } ∩ X = ∅} The ⊆ –smallest model of P X is denoted by Cn ( P X ) A set X of atoms is an stable model of a program P , if X = Cn ( P X ) Torsten Schaub (KRR@UP) ASP in Practice 10 / 69

  7. Introduction Formal Definition Syntax A rule, r , is an expression of the form a ← b 1 , . . . , b m , ∼ c 1 , . . . , ∼ c n , where 0 ≤ m , n and each a , b i , c j is an atom A logic program is a finite set of rules Semantics The reduct, P X , of a program P relative to a set X of atoms is defined by P X = { a ← b 1 , . . . , b m | r ∈ P and { c 1 , . . . , c n } ∩ X = ∅} The ⊆ –smallest model of P X is denoted by Cn ( P X ) A set X of atoms is an stable model of a program P , if X = Cn ( P X ) Torsten Schaub (KRR@UP) ASP in Practice 10 / 69

  8. Introduction Formal Definition Syntax A rule, r , is an expression of the form a ← b 1 , . . . , b m , ∼ c 1 , . . . , ∼ c n , where 0 ≤ m , n and each a , b i , c j is an atom A logic program is a finite set of rules Semantics The reduct, P X , of a program P relative to a set X of atoms is defined by P X = { a ← b 1 , . . . , b m | r ∈ P and { c 1 , . . . , c n } ∩ X = ∅} The ⊆ –smallest model of P X is denoted by Cn ( P X ) A set X of atoms is an stable model of a program P , if X = Cn ( P X ) Torsten Schaub (KRR@UP) ASP in Practice 10 / 69

  9. Introduction Formal Definition Syntax A rule, r , is an expression of the form a ← b 1 , . . . , b m , ∼ c 1 , . . . , ∼ c n , where 0 ≤ m , n and each a , b i , c j is an atom A logic program is a finite set of rules Semantics The reduct, P X , of a program P relative to a set X of atoms is defined by P X = { a ← b 1 , . . . , b m | r ∈ P and { c 1 , . . . , c n } ∩ X = ∅} The ⊆ –smallest model of P X is denoted by Cn ( P X ) A set X of atoms is an stable model of a program P , if X = Cn ( P X ) Torsten Schaub (KRR@UP) ASP in Practice 10 / 69

  10. Introduction Formal Definition Syntax A rule, r , is an expression of the form a ← b 1 , . . . , b m , ∼ c 1 , . . . , ∼ c n , where 0 ≤ m , n and each a , b i , c j is an atom A logic program is a finite set of rules Semantics The reduct, P X , of a program P relative to a set X of atoms is defined by P X = { a ← b 1 , . . . , b m | r ∈ P and { c 1 , . . . , c n } ∩ X = ∅} The ⊆ –smallest model of P X is denoted by Cn ( P X ) A set X of atoms is an stable model of a program P , if X = Cn ( P X ) Torsten Schaub (KRR@UP) ASP in Practice 10 / 69

  11. Introduction Formal Definition Syntax A rule, r , is an expression of the form a ← b 1 , . . . , b m , ∼ c 1 , . . . , ∼ c n , where 0 ≤ m , n and each a , b i , c j is an atom A logic program is a finite set of rules Semantics The reduct, P X , of a program P relative to a set X of atoms is defined by P X = { a ← b 1 , . . . , b m | r ∈ P and { c 1 , . . . , c n } ∩ X = ∅} The ⊆ –smallest model of P X is denoted by Cn ( P X ) A set X of atoms is an stable model of a program P , if X = Cn ( P X ) Torsten Schaub (KRR@UP) ASP in Practice 10 / 69

  12. Introduction Declarative problem solving “What is the problem?” instead of “How to solve the problem?” Problem Solution ✻ Modeling Interpreting ❄ ✲ Representation Output Solving Torsten Schaub (KRR@UP) ASP in Practice 11 / 69

  13. Introduction Declarative problem solving “What is the problem?” instead of “How to solve the problem?” Problem Solution ✻ Modeling Interpreting ❄ ✲ Representation Output Solving Torsten Schaub (KRR@UP) ASP in Practice 11 / 69

  14. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  15. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  16. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  17. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  18. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  19. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  20. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  21. Introduction Language Constructs Variables (over the Herbrand Universe) p(X) :- q(X) over constants { a , b , c } stands for p(a) :- q(a), p(b) :- q(b), p(c) :- q(c) Conditional Literals p :- q(X) : r(X) given r(a), r(b), r(c) stands for p :- q(a), q(b), q(c) Disjunction p(X) ; q(X) :- r(X) Integrity Constraints :- q(X), p(X) Choice 2 { p(X,Y) : q(X) } 7 :- r(Y) Aggregates s(Y) :- r(Y), 2 #count { p(X,Y) : q(X) } 7 also: #sum, #times, #avg, #min, #max, #even, #odd Torsten Schaub (KRR@UP) ASP in Practice 12 / 69

  22. Introduction Declarative problem solving “What is the problem?” instead of “How to solve the problem?” Problem Solution ✻ Modeling Interpreting ❄ ✲ Representation Output Solving Torsten Schaub (KRR@UP) ASP in Practice 13 / 69

  23. Introduction Declarative problem solving “What is the problem?” instead of “How to solve the problem?” Problem Solution ✻ Modeling Interpreting ❄ ✲ Representation Output Solving Torsten Schaub (KRR@UP) ASP in Practice 13 / 69

  24. Introduction Reasoning Modes Satisfiability Enumeration † Projection † Intersection ‡ Union ‡ Optimization † without solution recording ‡ without solution enumeration Torsten Schaub (KRR@UP) ASP in Practice 14 / 69

  25. Modeling by Example Outline 1 Motivation 2 Introduction 3 Modeling by Example Graph Coloring Queens Traveling Salesperson 4 Meta Programming 5 Conflict-Driven Answer Set Solving 6 Potassco 7 Summary Torsten Schaub (KRR@UP) ASP in Practice 15 / 69

  26. Modeling by Example Declarative problem solving “What is the problem?” instead of “How to solve the problem?” Problem Solution ✻ Modeling Interpreting ❄ ✲ Representation Output Solving Torsten Schaub (KRR@UP) ASP in Practice 16 / 69

  27. Modeling by Example Declarative problem solving “What is the problem?” instead of “How to solve the problem?” Problem Solution ✻ Modeling Interpreting ❄ ✲ Representation Output Solving Torsten Schaub (KRR@UP) ASP in Practice 16 / 69

  28. Modeling by Example ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 17 / 69

  29. Modeling by Example ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 17 / 69

  30. Modeling by Example ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 17 / 69

  31. Modeling by Example ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 17 / 69

  32. Modeling by Example ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 17 / 69

  33. Modeling by Example ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models ✻ Torsten Schaub (KRR@UP) ASP in Practice 17 / 69

  34. Modeling by Example Graph Coloring Outline 1 Motivation 2 Introduction 3 Modeling by Example Graph Coloring Queens Traveling Salesperson 4 Meta Programming 5 Conflict-Driven Answer Set Solving 6 Potassco 7 Summary Torsten Schaub (KRR@UP) ASP in Practice 18 / 69

  35. Modeling by Example Graph Coloring ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 19 / 69

  36. Modeling by Example Graph Coloring Graph Coloring node(1..6). edge(1,2). edge(1,3). edge(1,4). edge(2,4). edge(2,5). edge(2,6). edge(3,1). edge(3,4). edge(3,5). edge(4,1). edge(4,2). edge(5,3). edge(5,4). edge(5,6). edge(6,2). edge(6,3). edge(6,5). col(r). col(b). col(g). 1 { color(X,C) : col(C) } 1 :- node(X). :- edge(X,Y), color(X,C), color(Y,C). Torsten Schaub (KRR@UP) ASP in Practice 20 / 69

  37. Modeling by Example Graph Coloring Graph Coloring node(1..6). edge(1,2). edge(1,3). edge(1,4). edge(2,4). edge(2,5). edge(2,6). edge(3,1). edge(3,4). edge(3,5). edge(4,1). edge(4,2). edge(5,3). edge(5,4). edge(5,6). edge(6,2). edge(6,3). edge(6,5). col(r). col(b). col(g). 1 { color(X,C) : col(C) } 1 :- node(X). :- edge(X,Y), color(X,C), color(Y,C). Torsten Schaub (KRR@UP) ASP in Practice 20 / 69

  38. Modeling by Example Graph Coloring Graph Coloring node(1..6). edge(1,2). edge(1,3). edge(1,4). edge(2,4). edge(2,5). edge(2,6). edge(3,1). edge(3,4). edge(3,5). edge(4,1). edge(4,2). edge(5,3). edge(5,4). edge(5,6). edge(6,2). edge(6,3). edge(6,5). col(r). col(b). col(g). 1 { color(X,C) : col(C) } 1 :- node(X). :- edge(X,Y), color(X,C), color(Y,C). Torsten Schaub (KRR@UP) ASP in Practice 20 / 69

  39. Modeling by Example Graph Coloring Graph Coloring node(1..6). edge(1,2). edge(1,3). edge(1,4). edge(2,4). edge(2,5). edge(2,6). edge(3,1). edge(3,4). edge(3,5). edge(4,1). edge(4,2). edge(5,3). edge(5,4). edge(5,6). edge(6,2). edge(6,3). edge(6,5). col(r). col(b). col(g). 1 { color(X,C) : col(C) } 1 :- node(X). :- edge(X,Y), color(X,C), color(Y,C). Torsten Schaub (KRR@UP) ASP in Practice 20 / 69

  40. Modeling by Example Graph Coloring Graph Coloring node(1..6). edge(1,2). edge(1,3). edge(1,4). edge(2,4). edge(2,5). edge(2,6). edge(3,1). edge(3,4). edge(3,5). edge(4,1). edge(4,2). edge(5,3). edge(5,4). edge(5,6). edge(6,2). edge(6,3). edge(6,5). col(r). col(b). col(g). 1 { color(X,C) : col(C) } 1 :- node(X). :- edge(X,Y), color(X,C), color(Y,C). Torsten Schaub (KRR@UP) ASP in Practice 20 / 69

  41. Modeling by Example Graph Coloring ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 21 / 69

  42. Modeling by Example Graph Coloring Graph Coloring: Grounding $ gringo -t color.lp node(1). node(2). node(3). node(4). node(5). node(6). edge(1,2). edge(1,3). edge(1,4). edge(2,4). edge(2,5). edge(2,6). edge(3,1). edge(3,4). edge(3,5). edge(4,1). edge(4,2). edge(5,3). edge(5,4). edge(5,6). edge(6,2). edge(6,3). edge(6,5). col(r). col(b). col(g). 1 {color(1,r), color(1,b), color(1,g)} 1. 1 {color(2,r), color(2,b), color(2,g)} 1. 1 {color(3,r), color(3,b), color(3,g)} 1. 1 {color(4,r), color(4,b), color(4,g)} 1. 1 {color(5,r), color(5,b), color(5,g)} 1. 1 {color(6,r), color(6,b), color(6,g)} 1. :- color(1,r), color(2,r). :- color(2,g), color(5,g). ... :- color(6,r), color(2,r). :- color(1,b), color(2,b). :- color(2,r), color(6,r). :- color(6,b), color(2,b). :- color(1,g), color(2,g). :- color(2,b), color(6,b). :- color(6,g), color(2,g). :- color(1,r), color(3,r). :- color(2,g), color(6,g). :- color(6,r), color(3,r). :- color(1,b), color(3,b). :- color(3,r), color(1,r). :- color(6,b), color(3,b). :- color(1,g), color(3,g). :- color(3,b), color(1,b). :- color(6,g), color(3,g). :- color(1,r), color(4,r). :- color(3,g), color(1,g). :- color(6,r), color(5,r). :- color(1,b), color(4,b). :- color(3,r), color(4,r). :- color(6,b), color(5,b). :- color(1,g), color(4,g). :- color(3,b), color(4,b). :- color(6,g), color(5,g). :- color(2,r), color(4,r). :- color(3,g), color(4,g). :- color(2,b), color(4,b). :- color(3,r), color(5,r). :- color(2,g), color(4,g). :- color(3,b), color(5,b). :- color(2,r), color(5,r). :- color(3,g), color(5,g). Torsten Schaub (KRR@UP) ASP in Practice 22 / 69 :- color(2,b), color(5,b). :- color(4,r), color(1,r).

  43. Modeling by Example Graph Coloring Graph Coloring: Grounding $ gringo -t color.lp node(1). node(2). node(3). node(4). node(5). node(6). edge(1,2). edge(1,3). edge(1,4). edge(2,4). edge(2,5). edge(2,6). edge(3,1). edge(3,4). edge(3,5). edge(4,1). edge(4,2). edge(5,3). edge(5,4). edge(5,6). edge(6,2). edge(6,3). edge(6,5). col(r). col(b). col(g). 1 {color(1,r), color(1,b), color(1,g)} 1. 1 {color(2,r), color(2,b), color(2,g)} 1. 1 {color(3,r), color(3,b), color(3,g)} 1. 1 {color(4,r), color(4,b), color(4,g)} 1. 1 {color(5,r), color(5,b), color(5,g)} 1. 1 {color(6,r), color(6,b), color(6,g)} 1. :- color(1,r), color(2,r). :- color(2,g), color(5,g). ... :- color(6,r), color(2,r). :- color(1,b), color(2,b). :- color(2,r), color(6,r). :- color(6,b), color(2,b). :- color(1,g), color(2,g). :- color(2,b), color(6,b). :- color(6,g), color(2,g). :- color(1,r), color(3,r). :- color(2,g), color(6,g). :- color(6,r), color(3,r). :- color(1,b), color(3,b). :- color(3,r), color(1,r). :- color(6,b), color(3,b). :- color(1,g), color(3,g). :- color(3,b), color(1,b). :- color(6,g), color(3,g). :- color(1,r), color(4,r). :- color(3,g), color(1,g). :- color(6,r), color(5,r). :- color(1,b), color(4,b). :- color(3,r), color(4,r). :- color(6,b), color(5,b). :- color(1,g), color(4,g). :- color(3,b), color(4,b). :- color(6,g), color(5,g). :- color(2,r), color(4,r). :- color(3,g), color(4,g). :- color(2,b), color(4,b). :- color(3,r), color(5,r). :- color(2,g), color(4,g). :- color(3,b), color(5,b). :- color(2,r), color(5,r). :- color(3,g), color(5,g). Torsten Schaub (KRR@UP) ASP in Practice 22 / 69 :- color(2,b), color(5,b). :- color(4,r), color(1,r).

  44. Modeling by Example Graph Coloring ASP Solving Process ✲ ✲ ✲ Program Grounder Solver Models Torsten Schaub (KRR@UP) ASP in Practice 23 / 69

  45. Modeling by Example Graph Coloring Graph Coloring: Solving $ gringo color.lp | clasp 0 clasp version 2.1.0 Reading from stdin Solving... Answer: 1 edge(1,2) ... col(r) ... node(1) ... color(6,b) color(5,g) color(4,b) color(3,r) color(2,r) color(1,g) Answer: 2 edge(1,2) ... col(r) ... node(1) ... color(6,r) color(5,g) color(4,r) color(3,b) color(2,b) color(1,g) Answer: 3 edge(1,2) ... col(r) ... node(1) ... color(6,g) color(5,b) color(4,g) color(3,r) color(2,r) color(1,b) Answer: 4 edge(1,2) ... col(r) ... node(1) ... color(6,r) color(5,b) color(4,r) color(3,g) color(2,g) color(1,b) Answer: 5 edge(1,2) ... col(r) ... node(1) ... color(6,g) color(5,r) color(4,g) color(3,b) color(2,b) color(1,r) Answer: 6 edge(1,2) ... col(r) ... node(1) ... color(6,b) color(5,r) color(4,b) color(3,g) color(2,g) color(1,r) SATISFIABLE Models : 6 Time : 0.002s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s) CPU Time : 0.000s Torsten Schaub (KRR@UP) ASP in Practice 24 / 69

  46. Modeling by Example Graph Coloring Graph Coloring: Solving $ gringo color.lp | clasp 0 clasp version 2.1.0 Reading from stdin Solving... Answer: 1 edge(1,2) ... col(r) ... node(1) ... color(6,b) color(5,g) color(4,b) color(3,r) color(2,r) color(1,g) Answer: 2 edge(1,2) ... col(r) ... node(1) ... color(6,r) color(5,g) color(4,r) color(3,b) color(2,b) color(1,g) Answer: 3 edge(1,2) ... col(r) ... node(1) ... color(6,g) color(5,b) color(4,g) color(3,r) color(2,r) color(1,b) Answer: 4 edge(1,2) ... col(r) ... node(1) ... color(6,r) color(5,b) color(4,r) color(3,g) color(2,g) color(1,b) Answer: 5 edge(1,2) ... col(r) ... node(1) ... color(6,g) color(5,r) color(4,g) color(3,b) color(2,b) color(1,r) Answer: 6 edge(1,2) ... col(r) ... node(1) ... color(6,b) color(5,r) color(4,b) color(3,g) color(2,g) color(1,r) SATISFIABLE Models : 6 Time : 0.002s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s) CPU Time : 0.000s Torsten Schaub (KRR@UP) ASP in Practice 24 / 69

  47. Modeling by Example Queens Outline 1 Motivation 2 Introduction 3 Modeling by Example Graph Coloring Queens Traveling Salesperson 4 Meta Programming 5 Conflict-Driven Answer Set Solving 6 Potassco 7 Summary Torsten Schaub (KRR@UP) ASP in Practice 25 / 69

  48. Modeling by Example Queens The n-Queens Problem 5 Z0Z0Z Place n queens on an n × n 4 0Z0Z0 chess board Queens must not attack one 3 Z0Z0Z another 2 0Z0Z0 Q Q Q 1 Z0Z0Z Q Q 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 26 / 69

  49. Modeling by Example Queens Defining the Field queens.lp row (1..n). col (1..n). Create file queens.lp Define the field n rows n columns Torsten Schaub (KRR@UP) ASP in Practice 27 / 69

  50. Modeling by Example Queens Defining the Field Running . . . $ clingo queens.lp -c n=5 Answer: 1 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) SATISFIABLE Models : 1 Time : 0.000 Prepare : 0.000 Prepro. : 0.000 Solving : 0.000 Torsten Schaub (KRR@UP) ASP in Practice 28 / 69

  51. Modeling by Example Queens Placing some Queens queens.lp row (1..n). col (1..n). { queen(I,J) : row(I) : col(J) } . Guess a solution candidate by placing some queens on the board Torsten Schaub (KRR@UP) ASP in Practice 29 / 69

  52. Modeling by Example Queens Placing some Queens Running . . . $ clingo queens.lp -c n=5 3 Answer: 1 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) Answer: 2 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) queen(1,1) Answer: 3 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) queen(2,1) SATISFIABLE Models : 3+ ... Torsten Schaub (KRR@UP) ASP in Practice 30 / 69

  53. Modeling by Example Queens Placing some Queens: Answer 1 Answer 1 5 Z0Z0Z 4 0Z0Z0 3 Z0Z0Z 2 0Z0Z0 1 Z0Z0Z 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 31 / 69

  54. Modeling by Example Queens Placing some Queens: Answer 2 Answer 2 5 Z0Z0Z 4 0Z0Z0 3 Z0Z0Z 2 0Z0Z0 1 L0Z0Z 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 32 / 69

  55. Modeling by Example Queens Placing some Queens: Answer 3 Answer 3 5 Z0Z0Z 4 0Z0Z0 3 Z0Z0Z 2 QZ0Z0 1 Z0Z0Z 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 33 / 69

  56. Modeling by Example Queens Placing n Queens queens.lp row (1..n). col (1..n). { queen(I,J) : row(I) : col(J) }. :- not n { queen(I,J) } n. Place exactly n queens on the board Torsten Schaub (KRR@UP) ASP in Practice 34 / 69

  57. Modeling by Example Queens Placing n Queens Running . . . $ clingo queens.lp -c n=5 2 Answer: 1 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) \ queen(5,1) queen(4,1) queen(3,1) \ queen(2,1) queen(1,1) Answer: 2 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) \ queen(1,2) queen(4,1) queen(3,1) \ queen(2,1) queen(1,1) ... Torsten Schaub (KRR@UP) ASP in Practice 35 / 69

  58. Modeling by Example Queens Placing n Queens: Answer 1 Answer 1 5 L0Z0Z 4 QZ0Z0 3 L0Z0Z 2 QZ0Z0 1 L0Z0Z 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 36 / 69

  59. Modeling by Example Queens Placing n Queens: Answer 2 Answer 2 5 Z0Z0Z 4 QZ0Z0 3 L0Z0Z 2 QZ0Z0 1 LQZ0Z 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 37 / 69

  60. Modeling by Example Queens Horizontal and vertical Attack queens.lp row (1..n). col (1..n). { queen(I,J) : row(I) : col(J) }. :- not n { queen(I,J) } n. :- queen(I,J), queen(I,JJ), J != JJ. :- queen(I,J), queen(II,J), I != II. Forbid horizontal attacks Forbid vertical attacks Torsten Schaub (KRR@UP) ASP in Practice 38 / 69

  61. Modeling by Example Queens Horizontal and vertical Attack queens.lp row (1..n). col (1..n). { queen(I,J) : row(I) : col(J) }. :- not n { queen(I,J) } n. :- queen(I,J), queen(I,JJ), J != JJ. :- queen(I,J), queen(II,J), I != II. Forbid horizontal attacks Forbid vertical attacks Torsten Schaub (KRR@UP) ASP in Practice 38 / 69

  62. Modeling by Example Queens Horizontal and vertical Attack Running . . . $ clingo queens.lp -c n=5 Answer: 1 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) \ queen(5,5) queen(4,4) queen(3,3) \ queen(2,2) queen(1,1) ... Torsten Schaub (KRR@UP) ASP in Practice 39 / 69

  63. Modeling by Example Queens Horizontal and vertical Attack: Answer 1 Answer 1 5 Z0Z0L 4 0Z0L0 3 Z0L0Z 2 0L0Z0 1 L0Z0Z 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 40 / 69

  64. Modeling by Example Queens Diagonal Attack queens.lp row (1..n). col (1..n). { queen(I,J) : row(I) : col(J) }. :- not n { queen(I,J) } n. :- queen(I,J), queen(I,JJ), J != JJ. :- queen(I,J), queen(II ,J), I != II. :- queen(I,J), queen(II,JJ), (I,J) != (II,JJ), I-J == II-JJ. :- queen(I,J), queen(II,JJ), (I,J) != (II,JJ), I+J == II+JJ. Forbid diagonal attacks Torsten Schaub (KRR@UP) ASP in Practice 41 / 69

  65. Modeling by Example Queens Diagonal Attack Running . . . $ clingo queens.lp -c n=5 Answer: 1 row (1) row (2) row (3) row (4) row (5) \ col (1) col (2) col (3) col (4) col (5) \ queen(4,5) queen(1,4) queen(3,3) \ queen(5,2) queen(2,1) SATISFIABLE Models : 1+ Time : 0.000 Prepare : 0.000 Prepro. : 0.000 Solving : 0.000 Torsten Schaub (KRR@UP) ASP in Practice 42 / 69

  66. Modeling by Example Queens Diagonal Attack: Answer 1 Answer 1 5 ZQZ0Z 4 0Z0ZQ 3 Z0L0Z 2 QZ0Z0 1 Z0ZQZ 1 2 3 4 5 Torsten Schaub (KRR@UP) ASP in Practice 43 / 69

  67. Modeling by Example Queens Optimizing queens-opt.lp 1 { queen(I ,1..n) } 1 :- I = 1..n. 1 { queen (1..n,J) } 1 :- J = 1..n. :- 2 { queen(D-J,J) }, D = 2..2*n. :- 2 { queen(D+J,J) }, D = 1-n..n-1. Encoding can be optimized Much faster to solve Torsten Schaub (KRR@UP) ASP in Practice 44 / 69

  68. Modeling by Example Traveling Salesperson Outline 1 Motivation 2 Introduction 3 Modeling by Example Graph Coloring Queens Traveling Salesperson 4 Meta Programming 5 Conflict-Driven Answer Set Solving 6 Potassco 7 Summary Torsten Schaub (KRR@UP) ASP in Practice 45 / 69

  69. Modeling by Example Traveling Salesperson Traveling Salesperson node(1..6). edge(1,2;3;4). edge(2,4;5;6). edge(3,1;4;5). edge(4,1;2). edge(5,3;4;6). edge(6,2;3;5). cost(1,2,2). cost(1,3,3). cost(1,4,1). cost(2,4,2). cost(2,5,2). cost(2,6,4). cost(3,1,3). cost(3,4,2). cost(3,5,2). cost(4,1,1). cost(4,2,2). cost(5,3,2). cost(5,4,2). cost(5,6,1). cost(6,2,4). cost(6,3,3). cost(6,5,1). Torsten Schaub (KRR@UP) ASP in Practice 46 / 69

  70. Modeling by Example Traveling Salesperson Traveling Salesperson node(1..6). edge(1,2;3;4). edge(2,4;5;6). edge(3,1;4;5). edge(4,1;2). edge(5,3;4;6). edge(6,2;3;5). cost(1,2,2). cost(1,3,3). cost(1,4,1). cost(2,4,2). cost(2,5,2). cost(2,6,4). cost(3,1,3). cost(3,4,2). cost(3,5,2). cost(4,1,1). cost(4,2,2). cost(5,3,2). cost(5,4,2). cost(5,6,1). cost(6,2,4). cost(6,3,3). cost(6,5,1). Torsten Schaub (KRR@UP) ASP in Practice 46 / 69

  71. Modeling by Example Traveling Salesperson Traveling Salesperson node(1..6). edge(1,2;3;4). edge(2,4;5;6). edge(3,1;4;5). edge(4,1;2). edge(5,3;4;6). edge(6,2;3;5). cost(1,2,2). cost(1,3,3). cost(1,4,1). cost(2,4,2). cost(2,5,2). cost(2,6,4). cost(3,1,3). cost(3,4,2). cost(3,5,2). cost(4,1,1). cost(4,2,2). cost(5,3,2). cost(5,4,2). cost(5,6,1). cost(6,2,4). cost(6,3,3). cost(6,5,1). Torsten Schaub (KRR@UP) ASP in Practice 46 / 69

  72. Modeling by Example Traveling Salesperson Traveling Salesperson 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(X). 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(Y). reached(Y) :- cycle(1,Y). reached(Y) :- cycle(X,Y), reached(X). :- node(Y), not reached(Y). #minimize { cycle(X,Y) : cost(X,Y,C) = C }. Torsten Schaub (KRR@UP) ASP in Practice 47 / 69

  73. Modeling by Example Traveling Salesperson Traveling Salesperson 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(X). 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(Y). reached(Y) :- cycle(1,Y). reached(Y) :- cycle(X,Y), reached(X). :- node(Y), not reached(Y). #minimize { cycle(X,Y) : cost(X,Y,C) = C }. Torsten Schaub (KRR@UP) ASP in Practice 47 / 69

  74. Modeling by Example Traveling Salesperson Traveling Salesperson 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(X). 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(Y). reached(Y) :- cycle(1,Y). reached(Y) :- cycle(X,Y), reached(X). :- node(Y), not reached(Y). #minimize { cycle(X,Y) : cost(X,Y,C) = C }. Torsten Schaub (KRR@UP) ASP in Practice 47 / 69

  75. Modeling by Example Traveling Salesperson Traveling Salesperson 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(X). 1 { cycle(X,Y) : edge(X,Y) } 1 :- node(Y). reached(Y) :- cycle(1,Y). reached(Y) :- cycle(X,Y), reached(X). :- node(Y), not reached(Y). #minimize { cycle(X,Y) : cost(X,Y,C) = C }. Torsten Schaub (KRR@UP) ASP in Practice 47 / 69

  76. Modeling by Example Traveling Salesperson What is ASP good for? Combinatorial search problems in the realm of P , NP , and NP NP (some with substantial amount of data), like For instance, auctions, bio-informatics, computer-aided verification, configuration, constraint satisfaction, diagnosis, information integration, planning and scheduling, security analysis, semantic web, wire-routing, zoology and linguistics, and many more My favorite: Using ASP as a basis for a decision support system for NASA’s space shuttle (Gelfond et al., Texas Tech) Our own applications: Automatic synthesis of multiprocessor systems Inconsistency detection, diagnosis, repair, and prediction in large biological networks Home monitoring for risk prevention in ambient assisted living General game playing Torsten Schaub (KRR@UP) ASP in Practice 48 / 69

  77. Modeling by Example Traveling Salesperson What is ASP good for? Combinatorial search problems in the realm of P , NP , and NP NP (some with substantial amount of data), like For instance, auctions, bio-informatics, computer-aided verification, configuration, constraint satisfaction, diagnosis, information integration, planning and scheduling, security analysis, semantic web, wire-routing, zoology and linguistics, and many more My favorite: Using ASP as a basis for a decision support system for NASA’s space shuttle (Gelfond et al., Texas Tech) Our own applications: Automatic synthesis of multiprocessor systems Inconsistency detection, diagnosis, repair, and prediction in large biological networks Home monitoring for risk prevention in ambient assisted living General game playing Torsten Schaub (KRR@UP) ASP in Practice 48 / 69

  78. Modeling by Example Traveling Salesperson What is ASP good for? Combinatorial search problems in the realm of P , NP , and NP NP (some with substantial amount of data), like For instance, auctions, bio-informatics, computer-aided verification, configuration, constraint satisfaction, diagnosis, information integration, planning and scheduling, security analysis, semantic web, wire-routing, zoology and linguistics, and many more My favorite: Using ASP as a basis for a decision support system for NASA’s space shuttle (Gelfond et al., Texas Tech) Our own applications: Automatic synthesis of multiprocessor systems Inconsistency detection, diagnosis, repair, and prediction in large biological networks Home monitoring for risk prevention in ambient assisted living General game playing Torsten Schaub (KRR@UP) ASP in Practice 48 / 69

  79. Modeling by Example Traveling Salesperson What does ASP offer? Integration of KR, DB, and SAT techniques Succinct, elaboration-tolerant problem representations Rapid application development tool Easy handling of dynamic, knowledge intensive applications including: data, frame axioms, exceptions, defaults, closures, etc. Torsten Schaub (KRR@UP) ASP in Practice 49 / 69

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