Answer Set Solving in Practice

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1 Answer Set Solving in Practice Torsten Schaub University of Potsdam Potassco Slide Packages are licensed under a Creative Commons Attribution 3.0 Unported License. Torsten Schaub Answer Set Solving in Practice November 15, / 454

2 Motivation: Overview 1 Motivation 2 Nutshell 3 Shifting paradigms 4 Rooting ASP 5 ASP solving 6 Using ASP Torsten Schaub Answer Set Solving in Practice November 15, / 454

3 Motivation Outline 1 Motivation 2 Nutshell 3 Shifting paradigms 4 Rooting ASP 5 ASP solving 6 Using ASP Torsten Schaub Answer Set Solving in Practice November 15, / 454

4 Motivation Informatics What is the problem? versus How to solve the problem? Problem Solution Computer Output Torsten Schaub Answer Set Solving in Practice November 15, / 454

5 Motivation Informatics What is the problem? versus How to solve the problem? Problem Solution Computer Output Torsten Schaub Answer Set Solving in Practice November 15, / 454

6 Motivation Traditional programming What is the problem? versus How to solve the problem? Problem Solution Computer Output Torsten Schaub Answer Set Solving in Practice November 15, / 454

7 Motivation Traditional programming What is the problem? versus How to solve the problem? Problem Solution Programming Interpreting Program Executing Output Torsten Schaub Answer Set Solving in Practice November 15, / 454

8 Motivation Declarative problem solving What is the problem? versus How to solve the problem? Problem Solution Interpreting Computer Output Torsten Schaub Answer Set Solving in Practice November 15, / 454

9 Motivation Declarative problem solving What is the problem? versus How to solve the problem? Problem Solution Modeling Interpreting Representation Solving Output Torsten Schaub Answer Set Solving in Practice November 15, / 454

10 Motivation Declarative problem solving What is the problem? versus How to solve the problem? Problem Solution Modeling Interpreting Representation Solving Output Torsten Schaub Answer Set Solving in Practice November 15, / 454

11 Nutshell Outline 1 Motivation 2 Nutshell 3 Shifting paradigms 4 Rooting ASP 5 ASP solving 6 Using ASP Torsten Schaub Answer Set Solving in Practice November 15, / 454

12 Nutshell Answer Set Programming in a Nutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logic-based) knowledge representation and (nonmonotonic) reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NP NP ) in a uniform way ASP is versatile as reflected by the ASP solver clasp, winning first places at ASP, CASC, MISC, PB, and SAT competitions ASP embraces many emerging application areas Torsten Schaub Answer Set Solving in Practice November 15, / 454

13 Nutshell Answer Set Programming in a Nutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logic-based) knowledge representation and (nonmonotonic) reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NP NP ) in a uniform way ASP is versatile as reflected by the ASP solver clasp, winning first places at ASP, CASC, MISC, PB, and SAT competitions ASP embraces many emerging application areas Torsten Schaub Answer Set Solving in Practice November 15, / 454

14 Nutshell Answer Set Programming in a Nutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logic-based) knowledge representation and (nonmonotonic) reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NP NP ) in a uniform way ASP is versatile as reflected by the ASP solver clasp, winning first places at ASP, CASC, MISC, PB, and SAT competitions ASP embraces many emerging application areas Torsten Schaub Answer Set Solving in Practice November 15, / 454

15 Nutshell Answer Set Programming in a Nutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logic-based) knowledge representation and (nonmonotonic) reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NP NP ) in a uniform way ASP is versatile as reflected by the ASP solver clasp, winning first places at ASP, CASC, MISC, PB, and SAT competitions ASP embraces many emerging application areas Torsten Schaub Answer Set Solving in Practice November 15, / 454

16 Nutshell Answer Set Programming in a Nutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logic-based) knowledge representation and (nonmonotonic) reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NP NP ) in a uniform way ASP is versatile as reflected by the ASP solver clasp, winning first places at ASP, CASC, MISC, PB, and SAT competitions ASP embraces many emerging application areas Torsten Schaub Answer Set Solving in Practice November 15, / 454

17 Nutshell Answer Set Programming in a Nutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logic-based) knowledge representation and (nonmonotonic) reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NP NP ) in a uniform way ASP is versatile as reflected by the ASP solver clasp, winning first places at ASP, CASC, MISC, PB, and SAT competitions ASP embraces many emerging application areas Torsten Schaub Answer Set Solving in Practice November 15, / 454

18 Nutshell Answer Set Programming in a Hazelnutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities tailored to Knowledge Representation and Reasoning Torsten Schaub Answer Set Solving in Practice November 15, / 454

19 Nutshell Answer Set Programming in a Hazelnutshell ASP is an approach to declarative problem solving, combining a rich yet simple modeling language with high-performance solving capacities tailored to Knowledge Representation and Reasoning ASP = DB+LP+KR+SAT Torsten Schaub Answer Set Solving in Practice November 15, / 454

20 Shifting paradigms Outline 1 Motivation 2 Nutshell 3 Shifting paradigms 4 Rooting ASP 5 ASP solving 6 Using ASP Torsten Schaub Answer Set Solving in Practice November 15, / 454

21 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Automated planning, Kautz and Selman (ECAI 92) Represent planning problems as propositional theories so that models not proofs describe solutions Torsten Schaub Answer Set Solving in Practice November 15, / 454

22 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Automated planning, Kautz and Selman (ECAI 92) Represent planning problems as propositional theories so that models not proofs describe solutions Torsten Schaub Answer Set Solving in Practice November 15, / 454

23 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Automated planning, Kautz and Selman (ECAI 92) Represent planning problems as propositional theories so that models not proofs describe solutions Torsten Schaub Answer Set Solving in Practice November 15, / 454

24 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Automated planning, Kautz and Selman (ECAI 92) Represent planning problems as propositional theories so that models not proofs describe solutions Torsten Schaub Answer Set Solving in Practice November 15, / 454

25 Shifting paradigms Model Generation based Problem Solving Representation constraint satisfaction problem propositional horn theories propositional theories propositional theories propositional theories propositional programs propositional programs propositional programs first-order theories first-order theories first-order theories first-order theories auto-epistemic theories default theories. Solution assignment smallest model models minimal models stable models minimal models supported models stable models models minimal models stable models Herbrand models expansions extensions. Torsten Schaub Answer Set Solving in Practice November 15, / 454

26 Shifting paradigms Model Generation based Problem Solving Representation constraint satisfaction problem propositional horn theories propositional theories propositional theories propositional theories propositional programs propositional programs propositional programs first-order theories first-order theories first-order theories first-order theories auto-epistemic theories default theories. Solution assignment smallest model models minimal models stable models minimal models supported models stable models models minimal models stable models Herbrand models expansions extensions. Torsten Schaub Answer Set Solving in Practice November 15, / 454

27 Shifting paradigms Model Generation based Problem Solving Representation Solution constraint satisfaction problem assignment propositional horn theories smallest model propositional theories models SAT propositional theories minimal models propositional theories stable models propositional programs minimal models propositional programs supported models propositional programs stable models first-order theories models first-order theories minimal models first-order theories stable models first-order theories Herbrand models auto-epistemic theories expansions default theories extensions.. Torsten Schaub Answer Set Solving in Practice November 15, / 454

28 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Torsten Schaub Answer Set Solving in Practice November 15, / 454

29 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Torsten Schaub Answer Set Solving in Practice November 15, / 454

30 Shifting paradigms LP-style playing with blocks Prolog program on(a,b). on(b,c). above(x,y) :- on(x,y). above(x,y) :- on(x,z), above(z,y). Prolog queries?- above(a,c). true.?- above(c,a). no. Torsten Schaub Answer Set Solving in Practice November 15, / 454

31 Shifting paradigms LP-style playing with blocks Prolog program on(a,b). on(b,c). above(x,y) :- on(x,y). above(x,y) :- on(x,z), above(z,y). Prolog queries?- above(a,c). true.?- above(c,a). no. Torsten Schaub Answer Set Solving in Practice November 15, / 454

32 Shifting paradigms LP-style playing with blocks Prolog program on(a,b). on(b,c). above(x,y) :- on(x,y). above(x,y) :- on(x,z), above(z,y). Prolog queries?- above(a,c). true.?- above(c,a). no. Torsten Schaub Answer Set Solving in Practice November 15, / 454

33 Shifting paradigms LP-style playing with blocks Prolog program on(a,b). on(b,c). above(x,y) :- on(x,y). above(x,y) :- on(x,z), above(z,y). Prolog queries (testing entailment)?- above(a,c). true.?- above(c,a). no. Torsten Schaub Answer Set Solving in Practice November 15, / 454

34 Shifting paradigms Shuffled Prolog program on(a,b). on(b,c). above(x,y) :- above(x,z), on(z,y). above(x,y) :- on(x,y). LP-style playing with blocks Prolog queries?- above(a,c). Fatal Error: local stack overflow. Torsten Schaub Answer Set Solving in Practice November 15, / 454

35 Shifting paradigms Shuffled Prolog program on(a,b). on(b,c). above(x,y) :- above(x,z), on(z,y). above(x,y) :- on(x,y). LP-style playing with blocks Prolog queries?- above(a,c). Fatal Error: local stack overflow. Torsten Schaub Answer Set Solving in Practice November 15, / 454

36 Shifting paradigms Shuffled Prolog program on(a,b). on(b,c). above(x,y) :- above(x,z), on(z,y). above(x,y) :- on(x,y). LP-style playing with blocks Prolog queries (answered via fixed execution)?- above(a,c). Fatal Error: local stack overflow. Torsten Schaub Answer Set Solving in Practice November 15, / 454

37 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Torsten Schaub Answer Set Solving in Practice November 15, / 454

38 Shifting paradigms KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Torsten Schaub Answer Set Solving in Practice November 15, / 454

39 Shifting paradigms Formula on(a, b) on(b, c) (on(x, Y ) above(x, Y )) (on(x, Z) above(z, Y ) above(x, Y )) SAT-style playing with blocks Herbrand model { on(a, b), on(b, c), on(a, c), on(b, b), above(a, b), above(b, c), above(a, c), above(b, b), above(c, b) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

40 Shifting paradigms Formula on(a, b) on(b, c) (on(x, Y ) above(x, Y )) (on(x, Z) above(z, Y ) above(x, Y )) SAT-style playing with blocks Herbrand model { on(a, b), on(b, c), on(a, c), on(b, b), above(a, b), above(b, c), above(a, c), above(b, b), above(c, b) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

41 Shifting paradigms Formula on(a, b) on(b, c) (on(x, Y ) above(x, Y )) (on(x, Z) above(z, Y ) above(x, Y )) SAT-style playing with blocks Herbrand model { on(a, b), on(b, c), on(a, c), on(b, b), above(a, b), above(b, c), above(a, c), above(b, b), above(c, b) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

42 Shifting paradigms Formula on(a, b) on(b, c) (on(x, Y ) above(x, Y )) (on(x, Z) above(z, Y ) above(x, Y )) SAT-style playing with blocks Herbrand model { on(a, b), on(b, c), on(a, c), on(b, b), above(a, b), above(b, c), above(a, c), above(b, b), above(c, b) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

43 Shifting paradigms Formula on(a, b) on(b, c) (on(x, Y ) above(x, Y )) (on(x, Z) above(z, Y ) above(x, Y )) SAT-style playing with blocks Herbrand model (among 426!) { on(a, b), on(b, c), on(a, c), on(b, b), above(a, b), above(b, c), above(a, c), above(b, b), above(c, b) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

44 Rooting ASP Outline 1 Motivation 2 Nutshell 3 Shifting paradigms 4 Rooting ASP 5 ASP solving 6 Using ASP Torsten Schaub Answer Set Solving in Practice November 15, / 454

45 Rooting ASP KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Torsten Schaub Answer Set Solving in Practice November 15, / 454

46 Rooting ASP KR s shift of paradigm Theorem Proving based approach (eg. Prolog) 1 Provide a representation of the problem 2 A solution is given by a derivation of a query Model Generation based approach (eg. SATisfiability testing) 1 Provide a representation of the problem 2 A solution is given by a model of the representation Answer Set Programming (ASP) Torsten Schaub Answer Set Solving in Practice November 15, / 454

47 Rooting ASP Model Generation based Problem Solving Representation Solution constraint satisfaction problem assignment propositional horn theories smallest model propositional theories models propositional theories minimal models propositional theories stable models propositional programs minimal models propositional programs supported models propositional programs stable models first-order theories models first-order theories minimal models first-order theories stable models first-order theories Herbrand models auto-epistemic theories expansions default theories extensions.. Torsten Schaub Answer Set Solving in Practice November 15, / 454

48 Rooting ASP Answer Set Programming at large Representation Solution constraint satisfaction problem assignment propositional horn theories smallest model propositional theories models propositional theories minimal models propositional theories stable models propositional programs minimal models propositional programs supported models propositional programs stable models first-order theories models first-order theories minimal models first-order theories stable models first-order theories Herbrand models auto-epistemic theories expansions default theories extensions.. Torsten Schaub Answer Set Solving in Practice November 15, / 454

49 Rooting ASP Answer Set Programming commonly Representation Solution constraint satisfaction problem assignment propositional horn theories smallest model propositional theories models propositional theories minimal models propositional theories stable models propositional programs minimal models propositional programs supported models propositional programs stable models first-order theories models first-order theories minimal models first-order theories stable models first-order theories Herbrand models auto-epistemic theories expansions default theories extensions.. Torsten Schaub Answer Set Solving in Practice November 15, / 454

50 Rooting ASP Answer Set Programming in practice Representation Solution constraint satisfaction problem assignment propositional horn theories smallest model propositional theories models propositional theories minimal models propositional theories stable models propositional programs minimal models propositional programs supported models propositional programs stable models first-order theories models first-order theories minimal models first-order theories stable models first-order theories Herbrand models auto-epistemic theories expansions default theories extensions.. Torsten Schaub Answer Set Solving in Practice November 15, / 454

51 Rooting ASP Answer Set Programming in practice Representation Solution constraint satisfaction problem assignment propositional horn theories smallest model propositional theories models propositional theories minimal models propositional theories stable models propositional programs minimal models propositional programs supported models propositional programs stable models first-order theories models first-order theories minimal models first-order theories stable models first-order theories Herbrand models auto-epistemic theories expansions default theories extensions first-order programs stable Herbrand models Torsten Schaub Answer Set Solving in Practice November 15, / 454

52 Rooting ASP ASP-style playing with blocks Logic program on(a,b). on(b,c). above(x,y) :- on(x,y). above(x,y) :- on(x,z), above(z,y). Stable Herbrand model { on(a, b), on(b, c), above(b, c), above(a, b), above(a, c) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

53 Rooting ASP ASP-style playing with blocks Logic program on(a,b). on(b,c). above(x,y) :- on(x,y). above(x,y) :- on(x,z), above(z,y). Stable Herbrand model { on(a, b), on(b, c), above(b, c), above(a, b), above(a, c) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

54 Rooting ASP ASP-style playing with blocks Logic program on(a,b). on(b,c). above(x,y) :- on(x,y). above(x,y) :- on(x,z), above(z,y). Stable Herbrand model (and no others) { on(a, b), on(b, c), above(b, c), above(a, b), above(a, c) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

55 Rooting ASP ASP-style playing with blocks Logic program on(a,b). on(b,c). above(x,y) :- above(z,y), on(x,z). above(x,y) :- on(x,y). Stable Herbrand model (and no others) { on(a, b), on(b, c), above(b, c), above(a, b), above(a, c) } Torsten Schaub Answer Set Solving in Practice November 15, / 454

56 Rooting ASP ASP versus LP ASP Prolog Model generation Query orientation Bottom-up Top-down Modeling language Programming language Rule-based format Instantiation Unification Flat terms Nested terms (Turing +) NP( NP ) Turing Torsten Schaub Answer Set Solving in Practice November 15, / 454

57 Rooting ASP ASP Constructive Logic Closed (and open) world reasoning Model generation Bottom-up ASP versus SAT SAT Classical Logic Open world reasoning Modeling language Complex reasoning modes Satisfiability testing Satisfiability Satisfiability Enumeration/Projection Intersection/Union Optimization (Turing +) NP( NP ) NP Torsten Schaub Answer Set Solving in Practice November 15, / 454

58 ASP solving Outline 1 Motivation 2 Nutshell 3 Shifting paradigms 4 Rooting ASP 5 ASP solving 6 Using ASP Torsten Schaub Answer Set Solving in Practice November 15, / 454

59 ASP solving ASP solving Problem Solution Modeling Interpreting Logic Program Grounder Solver Solving Stable Models Torsten Schaub Answer Set Solving in Practice November 15, / 454

60 ASP solving SAT solving Problem Solution Programming Interpreting Formula (CNF) Solver Classical Models Solving Torsten Schaub Answer Set Solving in Practice November 15, / 454

61 ASP solving Rooting ASP solving Problem Solution Modeling Interpreting Logic Program Grounder Solver Solving Stable Models Torsten Schaub Answer Set Solving in Practice November 15, / 454

62 ASP solving Rooting ASP solving Problem Solution Modeling KR Interpreting Logic Program LP Grounder Solver DB Solving SAT Stable Models DB+KR+LP Torsten Schaub Answer Set Solving in Practice November 15, / 454

63 Using ASP Outline 1 Motivation 2 Nutshell 3 Shifting paradigms 4 Rooting ASP 5 ASP solving 6 Using ASP Torsten Schaub Answer Set Solving in Practice November 15, / 454

64 Using ASP Two sides of a coin ASP as High-level Language Express problem instance(s) as sets of facts Encode problem (class) as a set of rules Read off solutions from stable models of facts and rules ASP as Low-level Language Compile a problem into a logic program Solve the original problem by solving its compilation Torsten Schaub Answer Set Solving in Practice November 15, / 454

65 Using ASP What is ASP good for? Combinatorial search problems in the realm of P, NP, and NP NP (some with substantial amount of data), like Automated Planning Code Optimization Composition of Renaissance Music Database Integration Decision Support for NASA shuttle controllers Model Checking Product Configuration Robotics Systems Biology System Synthesis (industrial) Team-building and many many more Torsten Schaub Answer Set Solving in Practice November 15, / 454

66 Using ASP What is ASP good for? Combinatorial search problems in the realm of P, NP, and NP NP (some with substantial amount of data), like Automated Planning Code Optimization Composition of Renaissance Music Database Integration Decision Support for NASA shuttle controllers Model Checking Product Configuration Robotics Systems Biology System Synthesis (industrial) Team-building and many many more Torsten Schaub Answer Set Solving in Practice November 15, / 454

67 Using ASP What does ASP offer? Integration of DB, KR, 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 Answer Set Solving in Practice November 15, / 454

68 Using ASP What does ASP offer? Integration of DB, KR, 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 ASP = DB+LP+KR+SAT Torsten Schaub Answer Set Solving in Practice November 15, / 454

69 Using ASP What does ASP offer? Integration of DB, KR, 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 ASP = DB+LP+KR+SMT Torsten Schaub Answer Set Solving in Practice November 15, / 454

70 Using ASP [1] C. Anger, M. Gebser, T. Linke, A. Neumann, and T. Schaub. The nomore++ approach to answer set solving. In G. Sutcliffe and A. Voronkov, editors, Proceedings of the Twelfth International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 05), volume 3835 of Lecture Notes in Artificial Intelligence, pages Springer-Verlag, [2] C. Anger, K. Konczak, T. Linke, and T. Schaub. A glimpse of answer set programming. Künstliche Intelligenz, 19(1):12 17, [3] Y. Babovich and V. Lifschitz. Computing answer sets using program completion. Unpublished draft, [4] C. Baral. Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Torsten Schaub Answer Set Solving in Practice November 15, / 454

71 Using ASP [5] C. Baral, G. Brewka, and J. Schlipf, editors. Proceedings of the Ninth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 07), volume 4483 of Lecture Notes in Artificial Intelligence. Springer-Verlag, [6] C. Baral and M. Gelfond. Logic programming and knowledge representation. Journal of Logic Programming, 12:1 80, [7] S. Baselice, P. Bonatti, and M. Gelfond. Towards an integration of answer set and constraint solving. In M. Gabbrielli and G. Gupta, editors, Proceedings of the Twenty-first International Conference on Logic Programming (ICLP 05), volume 3668 of Lecture Notes in Computer Science, pages Springer-Verlag, [8] A. Biere. Adaptive restart strategies for conflict driven SAT solvers. Torsten Schaub Answer Set Solving in Practice November 15, / 454

72 Using ASP In H. Kleine Büning and X. Zhao, editors, Proceedings of the Eleventh International Conference on Theory and Applications of Satisfiability Testing (SAT 08), volume 4996 of Lecture Notes in Computer Science, pages Springer-Verlag, [9] A. Biere. PicoSAT essentials. Journal on Satisfiability, Boolean Modeling and Computation, 4:75 97, [10] A. Biere, M. Heule, H. van Maaren, and T. Walsh, editors. Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications. IOS Press, [11] G. Brewka, T. Eiter, and M. Truszczyński. Answer set programming at a glance. Communications of the ACM, 54(12):92 103, [12] K. Clark. Torsten Schaub Answer Set Solving in Practice November 15, / 454

73 Using ASP Negation as failure. In H. Gallaire and J. Minker, editors, Logic and Data Bases, pages Plenum Press, [13] M. D Agostino, D. Gabbay, R. Hähnle, and J. Posegga, editors. Handbook of Tableau Methods. Kluwer Academic Publishers, [14] E. Dantsin, T. Eiter, G. Gottlob, and A. Voronkov. Complexity and expressive power of logic programming. In Proceedings of the Twelfth Annual IEEE Conference on Computational Complexity (CCC 97), pages IEEE Computer Society Press, [15] M. Davis, G. Logemann, and D. Loveland. A machine program for theorem-proving. Communications of the ACM, 5: , [16] M. Davis and H. Putnam. A computing procedure for quantification theory. Torsten Schaub Answer Set Solving in Practice November 15, / 454

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