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 highperformance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logicbased) 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 highperformance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logicbased) 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 highperformance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logicbased) 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 highperformance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logicbased) 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 highperformance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logicbased) 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 highperformance solving capacities ASP has its roots in (deductive) databases logic programming (with negation) (logicbased) 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 highperformance 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 highperformance 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 firstorder theories firstorder theories firstorder theories firstorder theories autoepistemic 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 firstorder theories firstorder theories firstorder theories firstorder theories autoepistemic 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 firstorder theories models firstorder theories minimal models firstorder theories stable models firstorder theories Herbrand models autoepistemic 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 LPstyle 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 LPstyle 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 LPstyle 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 LPstyle 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). LPstyle 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). LPstyle 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). LPstyle 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 )) SATstyle 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 )) SATstyle 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 )) SATstyle 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 )) SATstyle 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 )) SATstyle 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 firstorder theories models firstorder theories minimal models firstorder theories stable models firstorder theories Herbrand models autoepistemic 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 firstorder theories models firstorder theories minimal models firstorder theories stable models firstorder theories Herbrand models autoepistemic 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 firstorder theories models firstorder theories minimal models firstorder theories stable models firstorder theories Herbrand models autoepistemic 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 firstorder theories models firstorder theories minimal models firstorder theories stable models firstorder theories Herbrand models autoepistemic 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 firstorder theories models firstorder theories minimal models firstorder theories stable models firstorder theories Herbrand models autoepistemic theories expansions default theories extensions firstorder programs stable Herbrand models Torsten Schaub Answer Set Solving in Practice November 15, / 454
52 Rooting ASP ASPstyle 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 ASPstyle 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 ASPstyle 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 ASPstyle 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 Bottomup Topdown Modeling language Programming language Rulebased 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 Bottomup 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 Highlevel 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 Lowlevel 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) Teambuilding 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) Teambuilding 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, elaborationtolerant 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, elaborationtolerant 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, elaborationtolerant 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 SpringerVerlag, [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. SpringerVerlag, [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 Twentyfirst International Conference on Logic Programming (ICLP 05), volume 3668 of Lecture Notes in Computer Science, pages SpringerVerlag, [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 SpringerVerlag, [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 theoremproving. 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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