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Answering the “why” in answer set programming – A survey of explanation approaches

Published online by Cambridge University Press:  15 January 2019

JORGE FANDINNO Open the ORCID record for JORGE FANDINNO [Opens in a new window]  and
CLAUDIA SCHULZ
JORGE FANDINNO
Affiliation:
Institut de Recherche en Informatique de Toulouse (IRIT), Université de Toulouse, CNRS, Toulouse, France (e-mail: jorge.fandinno@irit.fr)
CLAUDIA SCHULZ
Affiliation:
Ubiquitous Knowledge Processing (UKP) Lab, Technische Universität Darmstadt, Darmstadt, Germany (e-mail: schulz@ukp.informatik.tu-darmstadt.de)
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Abstract

Artificial intelligence (AI) approaches to problem-solving and decision-making are becoming more and more complex, leading to a decrease in the understandability of solutions. The European Union’s new General Data Protection Regulation tries to tackle this problem by stipulating a “right to explanation” for decisions made by AI systems. One of the AI paradigms that may be affected by this new regulation is answer set programming (ASP). Thanks to the emergence of efficient solvers, ASP has recently been used for problem-solving in a variety of domains, including medicine, cryptography, and biology. To ensure the successful application of ASP as a problem-solving paradigm in the future, explanations of ASP solutions are crucial. In this survey, we give an overview of approaches that provide an answer to the question of why an answer set is a solution to a given problem, notably off-line justifications, causal graphs, argumentative explanations, and why-not provenance, and highlight their similarities and differences. Moreover, we review methods explaining why a set of literals is not an answer set or why no solution exists at all.

Keywords

answer setexplanationjustificationdebugging
Type
Survey Article
Information
Theory and Practice of Logic Programming , Volume 19 , Special Issue 2: User-Oriented Logic Programming and Reasoning Paradigms , March 2019 , pp. 114 - 203
Copyright
Copyright © Cambridge University Press 2019 

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Footnotes

We are thankful to the anonymous reviewers for their valuable feedback, which helped to improve the paper. This study was funded by Centre International de Mathématiques et d’Informatique de Toulouse ANR-11-LABEX-0040-CIMI.

References

Albrecht, E., Krümpelmann, P. and Kern-Isberner, G. 2013. Construction of explanation graphs from extended dependency graphs for answer set programs. In Declarative Programming and Knowledge Management. INAP 2013, WLP 2013, WFLP 2013, Hanus, M. and Rocha, R., Eds. Lecture Notes in Computer Science, vol. 8439. Springer, Cham, 1–16.Google Scholar
Alviano, M., Dodaro, C., Faber, W., Leone, N. and Ricca, F. 2013. WASP: A native ASP solver based on constraint learning. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2013, Cabalar, P. and Son, T. C., Eds. Lecture Notes in Computer Science, vol. 8148. Springer, Berlin, Heidelberg, 54–66.CrossRefGoogle Scholar
Alviano, M., Dodaro, C., Leone, N. and Ricca, F. 2015. Advances in WASP. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2015, Calimeri, F., Ianni, G., and Truszczynski, M., Eds. Lecture Notes in Computer Science, vol. 9345. Springer, Cham, 40–54.CrossRefGoogle Scholar
Arora, T., Ramakrishnan, R., Roth, W. G., Seshadri, P. and Srivastava, D. 1993. Explaining program execution in deductive systems. In Deductive and Object-Oriented Databases. DOOD 1993, Ceri, S., Tanaka, K., and Tsur, S., Eds. Lecture Notes in Computer Science, vol. 760. Springer, Berlin, Heidelberg, 101–119.CrossRefGoogle Scholar
Balduccini, M. and Girotto, S. 2010. Formalization of psychological knowledge in answer set programming and its application. Theory and Practice of Logic Programming 10, 4–6, 725–740.CrossRefGoogle Scholar
Béatrix, C., Lefèvre, C., Garcia, L. and Stéphan, I. 2016. Justifications and blocking sets in a rule-based answer set computation. In Technical Communications of the 32nd International Conference on Logic Programming (ICLP’16), 6:1–6:15.Google Scholar
Boenn, G., Brain, M., De Vos, M. and Fitch, J. P. 2011. Automatic music composition using answer set programming. Theory and Practice of Logic Programming 11, 2–3, 397–427.CrossRefGoogle Scholar
Brain, M. and De Vos, M. 2005. Debugging logic programs under the answer set semantics. In Proceedings of the 3rd Workshop on Answer Set Programming, Advances in Theory and Implementation (ASP’05).Google Scholar
Brain, M. and De Vos, M. 2008. Answer set programming - a domain in need of explanation: A position paper. In Proceedings of the 3rd International Workshop on Explanation-Aware Computing (ExaCt’08), 37–48.Google Scholar
Brain, M., Gebser, M., Pührer, J., Schaub, T., Tompits, H. and Woltran, S. 2007a. Debugging ASP programs by means of ASP. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2007, Baral, C., Brewka, G., and Schlipf, J., Eds. Lecture Notes in Computer Science, vol. 4483. Springer, Berlin, Heidelberg, 31–43.CrossRefGoogle Scholar
Brain, M., Gebser, M., Pührer, J., Schaub, T., Tompits, H. and Woltran, S. 2007b. “That is illogical captain!” - The debugging support tool spock for answer-set programs: System description. In Proceedings of the 1st International Workshop on Software Engineering for Answer Set Programming (SEA’07), 71–85.Google Scholar
Bratko, I. 2001. Prolog Programming for Artificial Intelligence. Pearson Education.Google Scholar
Brewka, G., Eiter, T. and Truszczynski, M. 2011. Answer set programming at a glance. Communications of the ACM 54, 12, 92–103.CrossRefGoogle Scholar
Busoniu, P.-A., Oetsch, J., Pührer, J., Skocovsky, P. and Tompits, H. 2013. SeaLion: An eclipse-based IDE for answer-set programming with advanced debugging support. Theory and Practice of Logic Programming 13, 4–5, 657–673.CrossRefGoogle Scholar
Cabalar, P. 2011. Answer set; programming? In Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning - Essays Dedicated to Michael Gelfond on the Occasion of His 65th Birthday, 334–343.CrossRefGoogle Scholar
Cabalar, P. and Fandinno, J. 2013. An algebra of causal chains. CoRR abs/1312.6134.Google Scholar
Cabalar, P. and Fandinno, J. 2016. Justifications for programs with disjunctive and causal-choice rules. Theory and Practice of Logic Programming 16, 5–6, 587–603.CrossRefGoogle Scholar
Cabalar, P. and Fandinno, J. 2017. Enablers and inhibitors in causal justifications of logic programs. Theory and Practice of Logic Programming 17, 1, 49–74.CrossRefGoogle Scholar
Cabalar, P., Fandinno, J. and Fink, M. 2014. Causal graph justifications of logic programs. Theory and Practice of Logic Programming 14, 4–5, 603–618.CrossRefGoogle Scholar
Cabalar, P., Fandiño, J. and Fink, M. 2014. A complexity assessment for queries involving sufficient and necessary causes. In Proceedings of the 14th European Conference on Logics in Artificial Intelligence (JELIA’14). Lecture Notes in Computer Science, vol. 8761. Springer, 297–310.Google Scholar
Calimeri, F., Leone, N., Ricca, F. and Veltri, P. 2009. A visual tracer for DLV. In Proceedings of the 2nd International Workshop on Software Engineering for Answer Set Programming (SEA’09), 79–93.Google Scholar
Clark, K. L. 1978. Negation as failure. In Logic and Data Bases, Gallaire, H. and Minker, J., Eds. Springer, Boston, MA, 293–322.CrossRefGoogle Scholar
Costantini, S. 2001. Comparing different graph representations of logic programs under the answer set semantics. In Proceedings of the 1st International Workshop on Answer Set Programming: Towards Efficient and Scalable Knowledge Representation and Reasoning (ASP’01).Google Scholar
Costantini, S. 2006. On the existence of stable models of non-stratified logic programs. Theory and Practice of Logic Programming 6, 1–2, 169–212.CrossRefGoogle Scholar
Costantini, S., D’Antona, O. and Provetti, A. 2002. On the equivalence and range of applicability of graph-based representations of logic programs. Information Processing Letters 84, 5, 241–249.CrossRefGoogle Scholar
Costantini, S. and Provetti, A. 2010. Graph representations of logic programs: Properties and comparison. In Proceedings of the 6th Latin American Workshop on Non-Monotonic Reasoning, 1–14.Google Scholar
Costantini, S. and Provetti, A. 2011. Conflict, consistency and truth-dependencies in graph representations of answer set logic programs. In Revised Selected Papers of the 2nd International Workshop on Graph Structures for Knowledge Representation and Reasoning (GKR’11), 68–90.Google Scholar
Damásio, C. V., Analyti, A. and Antoniou, G. 2013. Justifications for logic programming. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2013, Cabalar, P. and Son, T. C., Eds. Lecture Notes in Computer Science, vol. 8148. Springer, Berlin, Heidelberg, 530–542.CrossRefGoogle Scholar
Damásio, C. V., Moura, J. and Analyti, A. 2015. Unifying justifications and debugging for answer-set programs. In Technical Communications of the 31st International Conference on Logic Programming (ICLP’15).Google Scholar
Damásio, C. V., Pires, J. M. and Analyti, A. 2015. Unifying justifications and debugging for answer-set programs. In Proceedings of the Technical Communications of the 31st International Conference on Logic Programming (ICLP’15), Vos, M. D., Eiter, T., Lierler, Y. and Toni, F., Eds. CEUR Workshop Proceedings, vol. 1433. CEUR-WS.org.Google Scholar
de Kleer, J. 1986. An assumption-based TMS. Artificial Intelligence 28, 2, 127–162.CrossRefGoogle Scholar
Denecker, M., Brewka, G., and Strass, H. 2015. A formal theory of justifications. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2015, Calimeri, F., Ianni, G., and Truszczynski, M., Eds. Lecture Notes in Computer Science, vol. 9345. Springer, Cham, 250–264.CrossRefGoogle Scholar
Denecker, M. and De Schreye, D. 1993. Justification semantics: A unifying framework for the semantics of logic programs. In Proceedings of the 2nd International Workshop on Logic Programming and Non-monotonic Reasoning (LPNMR’93), 365–379.Google Scholar
Dimopoulos, Y. 1996. On computing logic programs. Journal of Automated Reasoning 17, 3, 259–289.CrossRefGoogle Scholar
Dimopoulos, Y. and Torres, A. 1996. Graph theoretical structures in logic programs and default theories. Theoretical Computer Science 170, 1–2, 209–244.CrossRefGoogle Scholar
Dodaro, C., Gasteiger, P., Musitsch, B., Ricca, F. and Shchekotykhin, K. 2015. Interactive debugging of non-ground ASP programs. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2015, Calimeri, F., Ianni, G., and Truszczynski, M., Eds. Lecture Notes in Computer Science, vol. 9345. Springer, Cham, 279–293.CrossRefGoogle Scholar
Dung, P. M., Kowalski, R. A. and Toni, F. 2009. Assumption-based argumentation. In Argumentation in Artificial Intelligence, Simari, G. R. and Rahwan, I., Eds. Springer US, 199–218.CrossRefGoogle Scholar
El-Khatib, O., Pontelli, E. and Son, T. C. 2005. Justification and debugging of answer set programs in ASP - Prolog. In Proceedings of the 6th International Workshop on Automated Debugging (AADEBUG’05), 49–58.Google Scholar
Erdem, E. and Öztok, U. 2015. Generating explanations for biomedical queries. Theory and Practice of Logic Programming 15, 1, 35–78.CrossRefGoogle Scholar
Faber, W., Pfeifer, G. and Leone, N. 2011. Semantics and complexity of recursive aggregates in answer set programming. Artificial Intelligence 175, 1, 278–298.CrossRefGoogle Scholar
Fandinno, J. 2016a. Deriving conclusions from non-monotonic cause-effect relations. Theory and Practice of Logic Programming 16, 5–6, 670–687.CrossRefGoogle Scholar
Fandinno, J. 2016b. Towards deriving conclusions from cause-effect relations. Fundamenta Informaticae 147, 1, 93–131.CrossRefGoogle Scholar
Febbraro, O., Reale, K. and Ricca, F. 2010. A visual interface for drawing ASP Programs. In Proceedings of the 25th Italian Conference on Computational Logic (CILC’10).Google Scholar
Febbraro, O., Reale, K. and Ricca, F. 2011. ASPIDE: Integrated development environment for answer set programming. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2011, Delgrande, J. P. and Faber, W., Eds. Lecture Notes in Computer Science, vol. 6645. Springer, Berlin, Heidelberg, 317–330.CrossRefGoogle Scholar
Ferrand, G., Lesaint, W. and Tessier, A. 2006. Explanations and proof trees. Computers and Informatics 25, 2–3, 105–122.Google Scholar
Frühstück, M., Pührer, J. and Friedrich, G. 2013. Debugging answer-set programs with ouroboros? Extending the SeaLion Plugin. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2013, Cabalar, P. and Son, T. C., Eds. Lecture Notes in Computer Science, vol. 8148. Springer, Berlin, Heidelberg, 323–328.CrossRefGoogle Scholar
Gasteiger, P., Dodaro, C., Musitsch, B., Reale, K., Ricca, F. and Schekotihin, K. 2016. An integrated graphical user interface for debugging answer set programs. In Proceedings of the Workshop on Trends and Applications of Answer Set Programming (TAASP’16).Google Scholar
Gebser, M., Pührer, J., Schaub, T. and Tompits, H. 2008. A meta-programming technique for debugging answer-set programs. In Proceedings of the 23rd AAAI Conference on Artificial Intelligence (AAAI’18), Fox, D. and Gomes, C. P., Eds. AAAI Press, 448–453.Google Scholar
Gebser, M., Schaub, T., Thiele, S. and Veber, P. 2011. Detecting inconsistencies in large biological networks with answer set programming. Theory and Practice of Logic Programming 11, 2–3, 323–360.CrossRefGoogle Scholar
Gelfond, M. 2008. Answer sets. In Handbook of Knowledge Representation, Porter, B., van Harmelen, F., and Lifschitz, V., Eds. Elsevier, 285–316.CrossRefGoogle Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Logic Programming: Proceedings of the 5th International Conference and Symposium, vol. 2.Google Scholar
Gelfond, M. and Lifschitz, V. 1991. Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 3/4, 365–386.CrossRefGoogle Scholar
Goodman, B. and Flaxman, S. 2016. European union regulations on algorithmic decision-making and a “right to explanation.” arXiv preprint arXiv:1606.08813.Google Scholar
Green, T. J., Karvounarakis, G. and Tannen, V. 2007. Provenance semirings. In Proceedings of the 26th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Libkin, L., Ed. ACM, 31–40.Google Scholar
Hall, N. 2004. Two concepts of causation. In Causation and Counterfactuals, Collins, J., Hall, N. and Paul, L. A., Eds. MIT Press, Cambridge, MA, 225–276.Google Scholar
Hall, N. 2007. Structural equations and causation. Philosophical Studies 132, , 109–136.CrossRefGoogle Scholar
Halpern, J. Y. 2008. Defaults and normality in causal structures. In Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR’08), Brewka, G. and Lang, J., Eds. AAAI Press, 198–208.Google Scholar
Hitchcock, C. and Knobe, J. 2009. Cause and norm. Journal of Philosophy 11, 587–612.CrossRefGoogle Scholar
Inclezan, D. 2015. An application of answer set programming to the field of second language acquisition. Theory and Practice of Logic Programming 15, 01, 1–17.CrossRefGoogle Scholar
Konczak, K., Linke, T. and Schaub, T. 2006. Graphs and colorings for answer set programming. Theory and Practice of Logic Programming 6, 1–2, 61–106.CrossRefGoogle Scholar
Lee, J. 2005. A model-theoretic counterpart of loop formulas. In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI’05), 503–508.Google Scholar
Lefèvre, C., Béatrix, C., Stéphan, I. and Garcia, L. 2017. ASPeRiX, a first-order forward chaining approach for answer set computing. Theory and Practice of Logic Programming 17, 3, 266–310.CrossRefGoogle Scholar
Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S. and Scarcello, F. 2006. The DLV system for knowledge representation and reasoning. ACM Transactions on Computational Logic 7, 3, 499–562.CrossRefGoogle Scholar
Lewis, D. K. 1973. Causation. The Journal of Philosophy 70, 17, 556–567.CrossRefGoogle Scholar
Li, T., De Vos, M., Padget, J., Satoh, K. and Balke, T. 2015. Debugging ASP using ILP. In Proceedings of the Technical Communications of the 31st International Conference on Logic Programming (ICLP’15).Google Scholar
Lifschitz, V. 2008. What is answer set programming? In Proceedings of the 23rd AAAI Conference on Artificial Intelligence (AAAI’08), 1594–1597.Google Scholar
Lifschitz, V. 2010. Thirteen definitions of a stable model. In Fields of Logic and Computation, Essays Dedicated to Yuri Gurevich on the Occasion of His 70th Birthday, Blass, A., Dershowitz, N., and Reisig, W., Eds. Lecture Notes in Computer Science, vol. 6300. Springer, 488–503.Google Scholar
Lifschitz, V. 2017. Achievements in answer set programming. Theory and Practice of Logic Programming 17, 5–6, 961–973.CrossRefGoogle Scholar
Lin, F. and Zhao, Y. 2004. ASSAT: Computing answer sets of a logic program by SAT solvers. Artificial Intelligence 157, 1–2, 115–137.CrossRefGoogle Scholar
Linke, T. 2001. Graph theoretical characterization and computation of answer sets. In Proceedings of the 7th International Joint Conference on Artificial Intelligence (IJCAI’01), 641–648.Google Scholar
Linke, T. and Sarsakov, V. 2004. Suitable graphs for answer set programming. In Proceedings of the 11th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR’04), 154–168.Google Scholar
Lloyd, J. W. 1987. Declarative error diagnosis. New Generation Computing 5, 2, 133–154.CrossRefGoogle Scholar
Maudlin, T. 2004. Causation, counterfactuals, and the third factor. In Causation and Counterfactuals, Collins, J., Hall, E. J. and Paul, L. A., Eds. MIT Press.Google Scholar
McCarthy, J. 1977. Epistemological problems of artificial Intelligence. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI). MIT Press, Cambridge, MA, 1038–1044.Google Scholar
McCarthy, J. 1998. Elaboration tolerance. In Proceedings of the 4th Symposium on Logical Formalizations of Commonsense Reasoning (Commonsense’98), London, UK, 198–217. Updated version at http://www-formal.stanford.edu/jmc/elaboration.ps.Google Scholar
Oetsch, J., Pührer, J., Seidl, M., Tompits, H. and Zwickl, P. 2011. VIDEAS: A development tool for answer-set programs based on model-driven engineering technology. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2011, Delgrande, J. P. and Faber, W., Eds. Lecture Notes in Computer Science, vol. 6645. Springer, Berlin, Heidelberg, 382–387.CrossRefGoogle Scholar
Oetsch, J., Pührer, J. and Tompits, H. 2010. Catching the Ouroboros: On debugging non-ground answer-set programs. Theory and Practice of Logic Programming 10, 4–6, 513–529.CrossRefGoogle Scholar
Oetsch, J., Pührer, J. and Tompits, H. 2011. Stepping through an answer-set program. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2011, Delgrande, J. P. and Faber, W., Eds. Lecture Notes in Computer Science, vol. 6645. Springer, Berlin, Heidelberg, 134–147.CrossRefGoogle Scholar
Oetsch, J., Pührer, J. and Tompits, H. 2012. An FLP-style answer-set semantics for abstract-constraint programs with disjunctions. In Technical Communications of the 28th International Conference on Logic Programming (ICLP’12), 222–234.Google Scholar
Oetsch, J., Pührer, J. and Tompits, H. 2018. Stepwise debugging of answer-set programs. Theory and Practice of Logic Programming 18, 1, 30–80.CrossRefGoogle Scholar
Parliament and Council of the European Union 2016. Regulation (EU) 2016/679: General Data Protection Regulation.Google Scholar
Pemmasani, G., Guo, H. F., Dong, Y., Ramakrishnan, C. R. and Ramakrishnan, I. V. 2003. Online justification for tabled logic programs. In Functional and Logic Programming. FLOPS 2004, Kameyama, Y. and Stuckey, P. J., Eds. Lecture Notes in Computer Science, vol. 2998. Springer, Berlin, Heidelberg, 500–501.Google Scholar
Pereira, L. M. and Alferes, J. J. 1992. Well founded semantics for logic programs with explicit negation. In Proceedings of the 10th European conference on Artificial Intelligence (ECAI ’92), Neumann, B., Ed. John Wiley & Sons, Inc., New York, NY, USA, 102–106.Google Scholar
Pereira, L. M., Alferes, J. J. and Aparício, J. N. 1991. Contradiction removal semantics with explicit negation. In Knowledge Representation and Reasoning under Uncertainty. Logic at Work 1992. Masuch, M. and Polos, L., Eds. Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence), vol. 808. Springer, Berlin, Heidelberg.Google Scholar
Pereira, L. M., Aparcio, J. N. and Alferes, J. 1993. Non-monotonic reasoning with logic programming. The Journal of Logic Programming 17, 2, 227–263. Special Issue: Non-Monotonic Reasoning and Logic Programming.CrossRefGoogle Scholar
Pereira, L. M., Damásio, C. V. and Alferes, J. J. 1993. Debugging by diagnosing assumptions. In International Workshop on Automated and Algorithmic Debugging. Springer, 58–74.CrossRefGoogle Scholar
Perri, S., Ricca, F., Terracina, G., Cianni, D. and Veltri, P. 2007. An integrated graphic tool for developing and testing DLV programs. In Proceedings of the 1st International Workshop on Software Engineering for Answer Set Programming (SEA’07), 86–100.Google Scholar
Polleres, A., Frühstück, M., Schenner, G. and Friedrich, G. 2013. Debugging non-ground ASP programs with choice rules, cardinality and weight constraints. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2013, Cabalar, P. and Son, T. C., Eds. Lecture Notes in Computer Science, vol. 8148. Springer, Berlin, Heidelberg, 452–464.CrossRefGoogle Scholar
Pontelli, E. and Son, T. C. 2006. Justifications for logic programs under answer set semantics. In Proceedings of the 22nd International Conference on Logic Programming (ICLP’06), 196–210.Google Scholar
Pontelli, E., Son, T. C. and El-Khatib, O. 2009. Justifications for logic programs under answer set semantics. Theory and Practice of Logic Programming 9, 1, 1–56.CrossRefGoogle Scholar
Pührer, J. 2014. Stepwise Debugging in Answer-Set Programming: Theoretical Foundations and Practical Realisation. Ph.D. thesis, Vienna University of Technology.Google Scholar
Ricca, F., Grasso, G., Alviano, M., Manna, M., Lio, V., Iiritano, S. and Leone, N. 2012. Team-building with answer set programming in the Gioia-Tauro Seaport. Theory and Practice of Logic Programming 12, 3, 361–381.CrossRefGoogle Scholar
Roychoudhury, A., Ramakrishnan, C. R. and Ramakrishnan, I. V. 2000. Justifying proofs using memo tables. In Proceedings of the 2nd ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming (PPDP’00), 178–189.Google Scholar
Schulz, C. 2017. Developments in abstract and assumption-based argumentation and their application in logic programming. Ph.D. thesis, Imperial College London.Google Scholar
Schulz, C., Satoh, K. and Toni, F. 2015. Characterising and explaining inconsistency in logic programs. In Logic Programming and Nonmonotonic Reasoning. LPNMR 2015, Calimeri, F., Ianni, G., and Truszczynski, M., Eds. Lecture Notes in Computer Science, vol. 9345. Springer, Cham, 467–479.CrossRefGoogle Scholar
Schulz, C., Sergot, M. and Toni, F. 2013. Argumentation-based answer set justification. In Proceedings of the 11th International Symposium on Logical Formalizations of Commonsense Reasoning (Commonsense’13).Google Scholar
Schulz, C. and Toni, F. 2013. ABA-based answer set justification. Theory and Practice of Logic Programming 13, 4–5 Online-Supplement.Google Scholar
Schulz, C. and Toni, F. 2015. Logic programming in assumption-based argumentation revisited - Semantics and graphical representation. In Proceedings of the 29th AAAI Conference on Artificial Intelligence (AAAI’15), 1569–1575.Google Scholar
Schulz, C. and Toni, F. 2016. Justifying answer sets using argumentation. Theory and Practice of Logic Programming 16, 01, 59–110.CrossRefGoogle Scholar
Shapiro, E. Y. 1983. Algorithmic Program DeBugging. MIT Press, Cambridge, MA, USA.Google Scholar
Shchekotykhin, K. M. 2015. Interactive query-based debugging of ASP programs. In Proceedings of the 29th AAAI Conference on Artificial Intelligence (AAAI’15), 1597–1603.Google Scholar
Specht, G. 1993. Generating explanation trees even for negations in deductive database systems. In Proceedings of the 5th Workshop on Logic Programming Environments (LPE’93), Ducassé, M., Charlier, B. L., Lin, Y. and Yalçinalp, L. Ü., Eds. IRISA, Campus de Beaulieu, France, 8–13.Google Scholar
Sterling, L. and Lalee, M. 1986. An explanation shell for expert systems. Computational Intelligence 2, 1, 136–141.CrossRefGoogle Scholar
Sterling, L. and Shapiro, E. Y. 1994. The Art of Prolog: Advanced Programming Techniques. MIT press.Google Scholar
Sterling, L. and Yalcinalp, L. U. 1989. Explaining prolog based expert systems using a layered meta-interpreter. In Proceedings of the 11th international joint conference on Artificial intelligence - Volume 1 (IJCAI’89), Vol. 1. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 66–71.Google Scholar
Sureshkumar, A., De Vos, M., Brain, M. and Fitch, J. 2007. APE: An AnsProlog* environment. In Proceedings of the 1st International Workshop on Software Engineering for Answer Set Programming (SEA’07), 101–115.Google Scholar
Syrjänen, T. 2006. Debugging inconsistent answer set programs. In Proceedings of the 11th International Workshop on Non-Monotonic Reasoning (NMR’06), 77–84.Google Scholar
Syrjänen, T. and Niemelä, I. 2001. The smodels system. In Logic Programming and Nonmotonic Reasoning. LPNMR 2001, Eiter, T., Faber, W., and Truszczyński, M., Eds. Notes, Lecture in Computer Science, vol. 2173. Springer, Berlin, Heidelberg, 434–438.CrossRefGoogle Scholar
Ulbricht, M., Thimm, M. and Brewka, G. 2016. Measuring inconsistency in answer set programs. In Logics in Artificial Intelligence. JELIA 2016, Michael, L. and Kakas, A., Eds. Lecture Notes in Computer Science, vol. 10021. Springer, Cham, 577–583.Google Scholar
van Emden, M. H. and Kowalski, R. A. 1976. The semantics of predicate logic as a programming language. Journal of the ACM 23, 4, 733–742.CrossRefGoogle Scholar
Van Gelder, A. 1989. The alternating fixpoint of logic programs with negation. In Proceedings of the 8th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems. ACM, 1–10.Google Scholar
Van Gelder, A., Ross, K. and Schlipf, J. S. 1988. Unfounded sets and well-founded semantics for general logic programs. In Proceedings of the 7th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems. ACM, 221–230.Google Scholar
Van Gelder, A., Ross, K. A. and Schlipf, J. S. 1991. The well-founded semantics for general logic programs. Journal of the ACM (JACM) 38, 3, 619–649.CrossRefGoogle Scholar
You, J.-H. and Yuan, L. Y. 1994. A Three-valued semantics for deductive databases and logic programs. Journal of Computer and System Sciences 49, 2, 334–361.CrossRefGoogle Scholar