Abstract
Recent developments triggered by initiatives such as the Semantic Web, Linked Open Data, the Web of Things, and geographic information systems resulted in the wide and increasing availability of machine-processable data and knowledge in the form of data streams and knowledge bases. Applications building on such knowledge require reasoning with modal and intensional concepts, such as time, space, and obligations, that are defeasible. E.g., in the presence of data streams, conclusions may have to be revised due to newly arriving information. The current literature features a variety of domain-specific formalisms that allow for defeasible reasoning using specific intensional concepts. However, many of these formalisms are computationally intractable and limited to one of the mentioned application domains. In this paper, we define a general method for obtaining defeasible inferences over intensional concepts, and we study conditions under which these inferences are computable in polynomial time.
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Notes
- 1.
- 2.
Note that we often leave \(\mathcal {O}\) implicit as N allows to uniquely determine all elements from \(\mathcal {O}\). Also, to ease the presentation, we only consider unary intensional operators. Others can then often be represented using rules (see also [34]).
- 3.
We follow the usual notation in modal logic and interpretations explicitly include the corresponding frame.
- 4.
Since the intersection of an empty sequence of subsets of a set is the entire set, then, for n=0, i.e., when the body of the rule is empty, the satisfaction condition is just \(w\in \Vert A\Vert ^\dagger \) for any \(\dagger \in \{\top ,\mathsf{u}\}\).
- 5.
Due to space restrictions, we are not able to provide full details and examples of this procedure.
- 6.
Corresponding results for the data complexity of this problem for programs with variables can then be achieved in the usual way [20].
- 7.
This also aligns well with related work, e.g., for reasoning with time, such as stream reasoning where a finite timeline is often assumed, and avoids the exponential explosion on the number of worlds for satisfiability for some epistemic structures [40].
References
Abadi, M., Manna, Z.: Temporal logic programming. J. Symb. Comput. 8(3), 277–295 (1989)
Alberti, M., Gomes, A.S., Gonçalves, R., Leite, J., Slota, M.: Normative systems represented as hybrid knowledge bases. In: Leite, J., Torroni, P., Ågotnes, T., Boella, G., van der Torre, L. (eds.) CLIMA 2011. LNCS (LNAI), vol. 6814, pp. 330–346. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22359-4_23
Alberti, M., Knorr, M., Gomes, A.S., Leite, J., Gonçalves, R., Slota, M.: Normative systems require hybrid knowledge bases. In: AAMAS. IFAAMAS, pp. 1425–1426 (2012)
Allen, J.F.: Maintaining knowledge about temporal intervals. In: Readings in Qualitative Reasoning About Physical Systems, pp. 361–372. Elsevier, Amsterdam (1990)
Anicic, D., Rudolph, S., Fodor, P., Stojanovic, N.: Stream reasoning and complex event processing in ETALIS. Semant. Web 3(4), 397–407 (2012)
Arasu, A., Babu, S., Widom, J.: The CQL continuous query language: semantic foundations and query execution. VLDB J. 15(2), 121–142 (2006)
Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications, 2nd edn. Cambridge University Press, Cambridge (2007)
Barbieri, D.F., Braga, D., Ceri, S., Valle, E.D., Grossniklaus, M.: C-SPARQL: a continuous query language for RDF data streams. Int. J. Semant. Comput. 4(1), 3–25 (2010)
Bazoobandi, H.R., Beck, H., Urbani, J.: Expressive stream reasoning with laser. In: d’Amato, C., et al. (eds.) ISWC 2017. LNCS, vol. 10587, pp. 87–103. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68288-4_6
Beck, H., Dao-Tran, M., Eiter, T.: LARS: a logic-based framework for analytic reasoning over streams. Artif. Intell. 261, 16–70 (2018)
Beirlaen, M., Heyninck, J., Straßer, C.: Structured argumentation with prioritized conditional obligations and permissions. J. Logic Comput. 29(2), 187–214 (2019)
Brandt, S., Kalayci, E.G., Ryzhikov, V., Xiao, G., Zakharyaschev, M.: Querying log data with metric temporal logic. J. Artif. Intell. Res. 62, 829–877 (2018)
Brenton, C., Faber, W., Batsakis, S.: Answer set programming for qualitative spatio-temporal reasoning: Methods and experiments. In: Technical Communications of ICLP. OASICS, vol. 52, pp. 4:1–4:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)
Brewka, G., Ellmauthaler, S., Gonçalves, R., Knorr, M., Leite, J., Pührer, J.: Reactive multi-context systems: Heterogeneous reasoning in dynamic environments. Artif. Intell. 256, 68–104 (2018)
Cabalar, P., Dieguez, M., Schaub, T., Schuhmann, A.: Towards metric temporal answer set programming. Theory Pract. Logic Program. 20(5), 783–798 (2020)
Caminada, M., Sá, S., Alcântara, J., Dvořák, W.: On the equivalence between logic programming semantics and argumentation semantics. Int. J. Approx. Reasoning 58, 87–111 (2015)
del Cerro, L.F.: MOLOG: a system that extends PROLOG with modal logic. New Gener. Comput. 4(1), 35–50 (1986). https://doi.org/10.1007/BF03037381
Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)
Chen, Y., Wan, H., Zhang, Y., Zhou, Y.: dl2asp: implementing default logic via answer set programming. In: Janhunen, T., Niemelä, I. (eds.) JELIA 2010. LNCS (LNAI), vol. 6341, pp. 104–116. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15675-5_11
Dantsin, E., Eiter, T., Gottlob, G., Voronkov, A.: Complexity and expressive power of logic programming. ACM Comput. Surv. 33(3), 374–425 (2001)
Gelder, A.V.: The alternating fixpoint of logic programs with negation. In: Proceedings of SIGACT-SIGMOD-SIGART, pp. 1–10. ACM Press (1989)
Gelder, A.V., Ross, K.A., Schlipf, J.S.: The well-founded semantics for general logic programs. J. ACM 38(3), 620–650 (1991)
Gelfond, M.: Answer sets. In: Handbook of Knowledge Representation, Foundations of Artificial Intelligence, vol. 3, pp. 285–316. Elsevier, Amsterdam (2008)
Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Gener. Comput. 9(3–4), 365–385 (1991). https://doi.org/10.1007/BF03037169
Gonçalves, R., Alferes, J.J.: Specifying and reasoning about normative systems in deontic logic programming. In: Proceedings of AAMAS. IFAAMAS, pp. 1423–1424 (2012)
Gonçalves, R., Knorr, M., Leite, J.: Evolving multi-context systems. In: ECAI. Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 375–380. IOS Press, Amsterdam (2014)
Governatori, G., Rotolo, A., Riveret, R.: A deontic argumentation framework based on deontic defeasible logic. In: Miller, T., Oren, N., Sakurai, Y., Noda, I., Savarimuthu, B.T.R., Cao Son, T. (eds.) PRIMA 2018. LNCS (LNAI), vol. 11224, pp. 484–492. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03098-8_33
Izmirlioglu, Y., Erdem, E.: Qualitative reasoning about cardinal directions using answer set programming. In: Proceedings of AAAI, pp. 1880–1887. AAAI Press (2018)
Kasalica, V., Gerochristos, I., Alferes, J.J., Gomes, A.S., Knorr, M., Leite, J.: Telco network inventory validation with NoHR. In: Balduccini, M., Lierler, Y., Woltran, S. (eds.) Logic Programming and Nonmonotonic Reasoning. LPNMR 2019. Lecture Notes in Computer Science, vol. 11481, pp. 18–31. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-20528-7_2
Kasalica, V., Knorr, M., Leite, J., Lopes, C.: NoHR: An Overview. Künstl Intell, Heidelberg (2020)
Knorr, M., Alferes, J.J., Hitzler, P.: Local closed world reasoning with description logics under the well-founded semantics. Artif. Intell. 175(9–10), 1528–1554 (2011)
Knorr, M., Hitzler, P.: A comparison of disjunctive well-founded semantics. In: FAInt. CEUR Workshop Proceedings, vol. 277 (2007). CEUR-WS.org
Motik, B., Rosati, R.: Reconciling description logics and rules. J. ACM 57(5), 30:1–30:62 (2010)
Orgun, M.A., Wadge, W.W.: Towards a unified theory of intensional logic programming. J. Logic Program. 13(4), 413–440 (1992)
Pacuit, E.: Neighborhood Semantics for Modal Logic. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-319-67149-9
Panagiotidi, S., Nieves, J.C., Vázquez-Salceda, J.: A framework to model norm dynamics in answer set programming. In: MALLOW (2009)
Le-Phuoc, D., Dao-Tran, M., Xavier Parreira, J., Hauswirth, M.: A native and adaptive approach for unified processing of linked streams and linked data. In: Aroyo, L., et al. (eds.) ISWC 2011. LNCS, vol. 7031, pp. 370–388. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25073-6_24
Przymusinski, T.C.: Stable semantics for disjunctive programs. New Gener. Comput. 9(3/4), 401–424 (1991). https://doi.org/10.1007/BF03037171
Suchan, J., Bhatt, M., Walega, P.A., Schultz, C.P.L.: Visual explanation by high-level abduction: on answer-set programming driven reasoning about moving objects. In: Proceedings of AAAI, pp. 1965–1972. AAAI Press (2018)
Vardi, M.Y.: On the complexity of epistemic reasoning. In: Proceedings of LICS. pp. 243–252. IEEE Computer Society (1989)
Walega, P.A., Kaminski, M., Grau, B.C.: Reasoning over streaming data in metric temporal datalog. In: Proceedings of AAAI, pp. 3092–3099. AAAI Press (2019)
Walega, P.A., Schultz, C.P.L., Bhatt, M.: Non-monotonic spatial reasoning with answer set programming modulo theories. TPLP 17(2), 205–225 (2017)
Acknowledgements
The authors are indebted to the anonymous reviewers of this paper for helpful feedback. The authors were partially supported by FCT project RIVER (PTDC/CCI-COM/30952/2017) and by FCT project NOVA LINCS (UIDB/04516/2020). J. Heyninck was also supported by the German National Science Foundation under the DFG-project CAR (Conditional Argumentative Reasoning) KE-1413/11-1.
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Heyninck, J., Gonçalves, R., Knorr, M., Leite, J. (2021). Tractable Reasoning Using Logic Programs with Intensional Concepts. In: Faber, W., Friedrich, G., Gebser, M., Morak, M. (eds) Logics in Artificial Intelligence. JELIA 2021. Lecture Notes in Computer Science(), vol 12678. Springer, Cham. https://doi.org/10.1007/978-3-030-75775-5_22
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