Skip to main content
SpringerLink
Log in

Initial Steps Towards a Family of Regular-Like Plan Description Logics

  • Chapter
  • First Online:
Description Logic, Theory Combination, and All That

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11560))

Abstract

A wide range of ordinary Description Logics (DLs) have been explored by considering collections of concept/role constructors, and types of terminologies, yielding an array of complexity results. Representation and reasoning with plans is a very important topic in AI, yet there has been very little work on finding and studying DL constructors for plan concepts.

We start to remedy this problem here by considering Plan DLs where concept instances are sequences of action instances, and hence plan concepts can be viewed as analogues of formal languages, describing sets of strings. Inspired by the clasp system, we consider using regular-like expressions, obtaining a rich variety of Plan DLs based on combinations of regular-like expression constructors, including sequence (concatenation), alternation (union, disjunction), looping (Kleene star), conjunction (intersection), and complement. To model the important notion of concurrency, we also consider interleaving.

We present results from the formal language literature which have immediate bearing on the complexity of DL-like reasoning tasks. However, we also focus on succinctness of representation, and on expressive power, issues first studied by Franz Baader for ordinary DLs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See the table at http://www.cs.man.ac.uk/~ezolin/dl/, for example.

  2. 2.

    We assume the reader is only familiar with basic properties of regular expressions and finite automata, as taught in undergraduate CS courses.

  3. 3.

    Some mathematical formalisms such as quantifiers over variables in temporal DLs (e.g., [35]) do not appear to have an obvious representation in such a notation.

  4. 4.

    clasp actually does more, because it takes into account action concept taxonomies and the structure of actions.

  5. 5.

    For succinctness, we will frequently refer to \(Actions\) and \(Actions\) \(^*\) by their more usual formal language symbols \(\varSigma \) and \(\varSigma ^*\).

  6. 6.

    Recall that many space complexity classes are known to be closed under complement.

References

  1. Artale, A., Franconi, E.: A temporal description logic for reasoning about actions and plans. J. Artif. Intell. Res. 9, 463–506 (1998)

    Article  MathSciNet  Google Scholar 

  2. Baader, F.: A formal definition for the expressive power of knowledge representation languages. In: Proceedings of the ECAI, pp. 53–58 (1990)

    Google Scholar 

  3. Baader, F.: A formal definition for the expressive power of terminological knowledge representation languages. J. Log. Comput. 6(1), 33–54 (1996)

    Article  MathSciNet  Google Scholar 

  4. Baader, F., Brandt, S., Lutz, C.: Pushing the EL envelope. In: IJCAI 2005, Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, Edinburgh, Scotland, UK, 30 July–5 August 2005, pp. 364–369 (2005)

    Google Scholar 

  5. Baader, F., Calvanese, D., McGuinness, D., Patel-Schneider, P., Nardi, D.: The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press (2003)

    Google Scholar 

  6. Baader, F., Liu, H., ul Mehdi, A.: Verifying properties of infinite sequences of description logic actions. In: ECAI, Frontiers in Artificial Intelligence and Applications, vol. 215, pp. 53–58. IOS Press (2010)

    Google Scholar 

  7. Baader, F., Lutz, C., Milicic, M., Sattler, U., Wolter, F.: Integrating description logics and action formalisms: first results. In: AAAI, pp. 572–577. AAAI Press (2005)

    Google Scholar 

  8. Berglund, M., Björklund, H., Björklund, J.: Shuffled languages–representation and recognition. Theor. Comput. Sci. 489, 1–20 (2013)

    Article  MathSciNet  Google Scholar 

  9. Borgida, A.: Towards the systematic development of description logic reasoners: CLASP reconstructed. In: Proceedings of the KR 1992, Cambridge, MA, USA, pp. 259–269 (1992)

    Google Scholar 

  10. Borgida, A., Brachman, R.J., McGuinness, D.L., Resnick, L.A.: CLASSIC: a structural data model for objects. In: Proceedings of SIGMOD 1989, pp. 58–67 (1989)

    Article  Google Scholar 

  11. Brachman, R.J., Levesque, H.J.: The tractability of subsumption in frame-based description languages. In: AAAI, vol. 84, pp. 34–37 (1984)

    Google Scholar 

  12. De Giacomo, G., Lenzerini, M.: Boosting the correspondence between description logics and propositional dynamic logics. In: Proceedings of the AAAI 1994, pp. 205–212 (1994)

    Google Scholar 

  13. De Giacomo, G., Lenzerini, M.: Tbox and Abox reasoning in expressive description logics. In: Proceedings of the AAAI, pp. 37–48. AAAI Press (1996)

    Google Scholar 

  14. De Giacomo, G., Vardi, M.Y.: Linear temporal logic and linear dynamic logic on finite traces. In: IJCAI 2013, pp. 854–860. IJCAI/AAAI (2013)

    Google Scholar 

  15. Devanbu, P.T., Litman, D.J.: Taxonomic plan reasoning. Artif. Intell. 84(1–2), 1–35 (1996)

    Article  Google Scholar 

  16. Fürer, M.: The complexity of the inequivalence problem for regular expressions with intersection. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 234–245. Springer, Heidelberg (1980). https://doi.org/10.1007/3-540-10003-2_74

    Chapter  Google Scholar 

  17. Gelade, W., Neven, F.: Succinctness of the complement and intersection of regular expressions. ACM Trans. Comput. Logic 4(1), 1–19 (2012)

    Article  MathSciNet  Google Scholar 

  18. Gil, Y.: Description logics and planning. AI Mag. 26(2), 73–84 (2005)

    Google Scholar 

  19. Gruber, H., Holzer, M.: Tight bounds on the descriptional complexity of regular expressions. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 276–287. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02737-6_22

    Chapter  Google Scholar 

  20. Holzer, M., Kutrib, M.: The complexity of regular(-like) expressions. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 16–30. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14455-4_3

    Chapter  Google Scholar 

  21. Hunt III, H.B.: The equivalence problem for regular expressions with intersections is not polynomial in tape. Technical report, pp. 73–161, Department of Computer Science, Cornell University, Ithaca, New York (1973)

    Google Scholar 

  22. Jȩdrzejowicz, J., Szepietowski, A.: Shuffle languages are in P. Theor. Comput. Sci. 250(1–2), 31–53 (2001)

    Article  MathSciNet  Google Scholar 

  23. Jiang, T., Ravikumar, B.: A note on the space complexity of some decision problems for finite automata. Inf. Process. Lett. 40(1), 25–31 (1991)

    Article  MathSciNet  Google Scholar 

  24. Jones, N.D.: Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci. 11(1), 68–85 (1975)

    Article  MathSciNet  Google Scholar 

  25. Kozen, D.: Lower bounds for natural proof systems. In: 18th Annual Symposium on Foundations of Computer Science, Providence, Rhode Island, USA, 31 October–1 November 1977, pp. 254–266 (1977)

    Google Scholar 

  26. Liu, H., Lutz, C., Milicic, M., Wolter, F.: DL actions with GCIs: a pragmatic approach. In: CEUR Workshop Proceedings of Description Logics, vol. 189. CEUR-WS.org (2006)

    Google Scholar 

  27. Liu, H., Lutz, C., Miličić, M., Wolter, F.: Reasoning about actions using description logics with general TBoxes. In: Fisher, M., van der Hoek, W., Konev, B., Lisitsa, A. (eds.) JELIA 2006. LNCS (LNAI), vol. 4160, pp. 266–279. Springer, Heidelberg (2006). https://doi.org/10.1007/11853886_23

    Chapter  MATH  Google Scholar 

  28. Lutz, C., Wolter, F., Zakharyaschev, M.: Temporal description logics: a survey. In Proceedings of Temporal Representation and Reasoning, pp. 3–14. IEEE (2008)

    Google Scholar 

  29. Mayer, A.J., Stockmeyer, L.J.: The complexity of word problems-this time with interleaving. Inf. Comput. 115(2), 293–311 (1994)

    Article  MathSciNet  Google Scholar 

  30. Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: 13th Annual Symposium on Switching and Automata Theory, College Park, Maryland, USA, 25–27 October 1972, pp. 125–129 (1972)

    Google Scholar 

  31. Nebel, B.: Terminological reasoning is inherently intractable. Artif. Intell. 43(2), 235–249 (1990)

    Article  MathSciNet  Google Scholar 

  32. Petersen, H.: Decision problems for generalized regular expressions. In: Descriptional Complexity of Automata, Grammars and Related Structures, Proceedings, DCAGRS 2000, pp. 22–29 (2000)

    Google Scholar 

  33. Petersen, H.: The membership problem for regular expressions with intersection is complete in LOGCFL. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 513–522. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45841-7_42

    Chapter  Google Scholar 

  34. Schild, K.: A correspondence theory for terminological logics: preliminary report. In: IJCAI, pp. 466–471. Morgan Kaufmann (1991)

    Google Scholar 

  35. Schmiedel, A.: Temporal terminological logic. In: Proceedings of AAAI 1990, pp. 640–645 (1990)

    Google Scholar 

  36. Sipser, M.: Introduction to the Theory of Computation. PWS Publishing Company (1997)

    Google Scholar 

  37. Stockmeyer, L.J.: The complexity of decision problems in automata theory and logic. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (1974)

    Google Scholar 

  38. Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: Symposium on Theory of Computing (STOC 1973), pp. 1–9 (1973)

    Google Scholar 

  39. Weida, R.: Knowledge representation for plan recognition. In: IJCAI 1995 Workshop on the Next Generation of Plan Recognition Systems (1995)

    Google Scholar 

  40. Woods, W.A.: What’s important about knowledge representation. IEEE Comput. 16(10), 22–26 (1983)

    Article  Google Scholar 

Download references

Acknowledgement

I am very grateful to my colleague, Eric Allender for his patient guidance through the landscape of modern complexity theory, and various kinds of reductions. Grant Weddell and David Toman provided useful comments and probing questions about the goal of the entire enterprise.

Author information

Authors and Affiliations

  1. Department of Computer Science, Rutgers University, New Brunswick, NJ, USA

    Alexander Borgida

Authors
  1. Alexander Borgida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Borgida .

Editor information

Editors and Affiliations

  1. University of Bremen, Bremen, Germany

    Carsten Lutz

  2. University of Manchester, Manchester, UK

    Uli Sattler

  3. University of Iowa, Iowa City, IA, USA

    Cesare Tinelli

  4. TU Dresden, Dresden, Germany

    Anni-Yasmin Turhan

  5. University of Liverpool, Liverpool, UK

    Frank Wolter

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Borgida, A. (2019). Initial Steps Towards a Family of Regular-Like Plan Description Logics. In: Lutz, C., Sattler, U., Tinelli, C., Turhan, AY., Wolter, F. (eds) Description Logic, Theory Combination, and All That. Lecture Notes in Computer Science(), vol 11560. Springer, Cham. https://doi.org/10.1007/978-3-030-22102-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-22102-7_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22101-0

  • Online ISBN: 978-3-030-22102-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation