Skip to main content
SpringerLink
Log in

Expressivity and Complexity of Dependence Logic

  • Chapter
  • First Online:
Dependence Logic

Abstract

In this article we review recent results on expressivity and complexity of first-order, modal, and propositional dependence logic and some of its variants such as independence and inclusion logic. Dependence logic was introduced by Jouko Väänänen in [56]. On the syntactic side, it extends usual first-order logic by the so-called dependence atoms the meaning of which is that the value of x n is functionally determined by the values of x 1, …, x n−1. The semantics of dependence logic is defined using sets of assignments, teams, rather than single assignments as in first-order logic. Since the introduction of dependence logic in 2007, the area of team semantics has evolved into a general framework for logics in which various notions of dependence and independence can be formalized and studied. In this paper we mainly consider variants of dependence logic arising by replacing/supplementing dependence atoms with further dependency notions, and we also study propositional and modal variants.

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
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Abramsky, S., Väänänen, J.: From IF to BI: a tale of dependence and separation. Synthese 167 (2, Knowledge, Rationality & Action), 207–230 (2009). doi:10.1007/s11229-008-9415-6. http://dx.doi.org/10.1007/s11229-008-9415-6

    Google Scholar 

  2. Blackburn, P., de Rijke, M., Venema, Y.: Modal logics. In: Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  3. Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  4. Church, A.: A note on the Entscheidungsproblem. J. Symb. Log. 1 (1), 40–41 (1936)

    Article  MATH  Google Scholar 

  5. Cook, S.A.: A hierarchy for nondeterministic time complexity. In: Conference Record, Fourth Annual ACM Symposium on Theory of Computing, pp. 187–192. ACM, New York (1972)

    Google Scholar 

  6. Durand, A., Kontinen, J.: Hierarchies in dependence logic. ACM Trans. Comput. Log. 13 (4), 1–21 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Durand, A., Ebbing, J., Kontinen, J., Vollmer, H.: Dependence logic with a majority quantifier. In: Chakraborty, S., Kumar, A. (eds.) FSTTCS, LIPIcs, vol. 13, pp. 252–263. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2011)

    Google Scholar 

  8. Durand, A., Ebbing, J., Kontinen, J., Vollmer, H.: Dependence logic with a majority quantifier. J. Log. Lang. Inf. 24 (3), 289–305 (2015). doi:10.1007/s10849-015-9218-3. http://dx.doi.org/10.1007/s10849-015-9218-3

    Google Scholar 

  9. Ebbing, J., Lohmann, P.: Complexity of model checking for modal dependence logic. In: SOFSEM 2012: Theory and Practice of Computer Science. Lecture Notes in Computer Science, vol. 7147, pp. 226–237. Springer, Berlin, Heidelberg (2012)

    Google Scholar 

  10. Ebbing, J., Hella, L., Meier, A., Müller, J.S., Virtema, J., Vollmer, H.: Extended modal dependence logic. In: WoLLIC. Lecture Notes in Computer Science, vol. 8071, pp. 126–137. Springer, Berlin, Heidelberg (2013)

    Google Scholar 

  11. Ebbing, J., Kontinen, J., Müller, J., Vollmer, H.: A fragment of dependence logic capturing polynomial time. Log. Methods Comput. Sci. 10 (3) (2014). doi:10.2168/LMCS-10(3:3)2014. http://dx.doi.org/10.2168/LMCS-10(3:3)2014

  12. Engström, F.: Generalized quantifiers in dependence logic. J. Log. Lang. Inf. 21, 299–324 (2012). http://dx.doi.org/10.1007/s10849-012-9162-4. 10.1007/s10849-012-9162-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Engström, F., Kontinen, J.: Characterizing quantifier extensions of dependence logic. J. Symb. Log. 78, 0–9 (2013)

    Google Scholar 

  14. Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. In: Complexity of Computation. Proceedings of SIAM-AMS Symposium in Applied Mathematics, New York, 1973, pp. 43–73. SIAM–AMS Proceedings, vol. VII. American Mathematical Society, Providence, RI (1974)

    Google Scholar 

  15. Fagin, R.: Finite-model theory - a personal perspective. Theor. Comput. Sci. 116 (1&2), 3–31 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Galliani, P.: Inclusion and exclusion dependencies in team semantics - on some logics of imperfect information. Ann. Pure Appl. Log. 163 (1), 68–84 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Galliani, P., Hella, L.: Inclusion logic and fixed point logic. In: Rocca, S.R.D. (ed.) Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), vol. 23, pp. 281–295. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2013). doi:http://dx.doi.org/10.4230/LIPIcs.CSL.2013.281. http://drops.dagstuhl.de/opus/volltexte/2013/4203

  18. Galliani, P., Hannula, M., Kontinen, J.: Hierarchies in independence logic. In: Rocca, S.R.D. (ed.) Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), vol. 23, pp. 263–280. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2013). doi:http://dx.doi.org/10.4230/LIPIcs.CSL.2013.263. http://drops.dagstuhl.de/opus/volltexte/2013/4202

  19. Grädel, E.: Capturing complexity classes by fragments of second-order logic. Theor. Comput. Sci. 101 (1), 35–57 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grädel, E.: Model-checking games for logics of imperfect information. Theor. Comput. Sci. 493, 2–14 (2013). doi:10.1016/j.tcs.2012.10.033. http://dx.doi.org/10.1016/j.tcs.2012.10.033

    Google Scholar 

  21. Grädel, E., Väänänen, J.A.: Dependence and independence. Stud. Logica 101 (2), 399–410 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grandjean, E., Olive, F.: Graph properties checkable in linear time in the number of vertices. J. Comput. Syst. Sci. 68 (3), 546–597 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hannula, M.: Axiomatizing first-order consequences in independence logic. Ann. Pure Appl. Logic 166 (1), 61–91 (2015). doi:10.1016/j.apal.2014.09.002. http://dx.doi.org/10.1016/j.apal.2014.09.002

    Google Scholar 

  24. Hannula, M.: Hierarchies in inclusion logic with lax semantics. In: Banerjee, M., Krishna, S.N. (eds.) Proceedings of Logic and Its Applications - 6th Indian Conference, ICLA 2015, Mumbai, India, January 8–10, 2015. Lecture Notes in Computer Science, vol. 8923, pp. 100–118. Springer, Berlin (2015). doi:10.1007/978-3-662-45824-2_7

    Google Scholar 

  25. Hannula, M.: On variants of dependence logic: axiomatizability and expressiveness. Ph.D. thesis, University of Helsinki (2015)

    Google Scholar 

  26. Hannula, M., Kontinen, J.: Hierarchies in independence and inclusion logic with strict semantics. J. Log. Comput. 25 (3), 879–897 (2015). doi:10.1093/logcom/exu057. http://dx.doi.org/10.1093/logcom/exu057

    Google Scholar 

  27. Hannula, M., Kontinen, J., Virtema, J., Vollmer, H.: Complexity of propositional independence and inclusion logic. In: Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Proceedings, Part I. Lecture Notes in Computer Science, vol. 9234, pp. 269–280. Springer, Berlin (2015)

    Google Scholar 

  28. Hella, L., Stumpf, J.: The expressive power of modal logic with inclusion atoms. In: Esparza, J., Tronci, E. (eds.) Proceedings Sixth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2015, Genoa, Italy, 21–22nd September 2015, EPTCS, vol. 193, pp. 129–143 (2015). doi:10.4204/EPTCS.193.10

    Google Scholar 

  29. Hella, L., Luosto, K., Sano, K., Virtema, J.: The expressive power of modal dependence logic. In: Advances in Modal Logic 10, Invited and Contributed Papers from the Tenth Conference on “Advances in Modal Logic,” Held in Groningen, The Netherlands, August 5–8, 2014, pp. 294–312. College Publications, London (2014). http://www.aiml.net/volumes/volume10/Hella-Luosto-Sano-Virtema.pdf

  30. Hella, L., Kuusisto, A., Meier, A., Vollmer, H.: Modal inclusion logic: being lax is simpler than being strict. In: Italiano, G.F., Pighizzini, G., Sannella, D. (eds.) Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Milan, Italy, August 24–28, 2015, Proceedings, Part I. Lecture Notes in Computer Science, vol. 9234, pp. 281–292. Springer, Berlin (2015)

    Google Scholar 

  31. Hemaspaandra, E.: The complexity of poor man’s logic. J. Log. Comput. 11 (4), 609–622 (2001). Corrected version: [32]

    Google Scholar 

  32. Hemaspaandra, E.: The complexity of poor man’s logic. CoRR cs.LO/9911014v2 (2005). http://arxiv.org/abs/cs/9911014v2

  33. Henkin, L.: Logical Systems Containing Only a Finite Number of Symbols. Presses De l’Université De Montréal, Montreal (1967)

    MATH  Google Scholar 

  34. Hintikka, J., Sandu, G.: Informational independence as a semantical phenomenon. In: Logic, Methodology and Philosophy of Science, VIII (Moscow, 1987). Studies in Logic and the Foundations of Mathematics, vol. 126, pp. 571–589. North-Holland, Amsterdam (1989). doi:10.1016/S0049-237X(08)70066-1. http://dx.doi.org/10.1016/S0049-237X(08)70066-1

    Google Scholar 

  35. Hodges, W.: Compositional semantics for a language of imperfect information. Log. J. IGPL 5, 539–563 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Immerman, N.: Relational queries computable in polynomial time. Inf. Control. 68 (1-3), 86–104 (1986). doi:10.1016/S0019-9958(86)80029-8. http://dx.doi.org/10.1016/S0019-9958(86)80029-8

    Google Scholar 

  37. Kontinen, J.: Coherence and computational complexity of quantifier-free dependence logic formulas. Stud. Logica 101 (2), 267–291 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kontinen, J., Nurmi, V.: Team logic and second-order logic. Fundamenta Informaticae 106 (2–4), 259–272 (2011)

    MathSciNet  MATH  Google Scholar 

  39. Kontinen, J., Väänänen, J.: On definability in dependence logic. J. Log. Lang. Inf. 18 (3), 317–332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kontinen, J., Kuusisto, A., Lohmann, P., Virtema, J.: Complexity of two-variable dependence logic and if-logic. Inf. Comput. 239, 237–253 (2014). doi:10.1016/j.ic.2014.08.004. http://dx.doi.org/10.1016/j.ic.2014.08.004

    Google Scholar 

  41. Kontinen, J., Kuusisto, A., Virtema, J.: Decidable fragments of logics based on team semantics. CoRR abs/1410.5037 (2014). http://arxiv.org/abs/1410.5037

  42. Kontinen, J., Müller, J., Schnoor, H., Vollmer, H.: Modal independence logic. In: Advances in Modal Logic 10, Invited and Contributed Papers from the Tenth Conference on “Advances in Modal Logic,” Held in Groningen, The Netherlands, August 5–8, 2014, pp. 353–372. College Publications, London (2014). http://www.aiml.net/volumes/volume10/Kontinen-Mueller-Schnoor-Vollmer.pdf

  43. Kontinen, J., Müller, J.S., Schnoor, H., Vollmer, H.: A Van Benthem Theorem for Modal Team Semantics. In: 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), vol. 41, pp. 277–291. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2015)

    Google Scholar 

  44. Kuusisto, A.: A double team semantics for generalized quantifiers. J. Log. Lang. Inf. 24 (2), 149–191 (2015). doi:10.1007/s10849-015-9217-4. http://dx.doi.org/10.1007/s10849-015-9217-4

    Google Scholar 

  45. Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6 (3), 467–480 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lohmann, P., Vollmer, H.: Complexity results for modal dependence logic. Stud. Logica 101 (2), 343–366 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Müller, J.S.: Satisfiability and model checking in team based logics. Ph.D. thesis, Leibniz Universität Hannover (2014)

    Google Scholar 

  48. Nurmi, V.: Dependence logic: Investigations into higher-order semantics defined on teams. Ph.D. thesis, University of Helsinki (2009)

    Google Scholar 

  49. Peterson, G., Reif, J., Azhar, S.: Lower bounds for multiplayer noncooperative games of incomplete information. Comput. Math. Appl. 41 (7-8), 957–992 (2001). doi:DOI:10.1016/S0898-1221(00)00333-3. http://www.sciencedirect.com/science/article/B6TYJ-43P387P-19/2/cd72ba7ccb5f3c2a2b65eb3c45aa2ca7

    Google Scholar 

  50. Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. J. Log. Lang. Inf. 14, 369–395 (2005). doi:10.1007/s10849-005-5791-1. http://portal.acm.org/citation.cfm?id=1080942.1080949

    Google Scholar 

  51. Rönnholm, R.: Capturing k-ary existential second order logic with k-ary inclusion-exclusion logic. CoRR abs/1502.05632 (2015). http://arxiv.org/abs/1502.05632

  52. Sano, K., Virtema, J.: Axiomatizing propositional dependence logics. In: Kreutzer, S. (ed.) 24th EACSL Annual Conference on Computer Science Logic, CSL 2015, September 7–10, 2015, Berlin, Germany. LIPIcs, vol. 41, pp. 292–307. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015). doi:10.4230/LIPIcs.CSL.2015.292

  53. Sevenster, M.: Model-theoretic and computational properties of modal dependence logic. J. Log. Comput. 19 (6), 1157–1173 (2009). doi:10.1093/logcom/exn102. http://logcom.oxfordjournals.org/cgi/content/abstract/exn102v1

    Google Scholar 

  54. Tulenheimo, T., Sevenster, M.: On modal logic, IF logic, and IF modal logic. In: Advances in Modal Logic 6, Papers from the Sixth Conference on “Advances in Modal Logic,” Held in Noosa, Queensland, Australia, on 25–28 September 2006, pp. 481–501. College Publications, London (2006). http://www.aiml.net/volumes/volume6/Tulenheimo-Sevenster.ps

  55. Turing, A.: On computable numbers, with an application to the entscheidungsproblem. Proc. Lond. Math. Soc. Ser. 2 42, 230–265 (1936)

    MathSciNet  MATH  Google Scholar 

  56. Väänänen, J.: Dependence Logic. London Mathematical Society Student Texts, vol. 70. Cambridge University Press, Cambridge (2007). doi:10.1017/CBO9780511611193. http://dx.doi.org/10.1017/CBO9780511611193

  57. Väänänen, J.: Team Logic. In: Johan van Benthem Benedikt Löowe, D.G. (ed.) Interactive Logic. Texts in Logic and Games, vol. 1, pp. 281–302. Amsterdam University Press, Amsterdam (2007)

    Google Scholar 

  58. Väänänen, J.: Modal dependence logic. In: Apt, K.R., van Rooij, R. (eds.) New Perspectives on Games and Interaction. Texts in Logic and Games, vol. 4, pp. 237–254. Amsterdam University Press, Amsterdam (2008)

    Google Scholar 

  59. van Benthem, J.: Modal Logic and Classical Logic. Bibliopolis, Berkeley, CA (1985)

    MATH  Google Scholar 

  60. Vardi, M.Y.: The complexity of relational query languages (extended abstract). In: Lewis, H.R., Simons, B.B., Burkhard, W.A., Landweber, L.H. (eds.) Proceedings of the 14th Annual ACM Symposium on Theory of Computing, May 5–7, 1982, San Francisco, California, USA, pp. 137–146. ACM, New York (1982). doi:10.1145/800070.802186

  61. Virtema, J.: Complexity of validity for propositional dependence logics. In: Proceedings Fifth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2014, Verona, Italy, September 10–12, 2014. Electronic Proceedings in Theoretical Computer Science, vol. 161, pp. 18–31 (2014). doi:10.4204/EPTCS.161.5. http://dx.doi.org/10.4204/EPTCS.161.5

    Google Scholar 

  62. Yang, F.: Expressing second-order sentences in intuitionistic dependence logic. Stud. Logica 101 (2), 323–342 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  63. Yang, F.: On extensions and variants of dependence logic – a study of intuitionistic connectives in the team semantics setting. Ph.D. thesis, University of Helsinki (2014)

    Google Scholar 

  64. Yang, F., Väänänen, J.: Propositional logics of dependence. Annals of Pure and Applied Logic 167, 557–589 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referee for corrections and valuable comments. The second author was supported by grants 292767, 275241, and 264917 of the Academy of Finland.

Author information

Authors and Affiliations

  1. IMJ-PRG, CNRS UMR 7586, Université Paris Diderot, Case 7012, 75205, Paris cedex 13, France

    Arnaud Durand

  2. Department of Mathematics and Statistics, University of Helsinki, 68, 00014, Helsinki, Finland

    Juha Kontinen

  3. Institut für Theoretische Informatik, Leibniz Universität Hannover, Appelstr. 4, 30167, Hannover, Germany

    Heribert Vollmer

Authors
  1. Arnaud Durand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juha Kontinen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heribert Vollmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnaud Durand .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, United Kingdom

    Samson Abramsky

  2. Department of Mathematics & Statistics, University of Helsinki, Helsinki, Finland

    Juha Kontinen

  3. Dept of Mathematics & Statistics, University of Helsinki, Helsinki, Finland

    Jouko Väänänen

  4. Institut für Theoretische Informatik, Leibniz Universität Hannover, Hannover, Germany

    Heribert Vollmer

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Durand, A., Kontinen, J., Vollmer, H. (2016). Expressivity and Complexity of Dependence Logic. In: Abramsky, S., Kontinen, J., Väänänen, J., Vollmer, H. (eds) Dependence Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-31803-5_2

Download citation

Publish with us

Policies and ethics

Search

Navigation