Advertisement

An Algebra of Modular Systems: Static and Dynamic Perspectives

  • Eugenia TernovskaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11715)

Abstract

We introduce static and dynamic algebras for specifying combinations of modules communicating among them via shared second-order variables. In the static algebra, atomic modules are classes of structures. They are composed using operations of extended Codd’s relational algebra, or, equivalently, first-order logic with least fixed point. The dynamic algebra has essentially the same syntax, but with a specification of inputs and outputs in addition. In the dynamic setting, atomic modules are formalized in any framework that allows for the specification of their input-output behaviour by means of model expansion. Algebraic expressions are interpreted by binary relations on structures. We demonstrate connections of the dynamic algebra with a modal temporal logic and deterministic while programs.

Notes

Acknowledgements

Many thanks to Brett McLean, Jan Van den Bussche, Bart Bogaerts and other colleagues for useful discussions. The research is supported by NSERC.

References

  1. 1.
    Jónsson, B., Tarski, A.: Representation problems for relation algebras. Bull. Amer. Math. Soc. 74, 127–162 (1952)zbMATHGoogle Scholar
  2. 2.
    Pratt, V.R.: Origins of the calculus of binary relations. In: Proceedings of the Seventh Annual Symposium on Logic in Computer Science (LICS 1992), Santa Cruz, California, USA, 22–25 June 1992, pp. 248–254 (1992)Google Scholar
  3. 3.
    Surinx, D., den Bussche, J.V., Gucht, D.V.: The primitivity of operators in the algebra of binary relations under conjunctions of containments. In: LICS 2017 (2017)Google Scholar
  4. 4.
    Fletcher, G., et al.: Relative expressive power of navigational querying on graphs. Inf. Sci. 298, 390–406 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jackson, M., Stokes, T.: Modal restriction semigroups: towards an algebra of functions. IJAC 21(7), 1053–1095 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    McLean, B.: Complete representation by partial functions for composition, intersection and anti-domain. J. Log. Comput. 27(4), 1143–1156 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic (Foundations of Computing) (2000)Google Scholar
  8. 8.
    De Giacomo, G., Vardi, M.Y.: Linear temporal logic and linear dynamic logic on finite traces. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, IJCAI 2013, Beijing, China, 3–9 August 2013, pp. 854–860 (2013)Google Scholar
  9. 9.
    Reiter, R.: Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems. MIT Press, Cambridge (2001)CrossRefGoogle Scholar
  10. 10.
    Levesque, H., Reiter, R., Lespérance, Y., Lin, F., Scherl, R.: GOLOG: a logic programming language for dynamic domains. J Log. Program. Spec. Issue Actions 31(1–3), 59–83 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Tasharrofi, S., Ternovska, E.: A semantic account for modularity in multi-language modelling of search problems. In: Proceedings of the 8th International Symposium on Frontiers of Combining Systems (FroCoS), October 2011, pp. 259–274 (2011)CrossRefGoogle Scholar
  12. 12.
    Tasharrofi, S.: Arithmetic and modularity in declarative languages for knowledge representation. Ph.D. dissertation, School of Computing Science, Simon Fraser University, December 2013Google Scholar
  13. 13.
    Ternovska, E.: An algebra of combined constraint solving. In: Global Conference on Artificial Intelligence, GCAI 2015, Tbilisi, Georgia, 16–19 October 2015, pp. 275–295 (2015)Google Scholar
  14. 14.
    Ternovska, E.: Recent progress on the algebra of modular systems. In: Proceedings of the 11th Alberto Mendelzon International Workshop on Foundations of Data Management and the Web, Montevideo, Uruguay, 7–9 June 2017 (2017)Google Scholar
  15. 15.
    Henkin, L., Monk, J., Tarski, A.: Cylindric Algebras Part I. North-Holland, Amsterdam (1971)zbMATHGoogle Scholar
  16. 16.
    Bussche, J.: Applications of Alfred Tarski’s ideas in database theory. In: Fribourg, L. (ed.) CSL 2001. LNCS, vol. 2142, pp. 20–37. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44802-0_2CrossRefGoogle Scholar
  17. 17.
    Mitchell, D.G., Ternovska, E.: A framework for representing and solving NP search problems. In: Proceedings of AAAI 2005, pp. 430–435 (2005)Google Scholar
  18. 18.
    Vardi, M.Y.: The complexity of relational query language. In: 14th ACM Symposium Theory of Computing, Springer Verlag (Heidelberg, FRG and NewYork NY, USA)-Verlag, 1982Google Scholar
  19. 19.
    Denecker, M., Ternovska, E.: A logic of non-monotone inductive definitions. ACM Trans. Comput. Log. (TOCL) 9(2), 1–52 (2008)CrossRefGoogle Scholar
  20. 20.
    Kolokolova, A., Liu, Y., Mitchell, D., Ternovska, E.: On the complexity of model expansion. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 447–458. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-16242-8_32CrossRefGoogle Scholar
  21. 21.
    Ternovskaia, E.: Inductive definability and the situation calculus. In: Freitag, B., Decker, H., Kifer, M., Voronkov, A. (eds.) DYNAMICS 1997. LNCS, vol. 1472, pp. 227–248. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0055501CrossRefzbMATHGoogle Scholar
  22. 22.
    Abu Zaid, F., Grädel, E., Jaax, S.: Bisimulation safe fixed point logic. In: Invited and Contributed Papers from the Tenth Conference on Advances in Modal Logic 10. Advances in Modal Logic. Held in Groningen, The Netherlands, 5–8 August 2014, pp. 1–15 (2014)Google Scholar
  23. 23.
    Pratt, V.R.: Semantical considerations on Floyd-Hoare logic. In: 17th Annual Symposium on Foundations of Computer Science, Houston, Texas, USA, 25–27 October 1976, pp. 109–121 (1976)Google Scholar
  24. 24.
    Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. J. Comput. Syst. Sci. 18(2), 194–211 (1979)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Levesque, H., Reiter, R., Lespérance, Y., Lin, F., Scherl, R.: GOLOG: a logic programming language for dynamic domains. J. Log. Program. 31, 59–84 (1997)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mitchell, D., Ternovska, E.: Clause-learning for modular systems. In: Calimeri, F., Ianni, G., Truszczynski, M. (eds.) LPNMR 2015. LNCS (LNAI), vol. 9345, pp. 446–452. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-23264-5_37CrossRefGoogle Scholar
  27. 27.
    Bogaerts, B., Ternovska, E., Mitchell, D.: Propagators and solvers for the algebra of modular systems. In: Logic for Programming, Artificial Intelligence and Reasoning (LPAR) (2017)Google Scholar
  28. 28.
    Immerman, N.: Descriptive Complexity. Graduate Texts in Computer Science. Springer, Heidelberg (1999).  https://doi.org/10.1007/978-1-4612-0539-5CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Simon Fraser UniversityBurnabyCanada

Personalised recommendations