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Bireachability and Final Multialgebras

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Algebra and Coalgebra in Computer Science (CALCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3629))

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Abstract

Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational structures, representing relations as multivalued functions by selecting one argument as the “result”. This leads to strong algebraic properties missing in the case of relational structures. However, such strong properties can be obtained only by first choosing appropriate notion of homomorphism. We summarize earlier results on the possible notions of compositional homomorphisms of multialgebras and investigate in detail one of them, the outer-tight homomorphisms which yield rich structural properties not offered by other alternatives. The outer-tight homomorphisms are different from those obtained when relations are modeled as coalgebras and the associated congruence is the converse bisimulation equivalence. The category is cocomplete but initial objects are of little interest (essentially empty). On the other hand, the category does not, in general, possess final objects for the usual cardinality reasons. The main objective of the paper is to show that Aczel’s construction of final coalgebras for set-based functors can be modified and applied to multialgebras. We therefore extend the category admitting also structures over proper classes and show the existence of final objects in this category.

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References

  1. Aczel, P.: Non-well-founded sets. Technical Report 14, CSLI (1988)

    Google Scholar 

  2. Aczel, P., Mendler, N.: A final coalgebra theorem. In: Dybjer, P., Pitts, A.M., Pitt, D.H., Poigné, A., Rydeheard, D.E. (eds.) Category Theory and Computer Science. LNCS, vol. 389, pp. 357–365. Springer, Heidelberg (1989)

    Chapter  Google Scholar 

  3. Bošnjak, I., Madarász, R.: On power structures. Algebra and Discrete Mathematics 2, 14–35 (2003)

    Google Scholar 

  4. Brink, C.: Power structures. Algebra Universalis 30, 177–216 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cohn, P.M.: Universal Algebra. Mathematics and Its Applications, vol. 6. D.Reidel Publishing Company (1965)

    Google Scholar 

  6. de Roever, W.-P., Engelhardt, K.: Data Refinement: Model-Oriented Proof Methods and their Comparison. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  7. Grätzer, G.: A representation theorem for multialgebras. Arch. Math. 13 (1962)

    Google Scholar 

  8. Hagino, T.: A Categorical Programming Language. PhD thesis, Department of Computer Science, University of Edinburgh (1987)

    Google Scholar 

  9. Hermida, C.: A categorical outlook on relational modalities and simulations. In: Mendler, M., Goré, R.P., de Paiva, V. (eds.) Intuitionistic Modal Logic and Aplications, July 26. DIKU technical reports, vol. 02-15, pp. 17–34 (2002)

    Google Scholar 

  10. Hesselink, W.H.: A mathematical approach to nondeterminism in data types. ACM ToPLaS 10 (1988)

    Google Scholar 

  11. Hußmann, H.: Nondeterministic algebraic specifications and nonconfluent term rewriting. In: Grabowski, J., Wechler, W., Lescanne, P. (eds.) ALP 1988. LNCS, vol. 343. Springer, Heidelberg (1989)

    Google Scholar 

  12. Hußmann, H.: Nondeterministic algebraic specifications. PhD thesis, Fak. f. Mathematik und Informatik, Universitat Passau (1990)

    Google Scholar 

  13. Hußmann, H.: Nondeterminism in Algebraic Specifications and Algebraic Programs. Birkhäuser, Basel (1993); revised version of [12]

    MATH  Google Scholar 

  14. Jónsson, B., Tarski, A.: Boolean algebras with operators i. American J. Mathematics 73, 891–939 (1951)

    Article  MATH  Google Scholar 

  15. Jónsson, B., Tarski, A.: Boolean algebras with operators ii. American J. Mathematics 74, 127–162 (1952)

    Article  MATH  Google Scholar 

  16. Lamo, Y.: The institution of multialgebras – a general framework for algebraic software development. PhD thesis, Department of Informatics, University of Bergen (2002)

    Google Scholar 

  17. Madarász, R.: Remarks on power structures. Algebra Universalis 34(2), 179–184 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Meseguer, J.: Membership algebra as a logical framework for equational specification. In: Parisi-Presicce, F. (ed.) WADT 1997. LNCS, vol. 1376, pp. 18–61. Springer, Heidelberg (1998)

    Google Scholar 

  19. Pickert, G.: Bemerkungen zum homomorphie-begriff. Mathematische Zeitschrift 53 (1950)

    Google Scholar 

  20. Pickett, H.E.: Homomorphisms and subalgebras of multialgebras. Pacific J. of Mathematics 21, 327–342 (1967)

    MATH  MathSciNet  Google Scholar 

  21. Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249, 3–80 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schweigert, D.: Congruence relations on multialgebras. Discrete Mathematics 53, 249–253 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  23. Walicki, M., Hodzic, A., Meldal, S.: Compositional homomorphisms of relational structures. In: Freivalds, R. (ed.) FCT 2001. LNCS, vol. 2138., Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  24. Walicki, M., Meldal, S.: A complete calculus for the multialgebraic and functional semantics of nondeterminism. ACM ToPLaS 17(2) (1995)

    Google Scholar 

  25. Walicki, M., Meldal, S.: Multialgebras, power algebras and complete calculi of identities and inclusions. In: Reggio, G., Astesiano, E., Tarlecki, A. (eds.) Abstract Data Types 1994 and COMPASS 1994. LNCS, vol. 906. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  26. Walicki, M., Meldal, S.: Algebraic approaches to nondeterminism – an overview. ACM Computing Surveys 29(1) (March 1997)

    Google Scholar 

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Walicki, M. (2005). Bireachability and Final Multialgebras. In: Fiadeiro, J.L., Harman, N., Roggenbach, M., Rutten, J. (eds) Algebra and Coalgebra in Computer Science. CALCO 2005. Lecture Notes in Computer Science, vol 3629. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11548133_26

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  • DOI: https://doi.org/10.1007/11548133_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28620-2

  • Online ISBN: 978-3-540-31876-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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