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Main Steps in Defining Finitely Supported Mathematics

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Information and Communication Technologies in Education, Research, and Industrial Applications (ICTERI 2015)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 594))

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Abstract

This paper presents the main steps in defining a Finitely Supported Mathematics by using sets with atoms. Such a mathematics generalizes the classical Zermelo-Fraenkel mathematics, and represents an appropriate framework to work with (infinite) structures in terms of finitely supported objects. We focus on the techniques of translating the Zermelo-Fraenkel results into this Finitely Supported Mathematics over infinite (possibly non-countable) sets with atoms. Two general methods of proving the finite support property for certain algebraic structures are presented. Finally, we provide a survey on the applications of the Finitely Supported Mathematics in experimental sciences.

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Notes

  1. 1.

    A multiset on an alphabet \(\varSigma \) is a function from \(\varSigma \) to \(\mathbb {N}\) where each element in \(\varSigma \) has associated its multiplicity.

  2. 2.

    Let P be a predicate on A. We say that if P(a) is true for all but finitely many elements of A.

References

  1. Alexandru, A., Ciobanu, G.: Nominal event structures. Rom. J. Inf. Sci. Technol. 15, 79–90 (2012)

    Google Scholar 

  2. Alexandru, A., Ciobanu, G.: Nominal techniques for \(\pi I\)-calculus. Rom. J. Inf. Sci. Technol. 16, 261–286 (2013)

    Google Scholar 

  3. Alexandru, A., Ciobanu, G.: Nominal groups and their homomorphism theorems. Fundamenta Informaticae 131(3–4), 279–298 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Alexandru, A., Ciobanu, G.: On the development of the Fraenkel-Mostowski set theory. Bull. Polytech. Inst. Jassy LX, 77–91 (2014)

    Google Scholar 

  5. Alexandru, A., Ciobanu, G.: A nominal approach for fusion calculus. Rom. J. Inf. Sci. Technol. 17(3), 265–288 (2014)

    MathSciNet  Google Scholar 

  6. Alexandru, A., Ciobanu, G.: Mathematics of multisets in the Fraenkel-Mostowski framework. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie 58/106(1), 3–18 (2015)

    MathSciNet  Google Scholar 

  7. Alexandru, A., Ciobanu, G.: Defining finitely supported mathematics over sets with atoms. In: Batsakis, S., Bobalo, Y., Ermolayev, V., Kharchenko, V., Kobets, V., Kravtsov, H., Mayr, H.C., Nikitchenko, M., Peschanenko, V., Spivakovsky, A., Yakovyna, V., Zholtkevych, G. (eds.) 4th International Workshop on Algebraic, Logical, and Algorithmic Methods of System Modeling, Specification and Verification, vol. 1356, pp. 382–395 (2015). http://CEUR-WS.org

  8. Alexandru, A., Ciobanu, G.: Generalized multisets: from ZF to FSM. Comput. Inform. 34(5), 1133–1150 (2015)

    Google Scholar 

  9. Alexandru, A., Ciobanu, G.: Static analysis in finitely supported mathematics. In: 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. IEEE Computer Society Press (2015, in press)

    Google Scholar 

  10. Alexandru, A., Ciobanu, G.: Pawlak approximations in the framework of nominal sets. J. Multiple-Valued Logic Soft Comput. 26(3) (2016, in press)

    Google Scholar 

  11. Alexandru, A., Ciobanu, G.: Finitely supported subgroups of a nominal group. Mathematical Reports 18(2) (2016)

    Google Scholar 

  12. Banach, S., Tarski, A.: Sur la décomposition des ensembles de points en parties respectivement congruentes. Fundamenta Mathematicae 6, 244–277 (1924)

    Google Scholar 

  13. Barwise, J.: Admissible Sets and Structures: An Approach to Definability Theory. Perspectives in Mathematical Logic, vol. 7. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  14. Bojańczyk, M., Lasota, S.: Fraenkel-Mostowski sets with non-homogeneous atoms. In: Finkel, A., Leroux, J., Potapov, I. (eds.) RP 2012. LNCS, vol. 7550, pp. 1–5. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  15. Bojanczyk, M.: Nominal monoids. Theor. Comput. Syst. 53, 194–222 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bojanczyk, M., Braud, L., Klin, B., Lasota, S.: Towards nominal computation. In: 39th ACM Symposium on Principles of Programming Languages, pp. 401–412 (2012)

    Google Scholar 

  17. Bojanczyk, M., Klin, B., Lasota, S.: Automata with group actions. In: 26th Symposium on Logic in Computer Science, pp. 355–364. IEEE Computer Society Press (2011)

    Google Scholar 

  18. Bojanczyk, M., Klin, B., Lasota, S., Torunczyk, S.: Turing machines with atoms. In: 28th Symposium on Logic in Computer Science, pp. 183–192. IEEE Computer Society Press (2013)

    Google Scholar 

  19. Bojanczyk, M., Lasota, S.: A machine-independent characterization of timed languages. In: 39th International Colloquium on Automata, Languages and Programming, pp. 92–103 (2012)

    Google Scholar 

  20. Bojanczyk, M., Torunczyk, S.: Imperative programming in sets with atoms. In: D’Souza, D., Kavitha, T., Radhakrishnan, J. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, vol. 18, pp. 4–15. LIPIcs (2012)

    Google Scholar 

  21. Cohen, P.J.: The Independence of the Axiom of Choice. Stanford University, Mimeographed (1963)

    Google Scholar 

  22. Fraenkel, A.: Zu den grundlagen der Cantor-Zermeloschen mengenlehre. Mathematische Annalen 86, 230–237 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gabbay, M.J.: The pi-calculus in FM. In: Kamareddine, F.D. (ed.) Thirty Five Years of Automating Mathematics. Applied Logic Series, vol. 28, pp. 247–269. Springer, The Netherlands (2003)

    Chapter  Google Scholar 

  24. Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax with variable binding. Formal Aspects Comput. 13, 341–363 (2001)

    Article  Google Scholar 

  25. Gandy, R.: Church’s thesis and principles for mechanisms. In: Barwise, J., Keisler, H.J., Kunen, K. (eds.) The Kleene Symposium, pp. 123–148. North-Holland, Amsterdam (1980)

    Chapter  Google Scholar 

  26. Gödel, K.: The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Annals of Mathematics Studies. Princeton University Press, Princeton (1940)

    Google Scholar 

  27. Herrlich, H.: Axiom of Choice. Lecture Notes in Mathematics. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  28. Howard, P., Rubin, J.E.: Consequences of the Axiom of Choice. Mathematical Surveys and Monographs, vol. 59. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  29. Jech, T.J.: The Axiom of Choice. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  30. Krohn, K., Rhodes, J.: Algebraic theory of machines: prime decomposition theorem for finite semigroups and machines. Trans. Am. Math. Soc. 116, 450–464 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lindenbaum, A., Mostowski, A.: Uber die unabhangigkeit des auswahlsaxioms und einiger seiner folgerungen. Comptes Rendus des Seances de la Societe des Sciences et des Lettres de Varsovie. 31, 27–32 (1938)

    Google Scholar 

  32. Parrow, J., Victor, B.: The update calculus. In: Johnson, M. (ed.) Algebraic Methodology and Software Technology. LNCS, vol. 1349, pp. 409–423. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  33. Petrisan, D.: Investigations into algebra and topology over nominal sets. Ph.D. thesis, University of Leicester (2011)

    Google Scholar 

  34. Pitts, A.M.: Alpha-structural recursion and induction. J. ACM 53, 459–506 (2006)

    Article  MathSciNet  Google Scholar 

  35. Pitts, A.M.: Nominal Sets Names and Symmetry in Computer Science. Cambridge University Press, Cambridge (2013)

    Book  MATH  Google Scholar 

  36. Sangiorgi, D.: \(\pi \)-calculus, internal mobility, and agent-passing calculi. Rapport INRIA no.2539 (1995)

    Google Scholar 

  37. Shinwell, M.R.: The fresh approach: functional programming with names and binders. Ph.D. thesis, University of Cambridge (2005)

    Google Scholar 

  38. Tarski, A.: What are logical notions? Hist. Philos. Logic 7, 143–154 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Turner, D.: Nominal Domain Theory for Concurrency. Technical report no.751, University of Cambridge (2009)

    Google Scholar 

  40. Urban, C.: Nominal techniques in Isabelle/HOL. J. Autom. Reasoning 40, 327–356 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Romanian Academy, Institute of Computer Science, Iaşi, Romania

    Andrei Alexandru & Gabriel Ciobanu

Authors
  1. Andrei Alexandru
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  2. Gabriel Ciobanu
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Correspondence to Gabriel Ciobanu .

Editor information

Editors and Affiliations

  1. Lviv Polytechnic National University, Lviv, Ukraine

    Vitaliy Yakovyna

  2. Institute of Applied Informatics, Alpen-Adria-Universität Klagenfurt, Klagenfurt, Austria

    Heinrich C. Mayr

  3. Taras Shevchenko National Univ. Kyi, Kyiv, Ukraine

    Mykola Nikitchenko

  4. V.N. Karazin Kharkiv National University, Kharkiv, Ukraine

    Grygoriy Zholtkevych

  5. Kherson State University, Kherson, Ukraine

    Aleksander Spivakovsky

  6. University of Huddersfield, Huddersfiled, United Kingdom

    Sotiris Batsakis

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Alexandru, A., Ciobanu, G. (2016). Main Steps in Defining Finitely Supported Mathematics. In: Yakovyna, V., Mayr, H., Nikitchenko, M., Zholtkevych, G., Spivakovsky, A., Batsakis, S. (eds) Information and Communication Technologies in Education, Research, and Industrial Applications. ICTERI 2015. Communications in Computer and Information Science, vol 594. Springer, Cham. https://doi.org/10.1007/978-3-319-30246-1_5

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  • DOI: https://doi.org/10.1007/978-3-319-30246-1_5

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  • Online ISBN: 978-3-319-30246-1

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