Skip to main content
SpringerLink
Log in

Automorphisms of Types and Their Applications

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We outline recent results in the theory of type isomorphisms and automorphisms and present several practical applications of these results that can be useful in the contexts of programming and data security. Bibliography: 27 titles.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Babai, “Automorphism groups, isomorphism, reconstruction,” in: Handbook of Combinatorics, Vol. 2, Elsevier (1995), pp. 1447–1541.

  2. H. Barendregt, The Lambda Calculus: Its Syntax and Semantics, revised edition, North-Holland (1984).

  3. D. A. Basin, “Equality of terms containing associative-commutative functions and commutative binding operators is isomorphism complete,” in: M. E. Stickel (ed.), Proceedings of the 10th International Conference on Automated Deduction, Kaiserslautern, Germany, Lecture Notes in Artificial Intelligence, 449, Springer-Verlag (1990), pp. 251–260.

  4. R. Brown, Ph. G. Higgins, and R. Sivera, Nonabelian Algebraic Topology, European Math. Soc. (2011).

  5. K. Bruce and G. Longo, “Provable isomorphisms and domain equations in models of typed languages,” in: ACM Symposium on Theory of Computing (STOC 85) (1985), pp. 263–272.

  6. M. Dezani-Ciancaglini, “Characterization of normal forms possessing inverse in the λ-β-η-calculus,” Theoret. Comput. Sci., 2, 323–337 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Di Cosmo, Isomorphisms of Types: From λ-Calculus to Information Retrieval and Language Design, Birkhauser (1995).

  8. D. Gilles, “A complete proof synthesis method for the cube of type systems,” J. Logic Comput., 3, No. 3, 287–315 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Gilles, “Higher-order unification and matching,” in: A. Robinson and A. Voronkov (eds.), Handbook of Automated Reasoning, Elsevier (2001), pp. 1009–1062.

  10. J. Gil and Y. Zibin, “Efficient algorithms for isomorphisms of simple types,” Math. Struct. Comput. Sci., 15, No. 5, 917–957 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Goguen, “A typed operational semantics for type theory,” PhD thesis, University of Edinburgh (1994).

  12. M. Hall (Jr.), The Theory of Groups, The Macmillan Company (1959).

  13. C. Hankin, Lambda Calculi: A Guide for Computer Scientists, Clarendon Press, Oxford (1994).

    MATH  Google Scholar 

  14. J. Heather, G. Lowe, and S. Schneider, “How to prevent type flaw attacks on security protocols,” J. Comput. Sec., 11, No. 2, 217–244 (2003).

    Article  Google Scholar 

  15. J. Hoffstein, J. Pipher, and J. H. Silverman, An Introduction to Mathematical Cryptography, Springer, New York (2008).

    MATH  Google Scholar 

  16. G. Huet and D. C. Oppen, “Equations and rewrite rules: A survey,” technical report, Stanford University (1980).

  17. A. Mahalanobis, “The MOR cryptosystem and finite p-groups,” Contemp. Math., 633, 81–95 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Martelli and U. Montanari, “An efficient unification algorithm,” ACM Trans. Program. Lang. Syst., 4, No. 2, 258–282 (1982).

    Article  MATH  Google Scholar 

  19. F. Lindblad and M. Benke, “A tool for automated theorem proving in Agda,” in: C. Paulin-Mohring and B. Werner (eds.), Types for Proofs and Programs (TYPES 2004), Lect. Notes Comput. Sci., 3839, Springer, Berlin–Heidelberg (2006).

    Google Scholar 

  20. Z. Luo, Computation and Reasoning, Clarendon Press, Oxford (1994).

  21. N. Mitchell et al., “Hoogle: Haskell API search engine,” https://github.com/ndmitchell/hoogle.

  22. M. Rittri, “Using types as search keys in function libraries,” in: Proceedings of the Fourth International Conference on Functional Programming Languages and Computer Architecture (1989), pp. 174–183.

  23. M. Rittri, “Retrieving library functions by unifying types modulo linear isomorphism,” RAIRO-Theoret. Inform. Appl., 27, No. 6, 523–540 (1992).

    Article  MATH  Google Scholar 

  24. C. Runciman and I. Toyn, “Retrieving reusable software components by polymorphic type,” J. Funct. Progr., 1, No. 2, 191–211 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Soloviev, “On isomorphism of dependent products in a typed logical framework,” in: Post-Proceedings of TYPES 2014, LIPICS, Leibniz-Zentrum für Informatik, Schloss Dagstuhl (2015), pp. 275–288.

  26. S. Soloviev, “Automorphisms of types in certain type theories and representation of finite groups,” Math. Struct. Comput. Sci., 29, 511–551 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  27. A. T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam (1984).

Download references

Author information

Authors and Affiliations

  1. IRIT, Paul Sabatier University, Toulouse, France and ITMO University, St. Petersburg, Russia

    S. Soloviev & J. Malakhovski

Authors
  1. S. Soloviev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Malakhovski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Soloviev.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 468, 2018, pp. 287–308.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soloviev, S., Malakhovski, J. Automorphisms of Types and Their Applications. J Math Sci 240, 692–706 (2019). https://doi.org/10.1007/s10958-019-04386-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-019-04386-8

Search

Navigation