Journal of Mathematical Sciences

, Volume 240, Issue 5, pp 692–706 | Cite as

Automorphisms of Types and Their Applications

  • S. SolovievEmail author
  • J. Malakhovski

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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Babai, “Automorphism groups, isomorphism, reconstruction,” in: Handbook of Combinatorics, Vol. 2, Elsevier (1995), pp. 1447–1541.Google Scholar
  2. 2.
    H. Barendregt, The Lambda Calculus: Its Syntax and Semantics, revised edition, North-Holland (1984).Google Scholar
  3. 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.Google Scholar
  4. 4.
    R. Brown, Ph. G. Higgins, and R. Sivera, Nonabelian Algebraic Topology, European Math. Soc. (2011).Google Scholar
  5. 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.Google Scholar
  6. 6.
    M. Dezani-Ciancaglini, “Characterization of normal forms possessing inverse in the λ-β-η-calculus,” Theoret. Comput. Sci., 2, 323–337 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    R. Di Cosmo, Isomorphisms of Types: From λ-Calculus to Information Retrieval and Language Design, Birkhauser (1995).Google Scholar
  8. 8.
    D. Gilles, “A complete proof synthesis method for the cube of type systems,” J. Logic Comput., 3, No. 3, 287–315 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. Gilles, “Higher-order unification and matching,” in: A. Robinson and A. Voronkov (eds.), Handbook of Automated Reasoning, Elsevier (2001), pp. 1009–1062.Google Scholar
  10. 10.
    J. Gil and Y. Zibin, “Efficient algorithms for isomorphisms of simple types,” Math. Struct. Comput. Sci., 15, No. 5, 917–957 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    H. Goguen, “A typed operational semantics for type theory,” PhD thesis, University of Edinburgh (1994).Google Scholar
  12. 12.
    M. Hall (Jr.), The Theory of Groups, The Macmillan Company (1959).Google Scholar
  13. 13.
    C. Hankin, Lambda Calculi: A Guide for Computer Scientists, Clarendon Press, Oxford (1994).zbMATHGoogle Scholar
  14. 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).CrossRefGoogle Scholar
  15. 15.
    J. Hoffstein, J. Pipher, and J. H. Silverman, An Introduction to Mathematical Cryptography, Springer, New York (2008).zbMATHGoogle Scholar
  16. 16.
    G. Huet and D. C. Oppen, “Equations and rewrite rules: A survey,” technical report, Stanford University (1980).Google Scholar
  17. 17.
    A. Mahalanobis, “The MOR cryptosystem and finite p-groups,” Contemp. Math., 633, 81–95 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. Martelli and U. Montanari, “An efficient unification algorithm,” ACM Trans. Program. Lang. Syst., 4, No. 2, 258–282 (1982).CrossRefzbMATHGoogle Scholar
  19. 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. 20.
    Z. Luo, Computation and Reasoning, Clarendon Press, Oxford (1994).Google Scholar
  21. 21.
    N. Mitchell et al., “Hoogle: Haskell API search engine,”
  22. 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.Google Scholar
  23. 23.
    M. Rittri, “Retrieving library functions by unifying types modulo linear isomorphism,” RAIRO-Theoret. Inform. Appl., 27, No. 6, 523–540 (1992).CrossRefzbMATHGoogle Scholar
  24. 24.
    C. Runciman and I. Toyn, “Retrieving reusable software components by polymorphic type,” J. Funct. Progr., 1, No. 2, 191–211 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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.Google Scholar
  26. 26.
    S. Soloviev, “Automorphisms of types in certain type theories and representation of finite groups,” Math. Struct. Comput. Sci., 29, 511–551 (2019).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    A. T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam (1984).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.IRIT, Paul Sabatier University, Toulouse, France and ITMO UniversitySt. PetersburgRussia

Personalised recommendations