Java Type Unification with Wildcards

  • Martin Plümicke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5437)


With the introduction of Java 5.0 the type system has been extended by parameterized types, type variables, type terms, and wildcards. As a result very complex types can arise. The term

\(\tt{Vector}<{\tt{\texttt{?}\ extends\ Vector}<{\tt{AbstractList}<{Integer}>>>}}\)

is for example a correct type in Java 5.0.

In this paper we present a type unification algorithm for Java 5.0 type terms. The algorithm unifies type terms, which are in subtype relationship. For this we define Java 5.0 type terms and its subtyping relation, formally.

As Java 5.0 allows wildcards as instances of generic types, the subtyping ordering contains infinite chains. We show that the type unification is still finitary. We give a type unification algorithm, which calculates the finite set of general unifiers.


Type System Type Variable Simple Type Reduce1 Rule Type Inference 


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  1. 1.
    Gosling, J., Joy, B., Steele, G., Bracha, G.: The JavaTM Language Specification, 3rd edn. The Java series. Addison-Wesley, Reading (2005)Google Scholar
  2. 2.
    Plümicke, M., Bäuerle, J.: Typeless Programming in Java 5.0. In: Gitzel, R., Aleksey, M., Schader, M., Krintz, C. (eds.) 4th International Conference on Principles and Practices of Programming in Java. ACM International Conference Proceeding Series, pp. 175–181. Mannheim University Press (August 2006)Google Scholar
  3. 3.
    Plümicke, M.: Typeless Programming in Java 5.0 with wildcards. In: Amaral, V., Veiga, L., Marcelino, L., Cunningham, H.C. (eds.) 5th International Conference on Principles and Practices of Programming in Java. ACM International Conference Proceeding Series, pp. 73–82. ACM Press, New York (September 2007)Google Scholar
  4. 4.
    Damas, L., Milner, R.: Principal type-schemes for functional programs. In: Proc. 9th Symposium on Principles of Programming Languages (1982)Google Scholar
  5. 5.
    Smolka, G.: Logic Programming over Polymorphically Order-Sorted Types. PhD thesis, Department Informatik, University of Kaiserslautern, Kaiserslautern, Germany (May 1989)Google Scholar
  6. 6.
    Hanus, M.: Parametric order-sorted types in logic programming. In: Abramsky, S. (ed.) TAPSOFT 1991, CCPSD 1991, and ADC-Talks 1991. LNCS, vol. 494, pp. 181–200. Springer, Heidelberg (1991)Google Scholar
  7. 7.
    Hill, P.M., Topor, R.W.: A Semantics for Typed Logic Programs. In: Pfenning, F. (ed.) Types in Logic Programming, pp. 1–62. MIT Press, Cambridge (1992)Google Scholar
  8. 8.
    Beierle, C.: Type inferencing for polymorphic order-sorted logic programs. In: International Conference on Logic Programming, pp. 765–779 (1995)Google Scholar
  9. 9.
    Plümicke, M.: OBJ–P The Polymorphic Extension of OBJ–3. PhD thesis, University of Tuebingen, WSI-99-4 (1999)Google Scholar
  10. 10.
    Plümicke, M.: Type unification in Generic–Java. In: Kohlhase, M. (ed.) Proceedings of 18th International Workshop on Unification (UNIF 2004) (July 2004)Google Scholar
  11. 11.
    Martelli, A., Montanari, U.: An efficient unification algorithm. ACM Transactions on Programming Languages and Systems 4, 258–282 (1982)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Martin Plümicke
    • 1
  1. 1.University of Cooperative Education Stuttgart/HorbHorbGermany

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