Calculi of Meta-variables

  • Masahiko Sato
  • Takafumi Sakurai
  • Yukiyoshi Kameyama
  • Atsushi Igarashi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2803)


The notion of meta-variable plays a fundamental role when we define formal systems such as logical and computational calculi. Yet it has been usually understood only informally as is seen in most textbooks of logic. Based on our observations of the usages of meta-variables in textbooks, we propose two formal systems that have the notion of meta-variable.

In both calculi, each variable is given a level (non-negative integer), which classifies variables into object variables (level 0), meta-variables (level 1), metameta-variables (level 2) and so on. Then, simple arity systems are used to exclude meaningless terms like a meta-level function operating on the metameta-level. A main difference of the two calculi lies in the definitions of substitution. The first calculus uses textual substitution, which can often be found in definitions of quantified formulae: when a term is substituted for a meta-variable, free object-level variables in the term may be captured. The second calculus is based on the observation that predicates can be regarded as meta-level functions on object-level terms, hence uses capture-avoiding substitution.

We show both calculi enjoy a number of properties including Church-Rosser and Strong Normalization, which are indispensable when we use them as frameworks to define logical systems.


Meta-variable logical framework context λ-calculus 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dami, L.: A Lambda-Calculus for Dynamic Binding. Theoretical Computer Science 192, 201–231 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Davies, R.: A Temporal-Logic Approach to Binding-Time Analysis. In: 11th Annual IEEE Symposium on Logic in Computer Science (LICS 1996), pp. 184–195 (1996)Google Scholar
  3. 3.
    Geuvers, H., Jojgov, G.: Open Proofs and Open Terms: A Basis for Interactive Logic. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 537–552. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Glück, R., Jørgensen, J.: Efficient multi-level generating extensions for program specialization. In: Swierstra, S.D. (ed.) PLILP 1995. LNCS, vol. 982, pp. 259–278. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  5. 5.
    Harper, R., Honsell, F., Plotkin, G.: A Framework for Defining Logics. Journal of the Association for Computing Machinery 40(1), 143–194 (1993)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Hashimoto, M., Ohori, A.: A Typed Context Calculus. Theoretical Computer Science 266(1-2), 249–272 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)zbMATHGoogle Scholar
  8. 8.
    Mason, I.: Computing with Contexts. Higher-Order and Symbolic Computation 12, 171–201 (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    Sands, D.: Computing with Contexts - a Simple Approach. Electronic Notes in Theoretical Computer Science, vol. 10 (1998)Google Scholar
  10. 10.
    Sato, M.: Theory of Judgments and Derivations. In: Arikawa, S., Shinohara, A. (eds.) Progress in Discovery Science. LNCS (LNAI), vol. 2281, pp. 78–122. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Sato, M., Sakurai, T., Kameyama, Y.: A Simply Typed Context Calculus with First-Class Environments. Journal of Functional and Logic Programming 2002(4), 1–41 (2002)MathSciNetGoogle Scholar
  12. 12.
    Shoenfield, J.R.: Mathematical Logic. Addison-Wesley, Reading (1967)zbMATHGoogle Scholar
  13. 13.
    Talcott, C.: A Theory of Binding Structures and Applications to Rewriting. Theoretical Computer Science 112(1), 99–143 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Yamamoto, K., Okamoto, A., Sato, M., Igarashi, A.: A Typed Lambda Calculus with Quasi-quotation (in Japanese). In: Informal Proceedings of the 4th JSSST Workshop on Programming and Programming Languages, pp. 87–102 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Masahiko Sato
    • 1
  • Takafumi Sakurai
    • 2
  • Yukiyoshi Kameyama
    • 3
  • Atsushi Igarashi
    • 1
  1. 1.Graduate School of InformaticsKyoto University 
  2. 2.Department of Mathematics and InformaticsChiba University 
  3. 3.Institute of Information Sciences and ElectronicsUniversity of Tsukuba, and JST 

Personalised recommendations