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Calculi of meta-variables

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Abstract

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 metavariables 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 that 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.

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Authors and Affiliations

  1. Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Masahiko SATO & Atsushi Igarashi

  2. Department of Mathematics and Informatics, Chiba University, Chiba, 260-8522, Japan

    Takafumi Sakurai

  3. Department of Computer Science, University of Tsukuba, Tsukuba, 305-8573, Japan

    Yukiyoshi Kameyama

Authors
  1. Masahiko SATO
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  2. Takafumi Sakurai
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  3. Yukiyoshi Kameyama
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  4. Atsushi Igarashi
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Correspondence to Masahiko SATO.

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SATO, M., Sakurai, T., Kameyama, Y. et al. Calculi of meta-variables. Front. Comput. Sci. China 2, 12–21 (2008). https://doi.org/10.1007/s11704-008-0011-1

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  • DOI: https://doi.org/10.1007/s11704-008-0011-1

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