CSL 2003: Computer Science Logic pp 484-497

# Calculi of Meta-variables

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

## 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 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.

### Keywords

Meta-variable logical framework context λ-calculus

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© 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