### Abstract

Substitution is fundamental to the theory of logic and computation. Is substitution something that we define on syntax on a case-by-case basis, or can we turn the idea of substitution into a mathematical object? We give axioms for substitution and prove them sound and complete with respect to a canonical model. As corollaries we obtain a useful conservativity result, and prove that equality-up-to-substitution is a decidable relation on terms. These results involve subtle use of techniques both from rewriting and algebra. A special feature of our method is the use of nominal techniques. These give us access to a stronger assertion language, which includes so-called ‘freshness’ or ‘capture-avoidance’ conditions. This means that the sense in which we axiomatise substitution (and prove soundness and completeness) is particularly strong, while remaining quite general.

Original language | English |
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Pages (from-to) | 451-479 |

Journal | Formal Aspects of Computing |

Volume | 20 |

Issue number | 4-5 |

DOIs | |

Publication status | Published - 2008 |

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## Cite this

Gabbay, M. J., & Mathijssen, A. H. J. (2008). Capture-avoiding substitution as a nominal algebra.

*Formal Aspects of Computing*,*20*(4-5), 451-479. https://doi.org/10.1007/s00165-007-0056-1