Abstract clones for abstract syntax
Abstract
We give a formal treatment of simple type theories, such as the simply-typed $\lambda$-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed $\lambda$-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as $\beta$- and $\eta$-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2021
- DOI:
- 10.48550/arXiv.2105.00969
- arXiv:
- arXiv:2105.00969
- Bibcode:
- 2021arXiv210500969A
- Keywords:
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- Computer Science - Logic in Computer Science;
- Computer Science - Programming Languages;
- F.3.2;
- F.4.1
- E-Print:
- To appear in the proceedings of FSCD 2021