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A concrete model for a typed linear algebraic lambda calculus
Published online by Cambridge University Press: 21 November 2023
Abstract
We give an adequate, concrete, categorical-based model for Lambda-${\mathcal S}$, which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda-${\mathcal S}$ is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbid duplication of variables and to consider all lambda-terms as algebraic linear functions. The type system of Lambda-${\mathcal S}$ has a superposition constructor S such that a type A is considered as the base of a vector space, while SA is its span. Our model considers S as the composition of two functors in an adjunction relation between the category of sets and the category of vector spaces over $\mathbb C$. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.
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- © The Author(s), 2023. Published by Cambridge University Press
Footnotes
Partially founded by PICT projects 2019-1272 and 2021-I-A-00090, PIP project 1220200100368CO, and CSIC-UdelaR project 22520220100073UD.
This paper is the long journal version of (Díaz-Caro and Malherbe, 2019). In the present paper, the main new result is to revisit some rewrite rules in order to prove a theorem of adequacy.