A concrete model for a typed linear algebraic lambda calculus

ALEJANDRO DÍAZ-CARO; OCTAVIO MALHERBE

MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE

CAMBRIDGE UNIV PRESS

Año: 2023 vol. 34 p. 1 - 44

0960-1295

We give an adequate, concrete, categorical-based model for Lambda-S, which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda-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-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 C. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.