Reversible Monadic Computing

Abstract

We extend categorical semantics of monadic programming to reversible computing, by considering monoidal closed dagger categories: the dagger gives reversibility, whereas closure gives higher-order expressivity. We demonstrate that Frobenius monads model the appropriate notion of coherence between the dagger and closure by reinforcing Cayley's theorem; by proving that effectful computations (Kleisli morphisms) are reversible precisely when the monad is Frobenius; by characterizing the largest reversible subcategory of Eilenberg-Moore algebras; and by identifying the latter algebras as measurements in our leading example of quantum computing. Strong Frobenius monads are characterized internally by Frobenius monoids. and so T (I) is special.

Reversibility for Quantum Programming Language QML

“…reversibilidad en un modelo de circuito que utiliza compuertas cuánticas con las características mencionadas, podría resultar natural, porque hay un registro de las operaciones aplicadas y tal operación reversible ejecutada dos veces permitiría regresar al valor inicial. Sin embargo, en un lenguaje de programación donde las operaciones realizadas no están almacenadas explícitamente, la reversibilidad no es trivial [22], [34]- [36]. Para esto, varios autores han trabajado la reversibilidad aplicada a la computación clásica y cuántica, modelando a través de diagramas de flujo reversibles [31], almacenando en una pila las operaciones aplicadas [18], definiendo funciones inversas, entre otras [22], [26]- [29], [36], [37].…”

“…Sin embargo, en un lenguaje de programación donde las operaciones realizadas no están almacenadas explícitamente, la reversibilidad no es trivial [22], [34]- [36]. Para esto, varios autores han trabajado la reversibilidad aplicada a la computación clásica y cuántica, modelando a través de diagramas de flujo reversibles [31], almacenando en una pila las operaciones aplicadas [18], definiendo funciones inversas, entre otras [22], [26]- [29], [36], [37].…”

Reversible Effects as Inverse Arrows

“…In this case (12) also follows. If the underlying category is an inverse category, then µ • µ † • µ = µ, whence µ • µ † = id, and (13)- (14) follow. Thus, if T is a strong dagger Frobenius monad on a dagger/inverse category, then A is a dagger/inverse arrow.…”

A computational interpretation of compact closed categories: reversible programming with negative and fractional types

“…In the finitary setting, type isomorphisms provide a perfect, sound and complete, foundation for reversible programming languages [Fiore 2004;Fiore et al 2006;James and Sabry 2012a]. This simple model can be extended by relaxing the isomorphisms to be partial -giving models for Turing-complete reversible languages [Bowman et al 2011; James and Sabry 2012a; Kaarsgaard and Veltri 2019] and by incorporating reversible effects such as state [Heunen et al 2018;Heunen and Karvonen 2015]. The simple model of type isomorphisms was also recently extended by Chen et al [2020] with the same fractional types we use in this paper.…”