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\title{Topics in Compilers and Concurrency \\ Seminar Report on\\ Background Materials and Related Work}
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\author{Julius Fischer}
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\institute{Student at University of Freiburg (ALU)\\
    \email{julius.fischer@email.uni-freiburg.com}\\
    Matriculation Number: 5317202
    }


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\newcommand{\lad}{$\lambda_d$}
\newcommand{\dpsf}{DPS-$\tilde{F}$}

\section{Introduction}
In their work Bagrel and Spiwack \cite{bagrel_dp_2025} build on many prior contributions,
both directly in their calculus grammar and semantics, but also in their structural approach regarding typing and evaluation contexts.
From now on the calculus introduced in this specific paper will be referred to as \lad.
This report will highlight a number of select works, which are of significance to the $lambda_d$ calculus.

\section{A Functional Representation of Data Structures with a Hole - Y. Minamide\cite{minamide_holes_1998}}
This paper contributes fundamental work on holes in functional languages. 
It introduces a hole abstraction $\hat\lambda x. T$ to formalize data structures with a single hole. 
Which, while syntactically different, in principle remains similar to the \lad calculus.
Both utilize holes as the core features, where \lad has a type $T_1 \ltimes T_2$ to represent a
structure that is missing $T_1$ to complete a $T_2$, Minamide's calculus features $(T_1, T_2) hfun$.
In general Minamide focuses more on the similarity of his hole abstraction to the regular $\lambda$ abstraction
and the similarity of a structure containing a hole, to a function that returns an type $T_2$, when applied argument to an argument $T_1$.
Notably both calculi contain linearity constraints on holes, but Bagrel's work elevates some of those constraints by allowing for weakening.
Overall Minamide lays a lot of ground work, and influences that can be seen in the \lad formulation and in its discussion, as
similar benefits regarding tail recursion are addressed.

\section{Destination-Passing Style for Efficient Memory Management - Shaikhha et al. \cite{shaikhha_array_dps_2017}}
While Bagrel mostly theorizes on the advantages of the \lad calculus,
this paper give empirical evidence on runtime and memory improvements of Destination Passing Style (DPS) in a functional language.
Shaikhha et al. demonstrate the benefits of implementing a DPS-transformation step into the compilation of an array-programming language.
The authors chose to not give any direct memory control to the programmer, but their intermediate language '\dpsf' still feature
some similarity to \lad.
\dpsf is typed using a shape type, which contains the dimensions of the array, which will be written to a memory location/ destination.
Because of to the array-programming nature of the language, the shape type is fit only to arrays, but displays some flexibility, 
which, in a way, is more akin to the constructors used in \lad than to the holes used by Minamide \cite{minamide_holes_1998}.

\section{Tail Modulo Cons - \cite{bour_et_al_tmc_2021}}
This paper proposes a annotation controlled, compile time transformation of OCaml into a DPS intermediate language.
The intermediate language only features structures with single holes/ single destinations.
The driving goal of Tail Modulo Cons (TMC) is, as the name suggests, to elevate the issues of constructors wrapping a recursive call.
Particularly TMC allows for tail recursion, even if the recursive call is hidden behind a data constructor.
While TMC is an intermediate language it allows for some source level control, as only annotated functions are converted into DPS.


\section{Linear Haskell: practical linearity in a higher-order polymorphic language - Bernardy et al. \cite{lh_bernardy_2017}}
Linear logic/ typing stands at the core of the \lad type system, and its concrete form is massively influenced by Linear Haskell.
Bernardy et al. give deeper insight into linear typing as it is the sole focus of this paper. 
They focus on implementing linear typing in Haskell but give good intuition on linear types as a whole.
Though unrestricted types are not a core language feature but implemented in the language itself, the type system is
very similar to \lad, in fact the calculi even share syntax for the linear function.
Bernardy et al discuss many benefits of linear typing and \lad in it's whole bases on the idea of linear types to make
multiple writes on data impossible.

\section{A Unified View of Modalities in Type Systems - Abel and Bernardy \cite{abel_bernardy_2020}}
\lad includes a notion of modality using the exponential $!_mT$, of which the ground work is laid by Abel and Bernardy in this work.
They discuss modalities not exclusively for linear types in rigorous formality, diving into the precise algebraic and logical background.
In fact this paper shows a generalization of the modes used in \lad's type system, to ensure linearity.
The main strategy, which is used in \lad, is including the modality in an algebraic structure of the calculus (by indexing the modality $!_m$).
Bagrel claims this approach "greatly simplif[ies] [the] context management (especially in the computer-mechanized proofs)" \cite{bagrel_dp_2025} (Page 11).



\section{Formalization and Implementation of Safe Destination Passing in Pure Functional Programming Settings - \cite{bagrel_thesis_2025}}
Another honorable mention is Bagrel's dissertation, in which the author not only includes the findings of the original paper but 
also goes into further detail regarding linear typing and the entire intuition behind \lad.
It includes a more comprehensive explanation for most of which is covered in the article, as well as further on the usefulness of \lad
and an application of it.


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