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# Destination Calculus:
## A Linear 𝜆-Calculus for Purely Functional Memory Writes
### Topics in Compiler and Concurrency
#### Julius Fischer

###### 10. Juni 2026
$$
\def\leftcirc{{\triangleleft\kern-0.1em\circ}}
\def\myor{{\;|\;}}

$$

---
# What are Destinations?
- Destination Passing Style
    - Imperative
    - Function takes destination instead of returning structure
---
# Example in C
```
int* vec_add(int v1[], int v2[], int size) {
    int* res =  malloc(size * sizeof(int));
    for (int i = 0; i < size; i++) {
        res[i] = v1[i] + v2[i];
    }
    return res;
}
```
- Add an 'out parameter'
```
void dps_vec_add(int res[], int v1[], int v2[], int size) {
    for (int i = 0; i < size; i++) {
        res[i] = v1[i] + v2[i];
    }
}
```
---
# Motivation for DPS
- Typical functional map implementation
```
map f (x:xs) = (f x : map f xs1)
> (f x : map f xs) 
> (map f x1 : (f x2 : map f xs2)) 
> ...
```
- Not tail recursive 
    - Performance issue
    - Inefficient stack growth
```
map f (x:xs) dest = map f xs (push dest (f x))
> map f xs1 (push dest (f x1))
> map f xs2 (push dest (f x2))
```
--- 
# DPS in functional languages
- Just add pointers to mutable data?
    - $Int \mapsto Int \mapsto \textcolor{red}{T}$
    - $\textcolor{red}{T *}\mapsto Int \mapsto Int \mapsto \dots$
- Multiple issues with this
    - Memory safety
    - Multilpe writes on shared memory
    - Purity

---
# Destination Calculus
- Basic FP terms
$$
t, u := x \myor t' t \;|\; t ; t' \;|\; \dots
$$
- Case for all datatypes
$$
\begin{align}
    \dots \myor 
    &case_m\;t\;of\;(v_1, v_2) \mapsto u \myor\\
    &case_m\;t\;of\;(Inl\;v_1) \mapsto u \myor \dots
\end{align}
$$
- Additions for destination passing
$$
\begin{align}
    \dots \myor
    &
    new_\ltimes \myor
    from_\ltimes t \myor
    to_\ltimes t \myor
    upd_\ltimes \; t \; with \; x \mapsto t' \myor\\
    &
    t \blacktriangleleft t' \myor
    t  \leftcirc t' \myor
    t \triangleleft () \myor
    t \triangleleft Inl \myor
    t \triangleleft (,) \myor \dots
\end{align}
$$

--- 
# Referenes