Pi calculus can be transformed from lambda calculus by replacing a variable constructor with stream constructor. But the language and protocol accessing this stream could vary from backend to backend with respect to design requirements.

Stream calculus provides different disciplines for accessing underlying streams used in Pi calculus. For example: 1) linear types, or streams with constant direction without rollbacks where each element of stream is touched once; 2) random access arrays; 3) GPU sources; 4) runtime typed channels; 5) effect and coeffect streams (processes as streams). Such stream calculuses could be landed with such GPU languages as Futhark or AVX intrinsics languages, such as Julia. We could treat stream calculus as memory representation with different protocol accesses.

One type of disciplines is a stream calculus. While pi calculus could be imagined as lambda calculus where function arguments are channels or streams, stream calculus defines set of constructions over these streams. This calculus is needed to provide different forms of vectorization that can be used on GPU and AVX hardware.

## Language Axioms

```
Inductive StreamCalculus :=
| Map | Fold
| Scan | Iota
| SLoop | Transpose
| Rotate | SSplit
| Concat | Zip
| Reduce | StreamMap
| StreamMapPer | StreamRed
| StreamSeq | Partition
| Reshape | Shape
| Rearrange | Copy
| For | While.
```

# Calculus