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https://github.com/lamagraph/interaction-nets-in-fpga

Interaction nets based processor in Clash
https://github.com/lamagraph/interaction-nets-in-fpga

clash fpga haskell interaction-nets

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Interaction nets based processor in Clash

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README 

        
# Interaction nets based processor in Clash



## Required tools



### Pre-commit



We use [pre-commit](https://pre-commit.com/) for general tidy up of files.

To install pre-commit run:



```shell

pip install pre-commit # or install using your distro package manager

pre-commit install

```



To run pre-commit on all files run



```shell

pre-commit run --all-files

```



### Fourmolu



We use [Fourmolu](https://fourmolu.github.io/) as a formatter for Haskell source files with our custom config.

**Fourmolu must be explicitly enabled in VS Code!**



## Editor



Our editor of choice is [VS Code](https://code.visualstudio.com/) with following extensions:



- [Haskell](https://marketplace.visualstudio.com/items?itemName=haskell.haskell)



## Interaction nets



We use a variation of interaction nets that was proposer by Yves Lafont in 1989 in the paper ["Interaction nets"](https://dl.acm.org/doi/10.1145/96709.96718) as a base for our project.

Below we introduce basic definitions and discuss some assumptions that we use.



Let $\Sigma$ is an alphabet of **agents**.

Let $Ar: \Sigma \to \mathbb{N}$ is an **arity function**.

We suppose that $0 \in \mathbb{N}$.

Each agent $l \in \Sigma$ has a one **primary port** and $Ar(l)$ **secondary ports**: $Ar(l) + 1$ ports in total.



**Network** $\mathcal{n}$ over alphabet $\Sigma$ is an undirected graph where

- Each vertex is labelled with an agent and contains respective number of ports.

- Each edge is a connection between ports.

- Each port can be connected with not more then one port.



$\mathcal{N}_{\Sigma}$ is a set of all possible networks over $\Sigma$.



**Note**

- Network can contains ports that do not connected to other ports. The port without connection is a **free port**. $\mathcal{I}(\mathcal{n})$ is set of free ports of network $\mathcal{n}$ or an **interface of the network $\mathcal{n}$**.

- Network can consists of edges only. In this case, each end of edge is a free port.

- Network without vertices and edges is an **empty network**.



The pair of nodes $a$ and $b$ that connected via primary ports (there is an edge that connects primary port of $a$ with primary port of $b$) is an **active pair**.

We use $a \bowtie b$ to denote that $a$ and $b$ is an active pair.



Network $\mathcal{n}$ is in **normal form** if there is no active pairs in $\mathcal{n}$.



**Reduction rule** $r \in \Sigma \times \Sigma \times \mathcal{N}_{\Sigma}$ is graph rewriting rule.

$\mathcal{R}$ is a set of reduction rules.

- If $(l_1,l_2,\mathcal{n}) \in \mathcal{R}$ then $(l_2, l_1,\mathcal{n}) \in \mathcal{R}$.

- If $(l_1,l_2,\mathcal{n}_1) \in \mathcal{R}$ and $(m_1, m_2,\mathcal{n}_2) \in \mathcal{R}$ then $(l_1,l_2) \neq (m_1,m_2)$.

- For all $(l_1,l_2,\mathcal{n}) \in \mathcal{R}$, $\mathcal{n}$ is in normal form.

- For all $(l_1,l_2,\mathcal{n}) \in \mathcal{R}$, $Ar(l_1) + Ar(l_2) = |\mathcal{I}(\mathcal{n})|$.



Computation is an application of rewriting rules to active pairs.

If there is an active pair $a \bowtie b$ in network $\mathcal{n_0}$, where

- $a$ is labelled with $l_1$

- $b$ is labelled with $l_2$

- $r = (l_1, l_2, \mathcal{m}) \in \mathcal{R}$



then we can replace $a \bowtie b$ with $\mathcal{m}$ and get new network $\mathcal{n_1}$.

Thus, computation is a sequence of steps of the form $\mathcal{n_i} \xrightarrow{r} \mathcal{n_{i+1}}$.

Computation finishes when network in normal form.



Active pair $a \bowtie b$ is **realizable** if there is a sequence



$$\mathcal{n_0} \xrightarrow{r_0} \ldots \xrightarrow{r_{k-1}} \mathcal{n_k},$$



$r_i \in \mathcal{R}$, such that $\mathcal{n}_k$ contains $a \bowtie b$.

We assume that for the given network $\mathcal{n}$ and rules set $\mathcal{R}$, $\mathcal{R}$ contains rules for all **realizable** active pairs.