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Miloslav Ciz 2023-11-25 16:00:28 +01:00
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interaction_net.md
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Interaction nets are a way of performing computation by manipulating graphical diagrams according to a few simple rules. Interaction nets can be seen as one of many possible ways to create the lowest level basis for a [computer](computer.md), so we call them a *[model of computation](model_of_computation.md)*; other models of computation are for example [grammars](grammar.md), [Turing machines](turing_machine.md), [lambda calculus](lambda_calculus.md), [cellular automata](cellular_automaton.md) etc. -- all of these can be used to perform the same computations, though in different ways; interaction nets do this by representing a [program](program.md) with a [graph](graph.md)-like diagram which is then manipulated by some predefined rules and "reshaped" into another graph that represents the result of the performed computation. For this interaction nets can also be see as a kind of graphical [programming language](programming_language.md) (and even a textual one once we learn how to represent the graphical diagrams with text, which we call *interaction calculus*), though it's an extremely low level language (even things such as addition of natural numbers have to be programmed from scratch) that would be extremely impractical for everyday programming. The advantage of interaction nets is besides their simplicity usually said to be mainly their **[parallel nature](parallelism.md)**: as computation really means locally replacing parts of the network with predefined patterns, it is possible to simultaneously process multiple parts of the network at once, greatly speeding up the computation.
**WATCH OUT**: interaction nets are a bit confusing to newcomers because they look a bit like [logic circuits](logic_circuit.md) and so people think the computation happens by passing some data through the network -- THIS IS NOT THE CASE! The computation happens by CHANGING THE NETWORK itself, i.e. there is no data flowing, all that happens is REPLACING parts of the network with patterns defined by the rewriting rules. The idea is similar to that of [grammars](grammar.md).
**WATCH OUT**: interaction nets are a bit confusing to newcomers because they look a bit like [logic circuits](logic_circuit.md) and so people think the computation happens by passing some data through the network -- THIS IS NOT THE CASE! The computation happens by CHANGING THE NETWORK itself, i.e. there is no data flowing, all that happens is REPLACING parts of the network with patterns defined by the rewriting rules (similarly to how in lambda calculus there are really no function calls but just text replacement). The idea is similar to that of rewriting rules in [grammars](grammar.md). Think of the connections less like of electric wires and more like of strings with knots that you tie and untie to perform the computation.
A general interaction net consists of nodes, or **agents**, that are connected with wires, or **edges**. Alongside the net exist **interaction rules** that say how patterns in the net get replaced with other patterns. A concrete definition of specific agent types and interaction rules is called an **interaction system** (i.e. *interaction net* is the general idea, like for example a *[cellular automaton](cellular_automaton.md)*, while *interaction system* is a specific realization of that idea, like for example *[game of life](game_of_life.md)*).
@ -94,8 +94,39 @@ apply rule 1:
no more rules to apply
```
TODO: continue
One specific very important interaction system is called **"interaction combinators"** -- it is a very simple system, consisting of three predefined agents and three patterns of rewrite rules, which can simulate any other system, i.e. it is [Turing complete](turing_complete.md) (we say the model is capable of *universal computation*); we know this because it's possible to e.g. automatically convert any [lambda calculus](lambda_calculus.md) expression to interaction combinators. The following show the interaction combinator rules (`E`, `D` and `Y` are the three agents, `t` is a general subtree with possible additional inputs `{...}`):
One specific very important interaction system is called **interaction combinators** -- it is a very simple system, consisting of three predefined agents and three rewrite rules, which can simulate any other system, i.e. it is [Turing complete](turing_complete.md) (also *universal*).
```
TODO
erasing rule: _____
_____ _____ | |
x ---| | | | x <==| E |
{...}| t |==><==| E | rewrites to |_____|
y ---|_____| |_____| {...} _____
| |
y <==| E |
|_____|
duplication rule:
_____ _____
| |------| |
_____ _____ x <==| D | {...}| t |==> z
x ---| | | |--- z |_____|--. .-|_____|
{...}| t |==><==| D | rewrites to {...} _____ X _____
y ---|_____| |_____|--- w | |--' '-| |
y <==| D | {...}| t |==> w
|_____|------|_____|
non-termination rule:
_____ _____
| | _____ _____ | | .---.
| E |==>---| | | |---<==| E | rewrites to ( )
|_____| | D |==><==| Y | |_____| '---'
.---|_____| |_____|---.
( )
'--------------------------'
```
TODO: text representation, compare text representation of interaction nets with grammars?