The Interaction Calculus (IC) is a minimal programming language and model of computation obtained by "completing" the affine Lambda Calculus in a way that matches perfectly Lamping's optimal reduction algorithm. It can also be seen as a textual syntax for Symetric Interaction Combinators: both views are equivalent. As a model of computation, the IC has compeling characteristics:
It features higher-order functions, just like the Lambda Calculus.
It has a well-defined cost model, just like the Turing Machine.
It is inherently concurrent, making it prone to massive parallelism.
It is fully linear, making it garbage-collection free.
This repository contains a Rust reference implementation. Also check:
A formalization in Kind-Lang: IC.kind.
A typed version (a la Calculus of Constructions): Interaction Type Theory
git clone https://github.com/victortaelin/interaction_calculus
cd interaction_calculus
cargo install --path .def id = λx x
def c2 = λf λx (dup #b f0 f1 = f; (f0 (f1 x)))
(c2 id)ic main.icSee example.ic for a larger example.
Interaction Calculus terms are defined by the following grammar:
term ::=
| λx term -- abstraction
| (term term) -- application
| {term term}#N -- superposition
| dup #N {p q} = term; term -- duplication
| x -- variableWhere variable have global scope (can occur outside binding lambdas).
The IC has 4 primitive reduction rules:
((λx f) a)
---------- lambda application
x <- a
f
({u v}#i a)
---------------- superposition application
dup #i x0 x1 = a
{(u x0) (v x1)}#i
dup #i p q = λx f
body
----------------- lambda duplication
p <- λx0 r
q <- λx1 s
x <- {x0 x1}#i
dup #i r s = f
body
dup #i p q = {r s}#j
body
-------------------- superposition duplication
if #i == #j:
a <- fst
b <- snd
cont
else:
a <- {a0 a1}#j
b <- {b0 b1}#j
dup #i a0 a1 = fst;
dup #i b0 b1 = snd;
cont
Where, a <- b stands for a global, linear substitution of a by b. It can
be performed in O(1) by a simple array write, which, in turn, makes all
rewrite rules above O(1) too.
And that's all!
Consider the conventional Lambda Calculus, with pairs. It has two computational rules:
Lambda Application : (λx body) arg
Pair Projection : let {a b} = {fst snd} in cont
When compiling the Lambda Calculus to Interaction Combinators:
lams and apps can be represented as constructor nodes (γ)
pars and lets can be represented as duplicator nodes (δ)
As such, lambda applications and pair projections are just annihilations:
Lambda Application Pair Projection
(λx body) arg let {a b} = {fst snd} in cont
---------------- -----------------------------
x <- arg a <- fst
body b <- snd
cont
ret arg ret arg b a b a
| | | | | | | |
|___| | | |___| | |
app \ / \ / let \#/ \ /
| ==> \/ | ==> \/
| /\ | /\
lam /_\ / \ pair /#\ / \
| | | | | | | |
| | | | | | | |
x body x body fst snd fst snd
"The application of a lambda "The projection of a pair just
substitutes the lambda's var substitutes the projected vars
by the application's arg, and by each element of the pair, and
returns the lambda body." returns the continuation."But annihilations only happen when identical nodes interact. On interaction nets, it is possible for different nodes to interact, which triggers another rule, the commutation. That rule could be seen as handling the following expressions:
Lambda Projection : let {a b} = (λx body) in cont
Pair Application : ({fst snd} arg)
But how could we "project" a lambda or "apply" a pair? On the Lambda Calculus, these cases are undefined and stuck, and should be type errors. Yet, by interpreting the effects of the commutation rule on the interaction combinator point of view, we can propose a reasonable reduction for these lambda expressions:
Lambda Application Pair Application
let {a b} = (λx body) in cont ({fst snd} arg)
------------------------------ ---------------
a <- (λx0 b0) let {x0 x1} = arg in
b <- (λx1 b1) {(fst x0) (snd x1)}
x <- {x0 x1}
let {b0 b1} = body in
cont
ret arg ret arg ret arg ret arg
| | | | | | | |
|___| | | |___| | |
let \#/ /_\ /_\ app \ / /#\ /#\
| ==> | \/ | | ==> | \/ |
| |_ /\ _| | |_ /\ _|
lam /_\ \#/ \#/ pair /#\ \ / \ /
| | | | | | | |
| | | | | | | |
x body x body var body var body
"The projection of a lambda "The application of a pair is a pair
substitutes the projected vars of the first element and the second
by a copies of the lambda that element applied to projections of the
return its projected body, with application argument."
the bound variable substituted
by the new lambda vars paired."This, in a way, completes the lambda calculus; i.e., previously "stuck" expressions now have a meaningful computation. That system, as written, is Turing complete, yet, it is very limited, since it isn't capable of cloning pairs, or cloning cloned lambdas. There is a simple way to greatly increase its expressivity, though: by decorating lets with labels, and upgrading the pair projection rule to:
let #i{a,b} = #j{fst,snd} in cont
---------------------------------
if #i == #j:
a <- fst
b <- snd
cont
else:
a <- #j{a0,a1}
b <- #j{b0,b1}
let #i{a0,a1} = fst in
let #i{b0,b1} = snd in
contThat is, it may correspond to either an Interaction Combinator annihilation or
commutation, depending on the value of the labels #i and #j. This makes IC
capable of cloning pairs, cloning cloned lambdas, computing nested loops,
performing Church-encoded arithmetic up to exponentiation, expressing arbitrary
recursive functions such as the Y-combinators and so on. In other words, with
this simple extension, IC becomes extraordinarily powerful and expressive,
giving us:
A new model of computation that is similar to the lambda calculus, yet, can be reduced optimally.
A general purpose, higher-order "core language" that is lighter and faster than the lambda calculus.
A term-based view for interaction combinators, making it easier to reason about their graphs.
That said, keep in mind the IC is not equivalent to the Lambda Calculus. It is a
different model. There are λ-terms that IC can't compute, and vice-versa. For
example, the Lambda Calculus can perform self-exponentiation of church-nats as
λx (x x), which isn't possible on IC. Yet, on IC, we can have call/cc, direct
O(1) queues, and fully linear HOAS, which aren't possible on the Lambda
Calculus.
Finally, note that, in order to differentiate IC's "pairs" and "lets" from their λ-Calculus counterparts, which behave differently, we call them "sups" and "dups", respectivelly.
λu λv dup {a b} = {(λx x) (λy y)}; {(a u) (b v)}
------------------------------------------------ superposition-projection
λu λv {((λx x) u) ((λy y) v)}
----------------------------- lambda-application
λu λv {((λx x) u) v}
-------------------- lambda-application
λu λv {u v}dup {a b} = λx λy λz y; {a b}
----------------------------- lambda-projection
dup {a b} = λy λz y; {(λx0 a) (λx1 b)}
-------------------------------------- lambda-projection
dup {a b} = λz {y0 y1}; {(λx0 λy0 a) (λx1 λy1 b)}
------------------------------------------------- lambda-projection
dup {a b} = {y0 y1}; {(λx0 λy0 λz0 a) (λx1 λy1 λz1 b)}
------------------------------------------------------ superposition-projection
{(λx0 λy0 λz0 y0) (λx1 λy1 λz1 y1)}{{(λx x) (λy y)} (λt t)}
------------------------ superposition-application
dup {a0 a1} = λt t; {((λx x) a0) ((λy y) a1)}
--------------------------------------------- lambda-projection
dup {a0 a1} = {t0 t1}; {((λx x) (λt0 a0)) ((λy y) (λt1 a1))}
------------------------------------------------------------ superposition-projection
{((λx x) (λt0 t0)) ((λy y) (λt1 t1))}
------------------------------------- lambda-application
{((λx x) (λt0 t0)) (λt1 t1)}
---------------------------- lambda-application
{(λt0 t0) (λt1 t1)}This is equivalent to:
data Nat = S Nat | Z
add : Nat -> Nat -> Nat
add (S n) m = S (add n m)
add Z m = m
main : Nat
main = add (S (S (S Z))) (S (S Z))Here is a handwritten reduction of 2^(2^2).
The High-order Virtual Machine (HVM) is a high-performance practical implementation of the IC. Check it out!