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The Ultimate Calculus

An optimal, full λ-calculus evaluator that doesn't rely on interaction combinators and can be implemented in 150 lines.


The following program applies the not function 2^32 (4 billion) times to the boolean true and returns instantly, showing we can perform runtime fusion:

  ((4294967296n not) true)

The following program computes the normal form of (λs.λz.(s (s z)) λs.λz.(s (s z))), showing we can evaluate non-stratified terms:

main (({s} {z} (s (s z))) ({s} {z} (s (s z))))

For more examples, check example.js.

Wait, what?

It has been years since I started trying to implement a fast evaluator for λ-calculus terms. After an extensive search, I judged the best solution to be the "abstract algorithm", which compiles λ-terms to interaction nets, performes reductions, and then decompiles back. It is fast and allows us to do some neat optimizations like runtime fusion, but it has two main problems:

  • It is only practical for a subset of the λ-calculus. That's because of the slow book-keeping mechanism that is necessary to evaluate certain λ-terms such as (λs.λz.(s (s z)) λs.λz.(s (s z))) soundly.

  • Implementing interaction-net evaluation and compilation is annoying. While the implementation is short, its massively parallel, mutable nature makes it hard to reason about, and awkward to implement in pure functional languages.

The Ultimate Calculus is a superset of the λ-calculus that can be fully implemented in 150 lines of idiomatic Haskell. With it, you can still perform optimal, parallel, O(1) reductions by manually decorating your λ-terms with explicit duplications. If you don't, though, the algorithm gracefully falls back to traditional sharing techniques, allowing you to have full λ-terms.

How it works?

The Ultimate Calculus works by first extending the λ-calculus with pairs and let, which is very common, and then further extending it in two unusual ways: first, by allowing λ-bound variables to occur outside of their scopes, and second, by adding reduction rules to two undefined cases: "applying a pair to a value" and "projecting the elements of a lambda". And that's it.

While those extensions sound "heretic" since they remove scope boundaries and let us compute with intermediate values that aren't "real λ-terms", those terms eventually "cancel out": when the computation completes, we get a normal λ-term back. In a way, that is similar to how one must go outside the scope of real numbers to have solutions to all single-variable polynomials. While both things sound intuitively crazy, they're still mathematically meaningful.

The Ultimate Calculus is very similar to another system I published some time ago, the Symmetric Interaction Calculus, which started from the observation that interaction nets can be viewed as a syntax for a λ-calculus superset where variables can scape their scopes. The problem is that this system was still linear, so it couldn't be used for the evaluation of arbitrary λ-terms. The perfect calculus allows variables to occur more than once, giving us two copying options: with explicit duplications, which allows us to explore optimal reductions, or with non-linear variables, which falls back to (unspecified) traditional sharing techniques.

Note this is not a solution to the problem of evaluating full λ-calculus terms on interaction nets. It is merely a middle ground between interaction nets and textual calculi, allowing you to have a little bit of both, with the benefit of being implementable in traditional functional languages with a very small amount of code.


Ultimate calculus terms are defined by the following grammar:

term ::=
| {x} term               -- lambda
| (term term)            -- application
| [term,term]            -- pair
| let [x,y] = term; term -- projection
| x                      -- variable

Its reduction rules are the following:

(({x}f) a)
---------- (app-lam)
f [x <- a]

let [x,y] = [a,b]; t
-------------------- (let-par)
t [x <- a] [y <- b]

let [u,v] = {x} f; t
--------------------------------------------------------- (let-lam)
let [p,q] = f; t [u <- {x0}p][v <- {x1}q][x <- [x0,x1]]

([a,b] c)
-------------------------------- (app-par)
let [x0,x1] = c; [(a x0),(b x1)]

Here, [x <- a] stands for global, capture-avoiding substitution of x by a. As long as variables only occur once, all of those reduction rules are O(1), since they only perform a direct substitution of constructors and a single pointer swapping. When variables are used more than once, they require a deep copy of a term, replacing its variables with fresh names. That copy can then use alternative sharing strategies such as closures or just direct copies. This allows us to maintain full λ-calculus compatibility.

The let-lam rule, which performs the "pair projection" in a lambda is key for optimal reductions, since it splits the lambda in two separate, but "connected" terms. This allows us to perform a lazy, granular copy of them, which, in turn, allows us to express optimal sharing with careful placement of lets. Note that it can't be used indiscriminately to copy values, since using let to copy a term which itself has a let or a pair will be unsound w.r.t expected λ-calculus semantics. In other words, let should be either added carefuly, or by a compiler when it knows it is safe to do so. Of course, nothing prevents you from just using the Ultimate Calculus as the reference, in which case you can just ignore the expected λ-calculus semantics.


If you want to implement an interpreter or evaluator for a functional language or the λ-calculus, please consider using the Ultimate Calculus as a fundation or compilation target. It is very simple to implement and can be drastically faster than traditional reduction strategies such as Bruijn substitution or even HOAS with careful placement of lets.

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