> Here is a stack machine that [instead of addition] implements subtraction, based on the mode assignment i/o/i [without changing the code already used for addition]. (You might have heard people claim that logic programs can be "run backwards"; this is one thing that can mean.)

    k >> plus 0 _ P         |---->  k << P
    k >> plus (s N) _ (s P) |---->  k; _ >> plus N _ P
    k; _ << P               |---->  k << P

So if you're confused because of the slightly unusual notation, here's the same thing in Prolog syntax:

    % "Sum is the sum of S1 and S2"
    plus(S1, S2, Sum) :- Sum is S1 + S2.

    % "What is the sum of 3 and 5?"
    ?- plus(3, 5, S). 
    % Answer: S = 8

    % "Is 10 the sum of 3 and 5"
    ?- plus(3, 5, 10).
    % Answer: fail 

    % "What's the difference between 3 and 10?"
    ?- plus(3, X, 10).
    % throws an error 

It doesn't work this way in general because the Prolog is/2 predicate can only be used in one direction to evaluate the term on the right hand side where must all variable must be bound to a number in context. The article mentions Peano arithmetic as one finite/incomplete axiomatisation of natural numbers but doesn't elaborate on it.

> I've recently been curious about the abstract machines implied by this process for other kinds of programs.

Olivier Danvy's "Rational Reconstruction of the SECD Machine" [0] explores the idea of this transformation as well, but frames it as a relationship between operational and denotational semantics:

> This deconstruction–reconstruction is actually interesting in itself because it provides a bridge between small-step operational semantics (in the form of an abstract machine) and denotational semantics (in the form of a compositional evaluation function)

His work on (de/re)functionalization is super interesting.

[0]: https://link.springer.com/chapter/10.1007/11431664_4

[dead]