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But the converse is also true: there are things Absal can do that λ-terms can’t! In any case, there seems to be no other way: see this S. To have the first 3, closed scopes must be abandoned. In a sense, it is more powerful than the λ-calculus, because applications can move arguments to outside the function’s scope. For example, it has no problems expressing abstractions whose variables occur outside of their own bodies, which makes no sense on the λ-calculus. c) 1 2 3)--------------------------------- lambda applicationλt. Yet, it is much less powerful, because of the lack of duplication. The problem here is that, after copying a λ, the “same” bound variable will exist in two different places, but its occurrence remains in a single place. That is, since a duplication allows one lambda to be in two places at the same time, then we need a way to store two variables in one place at the same time.
That sounds like a reasonable solution, but doing so requires extending the graphical representation of λ-terms with new nodes and reduction rules; the so-called “oracle”.
As it turns out, after decades of research, nobody figured out an efficient oracle. term)----------------------------- lambda copyterm = (f0 & f1) (x0 & x1)λt.
This allows two optimizations: first, the λ can keep a pointer to the occurrence of its variable, so, there’s no need for a traversal.
Second, since the variable only occurs once, we can merely redirect the reference of (λp.
The proposal is to use the algorithm without an oracle (i.e., incomplete, but efficient), and restrict our term language to only be able to express terms that are compatible with it.
The most promising way to do it is to shift to a different underlying logic: Elementary Affine Logic, which essentially stratifies terms in “layers”, preventing a term to duplicate another term in the same level, making it Absal-compatible. This proposal is particularly interesting because of its nice normalization properties. I’ll develop a new language on which that mismatch doesn’t exist, the Symmetric Interaction Calculus (SIC). That means every Symmetric Interaction Net can be interpreted as an SIC term, and vice-versa; and each graph rewrite corresponds to one rule on the calculus.I’ll briefly explain how all terms of that language can be optimally reduced with the Abstract Algorithm, and how all intermediate steps of that reduction correspond to a SIC term. term)----------------------------- lambda copyterm = (f0 & f1) (x0 & x1)λt. Note that, since there is no rule to duplicate a superposed value, SIC corresponds to the abstract algorithm with only one “fan” node.YALE comes close, but its rewrite rules are much more complex, so it is still at least 10x slower in practice. That huge difference makes the whole algorithm too inefficient in practice. Here is an idea: what if, instead of restricting the λ-calculus to the subset that works with Absal, we fundamentally changed it, making both systems match perfectly? But, before, let’s forget the Abstract Algorithm altogether and try an exercise. 1) 2--------- lambda application1The language we have so far is the Affine λ-calculus, and it is a subset of both the λ-calculus and the SIC. In fact, it is not even Turing-complete and terminates after a constant number of applications. To solve this issue, we’ll introduce a last primitive to our language, superposition, which is represented by was defined to make our reduction complete and allow every term to have reduction. Imagine you took a time machine all the way to 1928 and was in charge of inventing λ-calculus, but, this time, with one constraint: . To restate our mission, we’re back to 1928 and we’ll try to implement the λ-calculus; that is, a general model of computation based on the notion of functions and substitutions; as if we were its original inventors, except we’re constrained to only use constant-time operations. Application, as defined, is not a constant-time operation. Now, paths will diverge, as we’ll do something the λ-calculus is not capable of. As such, it doesn’t work as a general model of computation, which is what we’re aiming for. Let’s first try extending our language with a primitive for global definitions:, we simply gave it a name, then used that name twice on the main term. Both constructs make no intuitive sense, but are nether less useful to make formulas work, and are eventually cancelled out. That is, the asymptotics are the same, but the practical performance of the algorithm goes downhill because rewrite counts are increased by large constants. Here is a visualization: This table compares the number of graph rewrite rules (thus, time) required to evaluate certain terms in 5 different implementations; 4 of them with some oracle, and one without, here called Absal. As you can see, the oracle-free version outperforms most implementations by 2 to 3 orders of magnitude. Finally, I’ll hint a direct isomorphism between the SIC and Symmetric Interaction Combinators, rendering the former as a mere textual interpretation of the later. That is, it is impossible to duplicate a duplication.If you want a short specification and explanation, check out the Rust implementation. To make it more powerful, we could just extend definitions and superpositions with labels.