# Calculus Term Paper

EAL is not only strongly normalizing, but its complexity is predictable, even for the untyped language. Because of that, a type theory based on EAL could even have type-level recursion without becoming inconsistent! It requires programmers to write explicit duplications with very unfamiliar restrictions.

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For some input programs, it, instead, simply fails to halt, or even returns downright wrong results; in this issue I present an example.

Moreover, even λ-terms that succeed to reduce pass through intermediate representations that correspond to no λ-term at all.

Beta-reduction does way too many things at the same time and, thus, isn’t allowed.

Here is a quick solution: let’s only allow variables to occur once. t 1 2)------------------------------- lambda application(λt.

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. We’ll completely ignore the abstract algorithm and develop such calculus by itself. First, because the substitution procedure must traverse over . Recall our definition of application: Does that break the world? It is still a computing model with unambiguous reduction rules. As soon as a function is applied to an argument, the occurrence of its variable is replaced by an argument, wherever that argument is, even outside itself. Obviously, we can’t actually do that, as we already agreed copying isn’t a constant-time operation. Let’s start by defining a syntax with functions, variables and function application: And, well, that’s it. If implemented naively, that’s a quadratic amount of work. With so many good things, one might wonder why it was not previously. All in all, I’m really glad to have figured out a way to use the abstract algorithm that is superior to “write a λ-term and hope it works”. One of the main reasons is it doesn’t actually evaluate every program of the λ-calculus. (Update: renamed to Symmetric Interaction Calculus.)This is another post about the Abstract Algorithm. term)--------------------------------------- superposed application K = x0 & x1term = f0 K & f1 Kλt. That’s because it is directly isomorphic to Symmetric Interaction Combinators.In it, I’ll explain why there is a mismatch between it and the λ-Calculus. If you haven’t read that paper, I’d strongly suggest you to, as that is, in my opinion, the most elegant model of computation I’ve seen.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.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.

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