# Calculus Term Paper

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Some common programming idioms are even impossible: nested for-loops, for example.

You must instead unroll all loops in your programs. t 1 2 3Notice how the first two applications of the inner function moved its argument to outside its own body. term)----------------------------- lambda copyterm = (f0 & f1) (x0 & x1)λt.

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.

(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.

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.

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