Necessity of the P combinator for evolving addition at larger scales

Determine whether the P combinator is necessary for the spontaneous emergence of the addition function in larger-scale AlChemy simulations. Specifically, assess if addition reliably appears without including the P combinator when the soups are initialized with S, K, and I combinators (and the same amplifier test regimen used in the paper) but scaled to larger system sizes and longer durations than those reported.

Background

AlChemy models chemical-like interactions using lambda expressions, and this work introduces amplifier test functions to endogenously select for target behaviors like Church numeral addition. In the experiments targeting addition, the initial soups included S, K, I, and P combinators alongside amplifier tests for addition and successor.

The authors observed that addition very rarely emerged in soups seeded only with S, K, and I, whereas including P increased the likelihood because composing P with successor can yield addition. They hypothesize that scaling up system size may obviate the need for P, but this remains to be tested.