Let’s continue from the post Chemical concrete machine, detailed (V) , by adding some more facts and remarks.
We have seen that in order to not have to apply a controlled sequence of CO-ASSOC molecules (i.e. moves), it is better to introduce a composite move, saying that a certain simple molecule is a propagator.
With this move the formalism of the chemical concrete machine is exactly as powerful as without it. The reason is that the move called “PROP+” can be seen as a sequence of CO-ASSOC moves and DIST+ moves.
Let’s accept this move (i.e. like if there is an associated enzyme to it) and let’s revisit the proof that the W molecule is a multiplier.
Besides the B, C, K, W molecules (which correspond to the B,C,K,W system ), other molecules are also interesting: in the following figure are presented the molecules F, I, S’ and S.
The molecule F may represent FALSE in the Church encoding of the booleans, along with K which encodes TRUE. The molecule I is the identity and the molecule S corresponds to the combinator S from the SKI combinator calculus. which is
a computational system that may be perceived as a reduced version of untyped lambda calculus. It can be thought of as a computer programming language, though it is not useful for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language.
These correspondences are established by using the algorithm for transforming lambda calculus terms into graphs in the graphic lambda calculus, then by using the dictionary between the gates from graphic lambda calculus to the essential molecules from the chemical concrete machine.
What about the S’ molecule? It is a version of the molecule S, i.e. it transforms into S by this reaction:
Also, there is a proof, analogous with the one for W, of the fact that S’ is a multiplier, by using the PROP+ move. This proof is better than the proof for S which was given at the end of the post Local FAN-IN eliminates global FAN-OUT (I) .