This post is partially a request for references, partially a place for possible discussions, in the comments, and partially a place for clarifications of the previous post Why build a chemical concrete machine, and how? .

I started to read Erik Winfree’s thesis Algorithmic self-assembly of DNA and, to my joy, I see that at least at the knowledge level of 1998, what I propose is different. Here is a short brief of what I got from Winfree’s thesis (along with my excuses for not representing things correctly and for misattributions) :

- Adleman, Lipton, propose a model of DNA computing which uses exponential space, i.e. all the candidates for a solution of a combinatorial problem, each one represented by a strand of DNA, which are then filtered by a sequence of physical, or chemical manipulations of test tubes, of one of the types: separate (a tube into two others, according to the value of the variable at “i” position), merge (two tubes into one), detect. Variants (not due to Adleman, Lipton, Karp) are to add an amplify operation (like a FAN-OUT with variables) which transform a tube into two copies of it. Or (Boneh), instead of amplify, an append operation which adds a new variable with a give value. All these models have variables and are based on the mechanism of selection of the solution from an exponential sea of candidates.
- Hagiya, single-strand DNA computation, using a mechanism called “whiplash PCR”, which I don’t understand yet, but which has the computational power of a GOTO program. Has variables.
- Winfree, single-strand DNA computation, but in 2D, where he proposes a “materialization” of a block cellular automaton (BCA) which has Turing universality. Has variables, tries to make a Turing machine.

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In the post Why build a chemical concrete machine, and how? I mentioned Turing machines, but this is obviously wrong, as can be seen by looking at the previous post A chemical concrete machine for lambda calculus. I don’t want to have a ” syringe with 10^9 nano geometrical turing machines”, no, that’s misleading, what I call a chemical concrete machine works with lambda calculus terms (among other graphs, more geometrical, from graphic lambda calculus), which are reduced by chemical reactions (using for example the graphic beta move enzyme). That’s different.

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At page 29 of Winfree’s thesis, there’s a picture illustrating various reactions and results of reactions between DNA strands. I find interesting the Holliday junction, (image taken from the wiki page)

because it’s relation to crossings in knot diagrams. Recall that in the -TANGLE sector of the graphic lambda calculus, the graphic beta move appears as a SPLICE move.

Compare with these images from 3D crossings in graphic lambda calculus:

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As that’s an exploratory post, kind of note to self, but open to anybody, take a look at this short course

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