Metabolism of loops (a chemical reaction network in the chemical concrete machine)

In the following figure you see a chemical reaction network in the chemical concrete machine which involves only “+” enzymes.

chem_eval_2

Comments:

  • the molecule from the left upper corner corresponds to combinator \Omega = (\lambda x . (xx)) (\lambda x . (xx))
  • for each molecule I marked the available reaction sites, with dashed red closed curves, along with the name in red of the enzyme which is interested in the respective reaction site,
  • for the chemical reactions, figured with blue arrows, I put the name of the move which corresponds to the respective chemical reaction, with an added “+” to indicate that the reaction is unidirectional, as if only “+” enzymes are available (that is I wrote “DIST^{+}” instead of “\delta^{+}” and so on, see for details the notations used in the chemical concrete machine),
  • whenever there are non-overlapping reaction sites, we can perform in parallel the moves (reactions),
  • but in some places we have overlapping reaction sites, like in the case of the molecule from the 3rd row, left, where a \beta reaction site and a DIST reaction site overlap.
  • In case of overlapping reaction sites there are multiple possibilities, which produce branches in the chemical reaction network,
  • I have not used any elimination of loops,
  • several molecules from the figure don’t correspond to lambda calculus terms, thus what  is figured is not a representation of the usual fact that the combinator \Omega does not have a normal form (one which cannot be reduced further),
  • in the middle of the figure we see a loop-producing cycle, hence the name,
  • curiously, by going outside lambda calculus, we can reduce \Omega (at left upper corner) to a “normal” form (right lower corner), i.e. a loop and a molecule which does not have any “+” reaction sites. (The green fork node of that molecule is a fan-out node and the molecule is like a fan-out gate with one output curling back to the input of the fan-out, but remember that no signals circulate through the arrows 🙂  )

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See also:

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UPDATE:  For clarity, here is the usual cycle of reductions of the combinator \Omega, seen in graphic lambda calculus (but with the drawing conventions of the chemical concrete machine):

chem_eval_4

(A similar figure appears after the proof of Theorem 3.1  arXiv:1305.5786v2 [cs.LO] . )

In graphic lambda calculus we have the GLOBAL FAN-OUT move, which, as the name says, is a global move. In the chemical concrete machine we have only local moves.  Here is how the same cycle is achieved in the chemical concrete machine.

chem_eval_3

You see a move (reaction) which is labelled “(multiple) CO-ASSOC”. The reason is that we need several CO-ASSOC moves to pass from the molecule at the right of the 2nd row to the one from the 3rd row. Alternatively, the same can be done with a SWITCH move, which is a succession of FAN-IN(+) , CO-COMM(+)  and FAN-IN(-), therefore unpleasant as well. Moreover, if we invent a SWITCH enzyme (i.e. if we accept SWITCH as a new move which does not modify the whole chemical machine, because SWITCH is a consequence of the other moves) then we have an explosion of places where SWITCH enzyme could act.

In conclusion, the usual endless reduction of \Omega in lambda calculus, is possible, but highly unlikely in the chemical concrete machine, and only in the presence of CO-ASSOC or SWITCH enzymes, and moreover under tight control of the places where these enzymes act.

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5 thoughts on “Metabolism of loops (a chemical reaction network in the chemical concrete machine)”

  1. ” tight control of the places where these enzymes act” What does this mean?
    Also: “a fan-out gate with one output curling back to the input of the fan-out” This is a single example of a “feedback loop” in cybernetics.

    1. Re: the “feedback loop”, you are right! thanks, nice, right?
      Re: “tight control” is because is like you want to change ((ac)(bd)) to ((ab)(cd)), you need certain well dfined sequences of assoc and comm rules to go from one another, you can’t just say, “by doing applying an arbitrary number of at most, say 5, associativity and commutativity properties of the operation, we can pass from ((ac)(bd)) to ((ab)(cd))”, because after 5 random applications of these properties starting with ((ac)(bd)), there is a small probability of ending with ((ab)(cd)).

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