3.4 Chemical Reaction Networks: Decidability ⇐⇒ No Probabilities
3.4.2 Probabilities = ⇒ Undecidable
3.4.2.2 Universality = ⇒ Probability Is Concentration Dependent . 143
It is straightforward to see that if one has a different clock design that exhibits even slightly super-linear slowdown – e.g., O(n1+∆1 ) – then this would also result in constant probability of output error. That is to say, the cost of reliable, efficient simulation of register machines could be made negligible with such a clock.
which it is possible to reach S. (Note that given any state, the question of whether it is possible to reach some state in S is computable, as discussed in section 3.4.1.) Note also that there is a boundb such that for any stateA ∈R, the length of the shortest sequence of reactions leading from Ainto Sis at most b. This means that there is some constant p0 such that for any state r ∈ R, the probability of entering S within b steps is at least p0. Thus, the probability of remaining inRmust decay at least exponentially.
This implies that the probability that the system will eventually enterSorQis 1, and so simply by computing the probabilities of the state tree for Rfar enough, one can compute the probability of entering Sto arbitrary precision.
3.4.2.3 Open Questions
Here we list some questions, along the lines of the results we have given, which we believe may be within reach and which (regardless of whether they are proven or disproven) would contribute to our understanding of what allows chemical reaction networks to be universal.
• Are continuous chemical reaction networks (using mass action kinetics) universal?
• Can one have a universal chemical reaction network which has constant probabilities (that don’t depend on concentrations) for all reactions except one, with the remaining reaction having a decaying probability that depends on time (but not on concentra- tions)?
• Can chemical reaction networks with reversible reactions be universal?
• Can the time and space requirements for stochastic chemical reaction networks, com- pared to a Turing Machine, be a simple polynomial slowdown in time, and an expo- nential increase in space?
The last question is almost easy: Our definition of a register machine can be augmented to to allow an instruction to multiply a register by 2 or to divide it by 2. This allows efficient simulation of a Turing machine, assuming the multiplication and division are efficient.
Multiplication can be efficiently performed by a chemical reaction network using a self- catalyzing doubling reaction (taking time logarithmic in the size of the register), but it turns out that dividing by 2 is the stumbling block. Simple approaches, like using a reaction of
the form Ri+Ri −→ Ri′, are very slow to finish, as the last few molecules being divided by 2 (i.e., being merged) have trouble finding each other. If an efficient way could be found to divide a given species by 2 (and note the remainder), the rest of the construction is comparatively easy. We have found this to be a very intriguing open problem.
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