Editing Open Problems (section)

=== Context ===

Models of computation are often presented either as (i) a single device that works for all inputs (e.g. a Turing machine, a Python program), or (ii) a family of devices where we have one device for each input length (e.g. a family of Boolean circuits with exactly one circuit $c_n$ for each input string length $n$). The relationship between standard models and CRNs where in both models we have (i) one device for all inputs is beginning to be well-understood (but still has interesting questions), a number of the questions above are best formalized in the (ii) family of devices setting. The purpose of this section is to describe how the latter setup might look. 

Families of devices can be uniform or nonuniform: we say that a family of devices is ''uniform'' if there is an algorithm that describes the entire family. For example, we say that a language $L$ is accepted by a uniform Boolean circuit family $\mathcal{C}$ if there exists a computable function $f : \{1\}^* \rightarrow \mathcal{C} $ such that $$\mathcal{C} = \left\{ c_n \mid \forall n\in\mathbb{N}, f(1^n) = c_n \text{ is a circuit that accepts } \{0,1\}^n \cap L \right\}.$$

The distinction between single-machine versus families of devices has appeared in work on CRNs. For example, here are two example ways for a CRN to compute a function of its input.
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<li>The input could be the initial count of some single species $X$. Then any natural number $n$ can be represented, or equivalently one could represent a binary string $x$ by using natural number $n$ as input, if $x$ is the $n$'th binary string in lexicographic order.

<li>The input could be a binary string $x$ of length $n$ represented by $n$ different input species $X_1,\ldots,X_n$. The presence or absence of each of these species is then able to represent any bit string of length $n$. Indeed, for each $n \in\mathbb{N}$ there is exactly one CRN, and we call the entire infinite set of CRNs a ''family''.  
</ul>