Given the sets
$K_1=\{\{a_0,b_1\},\{a_1\},\{b_0\}\}$.
$K_2=\{\{c_1,c_0,d_0,e_1\},\{d_1\},\{e_0\}\}$
$K_3=\{\{f_0,f_2,g_0,h_1\},\{f_1,f_3,g_2,h_3\},\{b_1\},\{b_3\},\{c_0\},\{c_2\}\}$
Every item of the form $letter_{digit}$ is a possible state for the system $K_i$. If two items are in the same set, say for $K_1$, $\{a_0,b_1\}$, then a transition between them is possible $(a_0\leftrightarrow b_1)$.
Using any finite amount of $K_i$'s, and an allowed operation of making sets consisting of two or more letters like this: $\{{a,b,f\}}$, which means a transition between $a_x,b_y,f_z$ is allowed for any $x,y,z$.
If we have two or more of one $K_i$, these are assigned different letters (ie none of a,b,c.. is used for two different $K_i$). When we transition from a state say $a_z$ in $K_j$, to that of another $b_y$ in $K_i$, and eventually return to a letter in $K_j$, then the digit $z$ is preserved of $K_i$ (it doesnt change state by itself).
A letter which is not identified with any other (by the method ${a,b,f}$ in above example), is called loose, initially all letters are loose (disconnected).
Is it possible to construct a composition using above rules to create a system acting like $X$ below, i.e. in total there is $3$ loose letters, every $K_i$ is assigned an initial state, and the resulting thing must follow the transition rules given by $X$:
$X=\{\{i_0,j_1\},\{i_1,k_0\},\{j_0\},\{k_1\}\}$
This thing has only two states, $0$ and $1$, this just means that the states of the $K_i$'s it is composed of, should only have two outcomes on the possible transitions between the loose letters (the internal state must be cyclic with period 2t for some t).
The state of this thing is then given completely by the state of each K_i (a single number for each), and exactly one letter in total. And is there a general method to decide given any $K_i$'s and $X$ to decide if $X$ is composable of some number of the $K_i$'s ?
Is there some finite atomic set of $K_i$'s by which any possible $X$ is composable?
Is there a more natural framework in which this question can be reformulated and answered?