Is it possible to have two (or more) non-universal Turing machines labeled $A_1$ and $A_2$, such that if $f(A_i)$ is the set of functions computable by $A_i$, and S={every computable function} then $S=f(A_1)\cup f(A_2)$? Is there any easy mathematical classification of which functions are computable by some given non-universal Turing machine? Or an algorithm to check if a function/formula f(x) is computable by a given Turing machine? Is it decidable which of the finite set of m-state, n-symbol (for say 20>m+n) Turing machines are universal turing machines?
I am mainly considering finite states, finite symbols, and finite symbols on starting tape.