Finite Automata with succinct representation of chains of states Consider a kind of automata similar to common DFAs or NFAs where it is possible to represent succinctly linear chains of states. In other words, an automaton like this:



could be represented in this way:



where the thick edge represent the chain of states, where each state is connected to the next by a single edge and all the edges are labeled in the same way, in this case by $a$.

So this is not really a counter or anything fancy, it is just a succinct representation of a very limited special case. By succinct, I mean that by representing the $k$ parameter in binary, the second automaton can be represented in logarithmically less space than the first. Let's call this kind of automata the "succinct automata", SA, so say DSA and NSA for short for the deterministic and nondeterministic variants.

Now, my question concerns the complexity of boolean operations over this kind of automata. 

In details:


Given two NSAs $\mathcal{A}$ and $\mathcal{B}$, is it possible to build the NSAs for $\mathcal{L}(\mathcal{A})\cup\mathcal{L}(\mathcal{B})$ and $\mathcal{L}(\mathcal{A})\cap\mathcal{L}(\mathcal{B})$, of size still polynomial in the size of $\mathcal{A}$ and $\mathcal{B}$ (i.e. without paying for the unrolling of the chains before computing the results)?
Is it possible to compute those operations on DSAs (deterministic) guaranteeing that the resulting automata stay deterministic (and still polynomial size)?
Is it possible to determinize an NSA with only a singly-exponential blowup (i.e. without paying for the unrolling of the chains before paying for the classic determinization)?


My feeling on all of these after having though about it a bit is that an exponential increase in size is needed in most of the cases, or that the results must be nondeterministic.

So the question is really: is anybody aware of a place where this kind of problems have been addressed? Has this variant of finite automata being studied before?
This is a very partial answer, but I have some ideas:


Clearly the union of NSAs can be taken without any blowup - just use the nondeterministic union of the initial states.
As for determinization, you'll have a double-exponential blowup. Consider the language $L_k=(a+b)^*a(a+b)^k$. That is, the $k$-th before last letter is $a$. You can easily construct an NSA of size $O(\log k)$ for it, but any DSA would need to "remember" the last $k-1$ letters (the same argument that works for DFAs would work here). So a DSA would need $2^{k-1}$ states, which is a double exponential blowup.
As for union of DSAs, this is more of a conjecture: consider the language $L_p=(a^p)^*$ for a prime number $p$. This can be recognized by a DSA with 2 states and $p-1$ written on one of the transitions. Now, consider the union $L_p\cup L_q$ for $p\neq q$. It seems to me that you'll need a lot of states (i.e. $p\cdot q$) in order to recognize this with a DSA. But I haven't thought this through completely.