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I'm currently doing some Formal Language research involving classes of languages above Regular but below Context Free. I'm looking at things like Reversal-Bounded Multicounter Machines, Single-stack counter machines, deterministic CFLs, etc.

I'm wondering if anybody knows of a good book or survey paper which outlines the properties of these languages. Most of what I'm looking at is too obscure or too new to be in my Hopcroft-Ullman book, even the 1979 edition.

Mainly I'm looking for which language classes are contained in one-another, the closure properties of these languages, and the decidability of basic problems (F-problems) on these languages.

Some example of things I'd look up in this reference:

  • Are all languages accepted by Reversal-bounded Multi-counter machines also accepted by non-reversal-bounded single counter machines?
  • Are the deterministic reversal-bounded MultiCounter languages closed under left and right concatenation?
  • Is universality decidable for single-counter machines.

These are just example questions, I have many others that come up in my day-to-day work.

As a starting point, I've tried tracing which papers cite Oscar Ibarra's "Reversal-Bounded Multicounter Machines and Their Decision Problems", but haven't found much.

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    $\begingroup$ Crossposted on CS.SE. $\endgroup$ – Juho May 28 '13 at 22:10
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    $\begingroup$ For a detailed analysis of one state multicounter machines see Hierarchies and Characterizations of Stateless Multicounter Machines $\endgroup$ – Marzio De Biasi May 30 '13 at 22:15
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    $\begingroup$ ... and I think that a lot of material/references can be found in recent (>2000) Ibarra's papers $\endgroup$ – Marzio De Biasi May 30 '13 at 22:38
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    $\begingroup$ Did you ask to Oscar Ibarra? $\endgroup$ – Abuzer Yakaryilmaz May 31 '13 at 4:43
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    $\begingroup$ @jmite There's no harm in trying :-) As a student myself, I've always gotten a response from a researcher when I've emailed them. In my experience, people are only happy to help out someone who is interested in their research. $\endgroup$ – Juho May 31 '13 at 16:12
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Not standard topics, no. And sorry, I have no general overview.

However, I would have a look at the PhD thesis of Klaus Reinhardt for at least a picture of the various families that live in this area. See page 64 for a diagram of the zoo. Motivated by Petri Nets with inhibitor arcs Reinhardt studies priority multicounters with restrictions on when to do zero tests. Non-trivial.

By the way, your last example question was discussed in this forum by user Sam Jones. Another Ibarra reference.

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