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I read here that a proof has been offered that a Turing Machine with 2 states and 3 symbols can be universal (in that it is capable of arbitrary finite computations). Even if this proof is accepted, it is based on a Turing machine that doesn't halt.

If we consider only computations that do halt (that end on an additional accepting state), the set of initial configurations that produce such computations could be considered a formal language. In that case, what would be the limitation to the class of languages that could be defined by those machines (with 2 states and 3 symbols)? Could they be specified using regular expressions?

Any help is appreciated.

The following TM halts and accepts iff the initial configuration is $(d_{1}01^*[1]d_{2} + d_{3}[0]d_{4})$, where $d_{i} \in \Sigma^\infty$, and $[x]$ (with $x \in \Sigma$) denotes the position of the tape head.

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André Souza Lemos
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    There can be no such (i.e. computable) characterisation of the set of always terminating Turing machines. That would solve the Halting problem. See also here. (Duplicate?) In particular, it's definitely not the same set as those for REG since REG $\subsetneq$ CSL $\subsetneq$ R. – Raphael Jun 12 '15 at 11:41
  • You mean it can only be found out if we took them one by one, right? – André Souza Lemos Jun 12 '15 at 11:43
  • Since I don't understand what you mean by that, I don't think that I mean that. – Raphael Jun 12 '15 at 11:43
  • A characterization of the kinds of languages that can be defined by machines from this finite (and approachable) set. That's what I mean. – André Souza Lemos Jun 12 '15 at 11:51
  • What do you mean by 'approachable'? Can you enumerate them? Can you find out, given a machine, whether it is one? – reinierpost Jun 12 '15 at 12:49
  • I've already narrowed the set of different MTs with 2 states and 3 symbols to less than $10^6$ (using symmetries, etc.). All machines I have sampled so far define languages that are regular. Given a particular machine, I can obviously study it as a particular problem, and find out if I can use, for instance, a regular expression to define the language for which it halts and accepts. What I want to know is if there is a general characterization for the whole set, i.e., if this problem has already been studied and solved. – André Souza Lemos Jun 12 '15 at 14:19
  • I'm struggling to understand what it means to "consider only computations that do halt", especially since (in general) we can't necessarily predict which computations will halt -- there's no general algorithm to do it. Are you asking about the set of languages accepted by such a TM? But the standard definition of the language accepted by a TM already only "considers" computations that eventually halt, so I'm not sure why you add this redundant restriction. Maybe your question is: If $M$ is a TM with 2 states and 3 symbols, is $L(M)$ guaranteed to be regular? – D.W. Jun 14 '15 at 06:45
  • Yes, $L(M)$ being defined as the set of initial configurations for which the machine stops. I've adapted the notation for reg. expressions to include a symbol for cells that won't be reached, so I'm always talking about the entire tape, not particular words. Hope this makes things clearer. I'll edit the question, if necessary. – André Souza Lemos Jun 14 '15 at 12:28

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