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The standard complexity classes such as P, NP are usually defined using Turing Machines. In finite model theory those classes can be defined via the classical first-order/second-order logics.

I am curious whether there exists an equivalent definition of such classes via $\mu$-recursive functions. What about register machines? lambda calculus? etc? In computability theory it is known that $\mu$-recursive functions, counter machines, lambda calculus and Turing Machines are all equivalent. (https://en.wikipedia.org/wiki/Computable_function)

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  • $\begingroup$ You can define PTIME using first-order logic? FO characterises LOGSPACE. For PTIME there is no such characterisation, only LFP on ordered structures. $\endgroup$
    – The User
    Jul 16, 2013 at 9:00
  • $\begingroup$ You are absolutely right. Actually I have in mind in general finite model theory characterises complexity classes via FO/MSO/Fixed-point. I don't don't really mean P,NP. Thank you for your answer though. $\endgroup$
    – Tony Tan
    Jul 16, 2013 at 9:33
  • $\begingroup$ I mean, actually it is unknown whether there is such a characterisation for PTIME, thus you would be famous if you could. ;) And I am sorry, FO is weaker than LOGSPACE, that was my point. $\endgroup$
    – The User
    Jul 16, 2013 at 9:38
  • $\begingroup$ It is true. Thanks anyway for the answer... :-) $\endgroup$
    – Tony Tan
    Jul 16, 2013 at 10:03

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Yes, there is. See [Cob64]. The idea is to replace primitive recursion in the definition of primitive recursive functions with bounded recursion on notion.

Another more delicate approach is taken in [BC92].

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