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PH is the complexity class that is the set of languages expressible by second-order logic. If so, is there any complexity class that is the set of languages expressible by first-order logic?

It seems likely that R is the complexity class that is the set of languages expressible by first-order logic, but I am not really sure.

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Did you check the Wikipedia page on Descriptive complexity theory? –  Yuval Filmus May 9 '12 at 1:25
    
@YuvalFilmus no.. and I got an answer! Thanks for help. –  user2346 May 9 '12 at 1:29
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up vote 2 down vote accepted

I'm just putting this here to remove it from the unanswered queue. From the wikipedia article on Descriptive Complexity Theory:

First-order logic defines the class FO, corresponding to $\mathsf{AC}^0$, the languages recognized by polynomial-size circuits of bounded depth, which equals the languages recognized by a concurrent random access machine in constant time.

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