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In the 1950's, Julius Büchi showed that $(\mathbb{N},S,+,0)$ is not merely a decidable structure as Presburger had shown, but an automatic structure, i.e. there is an encoding of the natural numbers (using finite binary strings) such that you can algorithmically assign to each formula $\phi(x_1,...,x_n)$ in the language of Presburger arithmetic a finite automaton $M_\phi$ which recognizes the set $\{(a_1,...,a_n):(\mathbb{N},S,+,0)\models \phi(a_1,...,a_n)\}$.

Now I believe that the ordered field of algebraic real numbers $(\mathbb{R}_{alg},+,*,<,0,1)$ is a decidable structure, i.e. there's an encoding of the algebraic real numbers such that the theory of the structure with constants added for all the algebraic real numbers is a decidable theory. But my question is, is $(\mathbb{R}_{alg},+,*,<,0,1)$ an automatic structure? That is, is there an encoding of the algebraic real numbers such that you can algorithmically assign to each formula $\phi(x_1,...,x_n)$ in the language of ordered fields a finite automaton $M_\phi$ which recognizes the set $\{(a_1,...,a_n):(\mathbb{R}_{alg},+,*,<,0,1)\models \phi(a_1,...,a_n)\}$?

Keshav Srinivasan
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    A very interesting question. Can you give a pointer to Büchi's work on this, please. – Rob Arthan Dec 02 '21 at 23:16
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    @RobArthan Here is Buchi’s original paper, it’s pretty unreadable: https://deepblue.lib.umich.edu/handle/2027.42/3930 Here’s a more modern presentation of Buchi’s proof: https://courses.engr.illinois.edu/cs498mv/fa2018/MSOonWords.pdf Buchi’s proof involves interpreting Presburger arithmetic within the weak monadic second-order theory of one successor (WS1S). But if you want to see a direct proof that Presburger is automatic without going through WS1S, check out this excerpt from Sipser’s book: https://gdurl.com/fZe8V – Keshav Srinivasan Dec 04 '21 at 05:57

1 Answers1

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The answer is no. Indeed, it is shown in [1, page 9, Theorem 3.10] that no infinite integral domain is automatic.

[1] B. Khoussainov, A. Nies, S. Rubin, F. Stephan. Automatic structures: richness and limitations. Log. Methods Comput. Sci. 3 (2007), no. 2, 2:2, 18 pp.

J.-E. Pin
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