Is Cantor's Theorem Automatic?

A regular language L together with a binary relation ≤L on L is an automatic presentation of the rationals iff (L,≤L) is isomorphic to (Q,≤) and ≤L can be accepted by a synchronous two-tape automaton. An automorphism of (L,≤L) is automatic iff it can be computed by a synchronous two-tape automaton. We show (1) that the canonical presentation from [5] is automatic-homogeneous (any tuple can be mapped to any other tuple by an automatic automorphism), (2) that there are presentations which are not automatic-homogeneous, and (3) that there are automatic linear orders that are not isomorphic to any regular subset of the canonical presentation. This last result disproves a conjecture from [5]. Classically, model theory as a branch of mathematical logic investigates structural properties of algebraic structures. A typical result is Cantor’s theorem: Any countable linear order (L,≤) is isomorphic to a set X of rational numbers (“the rational numbers are universal”). In effective model theory [7], interest shifts to the computational content of classical model theory. In the example of Cantor’s theorem, one is only interested in linear orders (L,≤) that are recursive (i.e., the set L is a recursive set of natural numbers and the relation ≤⊆ N × N is decidable). Then, one asks whether X can be chosen to be decidable under a suitable encoding of the rationals. If, in the example above, we restrict L, ≤ and X further to, say, sets decidable in a certain complexity class C, we arrive at complexity theoretic model theory. Here, Cantor’s theorem holds only under severe restrictions on the encoding of (L,≤) in the natural numbers and for certain complexity classes C (see [4] for a survey on this and many other results). Khoussainov and Nerode [11] initiated the investigation of automatic structures; in the light of the explanation above, this means that the complexity class C is the set of regular languages. Since these automatic structures are intimately linked with finite automata, the rich theory of finite automata is at the core of understanding this field. For instance, complementation and projection of finite automata allows to show that the first-order theory of any automatic structure is decidable [8]. This implies, e.g., the decidability of Presburger’s arithmetic [2] (using the same ideas, this decidability was also shown by Elgot [6]) or of the 1 Untypical, as far as difficulty of proof is concerned. 2 A thorough analysis of Cantor’s proof reveals that all the steps can be performed effectively; hence the effective version of Cantor’s Theorem holds.