A Generalization of Semenov’s Theorem to Automata over Real Numbers

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Abstract

This work studies the properties of finite automata recognizing vectors with real components, encoded positionally in a given integer numeration base. Such automata are used, in particular, as symbolic data structures for representing sets definable in the first-order theory 〈ℝ, ℤ, + , ≤ 〉, i.e., the mixed additive arithmetic of integer and real variables. They also lead to a simple decision procedure for this arithmetic.

In previous work, it has been established that the sets definable in 〈ℝ, ℤ, + , ≤ 〉 can be handled by a restricted form of infinite-word automata, weak deterministic ones, regardless of the chosen numeration base. In this paper, we address the reciprocal property, proving that the sets of vectors that are simultaneously recognizable in all bases, by either weak deterministic or Muller automata, are those definable in 〈ℝ, ℤ, + , ≤ 〉. This result can be seen as a generalization to the mixed integer and real domain of Semenov’s theorem, which characterizes the sets of integer vectors recognizable by finite automata in multiple bases. It also extends to multidimensional vectors a similar property recently established for sets of numbers.

As an additional contribution, the techniques used for obtaining our main result lead to valuable insight into the internal structure of automata recognizing sets of vectors definable in 〈ℝ, ℤ, + , ≤ 〉. This structure might be exploited in order to improve the efficiency of representation systems and decision procedures for this arithmetic.