Numeration Systems: a Link between Number Theory and Formal Language Theory

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and verification of infinite-state systems and present a list of open problems.

💡 Research Summary

The paper surveys the deep connections between numeration systems, number theory, and formal language theory, using Alan Cobham’s seminal results as a guiding framework. It begins by broadening the notion of numeration systems beyond the classical integer bases to include non‑standard representations such as Fibonacci, Zeckendorf, and more general linear recurrence based systems. Each system provides a way to encode natural numbers as strings over a finite alphabet, which then become the objects of study for recognizability.

The authors then focus on “recognizable sets of integers,” i.e., subsets of ℕ whose representations in a given base form a regular language accepted by a deterministic finite automaton (DFA). Cobham’s theorem—often called the multi‑base automaticity theorem—is presented in detail: if a set is recognizable in two multiplicatively independent bases (e.g., base p and base q with gcd(p,q)=1), then the set must be ultimately periodic. This result creates a bridge between the algebraic structure of numbers and the combinatorial structure of regular languages, showing that the only sets simultaneously regular in two such bases are those with a simple arithmetic description.

The third part of the survey deals with automatic sequences. An automatic sequence is generated by feeding the base‑b representation of the index n into a DFA and reading the output symbol. The paper explains how Cobham’s theorem restricts the complexity of such sequences and how these restrictions lead to transcendence results. In particular, real numbers whose digit expansions are produced by automatic sequences are often shown to be transcendental (e.g., the Baum‑Sweet, Thue‑Morse, and Rudin‑Shapiro numbers). The authors sketch proofs that rely on the low‑complexity nature of automatic sequences and on classical results such as the Mahler method.

Beyond pure theory, the paper highlights two concrete application domains. In combinatorial game theory, the Sprague‑Grundy function of impartial games can be expressed as an automatic sequence when the game’s move set has a regular structure. This representation enables the compression of the game’s state space into a finite automaton, allowing efficient computation of winning positions and strategy synthesis. In the verification of infinite‑state systems, the authors show how system configurations and transition relations can be encoded as recognizable sets in suitable numeration systems. Model‑checking problems then reduce to language inclusion or emptiness checks for DFAs, which are decidable and often tractable, offering a powerful alternative to traditional well‑structured or symbolic techniques.

The final section lists open problems that the authors deem fertile for future research. These include: (1) extending Cobham’s theorem to bases that are not multiplicatively independent, (2) developing algorithmic criteria for the transcendence of numbers defined by automatic sequences in arbitrary bases, (3) characterising the exact computational complexity of determining whether a given impartial game’s Grundy function is automatic, and (4) constructing fully automated verification pipelines that exploit automatic recognizability for a broader class of infinite‑state models, such as pushdown systems or timed automata.

Overall, the survey not only consolidates known results but also provides a coherent narrative that positions numeration systems as a unifying language for disparate areas of mathematics and computer science. By emphasizing both theoretical depth and practical relevance, it invites researchers to explore the rich interplay between number representations, automata theory, and algorithmic applications.