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Cobham's theorem

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Cobham's theorem is a theorem in combinatorics on words that has important connections with number theory, notably transcendental numbers, and automata theory. Informally, the theorem gives the condition for the members of a set S of natural numbers written in bases b1 and base b2 to be recognised by finite automata. Specifically, consider bases b1 and b2 such that they are not powers of the same integer. Cobham's theorem states that S written in bases b1 and b2 is recognised by finite automata if and only if S differs by a finite set from a finite union of arithmetic progressions. The theorem was proved by Alan Cobham in 1969[1] and has since given rise to many extensions and generalisations.[2][3]

Definitions

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Let be an integer. The representation of a natural number in base is the sequence of digits such that

where and . The word is often denoted , or more simply, .

A set of natural numbers S is recognisable in base or more simply -recognisable or -automatic if the set of the representations of its elements in base is a language recognisable by a finite automaton on the alphabet .

Two positive integers and are multiplicatively independent if there are no non-negative integers and such that . For example, 2 and 3 are multiplicatively independent, but 8 and 16 are not since . Two integers are multiplicatively dependent if and only if they are powers of a same third integer.

Problem statements

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Original problem statement

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More equivalent statements of the theorem have been given. The original version by Cobham is the following:[1]

Theorem (Cobham 1969) — Let be a set of non-negative integers and let and be multiplicatively independent positive integers. Then is recognizable by finite automata in both -ary and -ary notation if and only if it is ultimately periodic.

Another way to state the theorem is by using automatic sequences. Cobham himself calls them "uniform tag sequences.".[4] The following form is found in Allouche and Shallit's book:[5]

Theorem — Let and be two multiplicatively independent integers. A sequence is both -automatic and -automatic only if it is -automatic[6]

We can show that the characteristic sequence of a set of natural numbers S recognisable by finite automata in base k is a k-automatic sequence and that conversely, for all k-automatic sequences and all integers , the set of natural numbers such that is recognisable in base .

Formulation in logic

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Cobham's theorem can be formulated in first-order logic using a theorem proven by Büchi in 1960.[7] This formulation in logic allows for extensions and generalisations. The logical expression uses the theory[8]

of natural integers equipped with addition and the function defined by and for any positive integer , if is the largest power of that divides . For example, , and .

A set of integers is definable in first-order logic in if it can be described by a first-order formula with equality, addition, and .

Examples:

  • The set of odd numbers is definable (without ) by the formula
  • The set of the powers of 2 is definable by the simple formula .

Cobham's theorem reformulated — Let S be a set of natural numbers, and let and be two multiplicatively independent positive integers. Then S is first-order definable in and in if and only if S is ultimately periodic.

We can push the analogy with logic further by noting that S is first-order definable in Presburger arithmetic if and only if it is ultimately periodic. So, a set S is definable in the logics and if and only if it is definable in Presburger arithmetic.

Generalisations

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Approach by morphisms

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An automatic sequence is a particular morphic word, whose morphism is uniform, meaning that the length of the images generated by the morphism for each letter of its input alphabet is the same. A set of integers is hence k-recognisable if and only if its characteristic sequence is generated by a uniform morphism followed by a coding, where a coding is a morphism that maps each letter of the input alphabet to a letter of the output alphabet. For example, the characteristic sequence of the powers of 2 is produced by the 2-uniform morphism (meaning each letter is mapped to a word of length 2) over the alphabet defined by

which generates the infinite word

,

followed by the coding (that is, letter to letter) that maps to and leaves and unchanged, giving

.

The notion has been extended as follows:[9] a morphic word is -substitutive for a certain number if when written in the form

where the morphism , prolongable in , has the following properties:

  • all letters of occur in , and
  • is the dominant eigenvalue of the matrix of morphism , namely, the matrix , where is the number of occurrences of the letter in the word .

A set S of natural numbers is -recognisable if its characteristic sequence is -substitutive.

A last definition: a Perron number is an algebraic number such that all its conjugates belong to the disc . These are exactly the dominant eigenvalues of the primitive matrices of positive integers.

We then have the following statement:[9]

Cobham's theorem for substitutions — Let α et β be two multiplicatively independent Perron numbers. Then a sequence x with elements belonging to a finite set is both α-substitutive and β-substitutive if and only if x is ultimately periodic.

Logic approach

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The logic equivalent permits to consider more general situations: the automatic sequences over the natural numbers or recognisable sets have been extended to the integers , to the Cartesian products , to the real numbers and to the Cartesian products .[8]

Extension to

We code the base integers by prepending to the representation of a positive integer the digit , and by representing negative integers by followed by the number's -complement. For example, in base 2, the integer is represented as . The powers of 2 are written as , and their negatives (since is the representation of ).

Extension to

A subset of is recognisable in base if the elements of , written as vectors with components, are recognisable over the resulting alphabet.

For example, in base 2, we have and ; the vector is written as .

Semenov's theorem (1977)[10] — Let and be two multiplicatively independent positive integers. A subset of is -recognisable and -recognisable if and only if is describable in Presburger arithmetic.

An elegant proof of this theorem is given by Muchnik in 1991 by induction on .[11]

Other extensions have been given to the real numbers and vectors of real numbers.[8]

Proofs

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Samuel Eilenberg announced the theorem without proof in his book;[12] he says "The proof is correct, long, and hard. It is a challenge to find a more reasonable proof of this fine theorem." Georges Hansel proposed a more simple proof, published in the not-easily accessible proceedings of a conference.[13] The proof of Dominique Perrin[14] and that of Allouche and Shallit's book[15] contains the same error in one of the lemmas, mentioned in the list of errata of the book.[16] This error was uncovered in a note by Tomi Kärki,[17] and corrected by Michel Rigo and Laurent Waxweiler.[18] This part of the proof has been recently written.[19]

In January 2018, Thijmen J. P. Krebs announced, on Arxiv, a simplified proof of the original theorem, based on Dirichlet's approximation criterion instead of that of Kronecker; the article appeared in 2021.[20] The employed method has been refined and used by Mol, Rampersad, Shallit and Stipulanti.[21]

Notes and references

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  1. ^ a b Cobham, Alan (1969). "On the base-dependence of sets of numbers recognizable by finite automata". Mathematical Systems Theory. 3 (2): 186–192. doi:10.1007/BF01746527. MR 0250789.
  2. ^ Durand, Fabien; Rigo, Michel (2010) [Chapter originally written 2010]. "On Cobham's Theorem" (PDF). In Pin, J.-É. (ed.). Automata: from Mathematics to Applications. European Mathematical Society.
  3. ^ Adamczewski, Boris; Bell, Jason (2010) [Chapter originally written 2010]. "Automata in number theory" (PDF). In Pin, J.-É. (ed.). Automata: from Mathematics to Applications. European Mathematical Society.
  4. ^ Cobham, Alan (1972). "Uniform tag sequences". Mathematical Systems Theory. 6 (1–2): 164–192. doi:10.1007/BF01706087. MR 0457011.
  5. ^ Allouche, Jean-Paul [in French]; Shallit, Jeffrey (2003). Automatic Sequences: theory, applications, generalizations. Cambridge: Cambridge University Press. p. 350. ISBN 0-521-82332-3.
  6. ^ A "1-automatic" sequence is a sequence that is ultimately periodic
  7. ^ Büchi, J. R. (1990). "Weak Second-Order Arithmetic and Finite Automata". The Collected Works of J. Richard Büchi. Z. Math. Logik Grundlagen Math. Vol. 6. p. 87. doi:10.1007/978-1-4613-8928-6_22. ISBN 978-1-4613-8930-9.
  8. ^ a b c Bruyère, Véronique (2010). "Around Cobham's theorem and some of its extensions". Dynamical Aspects of Automata and Semigroup Theories. Satellite Workshop of Highlights of AutoMathA. Retrieved 19 January 2017.
  9. ^ a b Durand, Fabien (2011). "Cobham's theorem for substitutions". Journal of the European Mathematical Society. 13 (6): 1797–1812. arXiv:1010.4009. doi:10.4171/JEMS/294.
  10. ^ Semenov, Alexei Lvovich (1977). "Predicates regular in two number systems are Presburger". Sib. Mat. Zh. (in Russian). 18: 403–418. doi:10.1007/BF00967164. MR 0450050. S2CID 119658350. Zbl 0369.02023.
  11. ^ Muchnik (2003). "The definable criterion for definability in Presburger arithmetic and its applications" (PDF). Theoretical Computer Science. 290 (3): 1433–1444. doi:10.1016/S0304-3975(02)00047-6.
  12. ^ Eilenberg, Samuel (1974). Automata, Languages and Machines, Vol. A. Pure and Applied Mathematics. New York: Academic Press. pp. xvi+451. ISBN 978-0-12-234001-7..
  13. ^ Hansel, Georges (1982). "À propos d'un théorème de Cobham". In Perrin, D. (ed.). Actes de la Fête des mots (in French). Rouen: Greco de programmation, CNRS. pp. 55–59.
  14. ^ Perrin, Dominique (1990). "Finite Automata". In van Leeuwen, Jan (ed.). Handbook of Theoretical Computer Science. Vol. B: Formal Models and Semantics. Elsevier. pp. 1–57. ISBN 978-0444880741.
  15. ^ Allouche, Jean-Paul [in French]; Shallit, Jeffrey (2003). Automatic Sequences: theory, applications, generalizations. Cambridge: Cambridge University Press. ISBN 0-521-82332-3.
  16. ^ Shallit, Jeffrey; Allouche, Jean-Paul (31 March 2020). "Errata for Automatic Sequences: Theory, Applications, Generalizations" (PDF). Retrieved 25 June 2021.
  17. ^ Tomi Kärki (2005). "A Note on the Proof of Cobham's Theorem" (PDF). Rapport Technique n° 713. University of Turku. Retrieved 23 January 2017.
  18. ^ Michel Rigo; Laurent Waxweiler (2006). "A Note on Syndeticity, Recognizable Sets and Cobham's Theorem" (PDF). Bulletin of the EATCS. 88: 169–173. arXiv:0907.0624. MR 2222340. Zbl 1169.68490. Retrieved 23 January 2017.
  19. ^ Paul Fermé, Willy Quach and Yassine Hamoudi (2015). "Le théorème de Cobham" [Cobham's Theorem] (PDF) (in French). Archived from the original (PDF) on 2017-02-02. Retrieved 24 January 2017.
  20. ^ Krebs, Thijmen J. P. (2021). "A More Reasonable Proof of Cobham's Theorem". International Journal of Foundations of Computer Science. 32 (2): 203207. arXiv:1801.06704. doi:10.1142/S0129054121500118. ISSN 0129-0541. S2CID 39850911.
  21. ^ Mol, Lucas; Rampersad, Narad; Shallit, Jeffrey; Stipulanti, Manon (2019). "Cobham's Theorem and Automaticity". International Journal of Foundations of Computer Science. 30 (8): 1363–1379. arXiv:1809.00679. doi:10.1142/S0129054119500308. ISSN 0129-0541. S2CID 52156852.

Bibliography

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