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 is a finite union of arithmetic progressions. The theorem was proved by Alan Cobham in 1969 and has since given rise to many extensions and generalisations.
Definitions
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
Original problem statement
More equivalent statements of the theorem have been given. The original version by Cobham is the following: Another way to state the theorem is by using automatic sequences. Cobham himself calls them "uniform tag sequences.". The following form is found in Allouche and Shallit's book: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
Cobham's theorem can be formulated in first-order logic using a theorem proven by Büchi in 1960. This formulation in logic allows for extensions and generalisations. The logical expression uses the theory
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 . 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
Approach by morphisms
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: 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: