11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Cobham's theorem | 1/2 | https://en.wikipedia.org/wiki/Cobham's_theorem | reference | science, encyclopedia | 2026-05-05T14:40:12.466579+00:00 | kb-cron |
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 and has since given rise to many extensions and generalisations.
== Definitions == Let
n
>
0
{\displaystyle n>0}
be an integer. The representation of a natural number
n
{\textstyle n}
in base
b
{\textstyle b}
is the sequence of digits
n
0
n
1
⋯
n
h
{\displaystyle n_{0}n_{1}\cdots n_{h}}
such that
n
=
n
0
+
n
1
b
+
⋯
+
n
h
b
h
{\displaystyle n=n_{0}+n_{1}b+\cdots +n_{h}b^{h}}
where
0
≤
n
0
,
n
1
,
…
,
n
h
<
b
{\displaystyle 0\leq n_{0},n_{1},\ldots ,n_{h}<b}
and
n
h
>
0
{\displaystyle n_{h}>0}
. The word
n
0
n
1
⋯
n
h
{\displaystyle n_{0}n_{1}\cdots n_{h}}
is often denoted
⟨
n
⟩
b
{\displaystyle \langle n\rangle _{b}}
, or more simply,
n
b
{\displaystyle n_{b}}
. A set of natural numbers S is recognisable in base
b
{\textstyle b}
or more simply
b
{\textstyle b}
-recognisable or
b
{\textstyle b}
-automatic if the set
{
n
b
∣
n
∈
S
}
{\displaystyle \{n_{b}\mid n\in S\}}
of the representations of its elements in base
b
{\displaystyle b}
is a language recognisable by a finite automaton on the alphabet
{
0
,
1
,
…
,
b
−
1
}
{\displaystyle \{0,1,\ldots ,b-1\}}
. Two positive integers
k
{\displaystyle k}
and
ℓ
{\displaystyle \ell }
are multiplicatively independent if there are no non-negative integers
p
{\displaystyle p}
and
q
{\displaystyle q}
such that
k
p
=
ℓ
q
{\displaystyle k^{p}=\ell ^{q}}
. For example, 2 and 3 are multiplicatively independent, but 8 and 16 are not since
8
4
=
16
3
{\displaystyle 8^{4}=16^{3}}
. 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
u
{\displaystyle u}
and all integers
0
≤
i
<
k
{\displaystyle 0\leq i<k}
, the set
S
i
{\displaystyle S_{i}}
of natural numbers
s
{\displaystyle s}
such that
u
s
=
i
{\displaystyle u_{s}=i}
is recognisable in base
k
{\displaystyle k}
.
=== 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
⟨
N
,
+
,
V
r
⟩
{\displaystyle \langle N,+,V_{r}\rangle }
of natural integers equipped with addition and the function
V
r
{\displaystyle V_{r}}
defined by
V
r
(
0
)
=
1
{\displaystyle V_{r}(0)=1}
and for any positive integer
n
{\textstyle n}
,
V
r
(
n
)
=
r
m
{\displaystyle V_{r}(n)=r^{m}}
if
r
m
{\displaystyle r^{m}}
is the largest power of
r
{\displaystyle r}
that divides
n
{\textstyle n}
. For example,
V
2
(
20
)
=
4
{\displaystyle V_{2}(20)=4}
, and
V
3
(
20
)
=
1
{\displaystyle V_{3}(20)=1}
. A set of integers
S
{\displaystyle S}
is definable in first-order logic in
⟨
N
,
+
,
V
r
⟩
{\displaystyle \langle N,+,V_{r}\rangle }
if it can be described by a first-order formula with equality, addition, and
V
r
{\displaystyle V_{r}}
. Examples:
The set of odd numbers is definable (without
V
r
{\displaystyle V_{r}}
) by the formula
(
∃
y
)
(
x
=
y
+
y
+
1
)
{\displaystyle (\exists y)(x=y+y+1)}
The set
{
2
n
∣
n
≥
0
}
{\displaystyle \{2^{n}\mid n\geq 0\}}
of the powers of 2 is definable by the simple formula
V
2
(
x
)
=
x
{\displaystyle V_{2}(x)=x}
. 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
⟨
N
,
+
,
V
k
⟩
{\displaystyle \langle N,+,V_{k}\rangle }
and
⟨
N
,
+
,
V
ℓ
⟩
{\displaystyle \langle N,+,V_{\ell }\rangle }
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
B
=
{
a
,
0
,
1
}
{\displaystyle B=\{a,0,1\}}
defined by
a
↦
a
1
,
1
↦
10
,
0
↦
00
{\displaystyle a\mapsto a1\ ,\quad 1\mapsto 10\ ,\quad 0\mapsto 00}
which generates the infinite word