Abstract
Automatic sequences can be defined by DFAs with output (DFAO) in two natural ways. We propose to consider the minimal size of a corresponding DFAO as the complexity measure of the automatic sequence, for both variants. This paper compares these complexity measures and investigates their properties like the relationships with kernel and morphic sequences. There exist automatic sequences for which the one complexity is exponentially greater than the other one, in both directions. For both complexity measures we investigate the effect of taking basic operations on sequences like removing or adding an element in front, and observe that these operations may increase the complexity by at most a quadratic factor.
Introduction
Automatic sequences form an important class of infinite sequences over a finite alphabet; roughly speaking it is a first regular class going beyond ultimately periodic sequences. They have been extensively studied, in particular in the book [1] that serves as the main reference for research in this area. More recent references on the topic include [5, 9].
Automatic sequences depend on a base 
, with special interest for 
. Two well-known 2-automatic sequences are the Thue-Morse sequence and the regular paper folding sequence, to be defined in Sect. 2. Automatic sequences admit several equivalent characterizations, many of which are closely related to the following two. In the first one the ith element 
of the sequence a is the output of a DFAO when taking as input the k-ary notation of i. The second one is similar, but then the reverse of the k-ary notation of i is taken as input. It is natural to consider the minimal size of a corresponding DFAO as the complexity measure of the automatic sequence, for both variants, and we denote them by 
and 
. These complexity measures are the main topic of this paper. We show how they relate to other characterizations; in particular, 
is closely related to the size of the kernel of a, and 
is closely related to the size of the smallest alphabet needed to describe a as a morphic sequence with respect to a k-uniform morphism. In doing so, we follow constructions as presented in [1] for which we investigate the precise effect on the measures 
and 
.
A first result states that there is an exponential gap between both measures: there exist sequences of automatic sequences a, b for which 
is exponential in 
, and 
is exponential in 
.
A next natural question is about the effect of taking basic operations on sequences. For instance, for any sequence a its tail 
is obtained by removing its first element. We show that 
and 
for all k-automatic sequences, and that the last inequality is sharp. Similar results hold for adding an element in front rather than removing. Also other operations are considered, like pointwise combining two sequences and taking particular subsequences. About all of these basic operations f the main observation is that their sizes do not increase more than quadratically: 
and 
for all a.
Another interesting question is what happens for periodic sequences. In the current paper we only derive a quadratic upper bound for 
and a linear upper bound for 
, so opposite to the effect of 
. Whether and when these upper bounds are reached is a much more involved question that is investigated in [2]. The research project on this topic is a joined project of Wieb Bosma and the current author; as this analysis for periodic sequences requires arguments of a completely different combinatorial flavor than the automata based arguments in this paper, we decided to present the current paper and [2] separately.
Throughout the paper we make several claims about the exact values of 
and 
for particular sequences a. To compute these values we wrote a program to search for a DFAO of minimal size n having the corresponding property for 
for all 
for N being typically around 
. This was done by expressing the requirements as a satisfiability problem and then call a SAT solver. The smallest n for which the formula is satisfiable then is given. As only the requirements for 
are checked, this only yields a lower bound, but for N large enough it gives the exact value. According to [6], corollary 3.1 (page 59) two states in a DFAO of n states are equivalent are equivalent if and only if for every string of length 
they produce the same output. This can be improved to 
. Applying this for the union of the found automaton and the real automaton with bounds derived in this paper, this shows that for 
the exact value is obtained.
This paper is organized as follows. In Sect. 2 we give the basic definitions and a general lemma for proving lower bounds. In Sect. 3 we investigate the exponential gap between 
and 
. In Sect. 4 we define the kernel of an automatic sequence and investigate its relationship with 
. In Sect. 5 we present how to define automatic sequences as morphic sequences with respect to uniform morphisms, and investigate the relationship with 
. In Sect. 6 we investigate the effect of basic operations like 
on 
and 
. In Sect. 7 we give the upper bounds of 
and 
for periodic sequences. We conclude in Sect. 8.
Basic Definitions
Let 
and 
.
The set of infinite sequences 
over a finite alphabet 
is denoted by 
.
A DFA M with output (DFAO) is defined to be a tuple 
, where
Q is the finite set of states,
is the finite input alphabet,
is the transition function,
is the initial state,
is the finite output alphabet,
is the output function.
DFAOs are denoted by states and arrows just as is usual for DFAs; the extra information that 
is denoted by writing q/x in the state q.
As in DFAs, 
extends to 
by 
, 
. A DFAO M defines a function 
defined by 
. A function 
is called a finite state function if a DFAO M exists such that 
. For every finite state function f there exists a unique (up to renaming of states) DFAO M with a minimal number of states such that 
.
A DFAO of which the input alphabet 
is equal to 
, is called a k-DFAO.
Every natural number n has a unique representation 
, where 
and
 |
for 
. So 
and 
. Note that non-empty strings of which the leftmost symbol is 0 do not occur as 
for some number n.
Conversely, every 
represents a number 
:
 |
For any 
and any string 
the reverse 
of u is defined by 
.
An infinite sequence 
is called k-automatic if a k-DFAO 
exists such that 
for all 
. According to Theorem 5.2.1 from [1] a is k-automatic if and only if a k-DFAO 
exists such that 
for all 
. According to Theorem 5.2.3 from [1] a is k-automatic if and only if a k-DFAO 
exists such that 
for all 
.
Now we are ready to define the two natural measures 
, 
for k-automatic sequences that we investigate in this paper.
Definition 1
For any k-automatic sequence 
its size 
is defined to be the size of a smallest k-DFAO 
such that 
for all 
.
For any k-automatic sequence 
its reversed size 
is defined to be the size of a smallest k-DFAO 
such that 
for all 
.
Conversely, every k-DFAO 
defines two infinite sequences 
and 
over 
:
 |
for all 
. From the above definition it is immediate that 
and 
.
The Thue-Morse sequence
is defined by 
if the number of 1s in 
is even, and 
if the number of 1s in 
is odd, see, e.g., [1] Section 1.6, or OEIS A010060. We have 
, both justified by the DFAO on the right. 
The regular paper-folding sequence
(or dragon curve sequence is defined by 
for every 
for the unique representation 
, see, e.g., [1] Example 5.16., or OEIS A014577. We have 
, respectively justified by the following two DFAOs.
The following lemma is the basic tool for lower bounds on 
and 
.
Lemma 1
Let a be a k-automatic sequence, and 
such that for every 
there exists 
satisfying 
, then 
.
Let a be a k-automatic sequence, and 
such that for every 
there exists 
satisfying 
, then 
.
Proof
For the first claim let 
be a smallest k-DFAO such that 
for all 
. For 
define 
. For 
from the assumption we obtain 
, so 
. This shows 
, so 
.
The proof of the second claim is similar. 
The Exponential Gap
The following theorem shows that there can be an exponential gap between 
and 
, in both directions. Its proof is inspired by the folklore result that the language 
has an NFA of size 
, and its reverse has a DFA of size 
, but its smallest DFA has size at least 
. We found it in [8], Sect. 3.2, page 67, exercise 3. Many similar results on state complexity are known, e.g., in [7], it is proved that all values until 
can be reached as sizes.
Theorem 1
For every 
there exist k-automatic sequences a, b such that 
and 
, and 
and 
.
Proof
Define a by 
for 
, and 
if and only if the nth digit of 
is j, for 
, 
. The following DFAO satisfies 
by construction:
in which all unlabeled arrows are assumed to be labeled by all symbols 
. Since this DFAO has 
states we obtain 
.
For proving 
we apply Lemma 1. For 
define 
, so the numbers 
are exactly the numbers of k-ary length n, starting in a digit 
. For any two distinct such numbers 
and 
there is a position p on which they differ, so by choosing 
, the strings 
and 
differ in their n-th position. So the condition of Lemma 1 holds and we conclude 
.
Define b by 
for 
, and 
if and only if the nth element of 
is j, for 
, 
. A similar argument using the same automaton proves the claim for b. 
The k-kernel
For 
we define 
by 
for all 
. So for 
we have 
and 
.
For an infinite sequence 
over 
we define its k-kernel
to be the smallest set 
such that
,for every
and every 
we have 
.
We recall from [4], Prop. V.3.3, or [1], Theorem 6.6.2, that a is k-automatic if and only if 
is finite.
For a k-automatic sequence 
over the alphabet 
its k-kernel
has a natural DFAO structure: the DFAO 
, where
the input alphabet is
,
is the set of states,
is defined by 
,a is the initial state,
the output alphabet is
,the output function
is defined by 
.
Recall that for 
we have 
and 
, so in 
the 0-steps describe 
and the 1-steps describe 
. For 
the 2-kernel exactly coincides with the DFAO 
given in Sect. 2, in which 
coincides with 
and 
coincides with the sequence obtained from 
by swapping symbols 0 and 1. For 
the 2-kernel exactly coincides with the given DFAO 
, in which 
coincides with 
, 
with 
, 
with 
and 
with 
.
The following theorem is straightforwardly proved by induction on i:
Theorem 2
For every k-automatic sequence 
and every 
we have 
where 
refer to 
.
As a consequence, by only giving the DFAO 
the sequence a is fully defined.
Theorem 3
The DFAO 
is the unique DFAO of minimal size such that 
for every 
.
Proof
Let 
. Combining Theorem 2 with the fact that 
for all 
yields 
for every 
. Assume it is not of minimal size with this property. Then there are two distinct states 
such that 
for all 
. Since 
are sequences over 
, applying Theorem 2 to 
and 
yield 
for all 
. But then 
are equal as sequences, contradicting that they are distinct. 
Recall that 
is the minimal size |Q| for which a DFAO 
exists such that 
for every 
. We observe that a DFAO with this property does not need to be unique. For instance, for 
the DFAO 
is a minimal DFAO with this property, having two states a and 
, and 
, 
, 
. But the DFAO with the same two states a, b and 
, 
, 
produces the same sequence 
.
Next we observe that 
can be strictly smaller than 
, the size of the state space of 
. Define 
if the number of zeros in 
is odd, and 
if this number is even. Clearly it admits the following DFAO, in which as usual 
is denoted by q/x in the state q: 
Hence 
; we obtain 
since the sequence contains both 0 and 1. However, 
, since 
is the following DFAO:
The sequences a, b, c, d are as follows:
 |
 |
Observe that a and d differ only at the first position, and similarly for b and c. The next lemma states that this always occurs if 
is greater then 
.
Lemma 2
Let a be an infinite sequence over 
with kernel 
. Let 
such that 
for all 
. Assume that 
for 
. Then
 |
Proof
Let 
. For any 
define the numbers 
by 
; this is possible since 
does not start in 0 since 
. For any 
we obtain 
by considering 
. Hence
 |
 |
Theorem 4
Let a be a k-automatic sequence over an alphabet 
. Then
 |
Moreover, if a is periodic then 
.
Proof
The inequality 
holds since the automaton 
satisfies 
for every 
. For the other inequality let 
be a DFAO of minimal size 
such that 
for every 
. For every 
choose 
such that 
. Define 
on 
by 
.
According to Lemma 2
implies that 
for all 
, so the difference between b and c may only be caused by 
. Hence every equivalence class of 
has at most 
elements, while the number of equivalence classes is 
. This proves 
.
In case a is periodic then all elements of 
are periodic too, and 
for all 
implies 
. Hence in that case all equivalence classes consist of a single element, proving 
. 
Morphic Sequences
Recall that 
for the smallest 
being the set of states of a DFAO 
for which 
for every 
. Again this DFAO of minimal size is not unique: for 
the DFAO 
as given above also satisfies 
for all 
, but after changing 
to 
this property still holds, since 
never starts by 0.
Just like 
is strongly related to the kernel of a as described in Theorem 4, 
is strongly related to the number of symbols needed to describe a as a morphic sequence with respect to a k-uniform morphism. A sequence a over an alphabet 
is called morphic with respect to a morphism 
and a coding 
if 
for some 
satisfying 
, by which 
is a fixed point of h. The morphism 
is called k-uniform if the string 
has length k for every 
. It is well-known (Cobham [3], see also [1] Theorem 6.3.2) that a is k-automatic if and only if it is morphic with respect to a k-uniform morphism. For instance, 
for 
, and 
for 
, 
, 
.
Theorem 5
Let a be a k-automatic sequence. Let d(a) be the minimal size of the alphabet 
such that 
for a k-uniform morphism 
and a coding 
. Then 
.
Proof
The k-DFAO 
with 
and 
, where we write 
, satisfies 
for all 
as is showed in the proof of Theorem 6.3.2 of [1]. As 
is the smallest size of a k-DFAO with this property we obtain 
.
Conversely, if 
is a k-DFAO of size 
with 
for all 
, then by choosing a fresh state 
and defining 
, 
for 
, 
, 
for 
, 
, 
for 
, we obtain the k-DFAO 
of size 
with 
for all 
. Using the fact that 
we obtain 
for h defined by 
as is shown in the proof of Theorem 6.3.2 of [1]. Hence 
. 
The Effect of Basic Operations
For any sequence 
we define its tail 
by 
for all 
.
Theorem 6
For any k-automatic sequence a we have 
and 
. For every 
there exists a k-automatic sequence a such that 
and 
.
Proof
For the first claim take a DFAO 
of size 
with 
for all 
. Let 
be the smallest number 
such that 
exists with 
. Introduce fresh states 
and define the DFAO 
by
 |
 |
 |
 |
By construction we have 
for all 
, 
. So by defining 
for 
and 
for 
we obtain
 |
and
 |
for all 
, 
. Since 
, and 
, and every number in 
is either of the shape 
or 
, this proves that 
is a DFAO for 
. Since 
this yields 
.
For the second claim take a DFAO 
of size 
with 
for all 
. Define the DFAO 
of size 
by
 |
 |
 |
for all 
, 
. For every 
we have either 
or 
, for some 
, 
, 
. In the first case we have 
, in the second case 
. The DFAO 
has been constructed in such a way that 
and 
. Hence for all 
we have 
, proving the second claim.
As 
, for the last claim it suffices to prove 
. We define a by 
if the number of zeros in 
is divisible by n, and 
otherwise. A DFAO consisting of a single n-cycle easily produces a, so 
, and since a smaller one is not possible we obtain 
. Let 
, so 
for all 
. We prove 
by Lemma 1. Choose 
to be the numbers 
for 
. Let 
and 
for 
, then 
.
First we consider the case where 
and 
are distinct modulo n, choose r such that 
is divisible by n and 
is not. Choose 
. Then 
.
In the remaining case 
and 
are equal modulo n, and since 
we obtain that p and 
are distinct modulo n. Choose r such that 
is divisible by n and 
is not. Choose 
, then 
.
So the conditions of Lemma 1 hold, and 
. 
For our examples 
and 
we have 
, 
, 
and 
.
For any sequence 
over 
, and 
the sequence 
is defined by 
and 
for all 
. The next theorem states that the effect of 
is similar to 
.
Theorem 7
For any k-automatic sequence a over 
, and 
we have 
and 
. For every 
there exists a k-automatic sequence a such that 
and 
.
Proof
Similar to the proof of Theorem 6, with the roles of the symbols 0 and 
swapped, exploiting the property 
for any string v and any 
. 
For our examples 
and 
we have 
, 
, 
and 
.
Recall that for 
the operator 
on sequences a is defined by 
for all 
.
Theorem 8
For any k-automatic sequence a and 
we have 
and 
.
Proof
Let 
be a DFAO of size 
with 
for all 
. Define 
by 
for all 
. Then
 |
for all 
, so 
is a DFAO of size 
producing 
, so 
.
For the other claim let 
be a DFAO of size 
with 
for all 
. Define 
. Then
 |
for all 
, so 
is a DFAO of size 
producing 
, so 
. 
For our examples 
and 
we have 
, 
, 
and 
.
When applying an operator 
on two sequences 
, 
, by 
we mean the sequence defined by 
for all 
. For instance, 
applied on boolean sequences denotes the elementwise conjunction of the two boolean sequences.
Theorem 9
For any two k-automatic sequences 
, 
and every function 
we have 
and 
.
Proof
Let 
be a DFAO of size 
with 
for all 
. Let 
be a DFAO of size 
with 
for all 
. Then 
for 
defined by 
and 
for all 
, is a DFAO of size 
for f(a, b). The proof for the reversed version is similar. 
Combining our examples 
and 
we have 
and 
.
Periodic Sequences
Theorem 10
Let 
be a periodic sequence with 
. Then 
and 
.
Proof
Writing 
we obtain 
for all 
. Define 
by 
, 
, 
, 
, for all 
. Then by induction on the length of 
one proves that 
for every 
. Hence 
for all 
, proving that 
.
For the other claim we prove that 
, then the result follows from Theorem 4. The states of 
are sequences b for which there are numbers q, j such that 
for all 
. We have to show that there are at most 
such sequences b. This follows from the fact that this only i depends on the n values for 
and the at most 
values for 
. The latter follows since if k, n are relatively prime, then the values of 
are among the 
values 
, and otherwise there is some 
dividing both n and k, and the values are among the n/p multiples of p modulo n. 
A natural question is for which cases the bounds of Theorem 10 can be reached, in particular the quadratic bound for 
. This question is beyond the scope of this paper, but has been addressed in [2]. A main result of [2] is that if 
is prime and 2 is a primitive root modulo n (on which Artin’s conjecture states that this holds for infinitely many primes), then 
for 
.
Conclusions
We investigated two natural complexity measures for a k-automatic sequence a: 
closely related to the alphabet size required to present a as a morphic sequence with respect to a k-uniform morphism, and 
closely related to the size of the kernel of a. We saw how there can be an exponential gap between 
and 
, but basic operations like 
, adding an element in front, or applying a binary operator elementwise, never increases 
or 
by more than a quadratic factor. Many other operations, like changing the tenth element of a sequence, can be obtained by combining such basic operations, and hence yield a polynomial upper bound too. Probably these polynomial bounds can be improved strongly. Other open questions include a further investigation of when these upper bounds can be reached. Conversely, our SAT based tool provides values that are likely to be exact, but formally are only lower bounds. It would make sense to further investigate how to be sure to have the exact value, either depending on particular ways to define automatic sequences, or by giving general criteria for exactness depending on known upper bounds.
On periodic sequences this paper only contains some very basic observations; more involved observations are given in [2].
We want to thank Wieb Bosma for fruitful collaboration on this topic and careful proof reading. We want to thank Jeffrey Shallit for giving pointers to state complexity.
Contributor Information
Alberto Leporati, Email: alberto.leporati@unimib.it.
Carlos Martín-Vide, Email: carlos.martin@urv.cat.
Dana Shapira, Email: shapird@g.ariel.ac.il.
Claudio Zandron, Email: zandron@disco.unimib.it.
Hans Zantema, Email: h.zantema@tue.nl.
References
- 1.Allouche JP, Shallit J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge: Cambridge University Press; 2003. [Google Scholar]
- 2.Bosma, W.: Complexity of periodic sequences (2019). https://www.math.ru.nl/~bosma/pubs/periodic.pdf
- 3.Cobham A. Uniform tag sequences. Math. Systems Theory. 1972;6:164–192. doi: 10.1007/BF01706087. [DOI] [Google Scholar]
- 4.Eilenberg S. Automata, Languages and Machines. New York: Academic Press; 1974. [Google Scholar]
- 5.Endrullis J, Grabmayer C, Hendriks D. Mix-automatic sequences. In: Dediu A-H, Martín-Vide C, Truthe B, editors. Language and Automata Theory and Applications; Heidelberg: Springer; 2013. pp. 262–274. [Google Scholar]
- 6.Gill A. Introduction to the Theory of Finite-State Machines. New York: McGraw-Hill; 1962. [Google Scholar]
- 7.Jiraskova G. The ranges of state complexities for complement, star and reversal of regular languages. Int. J. Found. Comput. Sci. 2014;25(1):101–124. doi: 10.1142/S0129054114500063. [DOI] [Google Scholar]
- 8.Lawson MV. Finite Automata. Boca Raton: Chapman and Hall/CRC; 2004. [Google Scholar]
- 9.Shallit J. Decidability and enumeration for automatic sequences: a survey. In: Bulatov AA, Shur AM, editors. Computer Science – Theory and Applications; Heidelberg: Springer; 2013. pp. 49–63. [Google Scholar]