Morphic Sequences Research Articles

Morphic Sequences: Complexity and Decidability

Morphic Sequences: Complexity and Decidability

A conjecture of Dekking on the dimensions of the lower central series factors of a certain just infinite Lie algebra

Revisiting just infinite nil graded fractal Lie superalgebras studied by de Morais Costa and Petrogradsky, and more precisely a Lie algebra R in characteristic 2, we give a simple combinatorial construction of the sequence of dimensions of the lower central series factors of R in terms of automatic and morphic sequences. In particular we prove a conjecture of Dekking about this sequence of dimensions.

Two-block substitutions and morphic words

We consider in general two-block substitutions and their fixed points. We prove that some of them have a simple structure: their fixed points are morphic sequences. Others are intrinsically more complex, such as the Kolakoski sequence. We prove this for the Thue-Morse sequence in base 3/2.

How to prove that a sequence is not automatic

Automatic sequences have many properties that other sequences (in particular, non-uniformly morphic sequences) do not necessarily share. In this paper we survey a number of different methods that can be used to prove that a given sequence is not automatic. When the sequences take their values in a finite field Fq, this also permits proving that the associated formal power series are transcendental over Fq(X).

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue–Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue–Morse sequence.

Complexity of automatic sequences

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, such as 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 initial element, combining sequences, or taking arithmetic subsequences, and observe that these operations may increase the complexity at most polynomially. For periodic sequences we give sharp bounds for both complexity measures.

The sum of digits functions of the Zeckendorf and the base phi expansions

We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite words are letter-to-letter projections of fixed points of these morphisms. We characterize the first differences of both functions a) with generalized Beatty sequences, or unions of generalized Beatty sequences, and b) with morphic sequences.

Decidability, Arithmetic Subsequences and Eigenvalues of Morphic Subshifts

We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding whether, given $p\in \mathbb{N}$, there exists such a constant subsequence along an arithmetic progression of common difference $p$. In the special case of uniformly recurrent automatic sequences we explicitly describe the sets of such $p$ by means of automata.

On Factor Complexity of Morphic Sequences

On Factor Complexity of Morphic Sequences

On almost periodicity of morphic sequences

It is proved that the problem of determining whether a given morphic sequence is almost periodic is decidable.

TURTLE GRAPHICS OF MORPHIC SEQUENCES

The simplest infinite sequences that are not ultimately periodic are pure morphic sequences: fixed points of particular morphisms mapping single symbols to strings of symbols. A basic way to visualize a sequence is by a turtle curve: for every alphabet symbol fix an angle, and then consecutively for all sequence elements draw a unit segment and turn the drawing direction by the corresponding angle. This paper investigates turtle curves of pure morphic sequences. In particular, criteria are given for turtle curves being finite (consisting of finitely many segments), and for being fractal or self-similar: it contains an up-scaled copy of itself. Also space-filling turtle curves are considered, and a turtle curve that is dense in the plane. As a particular result we give an exact relationship between the Koch curve and a turtle curve for the Thue–Morse sequence, where until now for such a result only approximations were known.

A Counterexample to a Question of Hof, Knill and Simon

In this article, we give a negative answer to a question of Hof, Knill and Simon (1995) concerning purely morphic sequences obtained from primitive morphism containing an infinite number of palindromes. Their conjecture states that such palindromic sequences arise from substitutions that are in class $\mathcal{P}$. The conjecture was proven for the binary alphabet by B. Tan in 2007. We give here a counterexample for a ternary alphabet.

DECIDABILITY OF UNIFORM RECURRENCE OF MORPHIC SEQUENCES

We prove that the uniform recurrence of morphic sequences is decidable. For this we show that the number of derived sequences of uniformly recurrent morphic sequences is bounded. As a corollary we obtain that uniformly recurrent morphic sequences are primitive substitutive sequences.

ON UNIFORMLY RECURRENT MORPHIC SEQUENCES

A pure morphic sequence is a right-infinite, symbolic sequence obtained by iterating a letter-to-word substitution. For instance, the Fibonacci sequence and the Thue–Morse sequence, which play an important role in theoretical computer science, are pure morphic. Define a coding as a letter-to-letter substitution. The image of a pure morphic sequence under a coding is called a morphic sequence.A sequence x is called uniformly recurrent if for each finite subword u of x there exists an integer l such that u occurs in every l-length subword of x.The paper mainly focuses on the problem of deciding whether a given morphic sequence is uniformly recurrent. Although the status of the problem remains open, we show some evidence for its decidability: in particular, we prove that it can be solved in polynomial time on pure morphic sequences and on automatic sequences.In addition, we prove that the complexity of every uniformly recurrent, morphic sequence has at most linear growth: here, complexity is understood as the function that maps each positive integer n to the number of distinct n-length subwords occurring in the sequence.

Self-generating sets, integers with missing blocks, and substitutions

We give a new construction of the Kimberling sequence defined by: (a) 1 belongs to S; (b) if the positive integer x belongs to S, then 2 x and 4 x - 1 belong to S; and (c) nothing else belongs to S, hence, S = 1 2 3 4 6 7 8 11 12 14 15 16 … which is sequence A052499 in the Sloane's On-line Encyclopedia of Integer Sequences, by proving that this sequence is equal to sequence 1 + A003754, the sequence of integers whose binary expansion does not contain the block of digits 00. We give a general framework for this sequence and similar sequences, in relation to automatic or morphic sequences and to non-standard numeration systems such as the lazy Fibonacci expansion.