D0L Sequence Research Articles

Equality sets of binary D0L sequences

We study D0L sequences over a binary alphabet. If s=(s(n))n≥0 and t=(t(n))n≥0 are D0L sequences, their equality set E(s,t) is defined by E(s,t)={n≥0|s(n)=t(n)}. We show that if s and t are D0L sequences over a binary alphabet then their equality set E(s,t) is eventually periodic.

D0L Sequences and their Equality Sets

D0L Sequences and their Equality Sets

A new bound for the D0L language equivalence problem

We study the language equivalence problem for smooth and loop-free D0L systems. We show that the number of initial terms in the associated D0L sequences we have to consider to decide language equivalence depends only on the cardinality of the underlying alphabet.

EQUALITY SETS OF MORPHIC WORD SEQUENCES

We study equality sets of mappings. In particular, we study D0L equality sets. If s = (s(n))n≥0 and t = (t(n))n≥0 are D0L sequences, their equality set is defined by E(s,t) = {n ≥ 0 ∣ s(n) = t(n)}. We study various periodicity and decidability questions concerning these sets. We also study HD0L and DT0L equality sets.

The sequence equivalence problem for primitive D0L systems

The D0L sequence equivalence problem consists of deciding, given two morphisms g:X⁎→X⁎, h:X⁎→X⁎ and a word w∈X⁎, whether or not gi(w)=hi(w) for all i⩾0. We show that in case of primitive morphisms, to decide the D0L sequence equivalence problem, it suffices to consider the terms of the sequences with i<7n3nlogn, where n is the cardinality of X. A smaller bound is obtained for primitive looping morphisms.

Marked D0L systems and the [formula omitted]-conjecture

We show that to test the equivalence of two D0L sequences over an n-letter alphabet generated by marked morphisms it suffices to compare the first 2n+1 initial terms of the sequences. Under an additional condition it is enough to consider the 2n initial terms.

The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings

We review the concept of the number wall as an alternative to the traditional linear complexity profile (LCP), and sketch the relationship to other topics such as linear feedback shift-register (LFSR) and context-free Lindenmayer (D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is introduced having deficiency 2 modulo 3, and this property verified via the re-interpretation of the number wall as an aperiodic plane tiling.

COMPARING SUBWORD OCCURRENCES IN BINARY D0L SEQUENCES

We investigate binary words and ω-words, mainly those generated by D0L systems. We compare the numbers of occurrences of ab and ba as (scattered) subwords by introducing a 'difference function'. Of special interest are the 'balanced' or 'fair' words for which the function assumes the value 0. If the three first words in a D0L sequence are fair, so are all words in the sequence. Two kinds of matrices constitute a useful technical tool in the study. An explicit formula, leading to various consequences, is obtained for the difference function.

A new bound for the D0L sequence equivalence problem

The D0L sequence equivalence problem consists of deciding, given two morphisms $$g:X^{\ast}\longrightarrow X^{\ast}$$, $$h:X^{\ast}\longrightarrow X^{\ast}$$ and a word $$w\in X^{\ast}$$, whether or not g i (w) = hi(w) for all i ≥ 0. We show that in case of smooth and loop-free morphisms, to decide the D0L sequence equivalence problem, it suffices to consider the terms of the sequences with $$i < 7n^3\sqrt{n\log n}$$, where n is the cardinality of X.

Automatic graphs and D0L-sequences of finite graphs

The notion of end is a classical mean to understand the behavior of a graph at infinity. In this respect, we show that the problem of deciding whether an infinite automatic graph has more than one end is recursively undecidable. The proof involves the analysis of some global topological properties of the configuration graph of a self-stabilizing Turing machine. This result is applied to the graph transformation system theory: It allows us to set a picture of decidability issues concerning logic of D0 L-languages of finite graphs. We first show that the first-order theory of a D0 L-sequence of finite graphs is decidable. Secondly, we show that the first-order theory, when it is upgraded with a closure operator, becomes undecidable for D0 L-sequences of finite graphs.

Easy cases of the D0L sequence equivalence problem

To test the equivalence of two binary D0L sequences it suffices to compare the first four terms of the sequences. We introduce a larger class of D0L systems for which sequence equivalence can be decided by considering the first ten initial terms.

A Polynomial Bound for Certain Cases of the D0L Sequence Equivalence Problem

All known bounds for the D0L sequence equivalence problem in the general case are huge. We give a bound for polynomially bounded PD0L systems which is less than quartic with respect to the cardinality of the alphabet and the size of the systems.

A Polynomial Bound for Certain Cases of the D0L Sequence Equivalence Problem

A Polynomial Bound for Certain Cases of the D0L Sequence Equivalence Problem

On a bound for the D0L sequence equivalence problem

An explicit bound is given for a solution of the DO L sequence equivalence problem; that is, given two DO L systems G 1 and G 2 we compute explicitly positive integer constant G( G 1, G 2) such that the sequences generated by G 1 and G 2 are equal if and only if the corresponding first C( G 1, G 2) elements of the sequences are equal.

Elementary homomorphisms and a solution of the D0L sequence equivalence problem

This paper continues the research on elementary D0L systems. In particular we provide an alternative (and simpler than the one presented in [1]) proof that the D0L (sequence) equivalence problem is decidable.

D0L sequences

This paper investigates a family of sequences of strings (called DOL sequences) which originated from a study of some mathematical models for the development of biological organisms. Under investigation are: the role of erasing rules in systems generating DOL sequences, periodicities which may be observed in DOL sequences and some basic decision problems for DOL sequences.