The abelian complexity of the paperfolding word
We show that the abelian complexity function of the ordinary paperfolding word is a 2-regular sequence.
math.CO↗
arXiv subjects
Narad Rampersad
Publications and source records attributed to Narad Rampersad.
Enumeration and Decidable Properties of Automatic Sequences
We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems.
cs.FL↗
Fife's Theorem for (7/3)-Powers
We prove a Fife-like characterization of the infinite binary (7/3)-power-free words, by giving a finite automaton of 15 states that encodes all such words. As a consequence, we characterize all such words that are 2-automatic.
cs.FL↗
Fixed points avoiding Abelian $k$-powers
We show that the problem of whether the fixed point of a morphism avoids Abelian $k$-powers is decidable under rather general conditions
cs.FL↗
The growth function of S-recognizable sets
A set $X\subseteq\mathbb N$ is S-recognizable for an abstract numeration system S if the set $\rep_S(X)$ of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set is always either $Θ((\log(n))^{c-df}n^f)$ where $c,d\in\mathbb N$ and $f\ge 1$, or $Θ(n^r θ^{Θ(n^q)})$, where $r,q\in\mathbb Q$ with $q\le 1$. If the number of words of length n in the numeration language is bounded by a polynomial, then the growth function of an S-recognizable set is $Θ(n^r)$, where $r\in \mathbb Q$ with $r\ge 1$. Furthermore, for every $r\in \mathbb Q$ with $r\ge 1$, we can provide an abstract numeration system S built on a polynomial language and an S-recognizable set such that the growth function of X is $Θ(n^r)$. For all positive integers k and l, we can also provide an abstract numeration system S built on a exponential language and an S-recognizable set such that the growth function of X is $Θ((\log(n))^k n^l)$.
cs.FL↗
Multi-dimensional sets recognizable in all abstract numeration systems
We prove that the subsets of N^d that are S-recognizable for all abstract numeration systems S are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional setting.
cs.FL↗
Inverse Star, Borders, and Palstars
A language L is closed if L = L*. We consider an operation on closed languages, L-*, that is an inverse to Kleene closure. It is known that if L is closed and regular, then L-* is also regular. We show that the analogous result fails to hold for the context-free languages. Along the way we find a new relationship between the unbordered words and the prime palstars of Knuth, Morris, and Pratt. We use this relationship to enumerate the prime palstars, and we prove that neither the language of all unbordered words nor the language of all prime palstars is context-free.
cs.FL↗
State Complexity of Testing Divisibility
Under some mild assumptions, we study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m >= 2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of the multiples of m in the Fibonacci system is exactly 2m^2.
cs.FL↗
Abelian Primitive Words
We investigate Abelian primitive words, which are words that are not Abelian powers. We show that unlike classical primitive words, the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time. Also different from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration problems and the relation to the theory of codes.
cs.FL↗
Recurrent words with constant Abelian complexity
We prove the non-existence of recurrent words with constant Abelian complexity containing 4 or more distinct letters. This answers a question of Richomme et al.
math.CO↗
Automata and Reduced Words in the Free Group
We consider some questions about formal languages that arise when inverses of letters, words and languages are defined. The reduced representation of a language over the free monoid is its unique equivalent representation in the free group. We show that the class of regular languages is closed under taking the reduced representation, while the class of context-free languages is not. We also give an upper bound on the state complexity of the reduced representation of a regular language, and prove upper and lower bounds on the length of the shortest reducible string in a regular language. Finally we show that the set of all words which are equivalent to the words in a regular language can be nonregular, and that regular languages are not closed under taking a generalized form of the reduced representation.
cs.FL↗
Further applications of a power series method for pattern avoidance
In combinatorics on words, a word w over an alphabet Sigma is said to avoid a pattern p over an alphabet Delta if there is no factor x of w and no non-erasing morphism h from Delta^* to Sigma^* such that h(p) = x. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a certain wide class of patterns p there are exponentially many words of length n over a 4-letter alphabet that avoid p. We consider some further consequences of their work. In particular, we show that any pattern with k variables of length at least 4^k is avoidable on the binary alphabet. This improves an earlier bound due to Cassaigne and Roth.
math.CO↗
Dejean's conjecture holds for n>=27
We show that Dejean's conjecture holds for n>=27. This brings the final resolution of the conjecture by the approach of Moulin Ollagnier within range of the computationally feasible.
math.CO↗
On NFAs Where All States are Final, Initial, or Both
We examine questions involving nondeterministic finite automata where all states are final, initial, or both initial and final. First, we prove hardness results for the nonuniversality and inequivalence problems for these NFAs. Next, we characterize the languages accepted. Finally, we discuss some state complexity problems involving such automata.
cs.CC↗
Detecting patterns in finite regular and context-free languages
We consider variations on the following problem: given an NFA M and a pattern p, does there exist an x in L(M) such that p matches x? We consider the restricted problem where M only accepts a finite language. We also consider the variation where the pattern p is required only to match a factor of x. We show that both of these problems are NP-complete. We also consider the same problems for context-free grammars; in this case the problems become PSPACE-complete.
cs.FL↗
A proof of Dejean's conjecture
We prove Dejean's conjecture. Specifically, we show that Dejean's conjecture holds for the last remaining open values of n, namely 15 <= n <= 26.
math.CO↗
Periodicity, repetitions, and orbits of an automatic sequence
We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squareefree, or cubefree, etc.) We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope alpha, have automatic continued fraction expansions if alpha does.
cs.DM↗
Dejean's conjecture holds for n >= 30
We extend Carpi's results by showing that Dejean's conjecture holds for n >= 30.
math.CO↗