Search results for " Formal languages"
showing 10 items of 79 documents
A Classification of Trapezoidal Words
2011
Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us to characterize the special factors of Sturmian palindromes. We end with several open problems.
Abelian-Square-Rich Words
2017
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number of distinct abelian-square factors that is quadratic in $n$; and {\it uniformly abelian-sq…
On the least number of palindromes contained in an infinite word
2013
We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then investigate the same problem when the alphabet has size two.
Generating a Gray code for prefix normal words in amortized polylogarithmic time per word
2020
A prefix normal word is a binary word with the property that no substring has more $1$s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length $n$ as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in $\Oh(\log^2 n)$ amortized time per word, while the best generation algorithm hitherto has $\Oh(n)$ running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages.
On the Lie complexity of Sturmian words
2022
Bell and Shallit recently introduced the Lie complexity of an infinite word $s$ as the function counting for each length the number of conjugacy classes of words whose elements are all factors of $s$. They proved, using algebraic techniques, that the Lie complexity is bounded above by the first difference of the factor complexity plus one; hence, it is uniformly bounded for words with linear factor complexity, and, in particular, it is at most 2 for Sturmian words, which are precisely the words with factor complexity $n+1$ for every $n$. In this note, we provide an elementary combinatorial proof of the result of Bell and Shallit and give an exact formula for the Lie complexity of any Sturmi…
On the Structure of Bispecial Sturmian Words
2013
A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors of \emph{primitive} Christoffel words. We extend this result by showing that bispecial Sturmian wo…
Classical automata on promise problems
2015
Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Secon…
On prefix normal words and prefix normal forms
2016
A $1$-prefix normal word is a binary word with the property that no factor has more $1$s than the prefix of the same length; a $0$-prefix normal word is defined analogously. These words arise in the context of indexed binary jumbled pattern matching, where the aim is to decide whether a word has a factor with a given number of $1$s and $0$s (a given Parikh vector). Each binary word has an associated set of Parikh vectors of the factors of the word. Using prefix normal words, we provide a characterization of the equivalence class of binary words having the same set of Parikh vectors of their factors. We prove that the language of prefix normal words is not context-free and is strictly contai…
Primitive sets of words
2020
Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. We say that a submonoid $M$ generated by $k$ elements of $A^*$ is {\em $k$-maximal} if there does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X \subseteq A^*$ {\em primitive} if it is the basis of a $|X|$-maximal submonoid. This definition encompasses the notion of primitive word -- in fact, $\{w\}$ is a primitive set if and only if $w$ is a primitive word. By definition, for any set $X$, there exists a primitive set $Y$ such that $X \subseteq Y^*$. We therefore call $Y$…
Quantum Pushdown Automata
2001
Quantum finite automata, as well as quantum pushdown automata (QPA) were first introduced by C. Moore and J. P. Crutchfield. In this paper we introduce the notion of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of Kondacs and Watrous. It is established that the unitarity criteria of QPA are not equivalent to the corresponding unitarity criteria of quantum Turing machines. We show that QPA can recognize every regular language. Finally we present some simple languages recognized by QPA, not recognizable by deterministic pushdown automata.