Related papers: Generation of binary words avoiding alternating pa…
We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either…
When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct…
Recent language models, especially those based on recurrent neural networks (RNNs), make it possible to generate natural language from a learned probability. Language generation has wide applications including machine translation,…
In this article, we provide three generators of propositional formulae for arbitrary languages, which uniformly sample three different formulae spaces. They take the same three parameters as input, namely, a desired depth, a set of atomics…
The success of large language models (LLMs) has motivated formal theories of language generation and learning. We study the framework of \emph{language generation in the limit}, where an adversary enumerates strings from an unknown language…
We study the properties of the ternary infinite word p = 012102101021012101021012 ... , that is, the fixed point of the map h:0->01, 1->21, 2->0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We…
Cross-bifix-free sets are sets of words such that no prefix of any word is a sufix of any other word. In this paper, we introduce a general constructive method for the sets of cross-bifix-free q-ary words of fixed length. It enables us to…
We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the…
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 $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern…
Let $w$ be a finite word over the alphabet $\{0,1\}$. For any natural number $n$, let $s_w(n)$ denote the number of occurrence of $w$ in the binary expansion of $n$ as a scattered subsequence. We study the behavior of the partial sum…
Alphabetic codes and binary search trees are combinatorial structures that abstract search procedures in ordered sets endowed with probability distributions. In this paper, we design new linear-time algorithms to construct alphabetic codes,…
The Binary Two-Up Sequence is the lexicographically earliest sequence of distinct nonnegative integers with the property that the binary expansion of the n-th term has no 1-bits in common with any of the previous floor(n/2) terms. We show…
This paper addresses the uniform random generation of words from a context-free language (over an alphabet of size $k$), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional…
We propose a new approach to nondeterministic random number generation. In theory, the randomness originated from the uncorrelated nature of consecutive laser pulses with Poissonian photon number distribution and that of the consecutive…
A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$ (when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word is down-up). The study of alternating words avoiding classical permutation patterns was…
We study an interesting family of cooperating coroutines, which is able to generate all patterns of bits that satisfy certain fairly general ordering constraints, changing only one bit at a time. (More precisely, the directed graph of…
A Cayley permutation is a word of positive integers such that if a letter appears in this word, then all positive integers smaller than that letter also appear. We initiate a systematic study of pattern avoidance on Cayley permutations…
Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a…
Recently, Grytczuk, Kordulewski, and Niewiadomski defined an extremal word over an alphabet $\mathbb{A}$ to be a word with the property that inserting any letter from $\mathbb{A}$ at any position in the word yields a given pattern. In this…
We study language generation in the limit, where an algorithm observes an adversarial enumeration of strings from an unknown target language $K$ and must eventually generate new, unseen strings from $K$. Kleinberg and Mullainathan [KM24]…