Related papers: Binary Patterns in Binary Cube-Free Words: Avoidab…
We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior.…
We show how to enumerate words in $1^{m_1} \dots n^{m_n}$ that avoid the increasing consecutive pattern $12 \dots r$ for any $r \geq 2$. Our approach yields an $O(n^{s+1})$ algorithm to enumerate words in $1^s \dots n^s$, avoiding the…
We discuss inequalities holding between the vocabulary size, i.e., the number of distinct nonterminal symbols in a grammar-based compression for a string, and the excess length of the respective universal code, i.e., the code-based analog…
Given a language L and a nondeterministic finite automaton M, we consider whether we can determine efficiently (in the size of M) if M accepts at least one word in L, or infinitely many words. Given that M accepts at least one word in L, we…
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…
Human communication systems, such as language, evolve culturally; their components undergo reproduction and variation. However, a role for selection in cultural evolutionary dynamics is less clear. Often neutral evolution (also known as…
A natural number N is said to be palindromic if its binary representation reads the same forwards and backwards. In this paper we study the quotients of two palindromic numbers and answer some basic questions about the resulting sets of…
Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 2/3$. We improve that result by showing that…
We introduce a flexible class of well-quasi-orderings (WQOs) on words that generalizes the ordering of (not necessarily contiguous) subwords. Each such WQO induces a class of piecewise testable languages (PTLs) as Boolean combinations of…
We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -- the Thue-Morse word has the minimum possible subword complexity over all overlap-free…
Pattern matching is a popular feature in functional, imperative and object-oriented programming languages. Language designers should therefore invest effort in a good design for pattern matching. Most languages choose a first-match…
We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length…
This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting…
We investigate the properties of formal languages expressible in terms of formulas over quantifier-free theories of word equations, arithmetic over length constraints, and language membership predicates for the classes of regular, visibly…
Pretrained language models memorize vast amounts of information, including private and copyrighted data, raising significant safety concerns. Retraining these models after excluding sensitive data is prohibitively expensive, making machine…
We study the distribution and the popularity of left children on sets of treeshelves avoiding a pattern of size three. (Treeshelves are ordered binary increasing trees where every child is connected to its parent by a left or a right link.)…
Large language models are trained with tokenizers, and the resulting token distribution is highly imbalanced: a few words dominate the stream while most occur rarely. Recent practice favors ever-larger vocabularies, but it is unclear where…
A controversial test for Large Language Models concerns the ability to discern possible from impossible language. While some evidence attests to the models' sensitivity to what crosses the limits of grammatically impossible language, this…
A non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is privileged if…
This study presents a fascinating linguistic property related to the number of letters in words and their corresponding numerical values. By selecting any arbitrary word, counting its constituent letters, and subsequently spelling out the…