Related papers: The Complexity of Unavoidable Word Patterns
We describe a new non-constructive technique to show that squares are avoidable by an infinite word even if we force some letters from the alphabet to appear at certain occurrences. We show that as long as forced positions are at distance…
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…
We find finite-state recurrences to enumerate the words on the alphabet $[n]^r$ which avoid the patterns 123 and $1k(k-1)\dots2$, and, separately, the words which avoid the patterns 1234 and $1k(k-1)\dots2$.
We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this…
Machine learning researchers and practitioners steadily enlarge the multitude of successful learning models. They achieve this through in-depth theoretical analyses and experiential heuristics. However, there is no known general-purpose…
The subword complexity of a finite word $w$ of length $N$ is a function which associates to each $n\le N$ the number of all distinct subwords of $w$ having the length $n$. We define the \emph{maximal complexity} C(w) as the maximum of the…
Hard instances of natural computational problems are often elusive. In this note we present an example of a natural decision problem, the word problem for a certain finitely presented group, whose hard instances are easy to find. More…
Proving programs terminating is a fundamental computer science challenge. Recent research has produced powerful tools that can check a wide range of programs for termination. The analog for probabilistic programs, namely termination with…
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…
A correlation is a binary vector that encodes all possible positions of overlaps of two words, where an overlap for an ordered pair of words (u,v) occurs if a suffix of word u matches a prefix of word v. As multiple pairs can have the same…
In this paper we introduce and study a new complexity measure for finite words. For positive integer $d$ special scattered subwords, called super-$d$-subwords, in which the gaps are of length at least $(d-1)$, are defined. We give methods…
Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The…
For every pattern $p$ over the alphabet $\{x,y,x^R,y^R\}$, we specify the least $k$ such that $p$ is $k$-avoidable.
Human language defines the most complex outcomes of evolution. The emergence of such an elaborated form of communication allowed humans to create extremely structured societies and manage symbols at different levels including, among others,…
If no optimal propositional proof system exists, we (and independently Pudl\'ak) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts…
The recently confirmed Dejean's conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with…
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…
Bounds on the entropy of patterns of sequences generated by independently identically distributed (i.i.d.) sources are derived. A pattern is a sequence of indices that contains all consecutive integer indices in increasing order of first…
An infinite permutation is a linear ordering of the set of non-negative integers. Generally, the properties of infinite permutations analogous to those of infinite words show some resemblances and some differences between permutations and…
The difficulties of automatic extraction of definitions and methods from scientific documents lie in two aspects: (1) the complexity and diversity of natural language texts, which requests an analysis method to support the discovery of…