arXiv subjects

Dirk Nowotka

Publications and source records attributed to Dirk Nowotka.

41 records · Page 3Linked to original sources

arXiv2016

A note on Thue games

In this work we improve on a result from~\cite{GryKosZma15}. In particular, we investigate the situation where a word is constructed jointly by two players who alternately append letters to the end of an existing word. One of the players (Ann) tries to avoid (non-trivial) repetitions, while the other one (Ben) tries to enforce them. We show a construction that is closer to the lower bound showed in~\cite{GryKozMic13} using entropy compression, and building on the probabilistic arguments based on a version of the Lovász Local Lemma from~\cite{Peg11}. We provide an explicit strategy for Ann to avoid (non-trivial) repetitions over a $7$-letter alphabet.

cs.DM↗

arXiv2011

Pattern Avoidability with Involution

An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the size of the alphabet. For example, it is shown that the pattern a t(a) a can be avoided over three letters but not two letters, whereas it is well known that a a a is avoidable over two letters.

cs.FL↗

arXiv2010

Maximal Intersection Queries in Randomized Input Models

Consider a family of sets and a single set, called the query set. How can one quickly find a member of the family which has a maximal intersection with the query set? Time constraints on the query and on a possible preprocessing of the set family make this problem challenging. Such maximal intersection queries arise in a wide range of applications, including web search, recommendation systems, and distributing on-line advertisements. In general, maximal intersection queries are computationally expensive. We investigate two well-motivated distributions over all families of sets and propose an algorithm for each of them. We show that with very high probability an almost optimal solution is found in time which is logarithmic in the size of the family. Moreover, we point out a threshold phenomenon on the probabilities of intersecting sets in each of our two input models which leads to the efficient algorithms mentioned above.

cs.IR↗

arXiv2007

On a Non-Context-Free Extension of PDL

Over the last 25 years, a lot of work has been done on seeking for decidable non-regular extensions of Propositional Dynamic Logic (PDL). Only recently, an expressive extension of PDL, allowing visibly pushdown automata (VPAs) as a formalism to describe programs, was introduced and proven to have a satisfiability problem complete for deterministic double exponential time. Lately, the VPA formalism was extended to so called k-phase multi-stack visibly pushdown automata (k-MVPAs). Similarly to VPAs, it has been shown that the language of k-MVPAs have desirable effective closure properties and that the emptiness problem is decidable. On the occasion of introducing k-MVPAs, it has been asked whether the extension of PDL with k-MVPAs still leads to a decidable logic. This question is answered negatively here. We prove that already for the extension of PDL with 2-phase MVPAs with two stacks satisfiability becomes Σ_1^1-complete.

cs.LO↗

arXiv2003

Periodicity and Unbordered Words: A Proof of the Extended Duval Conjecture

The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered, if it has a proper prefix which is also a suffix of that word. Let f(w) denote the maximum length of all unbordered factors of w, and let p(w) denote the period of w. Clearly, f(w) < p(w)+1. We establish that f(w) = p(w), if w has an unbordered prefix of length f(w) and n > 2f(w)-2. This bound is tight and solves the stronger version of a 21 years old conjecture by Duval. It follows from this result that, in general, n > 3f(w)-3 implies f(w) = p(w) which gives an improved bound for the question asked by Ehrenfeucht and Silberger in 1979.

cs.DM↗