Johannes C. Schneider

UNAMBIGUOUS MORPHIC IMAGES OF STRINGS

We study a fundamental combinatorial problem on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism τ with τ(i) ≠ σ(i) for some symbol i in α and, nevertheless, τ(α) = σ(α). As a consequence of its elementary nature, this question shows a variety of connections to those topics in discrete mathematics which are based on finite strings and morphisms such as pattern languages, equality sets and, thus, the Post Correspondence Problem. Our studies demonstrate that the existence of unambiguous morphic images essentially depends on the structure of α: We introduce a partition of the set of all finite strings into those that are decomposable (referred to as prolix) in a particular manner and those that are indecomposable (called succinct). This partition, that is also known to be of major importance for the research on pattern languages and on finite fixed points of morphisms, allows to formulate our main result according to which a string α can be mapped by an injective morphism onto an unambiguous image if and only if α is succinct.

Restricted ambiguity of erasing morphisms

A morphism h is called ambiguous for a string s if there is another morphism that maps s to the same image as h; otherwise, it is called unambiguous. In this paper, we examine some fundamental problems on the ambiguity of erasing morphisms. We provide a detailed analysis of so-called ambiguity partitions, and our main result uses this concept to characterise those strings that have a morphism of strongly restricted ambiguity. Furthermore, we demonstrate that there are strings for which the set of unambiguous morphisms, depending on the size of the target alphabet of these morphisms, is empty, finite or infinite. Finally, we show that the problem of the existence of unambiguous erasing morphisms is equivalent to some basic decision problems for nonerasing multi-pattern languages.

ON THE DUAL POST CORRESPONDENCE PROBLEM

The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ,τ, one of which needs to be non-periodic, such that σ(α) = τ(α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only. Little is known about the Dual PCP for words α over larger than binary alphabets, especially for so-called ratio-primitive examples. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist.

Unambiguous morphic images of strings

Motivated by the research on pattern languages, we study a fundamental combinatorial question on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism ρ with

Unambiguous Erasing Morphisms in Free Monoids

\rho \not= \sigma

An e-Learning Tool for the Austrian Citizen Card

and ρ(α) = σ(α). Our main result shows that a rich and natural class of strings is provided with unambiguous morphic images.

Morphically primitive words

This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i. e. there exists no other morphism τ satisfying τ(α) = σ(α). While Freydenberger et al. (International Journal of Foundations of Computer Science 17, 2006) characterise those strings for which there exists an unambiguous non-erasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms which map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphisms for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given.

Unambiguous erasing morphisms in free monoids

Austria is one of the leading countries in the field of electronic identity. The Austrian Citizen Card System is presently going live; the next step is to inform the citizen about the functionalities and advantages, as well as on how to use the card. On authority of the Federal Chancellor’s Office, the Danube University has developed an e-learning tool that will be made available to all Austrian households. The target group is the citizen rather than the early technological adopter, and the aim is to explain the use of a citizen card rather than the technical background. The main pedagogical method therefore is to reduce technical complexity to an absolute minimum. This e-learning tool is positioned to be one piece of a sophisticated e-government communication strategy.