Thue's 1914 paper: a translation

This paper includes notes to accompany a reading of Thue's 1914 paper "Probleme uber Veranderungen von Zeichenreihen nach gegebenen Reglen", along with a translation of that paper. Thue's 1914 paper is mainly famous for proving an early example of an undecidable problem, cited prominently by Post. However, Post's paper principally makes use of the definition of Thue systems, described on the first two pages of Thue's paper, and does not depend on the more specific results in the remainder of Thue's paper. A closer study of the remaining parts of that paper highlight a number of important themes in the history of computing: the transition from algebra to formal language theory, the analysis of the "computational power" (in a pre-1936 sense) of rules, and the development of algorithms to generate rule-sets

Star-Free Languages are Church-Rosser Congruential

The class of Church-Rosser congruential languages has been introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential (belongs to CRCL), if there is a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S. To date, it is still open whether every regular language is in CRCL. In this paper, we show that every star-free language is in CRCL. In fact, we prove a stronger statement: For every star-free language L there exists a finite, confluent, and subword-reducing semi-Thue system S such that the total number of congruence classes modulo S is finite and such that L is a union of congruence classes modulo S. The construction turns out to be effective

Remarks on the Spectral Properties of Tight Binding and Kronig-Penney Models with Substitution Sequences

We comment on some recent investigations on the electronic properties of models associated to the Thue-Morse chain and point out that their conclusions are in contradiction with rigorously proven theorems and indicate some of the sources of these misinterpretations. We briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems.Comment: 15,CPT-94/P.3003,tex,

Bottom-up and top-down tree transformations - a comparison

The top-down and bottom-up tree transducer are incomparable with respect to their transformation power. The difference between them is mainly caused by the different order in which they use the facilities of copying and nondeterminism. One can however define certain simple tree transformations, independent of the top-down/bottom-up distinction, such that each tree transformation, top-down or bottom-up, can be decomposed into a number of these simple transformations. This decomposition result is used to give simple proofs of composition results concerning bottom-up tree transformations.\ud \ud A new tree transformation model is introduced which generalizes both the top-down and the bottom-up tree transducer

On some modifications and applications of the post correspondence problem

The Post Correspondence Problem was introduced by Emil Post in 1946. The problem considers pairs of lists of sequences of symbols, or words, where each word has its place on the list determined by its index. The Post Correspondence Problem asks does there exist a sequence of indices so that, when we write the words in the order of the sequence as single words from both lists, the two resulting words are equal. Post proved the problem to be undecidable, that is, no algorithm deciding it can exist. A variety of restrictions and modifications have been introduced to the original formulation of the problem, that have then been shown to be either decidable or undecidable. Both the original Post Correspondence Problem and its modifications have been widely used in proving other decision problems undecidable. In this thesis we consider some modifications of the Post Correspondence Problem as well as some applications of it in undecidability proofs. We consider a modification for sequences of indices that are infinite to two directions. We also consider a modification to the original Post Correspondence Problem where instead of the words being equal for a sequence of indices, we take two sequences that are conjugates of each other. Two words are conjugates if we can write one word by taking the other and moving some part of that word from the end to the beginning. Both modifications are shown to be undecidable. We also use the Post Correspondence Problem and its modification for injective morphisms in proving two problems from formal language theory to be undecidable; the first problem is on special shuffling of words and the second problem on fixed points of rational functions

Kinematic Diffraction from a Mathematical Viewpoint

Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Simultaneously, their relevance has grown in practice as well. In this context, the phenomenon of homometry shows various unexpected new facets. This is particularly so for systems with stochastic components. After the introduction to the mathematical tools, we briefly discuss pure point spectra, based on the Poisson summation formula for lattice Dirac combs. This provides an elegant approach to the diffraction formulas of infinite crystals and quasicrystals. We continue by considering classic deterministic examples with singular or absolutely continuous diffraction spectra. In particular, we recall an isospectral family of structures with continuously varying entropy. We close with a summary of more recent results on the diffraction of dynamical systems of algebraic or stochastic origin.Comment: 30 pages, invited revie