On the Order Type of Scattered Context-Free Orderings

We show that if a context-free grammar generates a language whose lexicographic ordering is well-ordered of type less than $\omega^2$, then its order type is effectively computable.Comment: In Proceedings GandALF 2019, arXiv:1909.05979. arXiv admin note: text overlap with arXiv:1907.1157

The FC-rank of a context-free language

We prove that the finite condensation rank (FC-rank) of the lexicographic ordering of a context-free language is strictly less than $\omega^\omega$

The Rank of Tree-Automatic Linear Orderings

We generalise Delhomm\'e's result that each tree-automatic ordinal is strictly below \omega^\omega^\omega{} by showing that any tree-automatic linear ordering has FC-rank strictly below \omega^\omega. We further investigate a restricted form of tree-automaticity and prove that every linear ordering which admits a tree-automatic presentation of branching complexity at most k has FC-rank strictly below \omega^k.Comment: 20 pages, 3 figure

On factorisation forests

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by well-orderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic second-order logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Event reconstruction in high resolution Compton telescopes

The development of germanium Compton telescopes for nuclear gamma-ray astrophysics (~0.2-20 MeV) requires new event reconstruction techniques to accurately determine the initial direction and energy of photon events, as well as to consistently reject background events. This paper describes techniques for event reconstruction, accounting for realistic instrument/detector performance and uncertainties. An especially important technique is Compton Kinematic Discrimination, which allows proper interaction ordering and background rejection with high probabilities. The use of these techniques are crucial for the realistic evaluation of the performance and sensitivity of any germanium Compton telescope configuration.Comment: Accepted for publication in A&AS