Wythoff Wisdom

International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems

Relaxed Wythoff has all Beatty Solutions

We find conditions under which the P-positions of three subtraction games arise as pairs of complementary Beatty sequences. The first game is due to Fraenkel and the second is an extension of the first game to non-monotone settings. We show that the P-positions of the second game can be inferred from the recurrence of Fraenkel\u27s paper if a certain inequality is satisfied. This inequality is shown to be necessary if the P-positions are known to be pairs of complementary Beatty sequences, and the family of irrationals for which this inequality holds is explicitly given. We highlight several games in the literature that have P-positions as pairs of complementary Beatty sequences with slope in this family. The third game we present is novel, and we show that the P-positions can be inferred from the same recurrence in any setting. It is shown that any pair of complementary Beatty sequences arises as the P-positions of some game in this family. We also provide background on some inverse problems which have appeared in the field over the last several years, in particular the Duchêne-Rigo conjecture. This paper presents a solution to the Fraenkel problem posed at the 2011 BIRS workshop, a modification of the Duchêne-Rigo conjecture

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Alan Turing: ¿personaje del siglo XX?

“Time” magazine chose Albert Einstein as “Person of the Century.” This was undoubtedly a reasonable choice, but as I will argue in this article, there are also good reasons for contending that Turing might have received this honor. One such reason I consider here is his purely scientific work, which stems from the greatest mathematical tradition, and how it affected the development of mathematics itself and was finally instrumental in shaping a new technological world. This is true both as regards the computation and treatment of information as well as the establishment of new forms of social relations. In relation to the foregoing, we have Turing’s contributions to the deciphering of secret codes during the Second World War, which in a somewhat metaphorical sense may be regarded as a new tool for undermining personal privacy, that civil right whose denial finally ruined his lifeEn diciembre de 1999, la revista Time eligió a Albert Einstein “The Person of the Century”. Fue, no cabe duda, una elección razonable, pero, como se argumenta en este artículo, existen también buenos argumentos para sostener que Turing podría haber recibido tal honor. En apoyo de semejante tesis están sus trabajos puramente científicos, que se esbozan aquí, trabajos que entroncan con la mejor tradición matemática, y cómo afectaron al desarrollo matemático, siendo finalmente instrumentales en la configuración de un nuevo mundo tecnológico, tanto en lo que al cálculo y manejo de información se refiere, como en lo relativo al establecimiento de nuevas formas de relaciones sociales. Relacionadas con lo anterior, se encuentran las aportaciones que hizo durante la Segunda Guerra Mundial al desciframiento de códigos secretos, que, en cierto sentido, metafórico, se pueden considerar como una nueva herramienta para socavar la privacidad, ese derecho civil cuya negación arruinó su propia vida

Opening opera: developing a framework that allows for the interactive creative processes of improvised theatre in the productions of new music-dramas

Opening Opera explores creative collaboration and dramatic improvisation in new music-dramas. Looking towards the art of improvised theatre, the aim is to achieve a dramatic process in opera which facilitates a flexible kind of dramaturgy, enabling singers and directors to lead and inform certain creative processes that are normally in the hands of the operatic composer and/or librettist alone. By developing particular unorthodox scoring methods, along with specific rehearsal schedule considerations that support these flexible processes, the composer attempts to create not improvised opera, but what could be called an open opera, where the compositional focus is working with a free vocal line with active accompaniment. This framework is one in which the composer provides the parameters and material for dramatic and compositional flexibility, and then ‘takes a step back’ during a collaborative and improvisational process, whilst retaining sufficient leadership and creative authority to realise the overarching structure satisfactorily. In order to develop said unorthodox scoring methods and processes, the creative team explored and informed structural, dramatic, technical and musical aspects of new material, utilising the performers’ specialised training and experience in an interactive creative process. This exploration brought up questions such as: how to allow an open process such as this one while still attempting to retain overarching artistic control; what are the parameters that the composer will need to determine (i.e. keep 'closed'); and what are the parameters that he must allow to be spontaneous, improvised or open? By opening up the process in this way, interesting genre specific problems were exposed that are more often than not left implicit rather than explicit by creators of opera. This exploration reveals knowledge beneficial to tutors, composers, librettists, singers, conductors and directors of music-dramas. This inquiry is primarily grounded in several methods extracted and modified from improvised theatre, opera, and open-scored compositions of the mid-20th century. The musical and qualitative data discussed in this commentary emerged during the preparation, writing and production of the author’s original music-drama studies. Studies include excerpts from two chamber operas; A Glacier’s Requiem (2013) (or “Bráð” in the original Icelandic), and Évariste (2014); a short monodrama study, Solitude 1 (2015); and a chamber opera presented in its entirety, After the Fall (2017)