**23 Jan 2012
Nijmegen, The Netherlands**

COIN - Coalgebra in the Netherlands, 23 January 2012, Nijmegen ============================================================== COIN, or Coalgebra in the Netherlands, is a seminar taking place alternating at the Radboud University Nijmegen and the CWI in Amsterdam. The aim of COIN is to bring together coalgebra researchers from various locations in the Netherlands, and share current results and questions in the world of coalgebra. We welcome presentations on any subject related to coalgebra. COIN is coordinated by Marcello Bonsangue (Leiden University/CWI), Bart Jacobs (Radboud Universiteit Nijmegen), and Jan Rutten (CWI/Radboud Universiteit Nijmegen), and its website can be found at http://homepages.cwi.nl/~winter/coin.html The next COIN meeting is scheduled for Monday, 23 January 2012, at the Huygens building of the Radboud Universiteit Nijmegen, in room HG00.086. The schedule is as follows: 14:30 - 15:15 Jörg Endrullis (VU) Part 1: Causality via productivity Part 2: Degree of undecidability of stream equality 15:15 - 16:00 Bart Jacobs (RU) Bases as coalgebras 16:15 - 17:00 Joost Winter (CWI) Generalized context-freeness Abstracts ========= Jörg Endrullis (VU) Part 1: Causality via productivity Part 2: Degree of undecidability of stream equality --------------------------------------------------- The talk consists of two parts: In part (1) we introduce a concept of relative causality that turns out to be useful in coinductive proofs, in particular using circular coinduction. Moreover, we will establish a link between productivity and causality. Thereby automated methods for proving productivity become applicable for proving (relative) causality. In part (2) we investigate the degree of undecidability of equality of streams specified by systems of equations. This question has been investigated by G. Rosu in 2006: "Equality of Streams is a Pi-0-2 complete problem". However, the results in this paper are based on a peculiar notion of equality. As it turns out, more conventional choices yield entirely different results of significantly higher complexity. Bart Jacobs (RU) Bases as coalgebras ------------------- The free algebra adjunction, between the category of algebras of a monad and the underlying category, induces a comonad on the category of algebras. The coalgebras of this comonad are the topic of study in this talk (following earlier work). It is illustrated how such coalgebras-on-algebras can be understood as bases, decomposing each element x into primitives elements from which x can be reconstructed via the operations of the algebra. This holds in particular for the free vector space monad, but also for other monads. For instance, continuous dcpos or stably continuous frames, where each element is the join of the elements way below it, can be described as such coalgebras. Further, it is shown how these coalgebras-on-algebras give rise to a comonoid structure for copy and delete, and thus to diagonalisation of endomaps like in linear algebra. Joost Winter (CWI) Generalized context-freeness ---------------------------- We provide a coinductive specification format of algebraic power series, which can be seen as a generalization of a number of existing notions, such as context-free languages, weighted context-free languages, and automatic sequences. This is done by means of systems of behavioural differential equations in a specific format, which yield exactly this class of sequences. Furthermore, connections are given with and between the various existing notions of algebraic power series. This is joint work with Jan Rutten (CWI/Radboud University) and Marcello Bonsangue (Leiden University/CWI).