Hidden Categories: a New Perspective on Lewin's Generalized Interval Systems and Klumpenhouwer Networks

In this work we provide a categorical formalization of several constructions found in transformational music theory. We first revisit David Lewin's original theoretical construction of Generalized Interval Systems (GIS) to show that it implicitly defines categories. When all the conditions in Lewin's definition are fullfilled, such categories coincide with the category of elements $\int_\mathbf{G} S$ for the group action $S \colon \mathbf{G} \to \mathbf{Sets}$ implied by the GIS structure. By focusing on the role played by categories of elements in such a context, we reformulate previous definitions of transformational networks in a $\mathbf{Cat}$-based diagrammatical perspective, and present a new definition of transformational networks (called CT-Nets) in general musical categories. We show incidently how such an approach provides a bridge between algebraic, geometrical and graph-theoretical approaches in transformational music analysis. We end with a discussion on the new perspectives opened by such a formalization of transformational theory, in particular with respect to $\mathbf{Rel}$-based transformational networks which occur in well-known music-theoretical constructions such as Douthett's and Steinbach's Cube Dance

Stochastic Relational Presheaves and Dynamic Logic for Contextuality

Presheaf models provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal and external choices, composition of systems, and so on, I will show how to extend those models and ideas by combining them with ideas from other category-theoretic approaches to relational models and to stochastic processes. It turns out that my extension yields a transitional formulation of sheaf-theoretic structures that Abramsky and Brandenburger proposed to characterize non-locality and contextuality. An alternative characterization of contextuality will then be given in terms of a dynamic modal logic of the models I put forward.Comment: In Proceedings QPL 2014, arXiv:1412.810

Weak bisimulations for labelled transition systems weighted over semirings

Weighted labelled transition systems are LTSs whose transitions are given weights drawn from a commutative monoid. WLTSs subsume a wide range of LTSs, providing a general notion of strong (weighted) bisimulation. In this paper we extend this framework towards other behavioural equivalences, by considering semirings of weights. Taking advantage of this extra structure, we introduce a general notion of weak weighted bisimulation. We show that weak weighted bisimulation coincides with the usual weak bisimulations in the cases of non-deterministic and fully-probabilistic systems; moreover, it naturally provides a definition of weak bisimulation also for kinds of LTSs where this notion is currently missing (such as, stochastic systems). Finally, we provide a categorical account of the coalgebraic construction of weak weighted bisimulation; this construction points out how to port our approach to other equivalences based on different notion of observability

How to kill epsilons with a dagger: a coalgebraic take on systems with algebraic label structure

We propose an abstract framework for modeling state-based systems with internal behavior as e.g. given by silent or ϵ-transitions. Our approach employs monads with a parametrized fixpoint operator † to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems.Funded by the ERDF through the Programme COMPETE and by the Portuguese Foundation for Science and Technology, project ref. FCOMP-01-0124-FEDER-020537 and SFRH/BPD/71956/2010. Acknowledge support by project ANR 12IS0 2001 PACE

Connector algebras for C/E and P/T nets interactions

A quite fourishing research thread in the recent literature on component based system is concerned with the algebraic properties of different classes of connectors. In a recent paper, an algebra of stateless connectors was presented that consists of five kinds of basic connectors, namely symmetry, synchronization, mutual exclusion, hiding and inaction, plus their duals and it was shown how they can be freely composed in series and in parallel to model sophisticated "glues". In this paper we explore the expressiveness of stateful connectors obtained by adding one-place buffers or unbounded buffers to the stateless connectors. The main results are: i) we show how different classes of connectors exactly correspond to suitable classes of Petri nets equipped with compositional interfaces, called nets with boundaries; ii) we show that the difference between strong and weak semantics in stateful connectors is reflected in the semantics of nets with boundaries by moving from the classic step semantics (strong case) to a novel banking semantics (weak case), where a step can be executed by taking some "debit" tokens to be given back during the same step; iii) we show that the corresponding bisimilarities are congruences (w.r.t. composition of connectors in series and in parallel); iv) we show that suitable monoidality laws, like those arising when representing stateful connectors in the tile model, can nicely capture concurrency aspects; and v) as a side result, we provide a basic algebra, with a finite set of symbols, out of which we can compose all P/T nets, fulfilling a long standing quest

Relational Presheaves as Labelled Transition Systems

International audienceWe show that viewing labelled transition systems as relational presheaves captures several recently studied examples. This approach takes into account possible algebraic structure on labels. Weak closure of a labelled transition system is characterised as a left (2-)adjoint to a change-of-base functor