Clafer: Lightweight Modeling of Structure, Behaviour, and Variability

Embedded software is growing fast in size and complexity, leading to intimate mixture of complex architectures and complex control. Consequently, software specification requires modeling both structures and behaviour of systems. Unfortunately, existing languages do not integrate these aspects well, usually prioritizing one of them. It is common to develop a separate language for each of these facets. In this paper, we contribute Clafer: a small language that attempts to tackle this challenge. It combines rich structural modeling with state of the art behavioural formalisms. We are not aware of any other modeling language that seamlessly combines these facets common to system and software modeling. We show how Clafer, in a single unified syntax and semantics, allows capturing feature models (variability), component models, discrete control models (automata) and variability encompassing all these aspects. The language is built on top of first order logic with quantifiers over basic entities (for modeling structures) combined with linear temporal logic (for modeling behaviour). On top of this semantic foundation we build a simple but expressive syntax, enriched with carefully selected syntactic expansions that cover hierarchical modeling, associations, automata, scenarios, and Dwyer's property patterns. We evaluate Clafer using a power window case study, and comparing it against other notations that substantially overlap with its scope (SysML, AADL, Temporal OCL and Live Sequence Charts), discussing benefits and perils of using a single notation for the purpose

Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams

Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github

Faithful Modeling of Product Lines with Kripke Structures and Modal Logic

Software product lines are now an established framework for software design. They are specified by special diagrams called feature models. For formal analysis, the latter are usually encoded by Boolean propositional theories. We discuss a major deficiency of this semantics, and show that it can be fixed by considering a product to be an instantiation process rather than its final result. We call intermediate states of this process partial products, and argue that what a feature model really defines is a poset of its partial products. We argue that such structures can be viewed as special Kripke structure that we call partial product Kripke structures, ppKS. To specify these Kripke structures, we propose a CTL-based logic, called partial product CTL, ppCTL. We show how to represent a feature model M by a ppCTL theory ML(M) (ML stands for modal logic) such that any ppKS satisfying the theory is equal to the partial product line determined by M. Hence, ML(M) can be considered a sound and complete representation of M. We also discuss several applications of the modal logic view in feature modeling, including refactoring of feature models

Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning

The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive operators (GENEOs), which are maps between function spaces associated with groups of transformations. We study the topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general strategies to initialise and compose operators. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of non-expansive operators. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show examples on the MNIST and fashion-MNIST datasets. By considering isometry-equivariant non-expansive operators, we describe a simple strategy to select and sample operators, and show how the selected and sampled operators can be used to perform both classical metric learning and an effective initialisation of the kernels of a convolutional neural network.Comment: Added references. Extended Section 7. Added 3 figures. Corrected typos. 42 pages, 7 figure

Dynamical and topological tools for (modern) music analysis

Is it possible to represent the horizontal motions of the melodic strands of a contrapuntal composition, or the main ideas of a jazz standard as mathematical entities? In this work, we suggest a collection of novel models for the representation of music that are endowed with two main features. First, they originate from a topological and geometrical inspiration; second, their low dimensionality allows to build simple and informative visualisations. Here, we tackle the problem of music representation following three non-orthogonal directions. We suggest a formalisation of the concept of voice leading (the assignment of an instrument to each voice in a sequence of chords) suggesting a horizontal viewpoint on music, constituted by the simultaneous motions of superposed melodies. This formalisation naturally leads to the interpretation of counterpoint as a multivariate time series of partial permutation matrices, whose observations are characterised by a degree of complexity. After providing both a static and a dynamic representation of counterpoint, voice leadings are reinterpreted as a special class of partial singular braids (paths in the Euclidean space), and their main features are visualised as geometric configurations of collections of 3-dimensional strands. Thereafter, we neglect this time-related information, in order to reduce the problem to the study of vertical musical entities. The model we propose is derived from a topological interpretation of the Tonnetz (a graph commonly used in computational musicology) and the deformation of its vertices induced by a harmonic and a consonance-oriented function, respectively. The 3-dimensional shapes derived from these deformations are classified using the formalism of persistent homology. This powerful topological technique allows to compute a fingerprint of a shape, that reflects its persistent geometrical and topological properties. Furthermore, it is possible to compute a distance between these fingerprints and hence study their hierarchical organisation. This particular feature allows us to tackle the problem of automatic classification of music in an innovative way. Thus, this novel representation of music is evaluated on a collection of heterogenous musical datasets. Finally, a combination of the two aforementioned approaches is proposed. A model at the crossroad between the signal and symbolic analysis of music uses multiple sequences alignment to provide an encompassing, novel viewpoint on the musical inspiration transfer among compositions belonging to different artists, genres and time. To conclude, we shall represent music as a time series of topological fingerprints, whose metric nature allows to compare pairs of time-varying shapes in both topological and in musical terms. In particular the dissimilarity scores computed by aligning such sequences shall be applied both to the analysis and classification of music

A filtering engine for large conceptual schemas

Postprint (published version