We present an application of strategy synthesis to enforce spatial properties. This is achieved by implementing a toolchain that enables the tools \texttt{CATLib} and \texttt{VoxLogicA} to interact in a fully automated way. The Contract Automata Library (\texttt{CATLib}) is aimed at both composition and strategy synthesis of games modelled in a dialect of finite state automata. The Voxel-based Logical Analyser (\texttt{VoxLogicA}) is a spatial model checker for the verification of properties expressed using the Spatial Logic of Closure Spaces on pixels of digital images. We provide examples of strategy synthesis on automata encoding motion of agents in spaces represented by images, as well as a proof-of-concept realistic example based on a case study from the railway domain. The strategies are synthesised with \texttt{CAT\-Lib}, while the properties to enforce are defined by means of spatial model checking of the images with \texttt{VoxLogicA}. The combination of spatial model checking with strategy synthesis provides a toolchain for checking and enforcing mobility properties in multi-agent systems in which location plays an important role, like in many collective adaptive systems. We discuss the toolchain's performance also considering several recent improvements.

This is the complementary material for our paper ``A Toolchain for Strategy Synthesis with Spatial Properties'' accepted for publication at the Journal of Software Tools and Technology Transfer. This repository contains a permanent snapshot of https://github.com/contractautomataproject/CATLib_PngConverter/tree/ec0938043146b008bbbcb4ed3ec79c06dd2e47d6

Spatial logic and spatial model checking have great poten- tial for traditional computer science domains and beyond. Reasoning about space involves two different conditional reachability modalities: a forward reachability, similar to that used in temporal logic, and a backward modality representing that a point can be reached from an- other point, under certain conditions. Since spatial models can be huge, suitable model minimisation techniques are crucial for efficient model checking. An effective minimisation method for the recent notion of spatial Compatible Path (CoPa)-bisimilarity is proposed, and shown to be correct. The core of our method is the encoding of Closure Models as La- belled Transition Systems, enabling minimisation algorithms for branching bisimulation to compute CoPa equivalence classes. Initial validation via benchmark examples demonstrates a promising speed-up in model checking of spatial properties for models of realistic size. Detailed proofs of all results are provided.

Topological Spatial Model Checking is a recent paradigm where model checking techniques are developed for the topological interpretation of Modal Logic. The Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability connectives that, in turn, can be used for expressing interesting spatial properties, such as "being near to" or "being surrounded by". SLCS constitutes the kernel of a solid logical framework for reasoning about discrete space, such as graphs and digital images, interpreted as quasi discrete closure spaces. Following a recently developed geometric semantics of Modal Logic, we propose an interpretation of SLCS in continuous space, admitting a geometric spatial model checking procedure, by resorting to models based on polyhedra. Such representations of space are increasingly relevant in many domains of application, due to recent developments of 3D scanning and visualisation techniques that exploit mesh processing. We introduce PolyLogicA, a geometric spatial model checker for SLCS formulas on polyhedra and demonstrate feasibility of our approach on two 3D polyhedral models of realistic size. Finally, we introduce a geometric definition of bisimilarity, proving that it characterises logical equivalence.

In these notes we prove that, in quasi-discrete closure models, the ISLCS forward (backword) conditional reachability operator can be expressed using a (possibly) infinite disjunction of nested formulas using only conjunction and the IMLC backward (forward) proximity operator.

We present two different extensions of the spatial logic for closure spaces (SLCS), and its spatio-temporal variant (?SLCS), with spa- tial quantification operators. The first concerns the existential quantifi- cation on individual points of a space. The second concerns the quantifi- cation on sets of points. The latter amounts to a form of quantification over atomic propositions, thus without the full power of second order logic. The spatial quantification operators are useful for reasoning about the existence of particular spatial objects in a space, their spatial rela- tion with respect to other spatial objects, and, in the spatio-temporal setting, to reason about the dynamic evolution of such spatial objects in time and space, including reasoning about newly introduced items. In this preliminary study we illustrate the expressiveness of the operators by means of several small, but representative, examples.

We present a feasibility study on the use of spatial logic model checking for real-time analysis of high-resolution video streams with the tool VoxLogicA. VoxLogicA is a voxel-based image analyser based on the Spatial Logic for Closure Spaces, a logic catered to deal with properties of spatial structures such as topological spaces, graphs and polyhedra. The underlying language includes operators to model proximity and reachability. We demonstrate, via the analysis of a series of video frames from a well-known video game, that it is possible to analyse high-resolution videos in real-time by exploiting the speed-up of VoxLogicA-GPU, a recently developed GPU-based version of the tool, which is 1-2 orders of magnitude faster than its previous iteration. Potential applications of real-time video analysis include medical imaging applications such as ultrasound exams, and other video-based diagnostic techniques. More broadly speaking, this work can be the first step towards novel information retrieval methods suitable to find information in a declarative way, in possibly large collections of video streams.

We investigate the application of strategy synthesis to enforce spatial properties. The Contract Automata Library (CATLib) performs both composition and strategy synthesis of games modelled in a dialect of finite state automata. The Voxel-based Logical Analyser (VoxLogicA) is a spatial model checker that allows the verification of properties expressed using the Spatial Logic of Closure Spaces on pixels of digital images. In this paper, we explore the integration of these two tools. We provide a basic example of strategy synthesis on automata encoding motion of agents in spaces represented by images. The strategy is synthesised with CATLib, whilst the properties to enforce are defined by means of spatial model checking of the images with VoxLogicA.

We adapt the standard notion of bisimilarity for topological models to closure models and refine it for quasi-discrete closure models. We also define an additional, weaker notion of bisimilarity that is based on paths in space and expresses a form of conditional reachability in a way that is reminiscent of Stuttering Equivalence on transition systems. For each bisimilarity we provide a characterisation with respect to a suitable spatial logic. In this report, the detailed proofs of all the results we present are also included.

We adapt the standard notion of bisimilarity for topological models to closure models and refine it for quasi-discrete closure models. We also define an additional, weaker notion of bisimilarity that is based on paths in space and expresses a form of conditional reachability in a way that is reminiscent of Stuttering Equivalence on transition systems. For each bisimilarity we provide a characterisation with respect to a suitable spatial logic.

In medical imaging, (semi-)automatic image analysis techniques have been proposed to support the current time-consuming and cognitively demanding practice of manual segmentation of regions of interest (ROIs). The recently proposed image query language ImgQL, based on spatial logic and model checking, represents segmentation methods as concise, domain-oriented, human-readable procedures aimed at domain experts rather than technologists, and has been validated in several case studies. Such efforts are directed towards a human-centred Artificial Intelligence methodology. To this aim, we complemented the ongoing research line with a study of the Human-Computer Interaction aspects. In this work we investigate the design of a graphical user interface (GUI) prototype that supports the analysis procedure with minimal impact on the focus and the memory load of domain experts.

Nowadays a plethora of health data is available for clinical and research usage. Such existing datasets can be augmented through artificial-intelligence-based methods by automatic, personalised annotations and recommendations. This huge amount of data lends itself to new usage scenarios outside the boundaries where it was created; just to give some examples: to aggregate data sources in order to make research work more relevant; to incorporate a diversity of datasets in training of Machine Learning algorithms; to support expert decisions in telemedicine. In such a context, there is a growing need for a paradigm shift towards means to interrogate medical databases in a semantically meaningful way, fulfilling privacy and legal requirements, and transparently with respect to ethical concerns. In the specific domain of Medical Imaging, in this paper we sketch a research plan devoted to the definition and implementation of query languages that can unambiguously express semantically rich queries on possibly multi-dimensional images, in a human-readable, expert-friendly and concise way. Our approach is based on querying images using Topological Spatial Logics, building upon a novel spatial model checker called VoxLogicA, to execute such queries in a fully automated way.

Logic based (semi-)automatic contouring in Medical Imaging has shown to be a very promising and versatile technique that can potentially greatly facilitate the work of different professionals in this domain while supporting explainability, easy replicability and exchange of medical image analysis methods. In such a context there is a clear need of a prototype Graphical User Interface (GUI) support for professionals which is usable, understandable and which reduces unnecessary cognitive load to the minimum, so that the focus of attention can remain on the main, critical, tasks such as image segmentation in support of planning of radiotherapy. In this paper we introduce a first proposal for a graphical user interface for the segmentation of medical images via the spatial logic based analyser VoxLogicA. Since both the logic approach to image analysis and its application in medical imaging are completely new, this is the first step in an iterative development process that will involve various analysis and development techniques, including empirical research and formal analysis. In the current work we analyse the GUI with a focus on the cognitive and memory load aspects which are critical in this domain of application.

In this position paper, we discuss the introduction of spatial verification techniques in an application scenario from smart stations, viz. analysing the user experience with respect to the lighting conditions of station areas. This is a case study in industrial projects. We discuss three challenging use cases for the application of spatial model checking in this setting. First, we envision how to use the spatial model checker VoxLogicA, which can analyse both 2D and 3D voxel-based maps, to explore the areas that users can visit in a station area and to characterise them with respect to their illumination conditions. This is aimed at monitoring a smart station. We also ideate statistical spatio-temporal model checking of the design of energy-saving protocols, exploiting the modelling of user preferences. Finally, we discuss the idea of quantifying the impact of design changes, based on the logs of smart stations, to identify and measure the incidence of undesired events (e.g. non-illuminated platforms where a train is passing by) before and after each change.

This paper provides a tutorial-style introduction, and a guide, to the recent advancements in spatial model checking that have made some relevant results possible. Among these, we mention fully automated segmentation of regions of interest in medical images by short, unambiguous spatial-logical specifications. This tutorial is aimed both at domain experts in medical imaging who would like to learn simple (scripting-alike) techniques for image analysis, making use of a modern, declarative language, and at experts in Formal Methods in Computer Science and Model Checking who would like to grasp how the theory of Spatial Logic and Model Checking has been turned into logic-based, dataset-oriented imaging techniques.

The tool VoxLogicA merges the state-of-the-art library of computational imaging algorithms ITK with the combination of declarative specification and optimised execution provided by spatial logic model checking. The analysis of an existing benchmark for segmentation of brain tumours via a simple logical specification reached very high accuracy. We introduce a new, GPU-based version of VoxLogicA and present preliminary results on its implementation, scalability, and applications.

Closure spaces are a generalisation of topological spaces obtained by removing the idempotence requirement on the closure operator. We adapt the standard notion of bisimilarity for topological models, namely Topo-bisimilarity, to closure models|we call the resulting equivalence CM-bisimilarity|and rene it for quasi-discrete closure models. We also dene two additional notions of bisimilarity that are based on paths on space, namely Path-bisimilarity and Compatible Path-bisimilarity, CoPa-bisimilarity for short. The former expresses (unconditional) reachability, the latter renes it in a way that is reminishent of Stuttering Equivalence on transition systems. For each bisimilarity we provide a logical characterisation, using variants of SLCS.We also address the issue of (space) minimisation via the three equivalences.

Topological Spatial Model Checking is a recent paradigm that combines Model Checking with the topological interpretation of Modal Logic. The Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability connectives that, in turn, can be used for expressing interesting spatial properties, such as "being near to" or "being surrounded by". SLCS constitutes the kernel of a solid logical framework for reasoning about discrete space, such as graphs and digital images, interpreted as quasi discrete closure spaces. In particular, the spatial model checker VoxLogicA, that uses an extended version of SLCS, has been used successfully in the domain of medical imaging. However, SLCS is not restricted to discrete space. Following a recently developed geometric semantics of Modal Logic, we show that it is possible to assign an interpretation to SLCS in continuous space, admitting a model checking procedure, by resorting to models based on polyhedra. In medical imaging such representations of space are increasingly relevant, due to recent developments of 3D scanning and visualisation techniques that exploit mesh processing. We demonstrate feasibility of our approach via a new tool, PolyLogicA, aimed at ecient verication of SLCS formulas on polyhedra, while inheriting some well-established optimization techniques already adopted in VoxLogicA. Finally, we cater for a geometric definition of bisimilarity, proving that it characterises logical equivalence.

Recently developed spatial and spatio-temporal model checking techniques have a wide range of application domains, among which large scale distributed systems as well as signal and image analysis. In the latter domain, automatic and semi-automatic contouring in Medical Imaging has shown to be a very promising and versatile application that may facilitate the work of professionals in this domain, while supporting explainability, easy replicability and exchange of medical image analysis methods. In recent work, spatial model-checking has been applied to the 3D contouring of brain tumours and related oedema in magnetic resonance images of the brain. In the present paper we address the contouring of 2D images of nevi. One of the challenges of contouring nevi is that they show considerable inhomogeneity in shape, colour, texture and size. In addition these images often include also extraneous elements such as hairs, patches and rulers. To deal with this challenge we explore the use of a texture similarity operator in combination with spatial logic operators. We investigate the feasibility of our technique on images of a large public database. We compare the results with associated ground truth segmentation provided by domain experts; the results are very promising, both from the quality and from the performance point of view.

The tool voxlogica merges the state-of-the-art library of computational imaging algorithms ITK with the combination of declarative specification and optimised execution provided by spatial logic model checking. The analysis of an existing benchmark for segmentation of brain tumours via a simple logical specification reached state-of-the-art accuracy. We present a new, GPU-based version of voxlogica and discuss its implementation, scalability, and applications.