Abstract
Spatial aspects of computation are increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification techniques are well suited to analyse the temporal evolution of concurrent systems; however, properties of space are typically not explicitly taken into account. This tutorial provides an introduction to recent work on a topology-inspired approach to formal verification of spatial properties depending upon (physical) space. A logic is presented, stemming from the tradition of topological interpretations of modal logics, dating back to earlier logicians such as Tarski, where modalities describe neighbourhood. These topological definitions are lifted to the more general setting of closure spaces, also encompassing discrete, graph-based structures. The present tutorial illustrates the extension of the framework with a spatial surrounded operator, leading to the spatial logic for closure spaces SLCS, and its combination with the temporal logic CTL, leading to STLCS. The interplay of space and time permits one to define complex spatio-temporal properties. Both for the spatial and the spatio-temporal fragment efficient model-checking algorithms have been developed and their use on a number of case studies and examples is illustrated.
Research partially funded by EU project QUANTICOL (nr. 600708).
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- 1.
When recovering the definition of a topological space via open sets from the Kuratowski definition, it is noteworthy that the preservation of binary unions is sufficient to prove that arbitrary unions of open sets are open.
- 2.
A minimal neighbourhood of x is a set that is a neighbourhood of x and is included in all other neighbourhoods of x.
- 3.
We leave open the possibility to change this notion, in chosen classes of closure spaces, practically making our theory dependent on such choice. The theoretical question of finding a uniform notion of path is left for future work.
- 4.
This notion of neighbourhood is also known as the von Neumann neighbourhood of radius 1.
- 5.
Web site: http://www.github.com/vincenzoml/topochecker.
- 6.
See http://ocaml.org.
- 7.
Actually one colour (yellow) could have been used, but in order to show multiple verification results combined in one picture, the orange points show the points that are yellow but that also satisfy the second property.
- 8.
- 9.
Note that the results may involve the same points, in which case the later result overwrites the previous result.
- 10.
Pisa: http://www.pisamo.it, Hangzhou: http://www.publicbike.net; Paris: http://www.velib.paris.fr, London: https://tfl.gov.uk/modes/cycling/santander-cycles.
- 11.
See also [9].
- 12.
See, e.g. http://bikes.oobrien.com/london.
- 13.
The results can be reproduced using the data and scripts, provided with the source code of the tool.
- 14.
The tool is a global model checker, therefore it is able to produce a graph for each state of the model, related to the truth value of formulas in that particular state, even if we only show results related to one specific state.
- 15.
More than one time step can be required. This can be achieved by repeated nesting of the EX operator. We did not do so for the sake of clarity in Fig. 23.
- 16.
We use artificial data for the sake of simplicity, but usage of the approach does not differ on real data.
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Acknowledgments
The authors like to thank Luca Bortolussi, Stephen Gilmore, Gianluca Grilletti, Laura Nenzi and Rytis Paškauskas who are involved in the Quanticol project and who are co-authors of the various articles on which this tutorial has been based. We like to thank Ezio Bartocci for sharing with us an earlier Matlab version of the Turing model.
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Ciancia, V., Latella, D., Loreti, M., Massink, M. (2016). Spatial Logic and Spatial Model Checking for Closure Spaces. In: Bernardo, M., De Nicola, R., Hillston, J. (eds) Formal Methods for the Quantitative Evaluation of Collective Adaptive Systems. SFM 2016. Lecture Notes in Computer Science(), vol 9700. Springer, Cham. https://doi.org/10.1007/978-3-319-34096-8_6
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