Wadge Degrees of ω\omega-Languages of Petri Nets

We prove that $\omega$-languages of (non-deterministic) Petri nets and $\omega$-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of $\omega$-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of $\omega$-languages of (non-deterministic) Turing machines which also form the class of effective analytic sets. In particular, for each non-null recursive ordinal $\alpha < \omega\_1^{{\rm CK}}$ there exist some ${\bf \Sigma}^0\_\alpha$-complete and some ${\bf \Pi}^0\_\alpha$-complete $\omega$-languages of Petri nets, and the supremum of the set of Borel ranks of $\omega$-languages of Petri nets is the ordinal $\gamma\_2^1$, which is strictly greater than the first non-recursive ordinal $\omega\_1^{{\rm CK}}$. We also prove that there are some ${\bf \Sigma}\_1^1$-complete, hence non-Borel, $\omega$-languages of Petri nets, and that it is consistent with ZFC that there exist some $\omega$-languages of Petri nets which are neither Borel nor ${\bf \Sigma}\_1^1$-complete. This answers the question of the topological complexity of $\omega$-languages of (non-deterministic) Petri nets which was left open in [DFR14,FS14].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326

Semantic Embedding of Petri Nets into Event-B

We present an embedding of Petri nets into B abstract systems. The embedding is achieved by translating both the static structure (modelling aspect) and the evolution semantics of Petri nets. The static structure of a Petri-net is captured within a B abstract system through a graph structure. This abstract system is then included in another abstract system which captures the evolution semantics of Petri-nets. The evolution semantics results in some B events depending on the chosen policies: basic nets or high level Petri nets. The current embedding enables one to use conjointly Petri nets and Event-B in the same system development, but at different steps and for various analysis.Comment: 16 pages, 3 figure

Measurements, Models, Systems and Design

531 s.

The Limit of Splitn-Language Equivalence

AbstractSplitting is a simple form of action refinement that may be used to express the duration of actions. In particular,splitnsubdivides each action intonphases. Petri netsNandN′ aresplitn-language equivalent ifsplitn(N) andsplitn(N′) are language equivalent. It is known that these equivalences get finer and finer with increasingn. This paper characterizes the limit of this sequence by a newly defined partial order semantics. This semantics is obtained from the interval-semiword semantics, which is fully abstract for action refinement and language equivalence, by closing it under a special swap operation. The new swap equivalence lies strictly between interval-semiword and step-sequence equivalence

A Nice Labelling for Tree-Like Event Structures of Degree 3 (Extended Version)

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the height. The main theorem we present in this paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors. Finally, we exemplify how to use this theorem to construct upper bounds for the labelling number of other event structures of degree 3

A flexible approach to the estimation of water budgets and its connection to the travel time theory.

The increasing impacts of climate changes on water related sectors are leading the scientists' attentions to the development of comprehensive models, allowing better descriptions of the water and solute transport processes. "Getting the right answers for the right reasons", in terms of hydrological response, is one of the main goals of most of the recent literature. Semi-distributed hydrological models, based on the partition of basins in hydrological response units (HRUs) to be connected, eventually, to describe a whole catchment, proved to be robust in the reproduction of observed catchment dynamics. 'Embedded reservoirs' are often used for each HRU, to allow a consistent representation of the processes. In this work, a new semi-disitrbuted model for runoff and evapotranspiration is presented: five different reservoirs are inter-connected in order to capture the dynamics of snow, canopy, surface flow, root-zone and groundwater compartments. The knowledge of the mass of water and solute stored and released through different outputs (e.g. discharge, evapotranspiration) allows the analysis of the hydrological travel times and solute transport in catchments. The latter have been studied extensively, with some recent benchmark contributions in the last decade. However, the literature remains obscured by different terminologies and notations, as well as model assumptions are not fully explained. The thesis presents a detailed description of a new theoretical approach that reworks the theory from the point of view of the hydrological storages and fluxes involved. Major aspects of the new theory are the 'age-ranked' definition of the hydrological variables, the explicit treatment of evaporative fluxes and of their influence on the transport, the analysis of the outflows partitioning coefficients and the explicit formulation of the 'age-ranked' equations for solutes. Moreover, the work presents concepts in a new systematic and clarified way, helping the application of the theory. To give substance to the theory, a small catchment in the prealpine area was chosen as an example and the results illustrated. The rainfall-runoff model and the travel time theory were implemented and integrated in the semi-distributed hydrological system JGrass-NewAge. Thanks to the environmental modelling framework OMS3, each part of the hydrological cycle is implemented as a component that can be selected, adopted, and connected at run-time to obtain a user-customized hydrological model. The system is flexible, expandable and applicable in a variety of modelling solutions. In this work, the model code underwent to an extensive revision: new components were added (coupled storages water budget, travel times components); old components were enhanced (Kriging, shortwave, longwave, evapotranspiration, rain-snow separation, SWE and melting components); documentation was standardized and deployed. Since the Thesis regards in wide sense the building of a collaborative system, a discussion of some general purpose tools that were implemented or improved for supporting the present research is also presented. They include the description and the verification of a software component dealing with the long-wave radiation budget and another component dealing with an implementation of some Kriging procedure