A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Reachability problems for hierarchical piecewise constant derivative systems

In this paper, we investigate the computability and complexity of reachability problems for two-dimensional hierarchical piecewise constant derivative (HPCD) systems. The main interest in HPCDs stems from the fact that their reachability problem is on the border between decidability and undecidability, since it is equivalent to that of reachability for one-dimensional piecewise affine maps (PAMs) which is a long standing open problem. Understanding the most expressive hybrid system models that retain decidability for reachability has generated a great deal of interest over the past few years. In this paper, we show a restriction of HPCDs (called RHPCDs) which leads to the reachability problem becoming decidable. We then study which additional powers we must add to the RHPCD model to render it 1D PAM-equivalent. Finally, we show NP-hardness of reachability for nondeterministic RHPCDs

Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.Comment: 16 page

ParaPlan: A Tool for Parallel Reachability Analysis of Planar Polygonal Differential Inclusion Systems

Andrei Sandler, and Olga Tveretina, ‘ParaPlan: A Tool for Parallel Reachability Analysis of Planar Polygonal Differential Inclusion Systems’, in Patricia Bouyer, Andrea Orlandini and Pierluigi San Pietro, eds. Proceedings Eight International Symposium on Games, Automata, Logics and Formal Verification (GandALF 2017), Rome, Italy, 20-22 September 2017, Electronic Proceedings in Theoretical Computer Science, Vol. 256: 283-296, September 2017. © 2017 The Author(s). This work is licensed under the Creative Commons Attribution License CC BY 4.0 https://creativecommons.org/licenses/by/4.0/We present the ParaPlan tool which provides the reachability analysis of planar hybrid systems defined by differential inclusions (SPDI). It uses the parallelized and optimized version of the algorithm underlying the SPeeDI tool. The performance comparison demonstrates the speed-up of up to 83 times with respect to the sequential implementation on various benchmarks. Some of the benchmarks we used are randomly generated with the novel approach based on the partitioning of the plane with Voronoi diagrams

O-Minimal Hybrid Reachability Games

In this paper, we consider reachability games over general hybrid systems, and distinguish between two possible observation frameworks for those games: either the precise dynamics of the system is seen by the players (this is the perfect observation framework), or only the starting point and the delays are known by the players (this is the partial observation framework). In the first more classical framework, we show that time-abstract bisimulation is not adequate for solving this problem, although it is sufficient in the case of timed automata . That is why we consider an other equivalence, namely the suffix equivalence based on the encoding of trajectories through words. We show that this suffix equivalence is in general a correct abstraction for games. We apply this result to o-minimal hybrid systems, and get decidability and computability results in this framework. For the second framework which assumes a partial observation of the dynamics of the system, we propose another abstraction, called the superword encoding, which is suitable to solve the games under that assumption. In that framework, we also provide decidability and computability results

Deciding Reachability for Piecewise Constant Derivative Systems on Orientable Manifolds

© 2019 Springer-Verlag. This is a post-peer-review, pre-copyedit version of a paper published in Reachability Problems: 13th International Conference, RP 2019, Brussels, Belgium, September 11–13, 2019, Proceedings. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-030-30806-3_14A hybrid automaton is a finite state machine combined with some k real-valued continuous variables, where k determines the number of the automaton dimensions. This formalism is widely used for modelling safety-critical systems, and verification tasks for such systems can often be expressed as the reachability problem for hybrid automata. Asarin, Mysore, Pnueli and Schneider defined classes of hybrid automata lying on the boundary between decidability and undecidability in their seminal paper ‘Low dimensional hybrid systems - decidable, undecidable, don’t know’ [9]. They proved that certain decidable classes become undecidable when given a little additional computational power, and showed that the reachability question remains unsolved for some 2-dimensional systems. Piecewise Constant Derivative Systems on 2-dimensional manifolds (or PCD2m) constitute a class of hybrid automata for which decidability of the reachability problem is unknown. In this paper we show that the reachability problem becomes decidable for PCD2m if we slightly limit their dynamics, and thus we partially answer the open question of Asarin, Mysore, Pnueli and Schneider posed in [9]

Relaxing goodness is still good for SPDIs

Polygonal hybrid systems (SPDIs) are planar hybrid systems, whose dynamics are defined in terms of constant differential inclusions, one for each of a number of polygonal regions partitioning the plane. The reachability problem for SPDIs is known to be decidable, but depends on the goodness assumption — which states that the dynamics do not allow a trajectory to both enter and leave a region through the same edge. In this paper we extend the decidability result to generalised SPDIs (GSPDI), SPDIs not satisfying the goodness property, and give an algorithmic solution to decide reachability of such systems.peer-reviewe

Relaxing goodness is still good

Polygonal hybrid systems (SPDIs) are planar hybrid systems, whose dynamics are defined in terms of constant differential inclusions, one for each of a number of polygonal regions partitioning the plane. The reachability problem for SPDIs is known to be decidable, but depends on the goodnessassumption -- which states that the dynamics do not allow a trajectory to both enter and leave a region through the same edge. In this paper we extend the decidability result to generalised SPDIs(GSPDI), SPDIs not satisfying the goodness assumption, and give an algorithmic solution to decide reachability of such systems.peer-reviewe