120,990 research outputs found
Breaking Dense Structures: Proving Stability of Densely Structured Hybrid Systems
Abstraction and refinement is widely used in software development. Such
techniques are valuable since they allow to handle even more complex systems.
One key point is the ability to decompose a large system into subsystems,
analyze those subsystems and deduce properties of the larger system. As
cyber-physical systems tend to become more and more complex, such techniques
become more appealing.
In 2009, Oehlerking and Theel presented a (de-)composition technique for
hybrid systems. This technique is graph-based and constructs a Lyapunov
function for hybrid systems having a complex discrete state space. The
technique consists of (1) decomposing the underlying graph of the hybrid system
into subgraphs, (2) computing multiple local Lyapunov functions for the
subgraphs, and finally (3) composing the local Lyapunov functions into a
piecewise Lyapunov function. A Lyapunov function can serve multiple purposes,
e.g., it certifies stability or termination of a system or allows to construct
invariant sets, which in turn may be used to certify safety and security.
In this paper, we propose an improvement to the decomposing technique, which
relaxes the graph structure before applying the decomposition technique. Our
relaxation significantly reduces the connectivity of the graph by exploiting
super-dense switching. The relaxation makes the decomposition technique more
efficient on one hand and on the other allows to decompose a wider range of
graph structures.Comment: In Proceedings ESSS 2015, arXiv:1506.0325
Proving Expected Sensitivity of Probabilistic Programs with Randomized Variable-Dependent Termination Time
The notion of program sensitivity (aka Lipschitz continuity) specifies that
changes in the program input result in proportional changes to the program
output. For probabilistic programs the notion is naturally extended to expected
sensitivity. A previous approach develops a relational program logic framework
for proving expected sensitivity of probabilistic while loops, where the number
of iterations is fixed and bounded. In this work, we consider probabilistic
while loops where the number of iterations is not fixed, but randomized and
depends on the initial input values. We present a sound approach for proving
expected sensitivity of such programs. Our sound approach is martingale-based
and can be automated through existing martingale-synthesis algorithms.
Furthermore, our approach is compositional for sequential composition of while
loops under a mild side condition. We demonstrate the effectiveness of our
approach on several classical examples from Gambler's Ruin, stochastic hybrid
systems and stochastic gradient descent. We also present experimental results
showing that our automated approach can handle various probabilistic programs
in the literature
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Combining centralised and distributed testing
Many systems interact with their environment at distributed interfaces (ports) and sometimes it is not possible to place synchronised local testers at the ports of the system under test (SUT). There are then two main approaches to testing: having independent local testers or a single centralised tester that interacts asynchronously with the SUT. The power of using independent testers has been captured using implementation relation \dioco. In this paper we define implementation relation \diococ for the centralised approach and prove that \dioco and \diococ are incomparable. This shows that the frameworks detect different types of faults and so we devise a hybrid framework and define an implementation relation \diocos for this. We prove that the hybrid framework is more powerful than the distributed and centralised approaches. We then prove that the Oracle problem is NP-complete for \diococ and \diocos but can be solved in polynomial time if we place an upper bound on the number of ports. Finally, we consider the problem of deciding whether there is a test case that is guaranteed to force a finite state model into a particular state or to distinguish two states, proving that both problems are undecidable for the centralised and hybrid frameworks
Formal methods for modeling and analysis of hybrid systems
A technique based on the use of a quantifier elimination decision procedure for real closed fields and simple theorem proving to construct a series of successively finer qualitative abstractions of hybrid automata is taught. The resulting abstractions are always discrete transition systems which can then be used by any traditional analysis tool. The constructed abstractions are conservative and can be used to establish safety properties of the original system. The technique works on linear and non-linear polynomial hybrid systems: the guards on discrete transitions and the continuous flows in all modes can be specified using arbitrary polynomial expressions over the continuous variables. An exemplar tool in the SAL environment built over the theorem prover PVS is detailed. The technique scales well to large and complex hybrid systems
Constructive Hybrid Games
Hybrid games are models which combine discrete, continuous, and adversarial
dynamics. Game logic enables proving (classical) existence of winning
strategies. We introduce constructive differential game logic (CdGL) for hybrid
games, where proofs that a player can win the game correspond to computable
winning strategies. This is the logical foundation for synthesis of correct
control and monitoring code for safety-critical cyber-physical systems. Our
contributions include novel static and dynamic semantics as well as soundness
and consistency.Comment: 60 pages, preprint, under revie
Convex Programs for Temporal Verification of Nonlinear Dynamical Systems
A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. The dual of this problem, i.e., the reachability problem, concerns proving the existence of a trajectory starting from the initial set that reaches another given set. Using insights from the linear programming duality appearing in the discrete shortest path problem, we show in this paper that reachability of continuous systems can also be verified through convex programming. Several convex programs for verifying safety and reachability, as well as other temporal properties such as eventuality, avoidance, and their combinations, are formulated. Some examples are provided to illustrate the application of the proposed methods. Finally, we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates
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