Simple closed-form upper and lower bounds are developed for the security of the Nakamoto consensus as a function of the confirmation depth, the honest and adversarial block mining rates, and an upper bound on the block propagation delay. The bounds are exponential in the confirmation depth and apply regardless of the adversary's attack strategy. The gap between the upper and lower bounds is small for Bitcoin's parameters. For example, assuming an average block interval of 10 minutes, a network delay bound of ten seconds, and 10% adversarial mining power, the widely used 6-block confirmation rule yields a safety violation between 0.11% and 0.35% probability.
more »
« less
Unbounded-Time Safety Verification of Stochastic Differential Dynamics.
In this paper, we propose a method for bounding the probability that a stochastic differential equation (SDE) system violates a safety specification over the infinite time horizon. SDEs are mathematical models of stochastic processes that capture how states evolve continuously in time. They are widely used in numerous applications such as engineered systems (e.g., modeling how pedestrians move in an intersection), computational finance (e.g., modeling stock option prices), and ecological processes (e.g., population change over time). Previously the safety verification problem has been tackled over finite and infinite time horizons using a diverse set of approaches. The approach in this paper attempts to connect the two views by first identifying a finite time bound, beyond which the probability of a safety violation can be bounded by a negligibly small number. This is achieved by discovering an exponential barrier certificate that proves exponentially converging bounds on the probability of safety violations over time. Once the finite time interval is found, a finite-time verification approach is used to bound the probability of violation over this interval. We demonstrate our approach over a collection of interesting examples from the literature, wherein our approach can be used to find tight bounds on the violation probability of safety properties over the infinite time horizon.
more »
« less
- Award ID(s):
- 1815983
- PAR ID:
- 10233219
- Date Published:
- Journal Name:
- Computer-Aided Verification (CAV)
- Volume:
- 12225
- Page Range / eLocation ID:
- 327-348
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This paper presents a model-free reinforcement learning (RL) algorithm for infinite-horizon average-reward Constrained Markov Decision Processes (CMDPs). Considering a learning horizon K, which is sufficiently large, the proposed algorithm achieves sublinear regret and zero constraint violation. The bounds depend on the number of states S, the number of actions A, and two constants which are independent of the learning horizon K.more » « less
-
Finkbeiner, B. ; Wies, T. (Ed.)Stochastic model checking (SMC) is a formal verification technique for the analysis of systems with probabilistic behavior. Scalability has been a major limiting factor for SMC tools to analyze real-world systems with large or infinite state spaces. The infinite-state Continuous-time Markov Chain (CTMC) model checker, STAMINA, tackles this problem by selectively exploring only a portion of a model’s state space, where a majority of the probability mass resides, to efficiently give an accurate probability bound to properties under verification. In this paper, we present two major improvements to STAMINA, namely, a method of calculating and distributing estimated state reachability probabilities that improves state space truncation efficiency and combination of the previous two CTMC analyses into one for generating the probability bound. Demonstration of the improvements on several benchmark examples, including hazard analysis of infinite-state combinational genetic circuits, yield significant savings in both run-time and state space size (and hence memory), compared to both the previous version of STAMINA and the infinite-state CTMC model checker INFAMY. The improved STAMINA demonstrates significant scalability to allow for the verification of complex real-world infinite-state systems.more » « less
-
In this paper we present a method based on linear programming that facilitates reliable safety verification of hybrid dynamical systems subject to perturbation inputs over the infinite time horizon. The verification algorithm applies the probably approximately correct (PAC) learning framework and consequently can be regarded as statistically formal verification in the sense that it provides formal safety guarantees expressed using error probabilities and confidences. The safety of hybrid systems in this framework is verified via the computation of so-called PAC barrier certificates, which can be computed by solving a linear programming problem. Based on scenario approaches, the linear program is constructed by a family of independent and identically distributed state samples. In this way we can conduct verification of hybrid dynamical systems that existing methods are not capable of dealing with. Some preliminary experiments demonstrate the performance of our approach.more » « less
-
This paper studies the satisfaction of a class of temporal properties for cyber-physical systems (CPSs) over a finite-time horizon in the presence of an adversary, in an environment described by discretetime dynamics. The temporal logic specification is given in safe−LTLF , a fragment of linear temporal logic over traces of finite length. The interaction of the CPS with the adversary is modeled as a two-player zerosum discrete-time dynamic stochastic game with the CPS as defender. We formulate a dynamic programming based approach to determine a stationary defender policy that maximizes the probability of satisfaction of a safe − LTLF formula over a finite time-horizon under any stationary adversary policy. We introduce secure control barrier certificates (S-CBCs), a generalization of barrier certificates and control barrier certificates that accounts for the presence of an adversary, and use S-CBCs to provide a lower bound on the above satisfaction probability. When the dynamics of the evolution of the system state has a specific underlying structure, we present a way to determine an S-CBC as a polynomial in the state variables using sum-of-squares optimization. An illustrative example demonstrates our approach.more » « less