Specification-Guided Verification and Abstraction Refinement of Mixed Monotone Stochastic Systems

Abstract

This paper addresses the problem of verifying discrete-time stochastic systems against omega-regular specifications using finite-state abstractions. Omegaregular properties allow specifying complex behavior and encompass, for example, linear temporal logic. We focus on a class of systems with mixed monotone dynamics. This class has been show to be amenable to efficient reachable set computation and models a wide-range of physically relevant systems. In general, finite-state abstractions of continuous state stochastic systems give rise to augmented Markov Chains wherein the probabilities of transition between states are restricted to an interval. We present a procedure to compute a finite-state Interval-valued Markov Chain abstraction of discrete-time, mixed-monotone stochastic systems subject to affine disturbances given a rectangular partition of the state-space. Then, we suggest an algorithm for performing verification against omega-regular properties in IMCs. Specifically, we aim to compute bounds on the probability of satisfying a specification from any initial state in the IMC. This is achieved by solving a reachability problem on sets of socalled winning and losing components in the Cartesian product between the IMC and a Rabin automaton representing the specification. Next, the verification of IMCs may yield a set of states whose acceptance status is undecided with respect to the specification, requiring a refinement of the abstraction. We describe a specification-guided approach that compares the best and worst-case behaviors of accepting paths in the IMC and targets the appropriate states accordingly. Finally, we show a case study.

Abstraction-based Synthesis for Stochastic Systems with Omega-Regular Objectives

“…As discussed in Section 2, for every such property, there exists a corresponding DRA representation A. Similar to [11, page 798] and [23] where the Cartesian product with a Markov Chain (MC) and an IMC are introduced, we define the product B ⊗ A between a BMDP and a DRA. Definition 7 (Product Bounded-Parameter Markov Decision Process).…”

“…Before presenting a solution to Subproblem 1.1, we first recall some basic results established in [23] for the purpose of verification in IMCs which we then extend to compute switching policies in BMDPs.…”

“…For a given policy µ of B and automaton A, the sets of accepting and nonaccepting BSCCs of the resulting product IMC B[µ] ⊗ A depend on the assumed values for the transitions with zero lower bound and non-zero upper bound which cause certain edges to be either "on" or "off". To resolve this, [23] showed that, for any product IMC, there exists a largest winning component and a largest losing component which can be created among all combinations of "on" and "off" transitions allowed by the transition bound functions of the product IMC.…”

“…Next, in [23, Theorem 1], it was shown that a lower bound on the probability of reaching an accepting BSCC in a product IMC I ⊗ A is achieved in an induced product MC (M A ⊗ ) with the smallest possible set of winning components admissible by I ⊗ A, which is the permanent winning component (W C) P of I ⊗ A, for all states of I ⊗ A. Furthermore, it was shown in [23,Lemma 9] that the probability of reaching an accepting BSCC in an induced product MC (M A ⊗ ) increases for all states of (M A ⊗ ) as more states are added to the set of winning components of (M A ⊗ ) while keeping all other transition probabilities identical. Therefore, for all states of B ⊗ A which are not in (W C) G P , a policy µ maximizing the lower bound probability of reaching a winning component has to belong to the set U (W C) G P and generates the largest possible permanent winning component in (B ⊗ A) [µ].…”

“…Furthermore, we introduce a quantitative measure capturing the degree of optimality of the switching policy designed in the BMDP abstraction when mapped onto the continuous abstracted states with respect to the objective of minimizing or maximizing the probability of fulfilling some specification in the original system. Finally, we propose a partition refinement technique inspired by our method in [23], which considered only the verification problem without inputs, in order to reach a desired level of optimality for the computed policy with respect to the continuous system states and progressively discard control actions which are guaranteed to be suboptimal.…”