Satisfiability Bounds for co-regular Properties in Interval-valued Markov Chains

Abstract

We derive an algorithm to compute satisfiability bounds for arbitrary ω-regular properties in an Interval-valued Markov Chain (IMC) interpreted in the adversarial sense. IMCs generalize regular Markov Chains by assigning a range of possible values to the transition probabilities between states. In particular, we expand the automata-based theory of ω-regular property verification in Markov Chains to apply it to IMCs.Any ω-regular property can be represented by a Deterministic Rabin Automata (DRA) with acceptance conditions expressed by Rabin pairs. Previous works on Markov Chains have shown that computing the probability of satisfying a given ω-regular property reduces to a reachability problem in the product between the Markov Chain and the corresponding DRA. We similarly define the notion of a product between an IMC and a DRA. Then, we show that in a product IMC, there exists a particular assignment of the transition values that generates a largest set of non-accepting states. Subsequently, we prove that a lower bound is found by solving a reachability problem in that refined version of the original product IMC. We derive a similar approach for computing a satisfiability upper bound in a product IMC with one Rabin pair. For product IMCs with more than one Rabin pair, we establish that computing a satisfiability upper bound is equivalent to lower-bounding the satisfiability of the complement of the original property. A search algorithm for finding the largest accepting and nonaccepting sets of states in a product IMC is proposed. Finally, we demonstrate our findings in a case study. arXiv:1809.06352v1 [cs.SY]

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

“…Preliminary results were reported in the conference papers [13] and [26]. The verification approach considered here is a significant improvement of these prior works.…”

“…Proof: We proved in [26] that any product IMC induces a set of MCs with a largest set of non-accepting BSCCs. Lemma 5 is deduced from this fact and Lemma 4.…”

“…There exists a set of product MCs induced by I ⊗ A such that U A 1 ∪ U A 2 are accepting BSCCs (see [26]), and thus winning components. If…”

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“…In [15], Dutreix and Coogan argue for computing minimum and maximum probabilities of satisfying an ω-regular property in an IMC interpreted as IMDP. In future work, they wish to apply the technique to solve the problem for BMDP, the controllable counterpart of IMC.…”

“…In this paper, we re-interpret their technique in a different light and using that perspective give a solution to BMDP, in both the uncertain and the adversarial understanding of the intervals. We consider both the upper bound (also called design choice of values in intervals [15]) and the lower bound (antagonistic in [15]). We present the results for controllers that try to maximize the probability to satisfy the ω-regular property; minimization is analogous as ω-regular languages are closed under complement.…”