Best Probabilistic Transformers

Abstract

Abstract. This paper investigates relative precision and optimality of analyses for concurrent probabilistic systems. Aiming at the problem at the heart of probabilistic model checking -computing the probability of reaching a particular set of states -we leverage the theory of abstract interpretation. With a focus on predicate abstraction, we develop the first abstract-interpretation framework for Markov decision processes which admits to compute both lower and upper bounds on reachability probabilities. Further, we describe how to compute and approximate such abstractions using abstraction refinement and give experimental results.

Advances and challenges of probabilistic model checking

“…The bounds also identify which parts of the model need to be refined. This process is repeated iteratively until the error falls below some prespecified tolerance ε. Quantitative abstraction refinement has been applied to the verification of PTAs [18], a probabilistic extension of ANSI-C [28] and PRISM models [29].…”

A Framework for Verification of Software with Time and Probabilities

“…These reachability values can be determined efficiently using value iteration [40] together with the corresponding adversary or strategy-pair which achieves the value. In [47], the game-based abstraction of Definition 3 above is phrased in terms of abstract interpretation [9] and the resulting bounds obtained are shown to coincide with the "best" values obtainable for a fixed abstraction of the MDP. [27,32,26] is an approach for automatically constructing abstractions of probabilistic models.…”

“…[27,32,26] is an approach for automatically constructing abstractions of probabilistic models. Using the notion of abstracting MDPs with stochastic games describe above, it has been successfully applied to the verification of a probabilistic extension of ANSI-C [26], probabilistic timed automata [32] and concurrent probabilistic systems [47].…”

“…In [20], MDP abstractions based on predicates are used to form a probabilistic CEGAR (counterexample-guided abstraction refinement) technique. Later work [47] adapted this to stochastic games. Other abstraction refinement frameworks for MDPs are put forward in [12] and [6].…”

Game-based Abstraction and Controller Synthesis for Probabilistic Hybrid Systems

“…This makes use of an abstraction introduced in [6], [7]. The idea in this case is to, as above, introduce a 3rd player to represent abstraction, but then to make this new player collaborate with the player representing the environment in the PHA.…”