Symbolic model checking for probabilistic processes

Abstract

Abstract.We introduce a symbolic model checking procedure for Probabilistic Computation Tree Logic PCTL over labelled Markov chains as models. Model checking for probabilistic logics typically involves solving linear equation systems in order to ascertain the probability of a given formula holding in a state. Our algorithm is based on the idea of representing the matrices used in the linear equation systems by Multi-Terminal Binary Decision Diagrams (MTBDDs) introduced in Clarke et al [14]. Our procedure, based on the algorithm used by Hansson and Jonsson [24], uses BDDs to represent formulas and MTBDDs to represent Markov chains, and is efficient because it avoids explicit state space construction. A PCTL model checker is being implemented in Verus [9].

Statistical Model Checking for Markov Decision Processes

“…There is extensive work on how the PMC problem can be solved through exact techniques [2], [3], [4], which compute correct probability bounds. Exact techniques do, however, rely on reasoning about the entire state space, which is widely considered to be the limiting factor in their applicability to large problems.…”

Faster and Symbolic CTMC Model Checking

“…MTBDDs are variants of BDDs that can efficiently deal with real matrices; they allow arbitrary real numbers in the terminal nodes instead of just 0 and 1. We show that CSL model checking can be treated using standard operations on MTBDDs, thus generalising the result for PCTL [5,18] to the continuous-time setting. This basically follows from the fact that CSL model checking amounts to the analysis of either the embedded discrete-time Markov chain (DTMC) -in the case of untimed until formulas and steady-state formulas -or the uniformised DTMC -in the case of time-bounded until -of the CTMC under consideration.…”

“…, b n ) where b i = 1 iff a i ∈ L(s). This encoding of states is standard [5,7,13]. Thus, S = { 0, 1 } n where each state s is identified with its encoding and R with the function F : { 0, 1 } 2n → IR where F (x 1 , y 1 , .…”

“…For each CSL formula Φ a BDD Sat[[ Φ ]] is defined that represents the characteristic function of the set Sat(Φ). By applying standard operators on MTBDDs we determine the MTBDDs P representing the transition probability matrix P of emb(C), and E the vector of exit rates E. Then: The steady state and unbounded until operators are treated symbolically as described in [5] and [7] respectively.…”

Abstract Interpretation of Programs as Markov Decision Processes

“…More recently, they have been studied from the angle of probabilistic computing systems [1-3, 9, 10, 18, 30]. Effective resolution techniques include linear programming [28, §7.2.7] [6] and newer data structures such as MTBDDs (multiterminal binary decision diagrams) [1]. However, the problem of large-or infinite-state systems has not been so well studied.…”