Symbolic Time and Space Tradeoffs for Probabilistic Verification

by   Krishnendu Chatterjee, et al.

We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal end-component (MEC) decomposition of Markov decision processes (MDPs). This problem generalizes the SCC decomposition problem of graphs and closed recurrent sets of Markov chains. The model of symbolic algorithms is widely used in formal verification and model-checking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of one-step neighborhood). For an input MDP with n vertices and m edges, the classical symbolic algorithm from the 1990s for the MEC decomposition requires O(n^2) symbolic operations and O(1) symbolic space. The only other symbolic algorithm for the MEC decomposition requires O(n √(m)) symbolic operations and O(√(m)) symbolic space. A main open question is whether the worst-case O(n^2) bound for symbolic operations can be beaten. We present a symbolic algorithm that requires O(n^1.5) symbolic operations and O(√(n)) symbolic space. Moreover, the parametrization of our algorithm provides a trade-off between symbolic operations and symbolic space: for all 0<ϵ≤ 1/2 the symbolic algorithm requires O(n^2-ϵ) symbolic operations and O(n^ϵ) symbolic space (O hides poly-logarithmic factors). Using our techniques we present faster algorithms for computing the almost-sure winning regions of ω-regular objectives for MDPs. We consider the canonical parity objectives for ω-regular objectives, and for parity objectives with d-priorities we present an algorithm that computes the almost-sure winning region with O(n^2-ϵ) symbolic operations and O(n^ϵ) symbolic space, for all 0 < ϵ≤ 1/2.


Symbolic Algorithms for Graphs and Markov Decision Processes with Fairness Objectives

Given a model and a specification, the fundamental model-checking proble...

Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

Solving parity games, which are equivalent to modal μ-calculus model che...

Symbolic Segmentation Using Algorithm Selection

In this paper we present an alternative approach to symbolic segmentatio...

Differentiation of the Cholesky decomposition

We review strategies for differentiating matrix-based computations, and ...

Near-Linear Time Algorithms for Streett Objectives in Graphs and MDPs

The fundamental model-checking problem, given as input a model and a spe...

Symbolic Generalization for On-line Planning

Symbolic representations have been used successfully in off-line plannin...

Fast Symbolic Algorithms for Omega-Regular Games under Strong Transition Fairness

We consider fixpoint algorithms for two-player games on graphs with ω-re...