
Symbolic Algorithms for Graphs and Markov Decision Processes with Fairness Objectives
Given a model and a specification, the fundamental modelchecking proble...
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Quasipolynomial SetBased Symbolic Algorithms for Parity Games
Solving parity games, which are equivalent to modal μcalculus model che...
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Differentiation of the Cholesky decomposition
We review strategies for differentiating matrixbased computations, and ...
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NearLinear Time Algorithms for Streett Objectives in Graphs and MDPs
The fundamental modelchecking problem, given as input a model and a spe...
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Symbolic Segmentation Using Algorithm Selection
In this paper we present an alternative approach to symbolic segmentatio...
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Symbolic Generalization for Online Planning
Symbolic representations have been used successfully in offline plannin...
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Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety, and Diameter
A model of computation that is widely used in the formal analysis of rea...
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Symbolic Time and Space Tradeoffs for Probabilistic Verification
We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal endcomponent (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 modelchecking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of onestep 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 worstcase 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 tradeoff between symbolic operations and symbolic space: for all 0<ϵ≤ 1/2 the symbolic algorithm requires O(n^2ϵ) symbolic operations and O(n^ϵ) symbolic space (O hides polylogarithmic factors). Using our techniques we present faster algorithms for computing the almostsure winning regions of ωregular objectives for MDPs. We consider the canonical parity objectives for ωregular objectives, and for parity objectives with dpriorities we present an algorithm that computes the almostsure winning region with O(n^2ϵ) symbolic operations and O(n^ϵ) symbolic space, for all 0 < ϵ≤ 1/2.
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