Michel de Rougemont

On the complexity of partially observed Markov decision processes

Abstract In the paper we consider the complexity of constructing optimal policies (strategies) for some type of partially observed Markov decision processes. This particular case of the classical problem deals with finite stationary processes, and can be represented as constructing optimal strategies to reach target vertices from a starting vertex in a graph with colored vertices and probabilistic deviations from an edge chosen to follow. The colors of the visited vertices is the only information available to a strategy. The complexity of Markov decision in the case of perfect information (bijective coloring of vertices) is known and briefly surveyed at the beginning of the paper. For the unobservable case (all the colors are equal) we give an improvement of the result of Papadimitriou and Tsitsiklis, namely we show that the problem of constructing even a very weak approximation to an optimal strategy is NP-hard. Our main results concern the case of a fixed bound on the multiplicity of coloring, that is a case of partially observed processes where some upper bound on the unobservability is supposed. We show that the problem of finding an optimal strategy is still NP-hard, but polytime approximations are possible. Some relations of our results to the Max-Word Problem are also indicated.

Probabilistic abstraction for model checking: An approach based on property testing

The goal of model checking is to verify the correctness of a given program, on all its inputs. The main obstacle, in many cases, is the intractably large size of the programs transition system. Property testing is a randomized method to verify whether some fixed property holds on individual inputs, by looking at a small random part of that input. We join the strengths of both approaches by introducing a new notion of probabilistic abstraction, and by extending the framework of model checking to include the use of these abstractions. Our abstractions map transition systems associated with large graphs to small transition systems associated with small random subgraphs. This reduces the original transition system to a family of small, even constant-size, transition systems. We prove that with high probability, “sufficiently” incorrect programs will be rejected (ϵ-robustness). We also prove that under a certain condition (exactness), correct programs will never be rejected (soundness). Our work applies to programs for graph properties such as bipartiteness, k-colorability, or any ∃∀ first order graph properties. Our main contribution is to show how to apply the ideas of property testing to syntactic programs for such properties. We give a concrete example of an abstraction for a program for bipartiteness. Finally, we show that the relaxation of the test alone does not yield transition systems small enough to use the standard model checking method. More specifically, we prove, using methods from communication complexity, that the OBDD size remains exponential for approximate bipartiteness.

Property Testing of Regular Tree Languages

Abstract We consider the edit distance with moves on the class of words and the class of ordered trees. We first exhibit a simple tester for the class of regular languages on words and generalize it to the class of ranked and unranked regular trees. We also show that this distance problem is

The reliability of queries (extended abstract)

\mathsf {NP}

Approximate Satisfiability and Equivalence

-complete on ordered trees.

Interactive Protocols on the Reals

We consider an unreliabable database as a random variable defined from a relational database with various probabilistic models. For a given query Q, we define its reliability on a database D15, pQ (lIll), as the probability that the answer to Q on an unreliable random instance coincides with the answer to Q on DB. We investigate the computational complexity of computing PQ (.DB), when Q is defined in various logic-based languages. We show that pQ (DB) is computable in polynomial time when Q is defined in first-order logic and that PQ (Ill?) is P#p computable when Q is defined in Datalog. We then discuss possible ways of estimating the reliability y for natural distributions.

Approximate verification and enumeration problems

Inspired by property testing, for every

Uniform generation in spatial constraint databases and applications

\varepsilon>0

StatsReduce in the cloud for approximate Analytics

we relax the classical satisfiability

A model of uncertainty for near-duplicates in document reference networks

U\models F