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Register a new userApproximate Weighted First-Order Model Counting: Exploiting Fast Approximate Model Counters and Symmetry
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Added by
Ondrej Kuzelka
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We study the symmetric weighted first-order model counting task and present ApproxWFOMC, a novel anytime method for efficiently bounding the weighted first-order model count in the presence of an unweighted first-order model counting oracle. The algorithm has applications to inference in a variety of first-order probabilistic representations, such as Markov logic networks and probabilistic logic programs. Crucially for many applications, we make no assumptions on the form of the input sentence. Instead, our algorithm makes use of the symmetry inherent in the problem by imposing cardinality constraints on the number of possible true groundings of a sentences literals. Realising the first-order model counting oracle in practice using the approximate hashing-based model counter ApproxMC3, we show how our algorithm outperforms existing approximate and exact techniques for inference in first-order probabilistic models. We additionally provide PAC guarantees on the generated bounds.
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In human-aware planning, a planning agent may need to provide an explanation to a human user on why its plan is optimal. A popular approach to do this is called model reconciliation, where the agent tries to reconcile the differences in its model and the humans model such that the plan is also optimal in the humans model. In this paper, we present a logic-based framework for model reconciliation that extends beyond the realm of planning. More specifically, given a knowledge base $KB_1$ entailing a formula $varphi$ and a second knowledge base $KB_2$ not entailing it, model reconciliation seeks an explanation, in the form of a cardinality-minimal subset of $KB_1$, whose integration into $KB_2$ makes the entailment possible. Our approach, based on ideas originating in the context of analysis of inconsistencies, exploits the existing hitting set duality between minimal correction sets (MCSes) and minimal unsatisfiable sets (MUSes) in order to identify an appropriate explanation. However, differently from those works targeting inconsistent formulas, which assume a single knowledge base, MCSes and MUSes are computed over two distinct knowledge bases. We conclude our paper with an empirical evaluation of the newly introduced approach on planning instances, where we show how it outperforms an existing state-of-the-art solver, and generic non-planning instances from recent SAT competitions, for which no other solver exists.
In 1998, Brassard, Hoyer, Mosca, and Tapp (BHMT) gave a quantum algorithm for approximate counting. Given a list of $N$ items, $K$ of them marked, their algorithm estimates $K$ to within relative error $varepsilon$ by making only $Oleft( frac{1}{varepsilon}sqrt{frac{N}{K}}right) $ queries. Although this speedup is of Grover type, the BHMT algorithm has the curious feature of relying on the Quantum Fourier Transform (QFT), more commonly associated with Shors algorithm. Is this necessary? This paper presents a simplified algorithm, which we prove achieves the same query complexity using Grover iterations only. We also generalize this to a QFT-free algorithm for amplitude estimation. Related approaches to approximate counting were sketched previously by Grover, Abrams and Williams, Suzuki et al., and Wie (the latter two as we were writing this paper), but in all cases without rigorous analysis.
We define a new measure of network symmetry that is capable of capturing approximate global symmetries of networks. We apply this measure to different networks sampled from several classic network models, as well as several real-world networks. We find that among the network models that we have examined, Erdos-Renyi networks have the least levels of symmetry, and Random Geometric Graphs are likely to have high levels of symmetry. We find that our network symmetry measure can capture properties of network structure, and help us gain insights on the structure of real-world networks. Moreover, our network symmetry measure is capable of capturing imperfect network symmetry, which would have been undetected if only perfect symmetry is considered.
We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is emph{nicely locally cwd-decomposable}. This notion generalizes that of a emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded emph{clique-width} with limited overlaps. We also consider such labelings for emph{bounded} first-order formulas on graph classes of emph{bounded expansion}. Some of these results are extended to counting queries.
The graphical structure of Probabilistic Graphical Models (PGMs) encodes the conditional independence (CI) relations that hold in the modeled distribution. Graph algorithms, such as d-separation, use this structure to infer additional conditional independencies, and to query whether a specific CI holds in the distribution. The premise of all current systems-of-inference for deriving CIs in PGMs, is that the set of CIs used for the construction of the PGM hold exactly. In practice, algorithms for extracting the structure of PGMs from data, discover approximate CIs that do not hold exactly in the distribution. In this paper, we ask how the error in this set propagates to the inferred CIs read off the graphical structure. More precisely, what guarantee can we provide on the inferred CI when the set of CIs that entailed it hold only approximately? It has recently been shown that in the general case, no such guarantee can be provided. We prove that such a guarantee exists for the set of CIs inferred in directed graphical models, making the d-separation algorithm a sound and complete system for inferring approximate CIs. We also prove an approximation guarantee for independence relations derived from marginal CIs.
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