Symmetric Weighted First-Order Model Counting

Beame, Paul; Broeck, Guy Van den; Gribkoff, Eric; Suciu, Dan

doi:10.1145/2745754.2745760

Cited by 46 publications

(91 citation statements)

References 42 publications

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“…For instance, the result for FO 2 -properties from [4] shows that all these properties can be associated with tractable enumeration functions. For discussions of the links between weighted model counting, the spectrum problem and 0-1 laws, see [3].In the current paper, we extend the result of [4] for FO 2 in two independent directions. We first consider FO 2 with a functionality axiom, that is, sentences of type ϕ ∧ ∀x∃ =1 y ψ(x, y) with ϕ and ψ in FO 2 .…”

supporting

confidence: 60%

“…We note that the proof of the Proposition makes use of the results and techniques of [4,3] in various ways, and thus much of the credit goes there. We sketch the proof-see Appendix C for more details.…”

Section: Counting and Prefix Classesmentioning

confidence: 99%

“…Firstly, [3] shows that there is an FO 3 -sentence ϕ with a #P 1 -complete model counting problem. We turn ϕ into a conjunction of prenex form sentences by eliminating quantified subformulae in a way resembling the Scott normal form procedure.…”

Section: Counting and Prefix Classesmentioning

confidence: 99%

“…We also identify a complete classification of first-order prefix classes according to whether WFOMC is in polynomial time or #P1-complete.The recent article [4] established the by now well-known result that the data complexity of WFOMC is in polynomial time for formulae of FO 2 , while [3] demonstrated that the three-variable fragment FO 3 contains formulae for which the problem is #P 1 -complete. We note that the non-symmetric variant of the problem is known to be #P-complete for some FO 2 -sentences [3].Weighted model counting problems have a range of well-known applications. For example, as pointed out in [3], WFOMC problems occur in a natural way in knowledge bases with soft constraints and are especially prominent in the area of Markov logic [6].…”

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confidence: 99%

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Weighted model counting beyond two-variable logic

Kuusisto

Lutz

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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It was recently shown by van den Broeck at al. that the symmetric weighted first-order model counting problem (WFOMC) for sentences of two-variable logic FO 2 is in polynomial time, while it is #P1-complete for some FO 3 -sentences. We extend the result for FO 2 in two independent directions: to sentences of the form ϕ ∧ ∀x∃ =1 y ψ(x, y) with ϕ and ψ formulated in FO 2 and to sentences of the uniform one-dimensional fragment U1 of FO, a recently introduced extension of two-variable logic with the capacity to deal with relation symbols of all arities. We note that the former generalizes the extension of FO 2 with a functional relation symbol. We also identify a complete classification of first-order prefix classes according to whether WFOMC is in polynomial time or #P1-complete.The recent article [4] established the by now well-known result that the data complexity of WFOMC is in polynomial time for formulae of FO 2 , while [3] demonstrated that the three-variable fragment FO 3 contains formulae for which the problem is #P 1 -complete. We note that the non-symmetric variant of the problem is known to be #P-complete for some FO 2 -sentences [3].Weighted model counting problems have a range of well-known applications. For example, as pointed out in [3], WFOMC problems occur in a natural way in knowledge bases with soft constraints and are especially prominent in the area of Markov logic [6]. For a recent comprehensive survey on these matters, see [5]. From a mathematical perspective, WFOMC offers a neat and general approach to elementary enumerative combinatorics. To give a simple illustration of this, consider WFOMC(ϕ, n, w,w) for the two-variable logic sentence ϕ = ∀x∀y(Rxy → (Ryx ∧ x = y)) with w(R) =w(R) = 1. The sentence states that R encodes a simple undirected graph and thus WFOMC(ϕ, n, w,w) = 2 ( n 2 ) , the number of graphs of order n (with the set n of vertices). Thus WFOMC provides a logic-based way of classifying combinatorial problems. For instance, the result for FO 2 -properties from [4] shows that all these properties can be associated with tractable enumeration functions. For discussions of the links between weighted model counting, the spectrum problem and 0-1 laws, see [3].In the current paper, we extend the result of [4] for FO 2 in two independent directions. We first consider FO 2 with a functionality axiom, that is, sentences of type ϕ ∧ ∀x∃ =1 y ψ(x, y) with ϕ and ψ in FO 2 . This extension is motivated, inter alia, by certain description logics with functional roles [1]. The connection of WFOMC to enumerative combinatorics also provides an important part of the motivation. Indeed, while FO 2 is a reasonable formalism for specifying properties of relations, adding functionality axioms allows us to also express properties of functions, possibly combined with relations. For example, applying WFOMC to the sentence ∀x¬Rxx ∧ ∀x∃ =1 yRxy gives the number of functions that do not have a fixed point. While the extension of FO 2 with a functionality axiom might appear simple at first sight, showing th...

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supporting

confidence: 60%

Section: Counting and Prefix Classesmentioning

confidence: 99%

Section: Counting and Prefix Classesmentioning

confidence: 99%

mentioning

confidence: 99%

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Weighted model counting beyond two-variable logic

Kuusisto

Lutz

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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“…Any combination of universal and existential quantification is liftable ). More tractable and intractable classes are discussed in Beame et al (2015). This list of known tractable classes is far from exhaustive, and many more complex MLNs are liftable, including ones with more than two variables.…”

Section: Lifted Probabilistic Inference and Tractabilitymentioning

confidence: 99%

Lifted generative learning of Markov logic networks

et al. 2015

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Markov logic networks (MLNs) are a well-known statistical relational learning formalism that combines Markov networks with first-order logic. MLNs attach weights to formulas in first-order logic. Learning MLNs from data is a challenging task as it requires searching through the huge space of possible theories. Additionally, evaluating a theory's likelihood requires learning the weight of all formulas in the theory. This in turn requires performing probabilistic inference, which, in general, is intractable in MLNs. Lifted inference speeds up probabilistic inference by exploiting symmetries in a model. We explore how to use lifted inference when learning MLNs. Specifically, we investigate generative learning where the goal is to maximize the likelihood of the model given the data. First, we provide a generic algorithm for learning maximum likelihood weights that works with any exact lifted inference approach. In contrast, most existing approaches optimize approximate measures such as the pseudo-likelihood. Second, we provide a concrete parameter learning algorithm based on first-order knowledge compilation. Third, we propose a structure learning algorithm that learns liftable MLNs, which is the first MLN structure learning algorithm that exactly optimizes the likelihood of the model. Finally, we perform an empirical evaluation on three real-world datasets. Our parameter learning algorithm results in more accurate models than several competing approximate approaches. It learns more accurate models in terms of test-set log-likelihood as well as prediction tasks. Furthermore, our tractable learner outperforms intractable models on prediction tasks suggesting that liftable models are a powerful hypothesis space, which may be sufficient for many standard learning problems.

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Uncertain Data Lineage

Roy¹

2018

Encyclopedia of Database Systems

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Symmetric Weighted First-Order Model Counting

Cited by 46 publications

References 42 publications

Weighted model counting beyond two-variable logic

Weighted model counting beyond two-variable logic

Lifted generative learning of Markov logic networks

Uncertain Data Lineage

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