Algebraic model counting

Kimmig, Angelika; Broeck, Guy Van den; Raedt, Luc De

doi:10.1016/j.jal.2016.11.031

Cited by 35 publications

(73 citation statements)

References 15 publications

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“…We ignore this for the sake of simplicity and stick to rational weights. (See also [12]. ) We now define, for technical purposes, some restricted versions of WFOMC and the operator W. First, if M is a class of models, we define WFOMC(ϕ, n, w,w) ↾ M to be the sum of the weights W(M, w,w) of models M ∈ M with domain n and vocabulary voc(ϕ) such that M |= ϕ.…”

Section: Preliminariesmentioning

confidence: 99%

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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Section: Preliminariesmentioning

confidence: 99%

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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“…While in practical applications the weights are positive real numbers, and the probabilities are numbers in [0, 1], in this paper we impose no restrictions on the values of the weights and probabilities. The definition (2) of WMC(F, w) applies equally well to negative weights, and, in fact, to any semiring structure for the weights [25]. There is, in fact, at least one application of negative probabilities [22], namely the particular reduction from MLNs to WFOMC described in Example 1.2: a newly introduced relation has weight 1/(w − 1), which is negative when Problem Weights for R, S, and T tuples Solution for Φ = ∀x∀y(R(x) ∨ S(x, y) ∨ T (y)) Table 1: Three variants of WFOMC, of increasing generality, illustrated on the sentence Φ = ∀x∀y(R(x) ∨ S(x, y) ∨ T (y)).…”

Section: Weighted Model Counting (Wmc) the Modelmentioning

confidence: 99%

Symmetric Weighted First-Order Model Counting

Beame

Broeck

Gribkoff

et al. 2015

Proceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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The FO Model Counting problem (FOMC) is the following: given a sentence Φ in FO and a number n, compute the number of models of Φ over a domain of size n; the Weighted variant (WFOMC) generalizes the problem by associating a weight to each tuple and defining the weight of a model to be the product of weights of its tuples. In this paper we study the complexity of the symmetric WFOMC, where all tuples of a given relation have the same weight. Our motivation comes from an important application, inference in Knowledge Bases with soft constraints, like Markov Logic Networks, but the problem is also of independent theoretical interest. We study both the data complexity, and the combined complexity of FOMC and WFOMC. For the data complexity we prove the existence of an FO 3 formula for which FOMC is #P1-complete, and the existence of a Conjunctive Query for which WFOMC is #P1-complete. We also prove that all γ-acyclic queries have polynomial time data complexity. For the combined complexity, we prove that, for every fragment FO k , k ≥ 2, the combined complexity of FOMC (or WFOMC) is #P-complete.

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“…WMC is traditionally used for probabilistic inference in Bayesian networks (Chavira and Darwiche 2008) and probabilistic programming (Fierens et al 2015) with a factorized weight function: W MC(φ, w|B) = b∈I(φ(B)) b i ∈b w(b i ). Algebraic model counting (Kimmig, Van den Broeck, and De Raedt 2017) generalizes WMC to commutative semirings. More formally, Definition 5.…”

Section: Algebraic Model Countingmentioning

confidence: 99%

“…The key contribution of this paper is that we show how to handle actual probability density functions instead of piecewise polynomials in the context of WMI by applying standard knowledge compilation techniques. To this end, we cast weighted model integration within the framework of algebraic model counting (AMC) (Kimmig, Van den Broeck, and De Raedt 2017). More specifically, we make the following contributions:…”

Section: Introductionmentioning

confidence: 99%

Exact and Approximate Weighted Model Integration with Probability Density Functions Using Knowledge Compilation

Martires

Dries

Raedt

2019

AAAI

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Weighted model counting has recently been extended to weighted model integration, which can be used to solve hybrid probabilistic reasoning problems. Such problems involve both discrete and continuous probability distributions. We show how standard knowledge compilation techniques (to SDDs and d-DNNFs) apply to weighted model integration, and use it in two novel solvers, one exact and one approximate solver. Furthermore, we extend the class of employable weight functions to actual probability density functions instead of mere polynomial weight functions.

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Algebraic model counting

Cited by 35 publications

References 15 publications

Weighted model counting beyond two-variable logic

Weighted model counting beyond two-variable logic

Symmetric Weighted First-Order Model Counting

Exact and Approximate Weighted Model Integration with Probability Density Functions Using Knowledge Compilation

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