In this paper we show that inference in 2-variable Markov logic networks (MLNs) with cardinality and function constraints is domain-liftable. To obtain this result we use existing domain-lifted algorithms for weighted first-order model counting (Van den Broeck et al, KR 2014) together with discrete Fourier transform of certain distributions associated to MLNs.
University of Trento（特伦托大学）；Fondazione Bruno Kessler（布鲁诺·凯塞勒基金会）
Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order theory on a given finite domain. WFOMC has emerged as a fundamental tool for probabilistic inference. Algorithms for WFOMC that run in polynomial time w.r.t. the domain size are called lifted inference algorithms. Such algorithms have been developed for multiple extensions of FO2(the fragment of first-order logic with two variables) for the special case of symmetric weight functions. We introduce the concept of lifted interpretations as a tool for formulating polynomials for WFOMC. Using lifted…
In this paper, we consider the problem of lifted inference in the context of Prism-like probabilistic logic programming languages. Traditional inference in such languages involves the construction of an explanation graph for the query and computing probabilities over this graph. When evaluating queries over probabilistic logic programs with a large number of instances of random variables, traditional methods treat each instance separately. For many programs and queries, we observe that explanations can be summarized into substantially more compact structures, which we call lifted explanation g…
Czech Technical University in Prague（布拉格捷克理工大学）；KU Leuven（鲁汶大学）
We study lifted weight learning of Markov logic networks. We show that there is an algorithm for maximum-likelihood learning of 2-variable Markov logic networks which runs in time polynomial in the domain size. Our results are based on existing lifted-inference algorithms and recent algorithmic results on computing maximum entropy distributions.
Fondazione Bruno Kessler（布鲁诺·凯斯勒基金会）；University of Trento（特伦托大学）
Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed-forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, …
We propose a new formalism for specifying and reasoning about problems that involve heterogeneous "pieces of information" -- large collections of data, decision procedures of any kind and complexity and connections between them. The essence of our proposal is to lift Codd's relational algebra from operations on relational tables to operations on classes of structures (with recursion), and to add a direction of information propagation. We observe the presence of information propagation in several formalisms for efficient reasoning and use it to express unary negation and operations used in grap…
Università di Ferrara（费拉拉大学）；Universidade do Porto（波尔图大学）
Lifted inference has been proposed for various probabilistic logical frameworks in order to compute the probability of queries in a time that depends on the size of the domains of the random variables rather than the number of instances. Even if various authors have underlined its importance for probabilistic logic programming (PLP), lifted inference has been applied up to now only to relational languages outside of logic programming. In this paper we adapt Generalized Counting First Order Variable Elimination (GC-FOVE) to the problem of computing the probability of queries to probabilistic lo…
TU Dresden（德累斯顿工业大学）；University of Wrocław（弗罗茨瓦夫大学）
We introduce and investigate the expressive description logic (DL) ALCSCC++, in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we prove that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, the satisfiability problem becomes undecidable if inverse roles are added to the languages. In addition, even without inverse roles, conjunctive query entailment in this DL turns out to be undecidable. We prove that decidability of querying can be regained if global and …
We study the complexity of the combination of the Description Logics ALCQ and ALCQI with a terminological formalism based on cardinality restrictions on concepts. These combinations can naturally be embedded into C^2, the two variable fragment of predicate logic with counting quantifiers, which yields decidability in NExpTime. We show that this approach leads to an optimal solution for ALCQI, as ALCQI with cardinality restrictions has the same complexity as C^2 (NExpTime-complete). In contrast, we show that for ALCQ, the problem can be solved in ExpTime. This result is obtained by a reduction …
Faculty of Electrical Engineering（电气工程学院）；Czech Technical University in Prague（布拉格捷克理工大学）
We consider the task of weighted first-order model counting (WFOMC) used for probabilistic inference in the area of statistical relational learning. Given a formula $ϕ$, domain size $n$ and a pair of weight functions, what is the weighted sum of all models of $ϕ$ over a domain of size $n$? It was shown that computing WFOMC of any logical sentence with at most two logical variables can be done in time polynomial in $n$. However, it was also shown that the task is $\texttt{#}P_1$-complete once we add the third variable, which inspired the search for extensions of the two-variable fragment that w…
First-order probabilistic models combine representational power of first-order logic with graphical models. There is an ongoing effort to design lifted inference algorithms for first-order probabilistic models. We analyze lifted inference from the perspective of constraint processing and, through this viewpoint, we analyze and compare existing approaches and expose their advantages and limitations. Our theoretical results show that the wrong choice of constraint processing method can lead to exponential increase in computational complexity. Our empirical tests confirm the importance of constra…
SRI International（SRI国际研究所）；U. of Texas at Dallas（德克萨斯大学达拉斯分校）；U. of California, Irvine（加州大学欧文分校）
We present SGDPLL(T), an algorithm that solves (among many other problems) probabilistic inference modulo theories, that is, inference problems over probabilistic models defined via a logic theory provided as a parameter (currently, propositional, equalities on discrete sorts, and inequalities, more specifically difference arithmetic, on bounded integers). While many solutions to probabilistic inference over logic representations have been proposed, SGDPLL(T) is simultaneously (1) lifted, (2) exact and (3) modulo theories, that is, parameterized by a background logic theory. This offers a foun…
KU Leuven（鲁汶大学）；University of California, Los Angeles（加州大学洛杉矶分校）
We propose an approach to lifted approximate inference for first-order probabilistic models, such as Markov logic networks. It is based on performing exact lifted inference in a simplified first-order model, which is found by relaxing first-order constraints, and then compensating for the relaxation. These simplified models can be incrementally improved by carefully recovering constraints that have been relaxed, also at the first-order level. This leads to a spectrum of approximations, with lifted belief propagation on one end, and exact lifted inference on the other. We discuss how relaxation…
Tufts University（塔夫茨大学）；University of Toronto（多伦多大学）
Lifted probabilistic inference (Poole, 2003) and symbolic dynamic programming for lifted stochastic planning (Boutilier et al, 2001) were introduced around the same time as algorithmic efforts to use abstraction in stochastic systems. Over the years, these ideas evolved into two distinct lines of research, each supported by a rich literature. Lifted probabilistic inference focused on efficient arithmetic operations on template-based graphical models under a finite domain assumption while symbolic dynamic programming focused on supporting sequential decision-making in rich quantified logical ac…
Spectra of Cardinality Queries over Description Logic Knowledge Bases
Leipzig University（莱比锡大学）；Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI)（可扩展数据分析与人工智能中心（ScaDS.AI））；University of Warsaw（华沙大学）
Lifting attempts to speed up probabilistic inference by exploiting symmetries in the model. Exact lifted inference methods, like their propositional counterparts, work by recursively decomposing the model and the problem. In the propositional case, there exist formal structures, such as decomposition trees (dtrees), that represent such a decomposition and allow us to determine the complexity of inference a priori. However, there is currently no equivalent structure nor analogous complexity results for lifted inference. In this paper, we introduce FO-dtrees, which upgrade propositional dtrees t…
University of Washington（华盛顿大学）；KU Leuven（鲁汶大学）；UCLA（加州大学洛杉矶分校）
In this paper we study lifted inference for the Weighted First-Order Model Counting problem (WFOMC), which counts the assignments that satisfy a given sentence in first-order logic (FOL); it has applications in Statistical Relational Learning (SRL) and Probabilistic Databases (PDB). We present several results. First, we describe a lifted inference algorithm that generalizes prior approaches in SRL and PDB. Second, we provide a novel dichotomy result for a non-trivial fragment of FO CNF sentences, showing that for each sentence the WFOMC problem is either in PTIME or #P-hard in the size of the …
KU Leuven（鲁汶大学）；University of Washington, Seattle（西雅图华盛顿大学）
Lifted probabilistic inference algorithms have been successfully applied to a large number of symmetric graphical models. Unfortunately, the majority of real-world graphical models is asymmetric. This is even the case for relational representations when evidence is given. Therefore, more recent work in the community moved to making the models symmetric and then applying existing lifted inference algorithms. However, this approach has two shortcomings. First, all existing over-symmetric approximations require a relational representation such as Markov logic networks. Second, the induced symmetr…
Lifted inference reduces the complexity of inference in relational probabilistic models by identifying groups of constants (or atoms) which behave symmetric to each other. A number of techniques have been proposed in the literature for lifting marginal as well MAP inference. We present the first application of lifting rules for marginal-MAP (MMAP), an important inference problem in models having latent (random) variables. Our main contribution is two fold: (1) we define a new equivalence class of (logical) variables, called Single Occurrence for MAX (SOM), and show that solution lies at extrem…
One of the big challenges in the development of probabilistic relational (or probabilistic logical) modeling and learning frameworks is the design of inference techniques that operate on the level of the abstract model representation language, rather than on the level of ground, propositional instances of the model. Numerous approaches for such "lifted inference" techniques have been proposed. While it has been demonstrated that these techniques will lead to significantly more efficient inference on some specific models, there are only very recent and still quite restricted results that show t…
Lifted Algorithms for Symmetric Weighted First-Order Model Sampling
Beihang University（北京航空航天大学）；Zhongfa Aviation Institute of Beihang University（北京航空航天大学中法航空学院）；ETH Zurich（苏黎世联邦理工学院）；Czech Technical University in Prague（捷克技术大学布拉格分校）
Czech Technical University in Prague（布拉格捷克理工大学）；Charles University（查理大学）；Cardiff University（卡迪夫大学）
We propose a method combining relational-logic representations with neural network learning. A general lifted architecture, possibly reflecting some background domain knowledge, is described through relational rules which may be handcrafted or learned. The relational rule-set serves as a template for unfolding possibly deep neural networks whose structures also reflect the structures of given training or testing relational examples. Different networks corresponding to different examples share their weights, which co-evolve during training by stochastic gradient descent algorithm. The framework…
We demonstrate a declarative differentiable programming framework based on the language of Lifted Relational Neural Networks, where small parameterized logic programs are used to encode relational learning scenarios. When presented with relational data, such as various forms of graphs, the program interpreter dynamically unfolds differentiable computational graphs to be used for the program parameter optimization by standard means. Following from the used declarative Datalog abstraction, this results into compact and elegant learning programs, in contrast with the existing procedural approache…
Lifted Model Construction without Normalisation: A Vectorised Approach to Exploit Symmetries in Factor Graphs
German Research Center for Artificial Intelligence (DFKI)（德国人工智能研究中心（DFKI））；Institute for Humanities-Centered Artificial Intelligence, University of Hamburg（汉堡大学人文中心人工智能研究所）
University of Edinburgh（爱丁堡大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of Toronto（多伦多大学）
Among the many approaches for reasoning about degrees of belief in the presence of noisy sensing and acting, the logical account proposed by Bacchus, Halpern, and Levesque is perhaps the most expressive. While their formalism is quite general, it is restricted to fluents whose values are drawn from discrete finite domains, as opposed to the continuous domains seen in many robotic applications. In this work, we show how this limitation in that approach can be lifted. By dealing seamlessly with both discrete distributions and continuous densities within a rich theory of action, we provide a very…
University of Huddersfield（哈德斯菲尔德大学）；Cardiff University（卡迪夫大学）；Università degli Studi di Brescia（布雷西亚大学）
Optimization - minimization or maximization - in the lattice of subsets is a frequent operation in Artificial Intelligence tasks. Examples are subset-minimal model-based diagnosis, nonmonotonic reasoning by means of circumscription, or preferred extensions in abstract argumentation. Finding the optimum among many admissible solutions is often harder than finding admissible solutions with respect to both computational complexity and methodology. This paper addresses the former issue by means of an effective method for finding subset-optimal solutions. It is based on the relationship between car…
The University of British Columbia（不列颠哥伦比亚大学）；KU Leuven（鲁汶大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Statistical relational models provide compact encodings of probabilistic dependencies in relational domains, but result in highly intractable graphical models. The goal of lifted inference is to carry out probabilistic inference without needing to reason about each individual separately, by instead treating exchangeable, undistinguished objects as a whole. In this paper, we study the domain recursion inference rule, which, despite its central role in early theoretical results on domain-lifted inference, has later been believed redundant. We show that this rule is more powerful than expected, a…
Probabilistic logic programming is a major part of statistical relational artificial intelligence, where approaches from logic and probability are brought together to reason about and learn from relational domains in a setting of uncertainty. However, the behaviour of statistical relational representations across variable domain sizes is complex, and scaling inference and learning to large domains remains a significant challenge. In recent years, connections have emerged between domain size dependence, lifted inference and learning from sampled subpopulations. The asymptotic behaviour of stati…
Fondazione Bruno Kessler（布鲁诺·凯塞勒基金会）；University of Trento（特伦托大学）
Markov Logic Networks (MLNs) define a probability distribution on relational structures over varying domain sizes. Many works have noticed that MLNs, like many other relational models, do not admit consistent marginal inference over varying domain sizes. Furthermore, MLNs learnt on a certain domain do not generalize to new domains of varied sizes. In recent works, connections have emerged between domain size dependence, lifted inference and learning from sub-sampled domains. The central idea to these works is the notion of projectivity. The probability distributions ascribed by projective mode…
State Key Lab of Novel Software Technology, Nanjing University（南京大学计算机软件新技术国家重点实验室）；University of Toronto（多伦多大学）；Birkbeck, University of London（伦敦大学伯贝克学院）
Czech Technical University in Prague（布拉格捷克理工大学）；ETH Zurich（苏黎世联邦理工学院）
We study computational aspects of relational marginal polytopes which are statistical relational learning counterparts of marginal polytopes, well-known from probabilistic graphical models. Here, given some first-order logic formula, we can define its relational marginal statistic to be the fraction of groundings that make this formula true in a given possible world. For a list of first-order logic formulas, the relational marginal polytope is the set of all points that correspond to the expected values of the relational marginal statistics that are realizable. In this paper, we study the foll…
