Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting

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 input domain; we prove that, in the first case our algorithm solves the WFOMC problem in PTIME, and in the second case it fails. Third, we present several properties of the algorithm. Finally, we discuss limitations of lifted inference for symmetric probabilistic databases (where the weights of ground literals depend only on the relation name, and not on the constants of the domain), and prove the impossibility of a dichotomy result for the complexity of probabilistic inference for the entire language FOL

Diffusion and Current of Brownian Particles in Tilted Piecewise Linear Potentials: Amplification and Coherence

Overdamped motion of Brownian particles in tilted piecewise linear periodic potentials is considered. Explicit algebraic expressions for the diffusion coefficient, current, and coherence level of Brownian transport are derived. Their dependencies on temperature, tilting force, and the shape of the potential are analyzed. The necessary and sufficient conditions for the non-monotonic behavior of the diffusion coefficient as a function of temperature are determined. The diffusion coefficient and coherence level are found to be extremely sensitive to the asymmetry of the potential. It is established that at the values of the external force, for which the enhancement of diffusion is most rapid, the level of coherence has a wide plateau at low temperatures with the value of the Peclet factor 2. An interpretation of the amplification of diffusion in comparison with free thermal diffusion in terms of probability distribution is proposed.Comment: To appear in PR

Brownian motors: noisy transport far from equilibrium

Transport phenomena in spatially periodic systems far from thermal equilibrium are considered. The main emphasize is put on directed transport in so-called Brownian motors (ratchets), i.e. a dissipative dynamics in the presence of thermal noise and some prototypical perturbation that drives the system out of equilibrium without introducing a priori an obvious bias into one or the other direction of motion. Symmetry conditions for the appearance (or not) of directed current, its inversion upon variation of certain parameters, and quantitative theoretical predictions for specific models are reviewed as well as a wide variety of experimental realizations and biological applications, especially the modeling of molecular motors. Extensions include quantum mechanical and collective effects, Hamiltonian ratchets, the influence of spatial disorder, and diffusive transport.Comment: Revised version (Aug. 2001), accepted for publication in Physics Report

Query processing on probabilistic data: a survey

Query Processing on Probabilistic Data: A Survey presents the main approaches developed in the literature on probabilistic relational data, reconciling concepts developed in parallel by the Database and Artificial Intelligence communities

Lifted probabilistic inference in relational models (UAI tutorial)

This tutorial explains the core ideas behind lifted probabilistic inference in statistical relational learning (SRL) and extensional query evaluation in probabilistic databases (PDBs). Both fields deal with relational representations of uncertainty and have realized that efficient inference is an enormous challenge. Both fields have also achieved remarkable results developing efficient algorithms for tasks previously thought to be intractable. SRL and PDBs have very recently started to connect through the common language of relational logic. We now understand their commonalities and differences. Typical inference tasks are different in nature, yet can be captured in the same weighted model counting framework. Theoretical complexity bounds from one field translate to the other. This tutorial covers several of these developments. Within SRL, we cover weighted model counting encodings of graphical models and Markov logic, the realization that symmetries enable tractable, lifted inference, as well as a deeper understanding of the theoretical strengths and limitations of lifting. Within PDBs, we study of the data complexity of probabilistic inference, where the FO formula is fixed, and one asks for the complexity of the problem as a function of size of the database. In particular, we discuss a dichotomy theorem for the data complexity stating that, for a large class of queries, weighted model counting is either efficient, or provably #P-hard. The tutorial will focus on the big ideas that set probabilistic inference with relations apart from more classical inference problems. We focus on why lifted algorithms work, which properties they exploit, how they can reduce complexity, but also why inference is still hard in the worst case.status: publishe

The most probable database problem

This paper proposes a novel inference task for probabilistic databases: the most probable database (MPD) problem. The MPD is the most probable deterministic database where a given query or constraint is true. We highlight two distinctive applications, in database repair of key and dependency constraints, and in finding most probable explanations in statistical relational learning. The MPD problem raises new theoretical questions, such as the possibility of a dichotomy theorem for MPD, classifying queries as being either PTIME or NP-hard. We show that such a dichotomy would diverge from dichotomies for other inference tasks. We then prove a dichotomy for queries that represent unary functional dependency constraints. Finally, we discuss symmetric probabilities and the opportunities for lifted inference.status: publishe