2,481 research outputs found

    Probabilistic Inference Modulo Theories

    Full text link
    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 foundation for extending it to rich logic languages such as data structures and relational data. By lifted, we mean algorithms with constant complexity in the domain size (the number of values that variables can take). We also detail a solver for summations with difference arithmetic and show experimental results from a scenario in which SGDPLL(T) is much faster than a state-of-the-art probabilistic solver.Comment: Submitted to StarAI-16 workshop as closely revised version of IJCAI-16 pape

    Exact Inference for Relational Graphical Models with Interpreted Functions: Lifted Probabilistic Inference Modulo Theories

    Full text link
    Probabilistic Inference Modulo Theories (PIMT) is a recent framework that expands exact inference on graphical models to use richer languages that include arithmetic, equalities, and inequalities on both integers and real numbers. In this paper, we expand PIMT to a lifted version that also processes random functions and relations. This enhancement is achieved by adapting Inversion, a method from Lifted First-Order Probabilistic Inference literature, to also be modulo theories. This results in the first algorithm for exact probabilistic inference that efficiently and simultaneously exploits random relations and functions, arithmetic, equalities and inequalities.Comment: Appeared in the Uncertainty in Artificial Intelligence Conference, August 201

    Structured Learning Modulo Theories

    Full text link
    Modelling problems containing a mixture of Boolean and numerical variables is a long-standing interest of Artificial Intelligence. However, performing inference and learning in hybrid domains is a particularly daunting task. The ability to model this kind of domains is crucial in "learning to design" tasks, that is, learning applications where the goal is to learn from examples how to perform automatic {\em de novo} design of novel objects. In this paper we present Structured Learning Modulo Theories, a max-margin approach for learning in hybrid domains based on Satisfiability Modulo Theories, which allows to combine Boolean reasoning and optimization over continuous linear arithmetical constraints. The main idea is to leverage a state-of-the-art generalized Satisfiability Modulo Theory solver for implementing the inference and separation oracles of Structured Output SVMs. We validate our method on artificial and real world scenarios.Comment: 46 pages, 11 figures, submitted to Artificial Intelligence Journal Special Issue on Combining Constraint Solving with Mining and Learnin

    Hybrid SRL with Optimization Modulo Theories

    Full text link
    Generally speaking, the goal of constructive learning could be seen as, given an example set of structured objects, to generate novel objects with similar properties. From a statistical-relational learning (SRL) viewpoint, the task can be interpreted as a constraint satisfaction problem, i.e. the generated objects must obey a set of soft constraints, whose weights are estimated from the data. Traditional SRL approaches rely on (finite) First-Order Logic (FOL) as a description language, and on MAX-SAT solvers to perform inference. Alas, FOL is unsuited for con- structive problems where the objects contain a mixture of Boolean and numerical variables. It is in fact difficult to implement, e.g. linear arithmetic constraints within the language of FOL. In this paper we propose a novel class of hybrid SRL methods that rely on Satisfiability Modulo Theories, an alternative class of for- mal languages that allow to describe, and reason over, mixed Boolean-numerical objects and constraints. The resulting methods, which we call Learning Mod- ulo Theories, are formulated within the structured output SVM framework, and employ a weighted SMT solver as an optimization oracle to perform efficient in- ference and discriminative max margin weight learning. We also present a few examples of constructive learning applications enabled by our method

    Hybrid Probabilistic Inference with Logical Constraints: Tractability and Message Passing

    Full text link
    Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of real-world hybrid scenarios where variables are heterogeneous in nature (both continuous and discrete) via the language of Satisfiability Modulo Theories (SMT); as well as computing probabilistic queries with arbitrarily complex logical constraints. Recent work has shown WMI inference to be reducible to a model integration (MI) problem, under some assumptions, thus effectively allowing hybrid probabilistic reasoning by volume computations. In this paper, we introduce a novel formulation of MI via a message passing scheme that allows to efficiently compute the marginal densities and statistical moments of all the variables in linear time. As such, we are able to amortize inference for arbitrarily rich MI queries when they conform to the problem structure, here represented as the primal graph associated to the SMT formula. Furthermore, we theoretically trace the tractability boundaries of exact MI. Indeed, we prove that in terms of the structural requirements on the primal graph that make our MI algorithm tractable - bounding its diameter and treewidth - the bounds are not only sufficient, but necessary for tractable inference via MI

    Scaling up Hybrid Probabilistic Inference with Logical and Arithmetic Constraints via Message Passing

    Full text link
    Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of real-world problems where variables are both continuous and discrete, via the language of Satisfiability Modulo Theories (SMT), as well as to compute probabilistic queries with complex logical and arithmetic constraints. Yet, existing WMI solvers are not ready to scale to these problems. They either ignore the intrinsic dependency structure of the problem at all, or they are limited to too restrictive structures. To narrow this gap, we derive a factorized formalism of WMI enabling us to devise a scalable WMI solver based on message passing, MP-WMI. Namely, MP-WMI is the first WMI solver which allows to: 1) perform exact inference on the full class of tree-structured WMI problems; 2) compute all marginal densities in linear time; 3) amortize inference inter query. Experimental results show that our solver dramatically outperforms the existing WMI solvers on a large set of benchmarks

    Constrained Sampling and Counting: Universal Hashing Meets SAT Solving

    Full text link
    Constrained sampling and counting are two fundamental problems in artificial intelligence with a diverse range of applications, spanning probabilistic reasoning and planning to constrained-random verification. While the theory of these problems was thoroughly investigated in the 1980s, prior work either did not scale to industrial size instances or gave up correctness guarantees to achieve scalability. Recently, we proposed a novel approach that combines universal hashing and SAT solving and scales to formulas with hundreds of thousands of variables without giving up correctness guarantees. This paper provides an overview of the key ingredients of the approach and discusses challenges that need to be overcome to handle larger real-world instances.Comment: Appears in proceedings of AAAI-16 Workshop on Beyond N

    Anytime Exact Belief Propagation

    Full text link
    Statistical Relational Models and, more recently, Probabilistic Programming, have been making strides towards an integration of logic and probabilistic reasoning. A natural expectation for this project is that a probabilistic logic reasoning algorithm reduces to a logic reasoning algorithm when provided a model that only involves 0-1 probabilities, exhibiting all the advantages of logic reasoning such as short-circuiting, intelligibility, and the ability to provide proof trees for a query answer. In fact, we can take this further and require that these characteristics be present even for probabilistic models with probabilities \emph{near} 0 and 1, with graceful degradation as the model becomes more uncertain. We also seek inference that has amortized constant time complexity on a model's size (even if still exponential in the induced width of a more directly relevant portion of it) so that it can be applied to huge knowledge bases of which only a relatively small portion is relevant to typical queries. We believe that, among the probabilistic reasoning algorithms, Belief Propagation is the most similar to logic reasoning: messages are propagated among neighboring variables, and the paths of message-passing are similar to proof trees. However, Belief Propagation is either only applicable to tree models, or approximate (and without guarantees) for precision and convergence. In this paper we present work in progress on an Anytime Exact Belief Propagation algorithm that is very similar to Belief Propagation but is exact even for graphical models with cycles, while exhibiting soft short-circuiting, amortized constant time complexity in the model size, and which can provide probabilistic proof trees.Comment: Submission to StaRAI-17 workshop at UAI-17 conferenc

    Trace Abstraction Modulo Probability

    Full text link
    We propose trace abstraction modulo probability, a proof technique for verifying high-probability accuracy guarantees of probabilistic programs. Our proofs overapproximate the set of program traces using failure automata, finite-state automata that upper bound the probability of failing to satisfy a target specification. We automate proof construction by reducing probabilistic reasoning to logical reasoning: we use program synthesis methods to select axioms for sampling instructions, and then apply Craig interpolation to prove that traces fail the target specification with only a small probability. Our method handles programs with unknown inputs, parameterized distributions, infinite state spaces, and parameterized specifications. We evaluate our technique on a range of randomized algorithms drawn from the differential privacy literature and beyond. To our knowledge, our approach is the first to automatically establish accuracy properties of these algorithms

    Using Quantum Computers to Learn Physics

    Full text link
    Since its inception at the beginning of the twentieth century, quantum mechanics has challenged our conceptions of how the universe ought to work; however, the equations of quantum mechanics can be too computationally difficult to solve using existing computers for even modestly large systems. Here I will show that quantum computers can sometimes be used to address such problems and that quantum computer science can assign formal complexities to learning facts about nature. Hence, computer science should not only be regarded as an applied science; it is also of central importance to the foundations of science.Comment: This article is designed as a popular article aimed at a general computer science audience and mostly reviews existing results, but it does contain several new results involving Hamiltonian inferenc
    • …
    corecore