62 research outputs found
Extracting the K best solutions from a valued and-or acyclic graph
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 117-118).In this thesis, we are interested in solving a problem that arises in model-based programming, specifically in the estimation of the state a system described by a probabilistic model. Some model-based estimators, such as the MEXEC algorithm and the DNNF-based Belief State Estimation algorithm, use a valued and-or acyclic graph to represent the possible estimates. These algorithms specifically use a valued smooth deterministic decomposable negation normal form (sd-DNNF) representation, a type of and-or acyclic graph. Prior work has focused on extracting either all or only the best solution from the sd-DNNF. This work develops an efficient algorithm that is able to extract the k best solutions, where k is a parameter to the algorithm. For a graph with -E- edges, -V - nodes and -Ev- children per non-leaf node, the algorithm presented in this thesis has a time complexity of O(-E-k log k +-E- log -Ev-+-V -k log -Ev-) and a space complexity O(-E-k).by Paul Harrison Elliott.M.Eng
On the Role of Canonicity in Bottom-up Knowledge Compilation
We consider the problem of bottom-up compilation of knowledge bases, which is
usually predicated on the existence of a polytime function for combining
compilations using Boolean operators (usually called an Apply function). While
such a polytime Apply function is known to exist for certain languages (e.g.,
OBDDs) and not exist for others (e.g., DNNF), its existence for certain
languages remains unknown. Among the latter is the recently introduced language
of Sentential Decision Diagrams (SDDs), for which a polytime Apply function
exists for unreduced SDDs, but remains unknown for reduced ones (i.e. canonical
SDDs). We resolve this open question in this paper and consider some of its
theoretical and practical implications. Some of the findings we report question
the common wisdom on the relationship between bottom-up compilation, language
canonicity and the complexity of the Apply function
The Language of Search
This paper is concerned with a class of algorithms that perform exhaustive
search on propositional knowledge bases. We show that each of these algorithms
defines and generates a propositional language. Specifically, we show that the
trace of a search can be interpreted as a combinational circuit, and a search
algorithm then defines a propositional language consisting of circuits that are
generated across all possible executions of the algorithm. In particular, we
show that several versions of exhaustive DPLL search correspond to such
well-known languages as FBDD, OBDD, and a precisely-defined subset of d-DNNF.
By thus mapping search algorithms to propositional languages, we provide a
uniform and practical framework in which successful search techniques can be
harnessed for compilation of knowledge into various languages of interest, and
a new methodology whereby the power and limitations of search algorithms can be
understood by looking up the tractability and succinctness of the corresponding
propositional languages
TP-Compilation for inference in probabilistic logic programs
We propose TP -compilation, a new inference technique for probabilistic logic programs that is based on forward reasoning. TP -compilation proceeds incrementally in that it interleaves the knowledge compilation step for weighted model counting with forward reasoning on the logic program. This leads to a novel anytime algorithm that provides hard bounds on the inferred probabilities. The main difference with existing inference techniques for probabilistic logic programs is that these are a sequence of isolated transformations. Typically, these transformations include conversion of the ground program into an equivalent propositional formula and compilation of this formula into a more tractable target representation for weighted model counting. An empirical evaluation shows that TP -compilation effectively handles larger instances of complex or cyclic real-world problems than current sequential approaches, both for exact and anytime approximate inference. Furthermore, we show that TP -compilation is conducive to inference in dynamic domains as it supports efficient updates to the compiled model
Algebraic model counting
Weighted model counting (WMC) is a well-known inference task on knowledge bases, and the basis for some of the most efficient techniques for probabilistic inference in graphical models. We introduce algebraic model counting (AMC), a generalization of WMC to a semiring structure that provides a unified view on a range of tasks and existing results. We show that AMC generalizes many well-known tasks in a variety of domains such as probabilistic inference, soft constraints and network and database analysis. Furthermore, we investigate AMC from a knowledge compilation perspective and show that all AMC tasks can be evaluated using sd-DNNF circuits, which are strictly more succinct, and thus more efficient to evaluate, than direct representations of sets of models. We identify further characteristics of AMC instances that allow for evaluation on even more succinct circuits
On the Relationship between Sum-Product Networks and Bayesian Networks
In this paper, we establish some theoretical connections between Sum-Product
Networks (SPNs) and Bayesian Networks (BNs). We prove that every SPN can be
converted into a BN in linear time and space in terms of the network size. The
key insight is to use Algebraic Decision Diagrams (ADDs) to compactly represent
the local conditional probability distributions at each node in the resulting
BN by exploiting context-specific independence (CSI). The generated BN has a
simple directed bipartite graphical structure. We show that by applying the
Variable Elimination algorithm (VE) to the generated BN with ADD
representations, we can recover the original SPN where the SPN can be viewed as
a history record or caching of the VE inference process. To help state the
proof clearly, we introduce the notion of {\em normal} SPN and present a
theoretical analysis of the consistency and decomposability properties. We
conclude the paper with some discussion of the implications of the proof and
establish a connection between the depth of an SPN and a lower bound of the
tree-width of its corresponding BN.Comment: Full version of the same paper to appear at ICML-201
- …