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AND/OR Search Spaces for Graphical Models
, 2004
"... The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the gr ..."
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Cited by 119 (44 self)
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The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the graphical model explicitly and may sometime reduce the search space exponentially. Indeed, most
Interactive Cost Configuration Over Decision Diagrams
 Journal of Artificial Intelligence Research (JAIR
"... Abstract In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrackf ..."
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Cited by 6 (1 self)
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Abstract In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrackfree user interaction online. In particular, binary decision diagrams (BDDs) have been successfully used as a compilation target for product and service configuration. In this paper we discuss how to extend BDDbased configuration to scenarios involving cost functions which express user preferences. We first show that an efficient, robust and easy to implement extension is possible if the cost function is additive, and feasible solutions are represented using multivalued decision diagrams (MDDs). We also discuss the effect on MDD size if the cost function is nonadditive or if it is encoded explicitly into MDD. We then discuss interactive configuration in the presence of multiple cost functions. We prove that even in its simplest form, multiplecost configuration is NPhard in the input MDD. However, for solving twocost configuration we develop a pseudopolynomial scheme and a fully polynomial approximation scheme. The applicability of our approach is demonstrated through experiments over realworld configuration models and productcatalogue datasets. Response times are generally within a fraction of a second even for very large instances.
Approximation by Quantization
"... Inference in graphical models consists of repeatedly multiplying and summing out potentials. It is generally intractable because the derived potentials obtained in this way can be exponentially large. Approximate inference techniques such as belief propagation and variational methods combat this by ..."
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Cited by 5 (1 self)
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Inference in graphical models consists of repeatedly multiplying and summing out potentials. It is generally intractable because the derived potentials obtained in this way can be exponentially large. Approximate inference techniques such as belief propagation and variational methods combat this by simplifying the derived potentials, typically by dropping variables from them. We propose an alternate method for simplifying potentials: quantizing their values. Quantization causes different states of a potential to have the same value, and therefore introduces contextspecific independencies that can be exploited to represent the potential more compactly. We use algebraic decision diagrams (ADDs) to do this efficiently. We apply quantization and ADD reduction to variable elimination and junction tree propagation, yielding a family of bounded approximate inference schemes. Our experimental tests show that our new schemes significantly outperform stateoftheart approaches on many benchmark instances. 1
An EM algorithm on BDDs with order encoding for logicbased probabilistic models
"... Logicbased probabilistic models (LBPMs) enable us to handle problems with uncertainty succinctly thanks to the expressive power of logic. However, most of LBPMs have restrictions to realize efficient probability computation and learning. We propose an EM algorithm working on BDDs with order encodin ..."
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Cited by 4 (3 self)
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Logicbased probabilistic models (LBPMs) enable us to handle problems with uncertainty succinctly thanks to the expressive power of logic. However, most of LBPMs have restrictions to realize efficient probability computation and learning. We propose an EM algorithm working on BDDs with order encoding for LBPMs. A notable advantage of our algorithm over existing approaches is that it copes with multivalued random variables without restrictions. The complexity of our algorithm is proportional to the size of a BDD representing observations. We utilize our algorithm to make a diagnoses of a logic circuit which contains stochastic error gates and show that restrictions of existing approaches can be eliminated by our algorithm.
AND/OR multivalued decision diagrams for constraint networks
 IN: PROCEEDINGS OF THE THIRTEENTH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2007
"... The paper is an overview of a recently developed compilation data structure for graphical models, with specific application to constraint networks. The AND/OR MultiValued Decision Diagram (AOMDD) augments well known decision diagrams (OBDDs, MDDs) with AND nodes, in order to capture function decomp ..."
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Cited by 2 (0 self)
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The paper is an overview of a recently developed compilation data structure for graphical models, with specific application to constraint networks. The AND/OR MultiValued Decision Diagram (AOMDD) augments well known decision diagrams (OBDDs, MDDs) with AND nodes, in order to capture function decomposition structure. The AOMDD is based on a pseudo tree of the network, rather than a linear ordering of its variables. The AOMDD of a constraint network is a canonical form given a pseudo tree. We describe two main approaches for compiling the AOMDD of a constraint network. The first is a top down, searchbased procedure, that works by applying reduction rules to the trace of the memory intensive AND/OR search algorithm. The second is a bottom up, inferencebased procedure, that uses a Bucket Elimination schedule. For both algorithms, the compilation time and the size of the AOMDD are, in the worst case, exponential in the treewidth of the constraint graph, rather than pathwidth as is known for ordered binary decision diagrams (OBDDs).
Existential Closures for Knowledge Compilation
 PROCEEDINGS OF THE TWENTYSECOND INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
"... We study the existential closures of several propositional languages L considered recently as target languages for knowledge compilation (KC), namely the incomplete fragments KROMC, HORNC, K/HC, renHC, AFF, and the corresponding disjunction closures KROMC[∨], ..."
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Cited by 2 (1 self)
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We study the existential closures of several propositional languages L considered recently as target languages for knowledge compilation (KC), namely the incomplete fragments KROMC, HORNC, K/HC, renHC, AFF, and the corresponding disjunction closures KROMC[∨],
Just Count the Satisfied Groundings: Scalable LocalSearch and Sampling Based Inference in MLNs
"... The main computational bottleneck in various sampling based and localsearch based inference algorithms for Markov logic networks (e.g., Gibbs sampling, MCSAT, MaxWalksat, etc.) is computing the number of groundings of a firstorder formula that are true given a truth assignment to all of its groun ..."
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Cited by 2 (1 self)
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The main computational bottleneck in various sampling based and localsearch based inference algorithms for Markov logic networks (e.g., Gibbs sampling, MCSAT, MaxWalksat, etc.) is computing the number of groundings of a firstorder formula that are true given a truth assignment to all of its ground atoms. We reduce this problem to the problem of counting the number of solutions of a constraint satisfaction problem (CSP) and show that during their execution, both sampling based and localsearch based algorithms repeatedly solve dynamic versions of this counting problem. Deriving from the vast amount of literature on CSPs and graphical models, we propose an exact junctiontree based algorithm for computing the number of solutions of the dynamic CSP, analyze its properties, and show how it can be used to improve the computational complexity of Gibbs sampling and MaxWalksat. Empirical tests on a variety of benchmarks clearly show that our new approach is several orders of magnitude more scalable than existing approaches.
Algorithms for Generating Ordered Solutions for Explicit AND/OR Structures
"... We present algorithmsfor generatingalternative solutions for explicit acyclic AND/OR structures in nondecreasing order of cost. The proposed algorithms use a best first search technique and report the solutions using an implicit representation ordered by cost. In this paper, wepresenttwoversionsoft ..."
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Cited by 1 (1 self)
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We present algorithmsfor generatingalternative solutions for explicit acyclic AND/OR structures in nondecreasing order of cost. The proposed algorithms use a best first search technique and report the solutions using an implicit representation ordered by cost. In this paper, wepresenttwoversionsofthesearchalgorithm–(a)aninitialversionofthebestfirst search algorithm, ASG, which may present one solution more than once while generating the ordered solutions, and (b) another version, LASG, which avoids the construction of the duplicate solutions. The actual solutions can be reconstructed quickly from the implicit compact representation used. We have applied the methods on a few test domains, some of them are synthetic while the others are based on well known problems including the search space of the 5peg Tower of Hanoi problem, the matrixchain multiplication problem and the problem of finding secondary structure of RNA. Experimental results show the efficacy of the proposed algorithms over the existing approach. Our proposed algorithms have potential use in various domains ranging from knowledge based frameworks to service composition, where the AND/OR structure is widely used for representing problems. 1.
The InclusionExclusion Rule and its Application to the Junction Tree Algorithm
, 2013
"... In this paper, we consider the inclusionexclusion rule – a known yet seldom used rule of probabilistic inference. Unlike the widely used sum rule which requires easy access to all joint probability values, the inclusionexclusion rule requires easy access to several marginal probability values. We ..."
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Cited by 1 (0 self)
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In this paper, we consider the inclusionexclusion rule – a known yet seldom used rule of probabilistic inference. Unlike the widely used sum rule which requires easy access to all joint probability values, the inclusionexclusion rule requires easy access to several marginal probability values. We therefore develop a new representation of the joint distribution that is amenable to the inclusionexclusion rule. We compare the relative strengths and weaknesses of the inclusionexclusion rule with the sum rule and develop a hybrid rule called the inclusionexclusionsum (IES) rule, which combines their power. We apply the IES rule to junction trees, treating the latter as a target for knowledge compilation and show that in many cases it greatly reduces the time required to answer queries. Our experiments demonstrate the power of our approach. In particular, at query time, on several networks, our new scheme was an order of magnitude faster than the junction tree algorithm.