Semiring induced valuation algebras: Exact and approximate local computation algorithms

Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra. There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non...

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Bibliographic Details
Published in:Artificial intelligence Vol. 172; no. 11; pp. 1360 - 1399
Main Authors: Kohlas, J., Wilson, N.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.07.2008
Elsevier
Subjects:
Applied sciences
Artificial intelligence
Combinatorics
Combinatorics. Ordered structures
Computer science; control theory; systems
Exact sciences and technology
Graph theory
Information retrieval. Graph
Join tree decompositions
Learning and adaptive systems
Local computation
Mathematics
Sciences and techniques of general use
Semirings
Soft constraints
Theoretical computing
Uncertainty
Valuation algebras
Valuation networks
Valuation networks
Uncertainty
Join tree decompositions
Valuation algebras
Local computation
Semirings
Soft constraints
Hypergraph
Algebraic structure
Probabilistic approach
Relational database
Network architecture
Constraint satisfaction
Semiring
Semigroup
Graph theory
Soft constraint
Modeling
Bayes network
Artificial intelligence
ISSN:0004-3702, 1872-7921
Online Access:Get full text
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Summary:Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra. There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non-classical logics. It turns out that many interesting models of valuation algebras may be derived from semiring valued mappings. In this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring. In particular, c-semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation. Also important are semirings whose multiplicative semigroup is embedded in a union of groups. They induce valuation algebras with a partially defined division. For these valuation algebras, the well-known architectures for Bayesian networks apply. We also extend the general computational framework to allow derivation of bounds and approximations, for when exact computation is not feasible.
ISSN:0004-3702
1872-7921
DOI:10.1016/j.artint.2008.03.003