The common order-theoretic structure of version spaces and ATMS's
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TL;DR
This paper exposes the common order-theoretic properties of the structures manipulated by the version space algorithm and the assumption-based truth maintenance systems by recasting them in the framework of convex spaces and reveals necessary and sufficient conditions for ensuring the preservation of an essential finite representability property in version space merging.
Abstract
This paper exposes the common order-theoretic properties of the structures manipulated by the version space algorithm [Mit78]and the assumption-based truth maintenance systems (ATMS) [dk86a,dk86b] by recasting them in the framework of convex spaces. Our analysis of version spaces in this framework reveals necessary and sufficient conditions for ensuring the preservation of an essential finite representability property in version space merging. This analysis is used to formulate several sufficient conditions for when a language will allow version spaces to be represented by finite sets of concepts (even when the universe of concepts may be infinite). We provide a new convex space based formulation of computation performs by an ATMS which extends the expressiveness of disjunctions in the systems. This approach obviates the need for hyper-resolution in dealing with disjunction and results in simpler label-update algorithms.