We demonstrate how order-theoretic abstractions can be useful in identifying, formalizing, and exploiting relationships between seemingly dissimilar AI algorithms that perform computations on partially-ordered sets. In particular, we show how the order-theoretic concept of an anti-chain can be used to provide an efficient representation for such sets when they satisfy certain special properties. We use anti-chains to identify and analyze the basic operations and representation optimizations in the version space learning algorithm [10] and the assumption-based truth maintenance system (ATMS) [2, 3]. Our analysis allows us to (1) extend the known theory [7, 10, 8] of admissibility of concept spaces for incremental version space merging, and (2) develop new, simpler label-update algorithms for ATMS's with DNF assumption formulas. Contents 1 Introduction 2 2 Representing Sets as Anti-Chains 4 3 Version Spaces 17 4 Assumption-Based Truth Maintenance Systems 32 5 Extended ATMS's 46 6 Ackno...

Truth maintenance systems have been studied by many authors and have become powerful tools in AI reasoning systems. From the viewpoint of commonsense reasoning, Doyle's TMS seems most interesting, for it allows nonmonotonic justifications. Its semantics, however, has remained unclear. In this paper, we shall give its declarative description in terms of autoepistemic logic, a kind of nonmonotonic logic. That is, we shall exhibit a one-to-one correspondence between states acceptable to the TMS and stable expansions of autoepistemic formulas attached to justifications. Thus, the TMS turns out to be a theorem prover of autoepistemic logic. For the practical interest, our result also suggests the possibility of implementing better TMS algorithms by using the theorem proving method of autoepistemic logic. 1