## The Common Order-Theoretic Structure of Version Spaces and ATMS's (1991)

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Venue: | In Proceedings of the National Conference on Artificial Intelligence |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Gunter91thecommon,

author = {Carl A. Gunter and Teow-hin Ngair and Devika Subramanian},

title = {The Common Order-Theoretic Structure of Version Spaces and ATMS's},

booktitle = {In Proceedings of the National Conference on Artificial Intelligence},

year = {1991},

pages = {500--505},

publisher = {AAAI Press}

}

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### Abstract

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...

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Citation Context ...tant to note that the elements ?; ? and the operations ;sare uniquely determined by the properties ascribed to them by the definition. Other basic properties of lattices can be found in a source like =-=[1]. Exa-=-mple 6 The poset Pwr(A); ` of environments is a lattice where ? = ; and ? = A. The meet is xsy = x " y and the join is xsy = x [ y. Let us now consider how to calculate intersections of lower and... |

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Citation Context ... 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 AT... |

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Citation Context ...tion. In other words, the computation is described as a mathematical entity: in particular, as the least fixed-point of a functional as one does in the denotational semantics of programming languages =-=[12, 6]-=-. The benefit of a mathematical treatment can be realized in proving correctness of algorithms for computing labels. Indeed a proof of correctness based on a fixed-point semantics was given in [5]. Wh... |

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Citation Context ...on 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 merg... |

158 |
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Citation Context ...iciently implementable manner. For instance, the functions in [2] that the ATMS is meant to compute can all be succinctly expressed using our interfaces. The same is true for most of the functions in =-=[4]-=-. To see some examples from the latter, consider the function node-consistent-with? from [4] (page 440). This takes as arguments an atom a 2 L and an environment x 2 E ; it returns the boolean value t... |

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Citation Context ...tion. In other words, the computation is described as a mathematical entity: in particular, as the least fixed-point of a functional as one does in the denotational semantics of programming languages =-=[12, 6]-=-. The benefit of a mathematical treatment can be realized in proving correctness of algorithms for computing labels. Indeed a proof of correctness based on a fixed-point semantics was given in [5]. Wh... |

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Citation Context ... ) d). The computation of the label LF[T (a), for any atom a 2 L is essentially the enumeration of all minimal "models" (restricted to atoms in A) of a in F [ T . This task is known to be #P=-=-complete [13]-=-. Thus, in the general case, we expect to perform computation exponential in the size of A. deKleer [3] uses hyper-resolution to incorporate disjunctions in T (the cause of the exponentiality) into la... |

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Citation Context ...but largely unappreciated importance for designing algorithms that manipulate partial information. We now illustrate this for what de Kleer called the `extended ATMS'. The extended ATMS is defined in =-=[3]-=- to be an ATMS where the input theory consists of Horn clauses over L and disjunctions of assumptions A 1s: : :sA n . The extended ATMS has the same interface (Table 3) as the basic ATMS, but permits ... |

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Citation Context ...heoretic formulation of an important property of compact subsets called `coherence' [15]) and in domain theory (where it is a necessary condition for the bases of domains with good closure properties =-=[14]-=-). Mellish identified these conditions in [8]; he too noted the need for finite quasimeets and quasi-joins of pairs of elements of P , and by fiat introduced a top and bottom element for P so that the... |

21 |
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Citation Context ...ns 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 Rep... |

15 |
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Citation Context ...�� be a poset x; y 2 P . The quasi-meet of x and y is defined by the equation x ~sy = maxfz 2 P j z �� x and z �� yg and the quasi-join of x and y is defined by x ~sy = minfz 2 P j zsx and=-= zsyg 23 In [8], # x-=- ~sy is called bb(x; y) and " x ~sy is called aa(x; y). If P is a lattice, then it is easy to check that x ~sy = fxsyg and x ~sy = fxsyg so, in the event that we are dealing with lattices, quasim... |

10 |
Convex Spaces as an Order-theoretic Basis for Problem Solving
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(Show Context)
Citation Context ... point of the closure operatorssOE. This can be done through the use of a minimal model for a collection of Horn clauses. Details sufficient to construct a proof of Theorem 22 can be found in [5] and =-=[11]. 41 -=-Table 4: Converting a Horn Clause OE to a Closure Operator OE. functionsOE(F)(b) = if b 6= a then F(b) else F(a) [ u (fold(" u , map(F)(S), f;g)) endif where OE is a Horn clause such that S is th... |

7 | de Kleer, An assumption-based tms - Johan - 1986 |

4 | de Kleer. Extending the ATMS - Johan - 1986 |

3 | On logical foundations of the ATMS - Fujiwara, Mizushima, et al. - 1990 |

3 |
Topology Via Logic, volume 5
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- 1989
(Show Context)
Citation Context ...as both properties is said to have property MW. Property M is familiar from ideas in topology (where it is an ordertheoretic formulation of an important property of compact subsets called `coherence' =-=[15]-=-) and in domain theory (where it is a necessary condition for the bases of domains with good closure properties [14]). Mellish identified these conditions in [8]; he too noted the need for finite quas... |

1 |
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(Show Context)
Citation Context ...o note that nothing in general is known about the structure of this poset; in particular, P will not typically be the collection of all subsets of UP (this deviation being the `representational bias' =-=[9] of -=-the concept space). If x �� y then we say that x is more specific than y or we say that y is more general than x. A training set 1 over a version space P is a pair (\Gamma; \Delta) where \Gamma ` ... |

1 | The description identi�cation problem - Mellish - 1991 |