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University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science August 1992 Search Through Systematic Set Enumeration Ron Rymon University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Ron Rymon, " Search Through Systematic Set Enumeration", . August 1992. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-92-66. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/297 For more information, please contact [email protected]. Search Through Systematic Set Enumeration Abstract In many problem domains, solutions take the form of unordered sets. We present the Set-Enumerations (SE)-tree - a vehicle for representing sets and/or enumerating them in a best-first fashion. eW demonstrate its usefulness as the basis for a unifying search-based framework for domains where minimal (maximal) elements of a power set are targeted, where minimal (maximal) partial instantiations of a set of variables are sought, or where a composite decision is not dependent on the order in which its primitive component-decisions are taken. Particular instantiations of SE-tree-based algorithms for some AI problem domains are used to demonstrate the general features of the approach. These algorithms are compared theoretically and empirically with current algorithms. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MS- CIS-92-66. This technical report is available at ScholarlyCommons: https://repository.upenn.edu/cis_reports/297 Search Through Systematic Set Enumeration MS-CIS-92-66 LINC LAB 234 Ron Rymon University of Pennsylvania School of Engineering and Applied Science Computer and Information Science Department Philadelphia, PA 19104-6389 August 1992 Search through Systematic Set Enumeration Ron Rymon* Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 In Proceedings KR-92, Cambridge HA, October 1992 Abstract More generally, variables can be instantiated from an arbitrary domain. In many problem domains, solutions take the Researchers in Artificial Intelligence (AI) have also form of unordered sets. We present the Set- made use of such abstract problems in their models. Enumeration (SE)-tree - a vehicle for repre- The HS problem, for example, was used by [Reiter 871 senting sets and/or enumerating them in a in his formalization of diagnosis. In a newer character- best-first fashion. We demonstrate its use- ization, diagnoses are viewed as partial assignments of fulness as the basis for a unifying search- state to components [de Kleer et al. 901. Many other based framework for domains where minimal A1 problems are, or could be, formulated so as to ad- (maximal) elements of a power set are tar- mit sets as solutions. geted, where minimal (maximal) partial in- stantiations of a set of variables are sought, Our goal in introducing the Set-Enumeration (SE)-tree or where a composite decision is not de- is to provide a unified search-based framework for solv- pendent on the order in which its primitive ing such problems, albeit their problem-independent component-decisions are taken. Particular solution criteria. SE-tree-based algorithms for differ- instantiations of SE-tree-based algorithms for ent problems will share their skeletal structure, but some A1 problem domains are used to demon- will each use additional domain-specific tactics. Fur- strate the general features of the approach. thermore, at a certain level of abstraction, even those These algorithms are compared theoretically tactics are general and can be shared across domains. and empirically with current algoritlims. General tactics identified here include pruning rules which exploit the SE-tree structure, exploration poli- cies, and problem decomposition methods. Incremen- 1 INTRODUCTION tal versions of SE-tree-based algorithms can be con- structed for some problem domains. In what follows, Many computer science problems admit solutions we use particular instantiations of SE-tree-based al- which are elements of a given power-set. Typically, gorithms to demonstrate the general features of the such sets are required to satisfy some problem-specific approach. criterion which designates them as solutions. In Consistency-based formulations of diagnosis will serve many cases, such criteria either include, or are aug- as a working example for most of this paper. Section 2 mented with, some minimality/maximality require- begins with a formal description of the basic SE-tree. ment. Consider, for example, the Hitting-Set (HS) Section 3 presents an SE-tree-based hitting-sets alge problem [Karp 721. Given a collection of sets, solu- rithm (SE-HS) for Reiter's original formulation. Being tions are required to have a non-empty intersection best-first, it can use any of a number of exploration with each member of the collection. In applications policies. This algorithm is first contrasted, as is, with of the HS problem, solutions are typically interesting the original algorithm. The SE-tree structure is then minimal with respect to set inclusion. In a more gen- used to improve SEHS by pruning away unpromis- eral class of ~roblems.solutions are ~artialinstantia- ing parts of the search space. We conclude with an tions of a sed of variables. A hitting-set, for example, empirical comparison of the two algorithms. can also be described as a membership-based mapping from the underlying set of primitive elements to {O,l). In Section 4, we extend the SE-tree to fit the more gen- eral class of problems, where solutions take the form 'Address for correspondence: Ron Rymon, Computer of partially instantiated sets of variables. Section 5 and Information Science, Room 423C, 3401 Walnut Street, a Philadelphia PA 19104, e-mail: [email protected]. presents an extended version of SE-HS for newer characterization of diagnoses [de Kleer et al. 901. Al- though derived from a very general search framework, this algorithm corresponds to a prime implicate gen- eration algorithm proposed by [Slagle et al. 701, and is empirically shown to perform quite well compared to a recent algorithm [Ngair 921. Unlike Slagle et al.'s algorithm, the extended SE-HS can work under diag- nostic theories with multiple fault modes, can use a variety of exploration policies for focusing purposes, and has an incremental version. Furthermore, we sub- sequently augment it with a problem decomposition tactic, thereby obtaining an improved version of Slagle et a1.k algorithm. Finally, we briefly review potential use of the SE-tree in abductive diagnostic frameworks. Figure 1: SE-tree for P({l,2,3,4)) In Section 6, we contrast features of the SE-tree with decision trees in the context of learning classification rules from examples. For lack of space, the scope of Notice also that the SE-tree can be used as a data this study is very limited and the reader is referred to structure for caching unordered sets, and as an ef- [Rymon 92b] for a more detailed analysis and empiri- fective means of checking whether a new set is sub- cal evaluation. sumed by any of those already cached. [de Kleer 921 has made such use of an SE-tree and reports significant 2 THE BASIC SE-TREE improvements in run-time. As a caching device, the SEtree is a special case of Knuth's trie data structure [Knuth 731, originally offered for ordered sets. While The Set-Enumeration (SE)-tree is a vehicle for repre- we too use the SE-tree for caching solutions and for senting and/or enumerating sets in a best-first fash- subsumption checking, our main objective in this pa- ion. The complete SEtree systematically enumerates per is its use in a search framework. elements of a power-set using a pre-imposed order on the underlying set of elements. In problems where the search space is a subset of that power-set that is (or 3 AN SE-TREE-BASED can be) closed under set-inclusion, the SE-tree induces a complete irredundant search technique. Let E be the HITTING-SET ALGORITHM underlying set of elements. We first index E's elements using a one-bone function ind : E + W. Then, given In this section, we demonstrate the use of the ba- any subset SGE, we define its SE-tree view: sic SE-tree structure for a hitting-set algorithm in the context of Reiter's theory of diagnosis. We open Definition 2.1 A Node's View with a brief introduction of Reiter's theory, to the point in which a hitting-set problem is formulated. An SE-tree-based algorithm (SE-HS) is then con- trasted with the dag-based algorithm proposed by Definition 2.2 A Basic Set Enumeration Tree [Reiter 87, Greiner et al. 891 to show that a large num- ber of calls to a subsumption checking procedure can Let F be a collection of sets that is closed under C be saved. Then, the SE-tree systematicity allows im- (i.e. for every SEF, if S'CS then S'EF). T is a Set proving SE-HS via a domain-specific pruning rule. Enumeration tree for F i$: Empirical comparison of the improved SE-HS with the dag-based implementation of [Greiner et al. 891 sup- 1. The mot of T is labeled by the empty set; ports our claims. 2. The children of a node labeled S in T are { S U{e} E F I eE View(ind,S) ). 3.1 REITER'S THEORY OF DIAGNOSIS Figure 1 illustrates an SE-tree for the complete power- Reiter's theory of diagnosis [Reiter 871 is among the set of {1,2,3,4). Note that restricting a node's ex- the most widely referenced logic-based approaches to pansion to its View, ensures that every set is uniquely model-based diagnosis.
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