Efficient Algorithms on the Family Associated to an Implicational System
An implication system (IS) on a finite set S is a set of rules called Σ -implications of the kind A →_Σ B, with A,B ⊆ S. A subset X ⊆ S satisfies A →_Σ B when ''A ⊆ X implies B ⊆ X'' holds, so ISs can be used to describe constraints on sets of elements, such as dependency or caus...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2004-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/330/pdf |
| _version_ | 1797270189514948608 |
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| author | Karell Bertet Mirabelle Nebut |
| author_facet | Karell Bertet Mirabelle Nebut |
| author_sort | Karell Bertet |
| collection | DOAJ |
| description | An implication system (IS) on a finite set S is a set of rules called Σ -implications of the kind A →_Σ B, with A,B ⊆ S. A subset X ⊆ S satisfies A →_Σ B when ''A ⊆ X implies B ⊆ X'' holds, so ISs can be used to describe constraints on sets of elements, such as dependency or causality. ISs are formally closely linked to the well known notions of closure operators and Moore families. This paper focuses on their algorithmic aspects. A number of problems issued from an IS Σ (e.g. is it minimal, is a given implication entailed by the system) can be reduced to the computation of closures φ _Σ (X), where φ _Σ is the closure operator associated to Σ . We propose a new approach to compute such closures, based on the characterization of the direct-optimal IS Σ _do which has the following properties: \beginenumerate ıtemit is equivalent to Σ ıtemφ _Σ _do(X) (thus φ _Σ (X)) can be computed by a single scanning of Σ _do-implications ıtemit is of minimal size with respect to ISs satisfying 1. and 2. \endenumerate We give algorithms that compute Σ _do, and from Σ _do closures φ _Σ (X) and the Moore family associated to φ _Σ . |
| first_indexed | 2024-04-25T02:00:19Z |
| format | Article |
| id | doaj.art-163762fbb33b41039991ea7d48a3413f |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:19Z |
| publishDate | 2004-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-163762fbb33b41039991ea7d48a3413f2024-03-07T15:06:37ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502004-01-01Vol. 6 no. 210.46298/dmtcs.330330Efficient Algorithms on the Family Associated to an Implicational SystemKarell Bertet0https://orcid.org/0000-0002-9741-4570Mirabelle Nebut1Laboratoire Informatique, Image et Interaction - EA 2118Laboratoire d'Informatique Fondamentale de LilleAn implication system (IS) on a finite set S is a set of rules called Σ -implications of the kind A →_Σ B, with A,B ⊆ S. A subset X ⊆ S satisfies A →_Σ B when ''A ⊆ X implies B ⊆ X'' holds, so ISs can be used to describe constraints on sets of elements, such as dependency or causality. ISs are formally closely linked to the well known notions of closure operators and Moore families. This paper focuses on their algorithmic aspects. A number of problems issued from an IS Σ (e.g. is it minimal, is a given implication entailed by the system) can be reduced to the computation of closures φ _Σ (X), where φ _Σ is the closure operator associated to Σ . We propose a new approach to compute such closures, based on the characterization of the direct-optimal IS Σ _do which has the following properties: \beginenumerate ıtemit is equivalent to Σ ıtemφ _Σ _do(X) (thus φ _Σ (X)) can be computed by a single scanning of Σ _do-implications ıtemit is of minimal size with respect to ISs satisfying 1. and 2. \endenumerate We give algorithms that compute Σ _do, and from Σ _do closures φ _Σ (X) and the Moore family associated to φ _Σ .https://dmtcs.episciences.org/330/pdflatticeordered setmoore familyimplicational systemclosure operatoralgorithm[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Karell Bertet Mirabelle Nebut Efficient Algorithms on the Family Associated to an Implicational System Discrete Mathematics & Theoretical Computer Science lattice ordered set moore family implicational system closure operator algorithm [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Efficient Algorithms on the Family Associated to an Implicational System |
| title_full | Efficient Algorithms on the Family Associated to an Implicational System |
| title_fullStr | Efficient Algorithms on the Family Associated to an Implicational System |
| title_full_unstemmed | Efficient Algorithms on the Family Associated to an Implicational System |
| title_short | Efficient Algorithms on the Family Associated to an Implicational System |
| title_sort | efficient algorithms on the family associated to an implicational system |
| topic | lattice ordered set moore family implicational system closure operator algorithm [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/330/pdf |
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