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Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure. / Oliver Kullmann

Fundamenta Informaticae, Volume: 109, Issue: 1, Pages: 83 - 119

Swansea University Author: Oliver, Kullmann

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DOI (Published version): 10.3233/FI-2011-429

Abstract

Concluding this mini-series of 2 articles on the foundations of generalised clause-sets, we study the combinatorial properties of non-boolean conjunctive normal forms (clause-sets), allowing arbitrary (but finite) sets of values for variables, while literals express that some variable shall not get...

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Published in: Fundamenta Informaticae
Published: 2011
URI: https://cronfa.swan.ac.uk/Record/cronfa5284
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spelling 2013-10-17T14:54:15.2413571 v2 5284 2012-02-23 Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure. 2b410f26f9324d6b06c2b98f67362d05 0000-0003-3021-0095 Oliver Kullmann Oliver Kullmann true false 2012-02-23 SCS Concluding this mini-series of 2 articles on the foundations of generalised clause-sets, we study the combinatorial properties of non-boolean conjunctive normal forms (clause-sets), allowing arbitrary (but finite) sets of values for variables, while literals express that some variable shall not get some (given) value. First we study the properties of the direct translation (or “encoding”) of generalised clause-sets into boolean clause-sets. Many combinatorial properties are preserved, and as a result we can lift fixed-parameter tractability of satisfiability in the maximal deficiency from the boolean case to the general case. Then we turn to irredundant clause-sets, which generalise minimally unsatisfiable clause-sets, and we prove basic properties. The simplest irredundant clause-sets are hitting clause-sets, and we provide characterisations and generalisations. Unsatisfiable irredundant clause-sets are the minimally unsatisfiable clause-sets, and we provide basic tools. These tools allow us to characterise the minimally unsatisfiable clause-sets of minimal deficiency. Finally we provide a new translation of generalised boolean clause-sets into boolean clause-sets, the nested translation, which preserves the conflict structure. As an application, we can generalise results for boolean clause-sets regarding the hermitian rank/defect, especially the characterisation of unsatisfiable hitting clause-sets where between every two clauses we have exactly one conflict. We conclude with a list of open problems, and a discussion of the “generic translation scheme”. Journal Article Fundamenta Informaticae 109 1 83 119 31 12 2011 2011-12-31 10.3233/FI-2011-429 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2013-10-17T14:54:15.2413571 2012-02-23T17:01:57.0000000 College of Science Computer Science Oliver Kullmann 0000-0003-3021-0095 1
title Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure.
spellingShingle Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure.
Oliver, Kullmann
title_short Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure.
title_full Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure.
title_fullStr Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure.
title_full_unstemmed Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure.
title_sort Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure.
author_id_str_mv 2b410f26f9324d6b06c2b98f67362d05
author_id_fullname_str_mv 2b410f26f9324d6b06c2b98f67362d05_***_Oliver, Kullmann
author Oliver, Kullmann
author2 Oliver Kullmann
format Journal article
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description Concluding this mini-series of 2 articles on the foundations of generalised clause-sets, we study the combinatorial properties of non-boolean conjunctive normal forms (clause-sets), allowing arbitrary (but finite) sets of values for variables, while literals express that some variable shall not get some (given) value. First we study the properties of the direct translation (or “encoding”) of generalised clause-sets into boolean clause-sets. Many combinatorial properties are preserved, and as a result we can lift fixed-parameter tractability of satisfiability in the maximal deficiency from the boolean case to the general case. Then we turn to irredundant clause-sets, which generalise minimally unsatisfiable clause-sets, and we prove basic properties. The simplest irredundant clause-sets are hitting clause-sets, and we provide characterisations and generalisations. Unsatisfiable irredundant clause-sets are the minimally unsatisfiable clause-sets, and we provide basic tools. These tools allow us to characterise the minimally unsatisfiable clause-sets of minimal deficiency. Finally we provide a new translation of generalised boolean clause-sets into boolean clause-sets, the nested translation, which preserves the conflict structure. As an application, we can generalise results for boolean clause-sets regarding the hermitian rank/defect, especially the characterisation of unsatisfiable hitting clause-sets where between every two clauses we have exactly one conflict. We conclude with a list of open problems, and a discussion of the “generic translation scheme”.
published_date 2011-12-31T03:16:42Z
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