{"title":"一阶公式的树宽度","authors":"Isolde Adler, M. Weyer","doi":"10.2168/LMCS-8(1:32)2012","DOIUrl":null,"url":null,"abstract":"We introduce tree-width for first order formulae \\phi, fotw(\\phi). We show\nthat computing fotw is fixed-parameter tractable with parameter fotw. Moreover,\nwe show that on classes of formulae of bounded fotw, model checking is fixed\nparameter tractable, with parameter the length of the formula. This is done by\ntranslating a formula \\phi\\ with fotw(\\phi)<k into a formula of the k-variable\nfragment L^k of first order logic. For fixed k, the question whether a given\nfirst order formula is equivalent to an L^k formula is undecidable. In\ncontrast, the classes of first order formulae with bounded fotw are fragments\nof first order logic for which the equivalence is decidable.\n Our notion of tree-width generalises tree-width of conjunctive queries to\narbitrary formulae of first order logic by taking into account the quantifier\ninteraction in a formula. Moreover, it is more powerful than the notion of\nelimination-width of quantified constraint formulae, defined by Chen and Dalmau\n(CSL 2005): for quantified constraint formulae, both bounded elimination-width\nand bounded fotw allow for model checking in polynomial time. We prove that\nfotw of a quantified constraint formula \\phi\\ is bounded by the\nelimination-width of \\phi, and we exhibit a class of quantified constraint\nformulae with bounded fotw, that has unbounded elimination-width. A similar\ncomparison holds for strict tree-width of non-recursive stratified datalog as\ndefined by Flum, Frick, and Grohe (JACM 49, 2002).\n Finally, we show that fotw has a characterization in terms of a cops and\nrobbers game without monotonicity cost.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2012-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Tree-width for first order formulae\",\"authors\":\"Isolde Adler, M. Weyer\",\"doi\":\"10.2168/LMCS-8(1:32)2012\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We introduce tree-width for first order formulae \\\\phi, fotw(\\\\phi). We show\\nthat computing fotw is fixed-parameter tractable with parameter fotw. Moreover,\\nwe show that on classes of formulae of bounded fotw, model checking is fixed\\nparameter tractable, with parameter the length of the formula. This is done by\\ntranslating a formula \\\\phi\\\\ with fotw(\\\\phi)<k into a formula of the k-variable\\nfragment L^k of first order logic. For fixed k, the question whether a given\\nfirst order formula is equivalent to an L^k formula is undecidable. In\\ncontrast, the classes of first order formulae with bounded fotw are fragments\\nof first order logic for which the equivalence is decidable.\\n Our notion of tree-width generalises tree-width of conjunctive queries to\\narbitrary formulae of first order logic by taking into account the quantifier\\ninteraction in a formula. Moreover, it is more powerful than the notion of\\nelimination-width of quantified constraint formulae, defined by Chen and Dalmau\\n(CSL 2005): for quantified constraint formulae, both bounded elimination-width\\nand bounded fotw allow for model checking in polynomial time. We prove that\\nfotw of a quantified constraint formula \\\\phi\\\\ is bounded by the\\nelimination-width of \\\\phi, and we exhibit a class of quantified constraint\\nformulae with bounded fotw, that has unbounded elimination-width. A similar\\ncomparison holds for strict tree-width of non-recursive stratified datalog as\\ndefined by Flum, Frick, and Grohe (JACM 49, 2002).\\n Finally, we show that fotw has a characterization in terms of a cops and\\nrobbers game without monotonicity cost.\",\"PeriodicalId\":49904,\"journal\":{\"name\":\"Logical Methods in Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2012-03-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Logical Methods in Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.2168/LMCS-8(1:32)2012\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Logical Methods in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.2168/LMCS-8(1:32)2012","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
We introduce tree-width for first order formulae \phi, fotw(\phi). We show
that computing fotw is fixed-parameter tractable with parameter fotw. Moreover,
we show that on classes of formulae of bounded fotw, model checking is fixed
parameter tractable, with parameter the length of the formula. This is done by
translating a formula \phi\ with fotw(\phi)
期刊介绍：
Logical Methods in Computer Science is a fully refereed, open access, free, electronic journal. It welcomes papers on theoretical and practical areas in computer science involving logical methods, taken in a broad sense; some particular areas within its scope are listed below. Papers are refereed in the traditional way, with two or more referees per paper. Copyright is retained by the author.
Topics of Logical Methods in Computer Science:
Algebraic methods
Automata and logic
Automated deduction
Categorical models and logic
Coalgebraic methods
Computability and Logic
Computer-aided verification
Concurrency theory
Constraint programming
Cyber-physical systems
Database theory
Defeasible reasoning
Domain theory
Emerging topics: Computational systems in biology
Emerging topics: Quantum computation and logic
Finite model theory
Formalized mathematics
Functional programming and lambda calculus
Inductive logic and learning
Interactive proof checking
Logic and algorithms
Logic and complexity
Logic and games
Logic and probability
Logic for knowledge representation
Logic programming
Logics of programs
Modal and temporal logics
Program analysis and type checking
Program development and specification
Proof complexity
Real time and hybrid systems
Reasoning about actions and planning
Satisfiability
Security
Semantics of programming languages
Term rewriting and equational logic
Type theory and constructive mathematics.
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