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Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them.

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  • Die deskriptive Komplexitätstheorie (beschreibende Komplexitätstheorie) ist ein Teilbereich der endlichen Modelltheorie, die den Zusammenhang der Ausdrucksstärke von Logiken und Komplexitätstheorie untersucht. Während Komplexitätsklassen wie NP oder PSPACE üblicherweise durch ein spezielles Maschinenmodell (üblicherweise Turingmaschinen) definiert werden, lassen sich mit Hilfe der deskriptiven Komplexitätstheorie diese Klassen auch durch „natürliche“ Logiken wie der Prädikatenlogik erster oder höherer Stufe oder Fixpunktlogiken charakterisieren. (de)
  • Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them. Specifically, each logical system produces a set of queries expressible in it. The queries – when restricted to finite structures – correspond to the computational problems of traditional complexity theory. The first main result of descriptive complexity was Fagin's theorem, shown by Ronald Fagin in 1974. It established that NP is precisely the set of languages expressible by sentences of existential second-order logic; that is, second order logic excluding universal quantification over relations, functions, and subsets. Many other classes were later characterized in such a manner, most of them by Neil Immerman: * First-order logic defines the class FO, corresponding to AC0, the languages recognized by polynomial-size circuits of bounded depth, which equals the languages recognized by a concurrent random access machine in constant time. * First-order logic with a commutative, transitive closure operator added yields SL, which equals L, problems solvable in logarithmic space. * First-order logic with a transitive closure operator yields NL, the problems solvable in nondeterministic logarithmic space. * In the presence of linear order, first-order logic with a least fixed point operator gives P, the problems solvable in deterministic polynomial time. * Existential second-order logic yields NP. * Universal second-order logic (excluding existential second-order quantification) yields co-NP. * Second-order logic corresponds to PH, as mentioned above. * Second-order logic with a transitive closure (commutative or not) yields PSPACE, the problems solvable in polynomial space. * Second-order logic with a least fixed point operator gives EXPTIME, the problems solvable in exponential time. * , logic with existential quantifier of order i followed by a formula of order is equal to * * HO is equal to ELEMENTARY (en)
  • En informatique théorique, la complexité descriptive est une branche de la théorie de la complexité et de la théorie des modèles, qui caractérise les classes de complexité en termes de logique qui permet de décrire les problèmes. La complexité descriptive donne un nouveau point de vue car on définit des classes de complexité sans faire appel à une notion de machines comme les machines de Turing. Par exemple la classe NP correspond à l'ensemble des problèmes exprimables en logique du second ordre existentielle : c'est le théorème de Fagin. (fr)
  • 記述計算量(きじゅつけいさんりょう、英: Descriptive complexity)は、有限モデル理論の一種であり、計算複雑性理論と数理論理学の一分野である。複雑性クラスを言語で表現するのに必要とされる論理の種類によって特徴付けることを目的とする。例えば、PHは二階述語論理の論理式で表現される言語のクラスと正確に対応している。このような複雑性と論理の繋がりによって、2つの分野の間で容易に変換が可能となり、新たな証明手法を生み出したり、ある複雑性クラスが本質的なものであって、特定の抽象機械に結びつくものではないことを示すことができる。 (ja)
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  • Die deskriptive Komplexitätstheorie (beschreibende Komplexitätstheorie) ist ein Teilbereich der endlichen Modelltheorie, die den Zusammenhang der Ausdrucksstärke von Logiken und Komplexitätstheorie untersucht. Während Komplexitätsklassen wie NP oder PSPACE üblicherweise durch ein spezielles Maschinenmodell (üblicherweise Turingmaschinen) definiert werden, lassen sich mit Hilfe der deskriptiven Komplexitätstheorie diese Klassen auch durch „natürliche“ Logiken wie der Prädikatenlogik erster oder höherer Stufe oder Fixpunktlogiken charakterisieren. (de)
  • En informatique théorique, la complexité descriptive est une branche de la théorie de la complexité et de la théorie des modèles, qui caractérise les classes de complexité en termes de logique qui permet de décrire les problèmes. La complexité descriptive donne un nouveau point de vue car on définit des classes de complexité sans faire appel à une notion de machines comme les machines de Turing. Par exemple la classe NP correspond à l'ensemble des problèmes exprimables en logique du second ordre existentielle : c'est le théorème de Fagin. (fr)
  • 記述計算量(きじゅつけいさんりょう、英: Descriptive complexity)は、有限モデル理論の一種であり、計算複雑性理論と数理論理学の一分野である。複雑性クラスを言語で表現するのに必要とされる論理の種類によって特徴付けることを目的とする。例えば、PHは二階述語論理の論理式で表現される言語のクラスと正確に対応している。このような複雑性と論理の繋がりによって、2つの分野の間で容易に変換が可能となり、新たな証明手法を生み出したり、ある複雑性クラスが本質的なものであって、特定の抽象機械に結びつくものではないことを示すことができる。 (ja)
  • Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them. (en)
rdfs:label
  • Deskriptive Komplexitätstheorie (de)
  • Descriptive complexity theory (en)
  • Complexité descriptive (fr)
  • 記述計算量 (ja)
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