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Fagin's Theorem
Fagin's theorem is the oldest result of descriptive complexity theory, a branch of computational complexity theory that characterizes complexity classes in terms of logic-based descriptions of their problems rather than by the behavior of algorithms for solving those problems. The theorem states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It was proven by Ronald Fagin in 1973 in his doctoral thesis, and appears in his 1974 paper. The arity required by the second-order formula was improved (in one direction) in Lynch's theorem, and several results of Grandjean have provided tighter bounds on nondeterministic ''random-access'' machines. Proof In addition to Fagin's 1974 paper, Immerman 1999 provides a detailed proof of the theorem. It is straightforward to show that every existential second-order formula can be recognized in NP, by nondeterministically choosing the value of all existentially-qualified v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Complexity ''Descriptive Complexity'' is a book in mathematical logic and computational complexity theory by Neil Immerman. It concerns descriptive complexity theory, an area in which the expressibility of mathematical properties using different types of logic is shown to be equivalent to their computability in different types of resource-bounded models of computation. It was published in 1999 by Springer-Verlag in their book series Graduate Texts in Computer Science. Topics The book has 15 chapters, roughly grouped into five chapters on first-order logic, three on second-order logic, and seven independent chapters on advanced topics. The first two chapters provide background material in first-order logic (including first-order arithmetic, the BIT predicate, and the notion of a first-order query) and complexity theory (including formal languages, resource-bounded complexity classes, and complete problems). Chapter three begins the connection between lo |