Abstract
Second-order logic consists of first-order logic plus the power to quantify over relations on the universe. We prove Fagin’s theorem which says that the queries computable in NP are exactly the second-order existential queries. A corollary due to Stockmeyer says that the second-order queries are exactly those computable in the polynomial-time hierarchy.
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Notes
A nondeterministic Turing machine can make one of at most a bounded number of choices at any step. By reducing this to a binary choice per step, we slow the machine down by a small constant factor and make the analysis simpler.
A Horn formula is a formula in conjunctive normal form with at most one positive literal per clause (Definition 9.26).
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© 1999 Springer Science+Business Media New York
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Immerman, N. (1999). Second-Order Logic and Fagin’s Theorem. In: Descriptive Complexity. Graduate Texts in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0539-5_8
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DOI: https://doi.org/10.1007/978-1-4612-0539-5_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6809-3
Online ISBN: 978-1-4612-0539-5
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