First-order reduction

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In computer science, a first-order reduction is a very weak type of reduction between two computational problems in computational complexity theory. A first-order reduction is a reduction where each component is restricted to be in the class FO of problems calculable in first-order logic.

Since we have \mbox{FO} \subsetneq \mbox{L}, the first-order reductions are weaker reductions than the logspace reductions.

Many important complexity classes are closed under first-order reductions, and many of the traditional complete problems are first-order complete as well (Immerman 1999 p. 49-50). For example, ST-connectivity is FO-complete for NL, and NL is closed under FO reductions (Immerman 1999, p. 51) (as are P, NP, and most other "well-behaved" classes).

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