On Canonical Forms of Complete Problems via First-order Projections

The class of problems complete for NP via first-order reductions is known to be characterized by existential second-order sentences of a fixed form. All such sentences are built around the so-called generalized IS-form of the sentence that defines IndependentSet. This result can also be understood as that every sentence that defines a NP-complete problem P can be decomposed in two disjuncts such that the first one characterizes a fragment of P as hard as IndependentSet and the second the rest of P . That is, a decomposition that divides every such sentence into a a “quotient and residue” modulo IndependentSet. In this paper, we show that this result can be generalized over a wide collection of complexity classes, including the so-called nice classes. Moreover, we show that such decomposition can be done for any complete problem with respect to the given class, and that two such decompositions are non-equivalent in general. Interestingly, our results are based on simple and well-known properties of first-order reductions.