2007/06/22 by Nerio Borges, Borges, Nerio, Blai Bonet +1
Computer Science · Mathematics · #Computational Complexity (cs.CC) #FOS: Computer and information sciences #Mathematics and Applications #Matrix Theory and Algorithms #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0706.3412
openalex publication_date 2007/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Independent-Set. 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 Independent-Set and the second the rest of P. That is, a decomposition that divides every such sentence into a quotient and residue modulo Independent-Set. 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.ow 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.