Abstract
The boolean hierarchy of k-partitions over NP for k ≥ 3 was introduced as a generalization of the well-known boolean hierarchy of sets (2-partitions). The classes of this hierarchy are exactly those classes of NP-partitions which are generated by finite labeled lattices. We generalize the boolean hierarchy of NP-partitions by studying partition classes which are defined by finite labeled posets. We give an exhaustive answer to the question of which relativizable inclusions between partition classes can occur depending on the relation between their defining posets. This provides additional evidence for the validity of the Embedding Conjecture for lattices. The study of the generalized boolean hierarchy is closely related to the issue of whether one can reduce the number of solutions of NP problems. For finite cardinality types, assuming the generalized boolean hierarchy of k-partitions over NP is strict, we give a complete characterization when such solution reductions are possible. This resolves in some sense an open question by Hemaspaandra et al.
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Kosub, S. NP-Partitions over Posets with an Application to Reducing the Set of Solutions of NP Problems. Theory Comput Systems 38, 83–113 (2005). https://doi.org/10.1007/s00224-004-1114-1
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DOI: https://doi.org/10.1007/s00224-004-1114-1