6 research outputs found
Disjoint NP-pairs from propositional proof systems
For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P). Moreover we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist
Tuples of disjoint NP-sets
Disjoint NP-pairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2. We define subclasses of the class of all disjoint k-tuples of NP-sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint NP-pairs exist if and only if complete disjoint k-tuples of NP-sets exist for all k ≥ 2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems
Classes of representable disjoint NP-pairs
For a propositional proof system P we introduce the complexity class of all disjoint -pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical -pairs associated with these proof systems hard or complete for . Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for and the reductions between the canonical pairs exist
Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP
We study the existence of optimal and p-optimal proof systems for classes in
the Boolean hierarchy over . Our main results concern
, i.e., the second level of this hierarchy:
If all sets in have p-optimal proof systems, then all sets in
have p-optimal proof systems. The analogous implication for
optimal proof systems fails relative to an oracle.
As a consequence, we clarify such implications for all classes
and in the Boolean hierarchy over : either we can
prove the implication or show that it fails relative to an oracle. Furthermore,
we show that the sets and have p-optimal proof
systems, if and only if all sets in the Boolean hierarchy over
have p-optimal proof systems which is a new characterization of a conjecture
studied by Pudl\'ak