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An Oracle Relative to Which NP = PSPACE but UP Has No Many-One Complete Sets


Core Concepts
The authors construct an oracle relative to which NP = PSPACE, but UP has no many-one complete sets.
Abstract
The authors construct an oracle Φ that satisfies the following properties: NP^Φ = PSPACE^Φ (property P1). This is achieved by encoding the PSPACE-complete problem QBFΦ into the oracle Φ. UP^Φ has no many-one complete sets (property P2). This is achieved by carefully constructing the oracle Φ in a way that for each NP^Φ-machine Ni, either: Ni is not an UP^Φ-machine (property P2.I), or there exists a witness language W^Φ_i that does not reduce to the language decided by Ni (property P2.II). The authors show that their oracle construction can be performed as stated and that the desired properties P1 and P2 hold relative to the constructed oracle Φ. The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. It answers several questions by Pudlák, e.g., the implications UP ⇒ CONN and SAT ⇒ TFNP are false relative to the oracle. Moreover, the oracle demonstrates that, in principle, it is possible that TFNP-complete problems exist, while at the same time SAT has no p-optimal proof systems.
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Deeper Inquiries

How can the construction of an oracle that satisfies TFNP ⇒ DisjCoNP be achieved, which remains as an open question in this work

To construct an oracle that satisfies TFNP ⇒ DisjCoNP, we need to carefully design the oracle construction process to ensure that the properties required for this implication are met. In the given context, the construction involves adding words to the oracle at specific stages based on the behavior of certain computations. By strategically choosing when and how to add words to the oracle, we can control the acceptance paths of the computations and ensure that the desired properties are satisfied. One approach could be to focus on the stages where the UP-machines do not have complete sets and manipulate the oracle additions to enforce the DisjCoNP property. This may involve adding specific words to the oracle at critical points to prevent reductions between disjoint pairs and maintain the separation between NP and coNP classes. By carefully selecting the words added to the oracle and considering the interactions between different computations, we can achieve the desired outcome of TFNP ⇒ DisjCoNP.

What are the implications of this oracle construction for the relationships between other complexity classes and conjectures beyond those discussed in the paper

The implications of the oracle construction presented in the context go beyond the specific conjectures discussed in the paper. By demonstrating the possibility of constructing an oracle where TFNP-complete problems exist while SAT has no p-optimal proof systems, we open up new avenues for exploring the relationships between different complexity classes and conjectures. One significant implication is the refutation of several previously open implications, such as UP ⇒ CONN, UP ⇒ DisjNP, CON ⇒ CONN, CON ⇒ NP ≠ coNP, SAT ⇒ DisjCoNP, SAT ⇒ TFNP, and SAT ⇒ NP ≠ coNP. These refutations challenge existing assumptions and highlight the complexity of the relationships between optimal proof systems, promise classes, and complete problems. Furthermore, the construction of this oracle provides insights into the interplay between different complexity classes and the potential existence of TFNP-complete problems. It offers a new perspective on the landscape of computational complexity and opens up possibilities for further research and exploration in this area.

Are there any practical applications or implications of the insights gained from this oracle construction, particularly in the context of cryptography or other areas where TFNP problems are relevant

The insights gained from this oracle construction have practical implications in various fields, particularly in cryptography and the design of secure cryptographic schemes. The existence of TFNP-complete problems has significant implications for the security and efficiency of cryptographic protocols. One practical application is in the development of cryptosystems based on the hardness of TFNP-complete problems. By leveraging the complexity of these problems, cryptographers can design secure encryption and authentication mechanisms that offer strong guarantees against attacks. The construction of an oracle demonstrating the existence of TFNP-complete problems provides a theoretical foundation for exploring the security implications of such problems in cryptographic applications. Additionally, the insights from this work can inform the design of cryptographic protocols that rely on the complexity of specific problems within the TFNP class. Understanding the relationships between different complexity classes and the existence of complete problems can lead to the development of more robust and secure cryptographic algorithms that resist attacks and ensure data confidentiality and integrity.
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