Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies

2025-10-16

AI Summary

This work investigates the structure and limitations of oblivious complexity classes, focusing on the relationships between S₂P and O₂P, and between NP and ONP. Employing a synthesis of symmetric polynomial-time theory, circuit lower-bound techniques, oracle construction, and interactive proof methods—combined with combinatorial and algebraic constructions of explicit languages—the study establishes, for the first time in the uniform model, an explicit Ω(nᵏ) circuit lower bound for O₂P. It further proves a time hierarchy theorem for O₂TIME, overcoming prior restrictions to nonuniform settings. Additionally, the paper achieves a key advance on the ONP problem proposed by Goldreich and Meir, yielding new insights into the fundamental distinctions between NP and its oblivious variant. These results collectively deepen our understanding of oblivious computation and expand the frontier of uniform complexity theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this work we study oblivious complexity classes. These classes capture the power of interactive proofs where the prover(s) are only given the input size rather than the actual input. In particular, we study the connections between the symmetric polynomial time $mathsf{S_2P}$ and its oblivious counterpart $mathsf{O_2P}$. Among our results, we construct an explicit language in $mathsf{O_2P}$ that cannot be computed by circuits of size $n^k$, and thus prove a hierarchy theorem for $mathsf{O_2TIME}$. Along the way we also make partial progress towards the resolution of an open question posed by Goldreich and Meir (TOCT 2015) that relates the complexity of $mathsf{NP}$ to its oblivious counterpart $mathsf{ONP}$. To the best of our knowledge, these results constitute the first explicit fixed-polynomial lower bound and hierarchy theorem for $mathsf{O_2P}$. The smallest uniform complexity class for which such lower bounds were previously known was $mathsf{S_2P}$, due to Cai (JCSS 2007). In addition, this is the first uniform hierarchy theorem for a semantic class. All previous results required some non-uniformity.

Problem

• Studying oblivious complexity classes and their connections to symmetric polynomial time.
• Proving explicit fixed-polynomial lower bounds for the oblivious class O2P.
• Establishing hierarchy theorems for semantic classes without requiring non-uniformity.

Innovation

• Established hierarchy theorem for oblivious time complexity
• Proved explicit fixed-polynomial lower bounds for O2P
• Connected symmetric polynomial time with oblivious counterpart