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Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies

Karthik Gajulapalli, Zeyong Li, Ilya Volkovich

TL;DR

The paper advances the study of oblivious complexity by linking oblivious symmetric time to fixed-polynomial circuit lower bounds and hierarchical structure. It introduces an explicit construction showing $\,\Ot \nsubseteq \SIZE[n^k]$ for all $k$ and proves a hierarchy for the oblivious time class $\Ott$, including parameterized bounds. Central to the approach is the Range Avoidance framework, together with prefix truth-table constructions that yield explicit hard languages while remaining in oblivious classes. The work also develops uniformly-sparse extensions to connect $\St$ and $\Ot$, establishing $\Se=\Oe$ and offering partial progress on Goldreich and Meir's open questions about NP and ONP, thereby marking the first uniform hierarchy results for a semantic class and broadening the toolkit for explicit lower bounds in oblivious settings.

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.

Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies

TL;DR

The paper advances the study of oblivious complexity by linking oblivious symmetric time to fixed-polynomial circuit lower bounds and hierarchical structure. It introduces an explicit construction showing for all and proves a hierarchy for the oblivious time class , including parameterized bounds. Central to the approach is the Range Avoidance framework, together with prefix truth-table constructions that yield explicit hard languages while remaining in oblivious classes. The work also develops uniformly-sparse extensions to connect and , establishing and offering partial progress on Goldreich and Meir's open questions about NP and ONP, thereby marking the first uniform hierarchy results for a semantic class and broadening the toolkit for explicit lower bounds in oblivious settings.

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 and its oblivious counterpart . Among our results, we construct an explicit language in that cannot be computed by circuits of size , and thus prove a hierarchy theorem for . 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 to its oblivious counterpart . To the best of our knowledge, these results constitute the first explicit fixed-polynomial lower bound and hierarchy theorem for . The smallest uniform complexity class for which such lower bounds were previously known was , 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.
Paper Structure (23 sections, 25 theorems, 6 equations, 1 figure)

This paper contains 23 sections, 25 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Symmetric Time
  4. Oblivious Complexity Classes
  5. Sparsity
  6. Range Avoidance
  7. Time Hierarchy Theorem
  8. Previous Results
  9. Our Results
  10. Proof Overview
  11. Avoid Framework for Circuit Lower bounds
  12. Hierarchy Theorems for $\Ott$
  13. Sparsity and Lower Bounds
  14. Preliminaries
  15. Complexity Classes
  16. ...and 8 more sections

Key Result

Theorem 1

For all $k \in \N$, $\Ot \not \subseteq \SIZE[n^k]$. Moreover, for each $k$ there exists an explicit language $L_k \in \Ot$ such that $L_k \not \in \SIZE[n^k]$.

Figures (1)

  • Figure 1: $\Ott[t(n)^2]$ Verifier for Language in $\Stt[t(n)]$ with $t(n)$-uniformly-sparse extension

Theorems & Definitions (57)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1.1: Grid Coloring AGL23
  • Definition 2.1: Deterministic Time
  • Definition 2.2: Symmetric Time
  • Definition 2.3: Single-valued $\FSt$ algorithm
  • Definition 2.4: Oblivious Symmetric Time
  • Definition 2.5
  • ...and 47 more