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The Polynomial Hierarchy does not collapse

Reiner Czerwinski

TL;DR

This paper addresses whether the Polynomial Hierarchy $\PH$ collapses by linking it to the Arithmetical Hierarchy $\AH$ through padding arguments and black-box search analyses. It shows that a $\Sigma_{k+1}^p$ language retains exponential query complexity relative to a $\Pi_k^p$ oracle, even after padding, implying no collapse of $\PH$. The core contributions include a padding-based reduction from $\AH$ to finite-input $\PH$ problems, a black-box search framework for oracle access, and a non-relativizing method that circumvents known relativization barriers. Taken together, these results strengthen the separation within the polynomial hierarchy and highlight the necessity of non-relativizing techniques in such complexity-theoretic proofs.

Abstract

The arithmetical hierarchy (AH) is similar to the polynomial hierarchy (PH). Unlike the PH, the AH does not collapse relative to any oracle. A language in the (k + 1)-st level of the AH is computable enumerable (c.e.) relative to the kth level. So, given an oracle in the kth level of the AH, we could use a black-box search to decide whether the input word is in the language. With very large padding arguments, i.e. the paddings grow faster than any relative to the level k of the AH computable function, we would construct a language contained in the k + 1 level of PH, if we use only a finite set of input words. From the oracle in AH, we would construct an analogue oracle at the kth level of PH. For the input words of the finite set, a word is in the language of AH, if and only if it is in the language of PH. And the input word is in the oracle set of AH, if and only if it is in the oracle of PH. As in the language of AH, we must apply a black-box search in the language of PH. So, we would also have exponentially many oracle queries in the language of PH. The PH does not collapse.

The Polynomial Hierarchy does not collapse

TL;DR

This paper addresses whether the Polynomial Hierarchy collapses by linking it to the Arithmetical Hierarchy through padding arguments and black-box search analyses. It shows that a language retains exponential query complexity relative to a oracle, even after padding, implying no collapse of . The core contributions include a padding-based reduction from to finite-input problems, a black-box search framework for oracle access, and a non-relativizing method that circumvents known relativization barriers. Taken together, these results strengthen the separation within the polynomial hierarchy and highlight the necessity of non-relativizing techniques in such complexity-theoretic proofs.

Abstract

The arithmetical hierarchy (AH) is similar to the polynomial hierarchy (PH). Unlike the PH, the AH does not collapse relative to any oracle. A language in the (k + 1)-st level of the AH is computable enumerable (c.e.) relative to the kth level. So, given an oracle in the kth level of the AH, we could use a black-box search to decide whether the input word is in the language. With very large padding arguments, i.e. the paddings grow faster than any relative to the level k of the AH computable function, we would construct a language contained in the k + 1 level of PH, if we use only a finite set of input words. From the oracle in AH, we would construct an analogue oracle at the kth level of PH. For the input words of the finite set, a word is in the language of AH, if and only if it is in the language of PH. And the input word is in the oracle set of AH, if and only if it is in the oracle of PH. As in the language of AH, we must apply a black-box search in the language of PH. So, we would also have exponentially many oracle queries in the language of PH. The PH does not collapse.
Paper Structure (14 sections, 8 theorems, 42 equations)

This paper contains 14 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Notations and Prerequisites
  3. Languages
  4. Relations
  5. Arithmetical Hierarchy
  6. Properties of $\AH$
  7. Polynomial Hierarchy
  8. Black-Box Search and the $\AH$
  9. Oracles and computable Permutations
  10. Black-Box Complexity with Oracles
  11. The Relationship between $\PH$ and $\AH$
  12. The $\PH$ does not collapse
  13. Notes on the Relativization Barrier
  14. Computability Theory in a Nutshell

Key Result

lemma 1

Let the language $L$ be defined with a c.e. or co-c.e. relation $R$, such that where $Q_1,Q_2,\dots,Q_k$ are alternating existential and universal quantifiers with If $Q_1$ is existential (universal), then $L$ is in $\Sigma_k$($\Pi_k$).

Theorems & Definitions (17)

  • remark 1
  • lemma 1
  • proof
  • theorem 1
  • proof
  • definition 1
  • theorem 2
  • proof
  • lemma 2
  • proof
  • ...and 7 more