Simulating Polynomial-Time Nondeterministic Turing Machines via Nondeterministic Turing Machines

TL;DR

The paper tackles the longstanding question of whether $\mathcal{NP}$ equals $\rm co\mathcal{NP}$ by constructing a language $L_s$ that is in $\mathcal{NP}$ but not in $\rm co\mathcal{NP}$, thereby establishing $\mathcal{NP}\neq{\rm co}\mathcal{NP}$ and implying $\mathcal{P}\neq\mathcal{NP}$. The core approach eschews traditional diagonalization in favor of a simulation-based scheme: enumerate polynomial-time nondeterministic Turing machines, build a universal nondeterministic TM $U$ to realize a language outside ${\rm co}\mathcal{NP}$, and then show $L_s$ is expressible as a union $\bigcup_i L_s^i$ with each $L_s^i\in\mathcal{NP}$, ensuring $L_s\in\mathcal{NP}$. The work further explores the relativization barrier, arguing that under certain rational assumptions, relativized separations may require non-enumerability results for ${\rm co}\mathcal{NP}$ machines, thus highlighting limits of palliative simulation techniques. Beyond the main separation, the paper derives consequences for proof complexity (no polynomially bounded Frege proofs) and discusses the existence of ${\rm co}\mathcal{NP}$-intermediate languages, providing a broader view of the structure of these classes. The synthesis links to oracle results and potential implications for related classes such as $\mathcal{NEXP}$ and $\mathcal{BPP}$, placing the findings within the wider landscape of complexity theory and proof complexity.

Abstract

We prove in this paper that there is a language $L_s$ accepted by some nondeterministic Turing machine that runs within time $O(n^k)$ for any positive integer $k\in\mathbb{N}_1$ but not by any ${\rm co}\mathcal{NP}$ machines. Then we further show that $L_s$ is in $\mathcal{NP}$, thus proving that $$\mathcal{NP}\neq{\rm co}\mathcal{NP}.$$ The main techniques used in this paper are simulation and the novel new techniques developed in the author's recent work. Our main result has profound implications, such as $\mathcal{P}\neq\mathcal{NP}$, etc. Further, if there exists some oracle $A$ such that $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$, we then explore what mystery lies behind it and show that if $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$ and under some rational assumptions, then the set of all ${\rm co}\mathcal{NP}^A$ machines is not enumerable, thus showing that the simulation techniques are not applicable for the first half of the whole step to separate $\mathcal{NP}^A$ from ${\rm co}\mathcal{NP}^A$. Finally, a lower bounds result for Frege proof systems is presented (i.e., no Frege proof systems can be polynomially bounded).

Figures (2)

• Figure 1: The most believed possibility between $\mathcal{NP}$ and ${\rm co}\mathcal{NP}$
• Figure 2: Cantor pairing function

Theorems & Definitions (54)

• theorem 1
• Corollary 2
• theorem 3
• Corollary 4
• Corollary 5
• Corollary 6
• theorem 7
• Corollary 8
• theorem 9
• Definition 2.1: $k$-tape nondeterministic Turing machine, AHU74