Simulating Polynomial-Time Nondeterministic Turing Machines via Nondeterministic Turing Machines

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 a groundbreaking result 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).

💡 Research Summary

The manuscript under review claims to resolve the long‑standing NP versus coNP problem by constructing a language Lₛ that is accepted by a nondeterministic Turing machine (NTM) running in O(nᵏ) time for any fixed k, yet cannot be recognized by any coNP machine. Moreover, the authors assert that Lₛ belongs to NP, thereby concluding NP ≠ coNP. From this central claim they derive a cascade of consequences: P ≠ NP, the non‑existence of polynomial‑size Frege proofs, and a relativization barrier result stating that, under the hypothesis Pᴬ ≠ NPᴬ = coNPᴬ for some oracle A, the set of all coNPᴬ machines is not enumerable.

Key components of the paper

1. Existence of Lₛ – The authors begin by noting that all polynomial‑time NTMs are enumerable and then sketch a diagonal construction that supposedly yields a language Lₛ not in coNP. The description is informal: they refer to “ordinary simulation techniques” and “novel new techniques” without giving a concrete definition of the machine, its encoding, or the verification procedure. No explicit NTM Mₖ is presented, nor is a certificate‑checking algorithm for Lₛ described.

2. Proof that Lₛ ∉ coNP – The paper claims that any coNP verifier would lead to a contradiction via a standard diagonal argument. However, the argument is left as a high‑level intuition; the authors never formalize the complement language, nor do they exhibit a reduction from an arbitrary coNP verifier to the diagonal language. The step that would require showing that a coNP machine cannot decide Lₛ is simply asserted.

3. Proof that Lₛ ∈ NP – To place Lₛ in NP the authors state that a nondeterministic certificate exists and can be verified by “simulation”. Again, the verification algorithm is not specified, and there is no bound on the certificate length or on the runtime of the verifier. In standard complexity theory, proving membership in NP demands an explicit polynomial‑time deterministic verifier, which is absent here.

4. Derived consequences – Assuming the two previous claims, the authors deduce NP ≠ coNP, and from that they claim P ≠ NP. While NP ≠ coNP indeed implies P ≠ NP (since P ⊆ NP ∩ coNP), the paper does not address the possibility that their proof of NP ≠ coNP could be flawed, making all downstream conclusions unreliable.

5. Relativization barrier – The manuscript references the classic Baker‑Gill‑Solovay (BGS) result that there exist oracles A with NPᴬ = coNPᴬ and oracles B with Pᴮ ≠ NPᴮ. It then posits a scenario where Pᴬ ≠ NPᴬ = coNPᴬ and, under “some rational assumptions”, argues that the collection of coNPᴬ machines cannot be enumerated. This claim contradicts the basic fact that Turing machines (deterministic or nondeterministic) are countable and can be listed by their descriptions. The paper does not clarify what encoding or restriction makes enumeration impossible, nor does it provide a rigorous proof.

6. Frege proof lower bounds – The authors claim to prove that no Frege proof system admits polynomial‑size proofs for all propositional tautologies, i.e., that Frege systems are not polynomially bounded. This is a major open problem in proof complexity. Existing literature (e.g., Razborov, Krajíček) only yields super‑polynomial lower bounds for restricted systems, while the general Frege lower bound remains unresolved. The manuscript offers no new combinatorial or algebraic technique, merely stating that “the novel new techniques” suffice. Consequently, the claim lacks any substantive evidence.

7. Structure of the paper – The document contains a typical introduction, a “contributions” section, background, and several numbered sections (enumerability, simulation, relativization, coNP structure, Frege systems). Appendices claim to prove the equivalence of two definitions of NP and to construct a variant language L′. However, these appendices are equally terse and do not contain formal proofs, leaving the central theorems unsupported.

Critical assessment

• Lack of formal constructions: The core language Lₛ is never defined with precision. Without an explicit NTM description, certificate format, or verification algorithm, the statements “Lₛ ∈ NP” and “Lₛ ∉ coNP” cannot be verified.

• Missing diagonal argument details: The classic diagonalization used to separate a language from a class requires careful handling of machine encodings and runtime bounds. The paper glosses over these details, making the argument non‑rigorous.

• Contradiction with known results: The claim that the set of coNPᴬ machines can be non‑enumerable directly contradicts the countability of Turing machine descriptions. The authors attempt to rescue the claim by invoking “rational assumptions” without specifying them, which is insufficient for a mathematical proof.

• Proof complexity overreach: Claiming a general Frege lower bound without providing a new combinatorial framework is a substantial overstatement. The community has recognized this as a deep open problem; any purported solution would need to survive extensive peer scrutiny.

• Citation and prior work: The manuscript references classic works (Cook‑Levin, Baker‑Gill‑Solovay, Razborov) but fails to engage with the substantial body of literature on NP vs coNP, relativization, and proof complexity. It also misinterprets some known implications (e.g., “NP ≠ coNP ⇒ P ≠ NP” is true, but the converse is not established).

Conclusion

While the paper’s title and abstract promise a breakthrough, the body does not deliver the rigorous constructions, detailed proofs, or novel techniques required to substantiate its claims. The central theorem that a language exists in NP but not in coNP is presented without a concrete definition or verification procedure, and the subsequent corollaries (P ≠ NP, Frege lower bounds, non‑enumerability of coNPᴬ machines) rely on this unproven foundation. Consequently, the manuscript does not meet the standards of a valid proof in computational complexity theory. Further work would need to provide explicit machine descriptions, formal diagonal arguments, and a clear, verifiable proof of the Frege lower bound before the community could accept any of the asserted results.