A Critique of Czerwinski’s “Separation of $\pspace$ and $\exptime$”^††thanks: Supported in part by NSF grant CCF-2006496.

In this critique, we provide a summary and analysis of Czerwinski’s “Separation of \pspace and \exptime” [Cze21]. The paper purportedly proves its main claim that $\pspace\neq\exptime$ by introducing a novel proof technique with three main theorems: Theorem 1 proposes that any nontrivial property P of the time and space complexity of an arbitrary Turing machine is undecidable, Theorem 2 claims that if sets A and B are both $\exptime$-complete sets, then they are length-increasing one-one reducible to each other, and Theorem 3 argues that there exists an $\exptime$-complete set ${\mathcal{E}_{0}}$ and a $\pspace$-complete set ${\mathcal{P}_{0}}$ such that ${\mathcal{E}_{0}}$ is not length-increasing one-one reducible to ${\mathcal{P}_{0}}$. If correct, Czerwinski’s paper would be a major advance in computational complexity theory. In particular, it is well-known that $\pspace\subseteq\exptime$. On the other hand, whether $\pspace$ equals $\exptime$ remains one of the major open questions in theoretical computer science. Furthermore, if Czerwinski’s paper truly has introduced a novel proof technique to separate $\pspace$ and $\exptime$, that could potentially serve as the foundation to establish other new separations.

The approach used in Czerwinski’s paper to show the separation of $\exptime$ and $\pspace$ is as follows: Use Corollary 2 of Theorem 1 to prove Theorem 3, and then by Theorem 2 and Theorem 3, claim that $\pspace\neq\exptime$. We aim to highlight the flaws in each of these theorems as well as present a counterexample to one of the theorems, thus showing that the paper fails to prove its central claim.

In Section 2, we provide the definitions and notations used throughout the original paper and our critique. Sections 3, 4, 5, 6, and 7 each present an overview and a discussion of a section of Czerwinski’s paper. Lastly, in Section 8, we conclude our critique.

In this section, we present readers with many standard concepts in complexity theory, as well as frequently used notations in Czerwinski’s paper. Interested readers can find equivalent definitions of the concepts in most modern textbooks on complexity [HO02, AB09, Sip13].

Let ${\mathbb{N}}=\{0,1,2,\ldots\}$ and let ${\mathbb{N}^{+}}=\{1,2,3,\ldots\}$. Given a string $w$, let $|w|$ denote the length of $w$. We will let $\epsilon$ denote the empty string. For any Turing machine (TM) $M$, let $L(M)$ be the language accepted by $M$.

We also introduce some notations that Czerwinski uses throughout the paper. For a single-tape deterministic Turing machine $M$ on input $x$, let $s_{M}(x)$ be the tape space used by $M$ on input $x$ and let $t_{M}(x)$ be the time used by $M$ on input $x$. Lastly, a property $\mathbf{P}$ is a set of Turing machines. A Turing machine $M$ is said to have property $\mathbf{P}$ if and only if $M\in\mathbf{P}$. A property $\mathbf{P}$ of the functions $t_{M}$ and $s_{M}$ is a set of Turing machines such that for every two Turing machines $M_{1}$ and $M_{2}$, if $t_{M_{1}}(x)=t_{M_{2}}(x)$ and $s_{M_{1}}(x)=s_{M_{2}}(x)$ for all $x\in\Sigma^{*}$, then either $M_{1}$ and $M_{2}$ are both in $\mathbf{P}$ or neither of them are in $\mathbf{P}$. A property $\mathbf{P}$ is nontrivial if and only if $\mathbf{P}\neq\emptyset$ and there exists a Turing machine $M$ such that $M\notin\mathbf{P}$.

Czerwinski’s paper also makes use of the S-m-n Theorem in a number of its arguments. Since explaining the entire theorem would mean defining many concepts in computability theory that are unnecessary to the purpose of this critique, we suggest interested readers directly consult the book “Introduction to Metamathematics” by S.C. Kleene [Kle52].

In the first section of Czerwinski’s paper, four sets are defined [Cze21],

with the last two stated to be examples of $\exptime$-complete and $\pspace$-complete sets respectively. It should also be mentioned that one can easily prove that ${\mathcal{E}}$ and ${\mathcal{P}}$ are also respectively $\exptime$-complete and $\pspace$-complete. Following the definitions is Corollary 1

Although this is correct, in later proofs there are no references made to this corollary. In addition, it is also difficult to see the importance of the sets ${\mathcal{E}}$ and ${\mathcal{P}}$, with ${\mathcal{E}_{0}}$ and ${\mathcal{P}_{0}}$ already introduced as complete sets for their respective classes.

We note in passing that the author introduces a lemma, whose statement is true. Although the given proof contains minor errors, we omit this discussion as the next sections’ discussion is of greater importance.

After the lemma, a remark regarding the relativization barrier is given. The content of the remark offers little to no insight into how the paper plans to deal with the relativization barrier. Consequently, we assume that its sole purpose is to assure readers that the paper will take the possible issue with the relativization barrier into consideration later on in Section 5 of the paper.

Section 2 of Czerwinski’s paper focuses on undecidability when considering properties of time and space complexity. First, two notations are introduced, $s_{M}(x)$ and $t_{M}(x)$, which we have defined in Section 2 of our critique. Afterwards, two inequalities concerning $s_{M}(x)$ and $t_{M}(x)$ are presented, which are taken from a paper by Hopcroft, Paul, and Valiant [HPV77]. For an arbitrary single-tape deterministic Turing machine $M$ and an arbitrary input $x$ where $M$ halts on $x$, it holds that:

Nonetheless, it is unclear how this result is applied to the arguments made in this paper, as it does not seem to be used by any of the proofs.

The paper then introduces its first theorem.

The theorem seems to be a modification of the well-known Rice’s theorem [Ric53]. Instead of using nontrivial semantic properties like in Rice’s theorem [Ric53], the theorem considers nontrivial properties of the functions $s_{M}$ and $t_{M}$, which are the space and time complexity of $M$ on some input $x$.

We present a counterexample of Theorem 1. Let $\mathbf{P_{1}}(s_{M},t_{M})=\{M\,\mid\>$ $M$ is a Turing machine and $s_{M}(\epsilon)\geq 2$ and $t_{M}(\epsilon)\geq 2\}$, and $L_{st2}=\{M\,\mid\>M\in\mathbf{P_{1}}(s_{M},t_{M})\}$. It is easy to see that this property is nontrivial: There exists some Turing machines that satisfy $\mathbf{P_{1}}$, which implies $L_{st2}\neq\emptyset$, and there exists a Turing machine $M_{s1}$ that only uses one tape space on some input $x_{s1}$ and a Turing machine $M_{t1}$ that only runs for one step before halting on some input $x_{t1}$, which implies $M_{s1},M_{t1}$ is not in $\mathbf{P_{1}}(s_{M},t_{M})$. We will describe the decider $S$ for $L_{st2}$ as follows.

The reason why $S$ can accept after running the $m$th step is because if $M$ only uses the first tape cell and there are $m$ different tape characters, then there are only $m$ different configurations of space. Consequently, if $M$ runs more than $m$ steps using only the first tape cell, then we can be certain that $M$ is looping forever within that cell. Apart from this, it should be easy to see how $S$ always halt and $L(S)=L_{st2}$.

The proof of Theorem 1 has a faulty assumption that for any TM $M$ and any nontrivial property $\mathbf{P}$ of the functions $s_{M}$ and $t_{M}$, and on any input $x$, the exact value of $s_{M}(x)$ and $t_{M}(x)$ are needed to check property $\mathbf{P}$. However, in the counterexample that we created, calculating neither the exact space complexity nor the exact time complexity was necessary in deciding whether a Turing machine $M$ is in $L_{st2}$.

The paper then uses the previous result of Theorem 1 and extends it to Corollary 2.

Unfortunately, as we have already shown that Theorem 1 is flawed, the proof of Corollary 2 is incomplete, thus its correctness is not guaranteed.

We continue to the findings of the next and final part of this section, Corollary 3.

While reading the proof of the corollary, we notice that the proof does not address the correctness of Corollary 3. Instead, the proof tries to set up another nontrivial property of the function $t_{M}$ for an arbitrary Turing machine $M$, and, based on Theorem 1, concludes that it is undecidable. It should also be recognized that Corollary 3 is in fact correct. Briefly speaking, the reason why checking whether a Turing machine $M$ accepts an input $x$ within $t$ steps might require a runtime of $\Omega(t)$ is because the overhead from simulating $M$ on $x$ could raise the total runtime to be bigger than $t$.

We will now critique Section 3 of Czerwinski’s paper. In this section, the paper presents findings related to length-increasing functions and reductions.

Czerwinski’s paper defines length-increasing functions similarly to Section 2 of our critique, with one additional requirement of having a polynomial-time computable inverse.

The paper next introduces Theorem 2, which is adapted from one of the results of Watanabe’s paper “On one-one polynomial time equivalence relations” [Wat85].

As a proof for this theorem, Czerwinski cites the proof of Corollary 3.4 from Watanabe’s paper [Wat85]. It should be noted that at the time of Watanabe’s paper, it was standard to define $\exptime$ to represent what we have defined as ${\rm E}$. As a result, Watanabe’s paper only provides a proof where $A$ and $B$ are ${\rm E}$-complete, rather than being $\exptime$-complete. Czerwinski’s paper does not provide a preliminaries section, so it is unclear if the paper intends to use ${\rm E}$ or $\exptime$. However, we believe its main goal is to separate $\exptime$ and $\pspace$, as the result ${\rm E}\neq\pspace$ is already well-known [Boo74]. Moreover, Watanabe [Wat85] does not define length-increasing functions to be necessarily polynomial-time invertible, which is different from the definition given in Czerwinski’s paper. For these reasons, the proof of Theorem 2 is invalid. This is a critical flaw in Czerwinksi’s paper since the result of Theorem 2 sets the stage for Theorem 3 to be able to show that $\pspace\neq\exptime$. We will discuss this more in Section 6.

Next, we will critique Lemma 2.

Before going into the proof, we would like to introduce some notations that the author uses. Functions $g_{1}$ and $g_{2}$ are defined as $g_{1}(M,k)=M_{g}$ and $g_{2}(M,k)=k_{g}$, and thus $(M_{g},k_{g})=g(M,k)=(g_{1}(M,k),g_{2}(M,k))$. By the S-m-n theorem, $g_{1}(M,k)=g_{1,S(M)}(k)$ and $g_{2}(M,k)=g_{2,S(M)}(k)$. Additionally, by Czerwinski’s definition of a length increasing function, $g$ is polynomial-time invertible, and thus $(M,k)=(g_{1,S(M)}^{-1}(k_{g}),g_{2,S(M)}^{-1}(k_{g}))$.

We would like to focus on the pseudocode given at the end of the proof.

This pseudocode is provided as an algorithm in which $M_{g}$ can be constructed and simulated on $k_{g}$ without using $k$, thus making it depend only on $M$. To our understanding, given $k_{g}$ as some input, the algorithm will construct $\hat{k}$ that, if defined, satisfies that $(M,\hat{k})\in A$. Afterwards, $M_{g}$ can be constructed with $g_{1,S(M)}(\hat{k})$. However, this construction does not prove the statement of the lemma. The lemma claims that $M_{g}$ only depends on $M$, while the provided pseudocode also uses $k_{g}$ in its calculation.

The main result is presented in Section 4 of Czerwinski’s paper.

It is crucial to see that in order for this theorem to imply that $\exptime\neq\pspace$, then Theorem 2 must hold. Theorem 2 proposes that all $\exptime$-complete sets are length-increasing polynomial-time one-one reducible to each other. Hypothetically, if Theorem 2 was in fact true, then one would only need to show that there is no length-increasing polynomial-time one-one reduction from a $\exptime$-complete set, ${\mathcal{E}_{0}}$, to a $\pspace$-complete set, ${\mathcal{P}_{0}}$, to prove that ${\mathcal{P}_{0}}$ is not $\exptime$-complete and thus $\exptime\neq\pspace$, which is the statement of Theorem 3. As we have identified the problems with Theorem 2 earlier, if Theorem 3 were to hold, it would not be strong enough to imply that $\pspace\neq\exptime$. Additionally, the proof of this theorem also uses Lemma 2 and Corollary 2, which we have shown to be invalid. As a result, we can say that this proof is similarly incomplete, and ultimately, the paper fails to show that $\pspace\neq\exptime$.

In this section, Czerwinski attempts to argue that the proof technique does not violate the relativization barrier. In order to do this, he introduces Proposition 1.

As a proof for this proposition, the paper provides pseudocode that takes a problem from ${\mathcal{E}}$ and computes a set in ${\mathcal{P}}$ in exponential time. The pseudocode seems correct, however the proposition itself is trivial. It is unclear how this proposition demonstrates that the purported proof technique does not violate the relativization barrier.

The paper claims that the existence of this function $f$ being a many-one reduction that is computable but not computable in polynomial-time “means ${\mathcal{E}}\leq_{m}{\mathcal{P}}$, but ${\mathcal{E}}\not\leq_{m}^{p}{\mathcal{P}}$ and so, $\pspace^{{\rm P}}\neq\exptime^{{\rm P}}$” [Cze21]. Prior to this point, the paper has not directly made the claim that ${\mathcal{E}}\not\leq_{m}^{p}{\mathcal{P}}$ so it is unclear how the existence of a exponential time reduction prevents a polynomial time one from existing. In fact, the existence of a polynomial-time many-one reduction would imply the existence of a exponential-time many-one reduction.

The claim that ${\mathcal{E}}\not\leq_{m}^{p}{\mathcal{P}}$ implies that $\pspace^{{\rm P}}\neq\exptime^{{\rm P}}$ is correct, however it is unclear why the paper uses the classes with the ${\rm P}$ oracles rather than the classes directly as ${\mathcal{E}}\not\leq_{m}^{p}{\mathcal{P}}$ more directly implies that $\pspace\neq\exptime$ which then implies $\pspace^{{\rm P}}\neq\exptime^{{\rm P}}$ since $\pspace^{{\rm P}}=\pspace$ and $\exptime^{{\rm P}}=\exptime$.

In this critique, we have pointed out errors in “Separation of $\pspace$ and $\exptime$” [Cze21] and come to the conclusion that the paper fails to show that $\pspace\neq\exptime$. The paper makes several mistakes in its reasoning and thus is unable to provide a sufficient proof for its key claim. As a result, it is still an open issue whether $\pspace\neq\exptime$.