On equivalence, languages equivalence and minimization of multi-letter and multi-letter measure-many quantum automata

We first show that given a $k_1$-letter quantum finite automata $\mathcal{A}_1$ and a $k_2$-letter quantum finite automata $\mathcal{A}_2$ over the same input alphabet $\Sigma$, they are equivalent if and only if they are $(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k$-equivalent where $n_1$, $i=1,2$, are the numbers of state in $\mathcal{A}_i$ respectively, and $k=\max\{k_1,k_2\}$. By applying a method, due to the author, used to deal with the equivalence problem of {\it measure many one-way quantum finite automata}, we also show that a $k_1$-letter measure many quantum finite automaton $\mathcal{A}_1$ and a $k_2$-letter measure many quantum finite automaton $\mathcal{A}_2$ are equivalent if and only if they are $(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k$-equivalent where $n_i$, $i=1,2$, are the numbers of state in $\mathcal{A}_i$ respectively, and $k=\max\{k_1,k_2\}$. Next, we study the language equivalence problem of those two kinds of quantum finite automata. We show that for $k$-letter quantum finite automata, the non-strict cut-point language equivalence problem is undecidable, i.e., it is undecidable whether $L_{\geq\lambda}(\mathcal{A}_1)=L_{\geq\lambda}(\mathcal{A}_2)$ where $0<\lambda\leq 1$ and $\mathcal{A}_i$ are $k_i$-letter quantum finite automata. Further, we show that both strict and non-strict cut-point language equivalence problem for $k$-letter measure many quantum finite automata are undecidable. The direct consequences of the above outcomes are summarized in the paper. Finally, we comment on existing proofs about the minimization problem of one way quantum finite automata not only because we have been showing great interest in this kind of problem, which is very important in classical automata theory, but also due to that the problem itself, personally, is a challenge. This problem actually remains open.Comment: 30 pages, conclusion section correcte

Undecidability of L(A)=L(B)L(\mathcal{A})=L(\mathcal{B}) recognized by measure many 1-way quantum automata

Let $L_{>\lambda}(\mathcal{A})$ and $L_{\geq\lambda}(\mathcal{A})$ be the languages recognized by {\em measure many 1-way quantum finite automata (MMQFA)} (or,{\em enhanced 1-way quantum finite automata(EQFA)}) $\mathcal{A}$ with strict, resp. non-strict cut-point $\lambda$. We consider the languages equivalence problem, showing that \begin{itemize} \item {both strict and non-strict languages equivalence are undecidable;} \item {to do this, we provide an additional proof of the undecidability of non-strict and strict emptiness of MMQFA(EQFA), and then reducing the languages equivalence problem to emptiness problem;} \item{Finally, some other Propositions derived from the above results are collected.} \end{itemize}Comment: Readability improved, title change

Undecidability of model-checking branching-time properties of stateless probabilistic pushdown process

In this paper, we settle a problem in probabilistic verification of infinite--state process (specifically, {\it probabilistic pushdown process}). We show that model checking {\it stateless probabilistic pushdown process} (pBPA) against {\it probabilistic computational tree logic} (PCTL) is undecidable.Comment: Author's comments on referee's report added, Interestin

Simple characterizations for commutativity of quantum weakest preconditions

In a recent letter [Information Processing Letters~104 (2007) 152-158], it has shown some sufficient conditions for commutativity of quantum weakest preconditions. This paper provides some alternative and simple characterizations for the commutativity of quantum weakest preconditions, i.e., Theorem 3.1, Theorem 3.2 and Proposition 3.3 in what follows. We also show that to characterize the commutativity of quantum weakest preconditions in terms of $[M,N]$ ($=MN-NM$) is hard in the sense of Proposition 4.1 and Proposition 4.2.Comment: Re-written, comments are welcom

Resolution of the Linear-Bounded Automata Question

This work resolve a longstanding open question in automata theory, i.e. the {\it linear-bounded automata question} ( shortly, {\it LBA question}), which can also be phrased succinctly in the language of computational complexity theory as $NSPACE[n]\overset{?}{=}DSPACE[n]$. We prove that $NSPACE[n]\neq DSPACE[n]$. Our proof technique is based on diagonalization against all deterministic Turing machines working in $O(n)$ space by an universal nondeterministic Turing machine running in $O(n)$ space. Our proof also implies the following consequences: (1) There exists no deterministic Turing machine working in $O(\log n)$ space deciding the $st$-connectivity question (STCON); (2) $L\neq NL$; (3) $L\neq P$.Comment: The definition of enumeration supplemented, feedbacks are welcome. arXiv admin note: text overlap with arXiv:2106.1188

Diagonalization of Polynomial-Time Deterministic Turing Machines Via Nondeterministic Turing Machine

The diagonalization technique was invented by Georg Cantor to show that there are more real numbers than algebraic numbers and is very important in computer science. In this work, we enumerate all polynomial-time deterministic Turing machines and diagonalize over all of them by a universal nondeterministic Turing machine. As a result, we obtain that there is a language $L_d$ not accepted by any polynomial-time deterministic Turing machines but accepted by a nondeterministic Turing machine working within $O(n^k)$ for any $k\in\mathbb{N}_1$. By this, we further show that $L_d\in \mathcal{NP}$ . That is, we present a proof that $\mathcal{P}$ and $\mathcal{NP}$ differ.Comment: (v12/3/4/5/6): The review report from "Annals of Mathematics" attached. As of today, no experts deny that $L_d\in\mathcal{NP}$ from the report; typos corrected. Additional open question added. [v16] JACM's area editor completely unable to present counterexamples from the report (But surprising, the referee's comments being masked by area editor

The Separation of NP\mathcal{NP} and PSPACE\mathcal{PSPACE}

There is a conjecture on $\mathcal{NP}\overset{?}{=}\mathcal{PSPACE}$ in computational complexity. It is a widespread belief that $\mathcal{NP}\neq \mathcal{PSPACE}$. In this paper, we show that $\mathcal{NP}\neq \mathcal{PSPACE}$ via the premise of NTIME[$S(n)] \subseteq$ DSPACE[$S(n)$], and then by diagonalization against all polynomial-time nondeterministic Turing machines via a universal nondeterministic Turing machine $M_0$ running in space $O(n^k)$ for any $k\in\mathbb{N}_1$. Thus, we obtain a language $L_d$ not accepted by any polynomial-time nondeterministic Turing machines but accepted by $M_0$. We also show that $L_d\in \mathcal{PSPACE}$, which results in the conclusion that $\mathcal{NP}\neq \mathcal{PSPACE}$.Comment: v12: typos corrected; 11 page

Quantum and Probabilistic Computers Rigorously Powerful than Traditional Computers, and Derandomization

In this paper, we extend the techniques used in our previous work to show that there exists a probabilistic Turing machine running within time $O(n^k)$ for all $k\in\mathbb{N}_1$ accepting a language $L_d$ which is different from any language in $\mathcal{P}$, and then to show that $L_d\in\mathcal{BPP}$, thus separating the complexity classes $\mathcal{P}$ and $\mathcal{BPP}$ (i.e., $\mathcal{P}\subsetneq\mathcal{BPP}$). Since the complexity class of {\em bounded error quantum polynomial-time} $\mathcal{BQP}$ contains the complexity class $\mathcal{BPP}$, i.e., $\mathcal{BPP}\subseteq\mathcal{BQP}$, we thus obtain the result that quantum computers are {\em rigorously powerful than} traditional computers. Namely, $\mathcal{P}\subsetneq\mathcal{BQP}$. We further show that (1). $\mathcal{P}\subsetneq\mathcal{RP}$; (2). $\mathcal{P}\subsetneq\text{co-}\mathcal{RP}$; (3). $\mathcal{P}\subsetneq\mathcal{ZPP}$. The result of $\mathcal{P}\subsetneq\mathcal{BPP}$ shows that {\em randomness} plays an essential role in probabilistic algorithm design. Specifically, we show that: (1). The number of random bits used by any probabilistic algorithm which accepts the language $L_d$ can not be reduced to $O(\log n)$; (2). There exits no efficient (complexity-theoretic) {\em pseudorandom generator} (PRG) $G:\{0,1\}^{O(\log n)}\rightarrow \{0,1\}^n$; (3). There exists no quick HSG $H:k(n)\rightarrow n$ such that $k(n)=O(\log n)$.Comment: [v3] references added; minor revised; 31 pages. arXiv admin note: text overlap with arXiv:2110.0621

Model-Checking Branching-Time Properties of Stateless Probabilistic Pushdown Systems and Its Quantum Extension

In this work, we first resolve a question in the probabilistic verification of infinite-state systems (specifically, the probabilistic pushdown systems). We show that model checking stateless probabilistic pushdown systems (pBPA) against probabilistic computational tree logic (PCTL) is generally undecidable. We define the quantum analogues of the probabilistic pushdown systems and Markov chains and investigate whether it is necessary to define a quantum analogue of probabilistic computational tree logic to describe the branching-time properties of the quantum Markov chain. We also study its model-checking problem and show that the model-checking of stateless quantum pushdown systems (qBPA) against probabilistic computational tree logic (PCTL) is generally undecidable, too. The immediate corollaries of the above results are summarized in the work.Comment: Obvious typos corrected in new version; this work is a quantum extension of arXiv:1405.4806, [v13]; 30 pages; comments are welcom