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Tianrong Lin

Publications and source records attributed to Tianrong Lin.

15 recordsLinked to original sources
arXiv2025

Computational Complexity of Model-Checking Quantum Pushdown Systems

In this paper, we study the problem of model-checking quantum pushdown systems from a computational complexity point of view. We arrive at the following equally important, interesting new results: We first extend the notions of the {\it probabilistic pushdown systems} and {\it Markov chains} to their quantum counterparts, i.e., {\em quantum pushdown system (qPDS)} and {\em quantum Markov chains}, and prove a necessary and sufficient condition for a qPDS to be well formed, also presenting a method to extend the local transition function of a well-formed qPDS to a unitary local time evolution operator. Next, we investigate the question of whether it is necessary to define a quantum analogue of {\it probabilistic computational tree logic} to describe the probabilistic and branching-time properties of the {\it quantum Markov chain}. We study its model-checking question and show that model-checking of {\it stateless quantum pushdown systems (qBPA)} against {\it probabilistic computational tree logic (PCTL)} is generally undecidable, i.e., there exists no algorithm for model-checking {\it stateless quantum pushdown systems (qBPA)} against {\it probabilistic computational tree logic}. We then study in which case there exists an algorithm for model-checking {\it stateless quantum pushdown systems} and show that the problem of model-checking {\it stateless quantum pushdown systems (qBPA)} against {\it bounded probabilistic computational tree logic} (bPCTL) is decidable, and further show that this problem is in $\mathit{NP}$-hard. Our reduction is from the {\it bounded Post Correspondence Problem} for the first time, a well-known $\mathit{NP}$-complete problem.

cs.LO↗
arXiv2024

Simulating Polynomial-Time Nondeterministic Turing Machines via Nondeterministic Turing Machines

We prove in this paper that there exists 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 accepted by any ${\rm co}\mathcal{NP}$ machines. We further show that $L_s$ is in $\mathcal{NP}$, thereby proving the groundbreaking result that $$\mathcal{NP}\neq{\rm co}\mathcal{NP}. $$ The main techniques used in this paper are simulation together with the novel techniques developed in the author's recent work. Our main result has profound implications, such as $\mathcal{P}\neq\mathcal{NP}$. Furthermore, if there exists some oracle $A$ such that $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$, we explore the underlying reasons and show that, under this condition and some reasonable assumptions, the set of all ${\rm co}\mathcal{NP}^A$ machines is not enumerable. This implies that simulation techniques cannot be applied to the first part of the separation of $\mathcal{NP}^A$ 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).

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arXiv2023

Probabilistic Computers (and Hence Quantum Computers) Are Rigorously More Powerful Than Classical Deterministic Computers, and Derandomization

In this paper, we extend the techniques developed in our previous work to construct a probabilistic Turing machine that runs within time $O(n^k)$ for every $k\in\mathbb{N}_1$ and accepts a language $L_d\notin\mathcal{P}$. We further show that $L_d\in\mathcal{BPP}$, thereby separating $\mathcal{BPP}$ from $\mathcal{P}$ (i.e., $\mathcal{P}\subsetneqq\mathcal{BPP}$). Since the complexity class $\mathcal{BQP}$ of {\em bounded error quantum polynomial-time computation} contains $\mathcal{BPP}$ (i.e., $\mathcal{BPP}\subseteq\mathcal{BQP}$), our result confirms the long-standing conjecture that quantum computers are {\em rigorously more powerful} than classical deterministic computers (i.e., $\mathcal{P}\subsetneqq\mathcal{BQP}$). As an important consequence of the above results, we disprove the {\bf Extended Church-Turing Thesis}. Furthermore, we establish the following separations: (1) $\mathcal{P}\subsetneqq\mathcal{RP}$; (2) $\mathcal{P}\subsetneqq{\rm co}\mathcal{RP}$; (3) $\mathcal{P}\subsetneqq\mathcal{ZPP}$. These relationships were long-standing open questions in complexity theory. In addition, the separation $\mathcal{P}\subsetneqq\mathcal{BPP}$ demonstrates that {\em randomness} plays an essential role in probabilistic computation. In particular, we prove the following: (4) The number of random bits used by any probabilistic algorithm accepting $L_d$ cannot be reduced to $O(\log n)$; (5) There exists no efficient (complexity-theoretic) {\em pseudorandom generator} (PRG): $$ G:\{0,1\}^{O(\log n)}\rightarrow \{0,1\}^n;$$ (6) There exists no quick HSG $H:k(n)\rightarrow n$ with $k(n)=O(\log n)$.

cs.CC↗
arXiv2022

On Probabilistic $\omega$-Pushdown Systems, and $\omega$-Probabilistic Computational Tree Logic

In this paper, we define the notion of a {\em probabilistic $\omega$-pushdown automaton} and study its model-checking problem against $\omega$-probabilistic computational tree logic ($\omega$-PCTL) and its bounded version from a computational complexity perspective. Specifically, we obtain the following important new results: (1) We first discuss the expressiveness of the logics PCTL, PCTL$^*$, $\omega$-${\rm PCTL}$, and $\omega$-${\rm PCTL}^*$ and study how B\"uchi conditions of probabilistic $\omega$-pushdown systems influence $\omega$-PCTL formulas. We then investigate the model-checking problem for {\em stateless probabilistic $\omega$-pushdown system ($\omega$-pBPA)} against $\omega$-PCTL (as defined by Chatterjee, Sen, and Henzinger in \cite{CSH08}). By constructing $\omega$-PCTL formulas that encode the {\em Post Correspondence Problem}, we show that this model-checking problem is generally undecidable. (2) We then study under which conditions there exists an algorithm for model-checking {\it stateless probabilistic $\omega$-pushdown systems} against $\omega$-PCTL-like logic. In particular, we show that the model-checking problem for {\it stateless probabilistic $\omega$-pushdown systems} against $\omega$-{\it bounded probabilistic computational tree logic} ($\omega$-bPCTL) is decidable and $\mathit{NP}$-hard. Currently, there is no known lower bound for this problem that is better than ours. (3) Finally, we investigate an upper bound for the model-checking problem for {\em stateless probabilistic $\omega$-pushdown systems} against $\omega$-bounded probabilistic computational tree logic ($\omega$-bPCTL). We propose a potential approach to solving it by establishing a conditional upper bound and analyze the challenges of this method.

cs.LO↗
arXiv2021

On Baker-Gill-Solovay Oracle Turing Machines and Relativization Barrier

This work analyses the so-called "Relativization Barrier" with respect to the Baker-Gill-Solovay oracle Turing machine. We show that the {\em diagonalization} technique is a valid mathematical proof technique, but it has some prerequisites when referring to the "relativization barrier."

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arXiv2021

Diagonalization of Polynomial-Time Deterministic Turing Machines via Nondeterministic Turing Machines

The {\em diagonalization technique} was invented by Georg Cantor to show that there are more real numbers than algebraic numbers and is very crucial in {\em theoretical computer science}. In this work, we enumerate all of the polynomial-time deterministic Turing machines and diagonalize against 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 running within time $O(n^k)$ for any $k\in\mathbb{N}_1$. Based on these, we further show that $L_d\in\mathcal{NP}$. That is, in this work, we present a proof that $\mathcal{P}$ and $\mathcal{NP}$ differ. Meanwhile, we show that there exists a language $L_s$ in $\mathcal{P}$, but the machine accepting it also runs within time $O(n^k)$ for all $k\in\mathbb{N}_1$. Lastly, we show that if $\mathcal{P}^O=\mathcal{NP}^O$ and on some rational base assumptions, then the set $P^O$ of all polynomial-time deterministic oracle Turing machines with oracle $O$ is not enumerable, thus demonstrating that the diagonalization technique ({\em via a universal nondeterministic oracle Turing machine}) will generally {\em not} apply to the relativized versions of the $\mathcal{P}$ versus $\mathcal{NP}$ problem.

cs.CC↗
arXiv2021

Resolution of The Linear-Bounded Automata Question

This paper resolves a famous and longstanding open question in automata theory, i.e., the {\it linear-bounded automata question} (or, for short, the LBA question), which can also be phrased succinctly in the language of computational complexity theory as $${\rm NSPACE}[n]\overset{?}{=}{\rm DSPACE}[n]. $$ In fact, we prove a more general result that $${\rm DSPACE}[S(n)]\subsetneqq {\rm NSPACE}[S(n)], $$where $S(n)\geq n$ is a space-constructible function. Our proof technique is based on diagonalization against deterministic $S(n)$ space-bounded Turing machines by means of a universal nondeterministic Turing machine, together with other novel and interesting new techniques developed in this paper. Our proof also implies the following consequences, which resolve some famous open questions in complexity theory: (1). ${\rm DSPACE}[n]\subsetneqq {\rm NSPACE}[n]$; (2). $L\subsetneqq NL$; (3). $L\subsetneqq P$; (4). There exists no deterministic Turing machine working in $O(\log n)$ space that decides the $st$-connectivity question (STCON).

cs.CC↗
arXiv2021

The Separation of $\mathit{NP}$ and $\mathit{PSPACE}$

There is an important and interesting open question in computational complexity on the relation between the complexity classes $\mathcal{NP}$ and $\mathcal{PSPACE}$. It is a widespread belief that $\mathcal{NP}\ne\mathcal{PSPACE}$. In this paper, we confirm this conjecture affirmatively by showing that there is a language $L_d$ accepted by no polynomial-time nondeterministic Turing machines but accepted by a nondeterministic Turing machine running within space $O(n^k)$ for all $k\in\mathbb{N}_1$. We achieve this by virtue of the prerequisite of $$ {\rm NTIME}[S(n)]\subseteq{\rm DSPACE}[S(n)], $$ and then by diagonalization against all polynomial-time nondeterministic Turing machines via a universal nondeterministic Turing machine $M_0$. We further show that $L_d\in \mathcal{PSPACE}$, which leads to the conclusion $$ \mathcal{NP}\subsetneqq\mathcal{PSPACE}. $$ Our approach is based on standard diagonalization and novel new techniques developed in the author's recent works \cite{Lin21a,Lin21b} with some new refinement.

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arXiv2020

A "Proof" of $P\neq NP$

This short note present a "proof" of $P\neq NP$. The "proof" with double quotation marks is to indicate that we do not know whether the proof is correct or not (We're confused because we do know in which we make the mistakes).

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arXiv2013

Undecidability of MM-QFAs Language Equivalence Problem

Let $L_{>\lambda}(\mathcal{A})$ and $L_{\geq\lambda}(\mathcal{A})$ be the languages recognized by {\em measure many 1-way quantum finite automata (MM-QFA)} (or,{\em enhanced 1-way quantum finite automata(EQFA)}) $\mathcal{A}$ with strict and non-strict cut-point $\lambda$, respectively. We consider the language equivalence problem and show the following 1. both strict and non-strict language equivalence are undecidable; 2. we provide an another proof of the undecidability of non-strict and strict emptiness of MM-QFA and EQFA, and then reducing the language equivalence problem to emptiness problem; 3. lastly, we obtain some other properties which can be derived from the above results.

cs.FL↗
arXiv2012

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.

cs.CC↗
arXiv2011

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.

cs.LO↗
arXiv2011

Some results on equivalence of multi-letter quantum finite automata

Two quantum finite automata are equivalent if for all input string $\omega$ over the input alphabet the two automata accept $\omega$ with equal probability. In [Theoret. Comput. Sci. 410 (2009) 3006-3017], it was shown that a $k_1$-letter QFA $\mathcal{A}_1$ and a $k_2$-letter QFA $\mathcal{A}_2$ over $\Sigma=\{\sigma\}$, are equivalent if and only if they are $(n_1+n_2)^4+k-1$-equivalent where $n_i$ is the number of states of $\mathcal{A}_i$, $i=1,2$, and $k=\max\{k_1,k_2\}$. In this letter, we improve the above upper-bound to $(n_1^2+n_2^2-1)+k$. This also answers an open problem of Qiu et al. [Acta Informatica 48 (2011) 271-290]. Further, we show that, in the case of $\Sigma=\{\sigma_1,...,\sigma_t\}$ with $2\leq t<\infty$, there exists an integer $z$ such that $\mathcal{A}_1$ and $\mathcal{A}_2$ are equivalent if and only if they satisfy $z$-equivalent.

cs.CC↗
arXiv2011

Another approach to the equivalence of measure-many one-way quantum finite automata and its application

In this paper, we present a much simpler, more direct, and more elegant approach to the equivalence problem for {\it measure-many one-way quantum finite automata} (MM-1QFAs). The approach is essentially a generalization of the work of Carlyle [J.~Math.~Anal.~Appl.~7 (1963) 167--175]. Specifically, we reduce the equivalence problem for MM-1QFAs to the equivalence of two (initial) vectors. As an application of this approach, we use it to solve the equivalence problem for {\it enhanced one-way quantum finite automata} (E-1QFAs) introduced by Nayak [Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp.~369--376]. We prove that two E-1QFAs $\mathcal{A}_1$ and $\mathcal{A}_2$ over $\Sigma$ are equivalent if and only if they are $(n_1^2 + n_2^2 - 1)$-equivalent, where $n_1$ and $n_2$ are the numbers of states in $\mathcal{A}_1$ and $\mathcal{A}_2$, respectively.

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