Communication Complexity Method for Measuring Nondeterminism in Finite Automata

While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfa's) remain open. One such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata. In this paper the concept of communication complexity is applied in order to achieve progress in this problem area. The main results are as follows:1. Deterministic communication complexity provides lower bounds on the size of nfa's with bounded unambiguity. Applying this fact, the proofs of several results about nfa's with limited ambiguity can be simplified and presented in a uniform way.2. There is a family of languages KON_k² with an exponential size gap between nfa's with polynomial leaf number/ambiguity and nfa's with ambiguity k. This partially provides an answer to the open problem posed by B. Ravikumar and O. Ibarra (1989, SIAM J. Comput.18, 1263–1282) and H. Leung (1998, SIAM J. Comput.27, 1073–1082).     

有关时间自动机重置的若干问题的计算复杂性

自动机的重置序列也称为同步序列,具有以下特性：有限自动机通过运行重置序列w,可从任意一个未知的或无法观测到的状态q₀到达某个特定状态q_w.这仅依赖于w,而与开始运行w时的状态q₀无关.这一特性可用于部分可观察的复杂系统的自动恢复,而无需重启,甚至有时不能重启.基于此,重置问题自出现以来便得到关注和持续研究.最近几年,它被扩展到可以描述诸如分布式、嵌入式实时系统等复杂系统的无限状态模型上,比如时间自动机和寄存器自动机等.以时间自动机的重置问题的计算复杂性为研究对象,发现重置问题与可达性问题有着紧密的联系.主要贡献是：（1）利用时间自动机可达性问题的最新成果,完善完全的确定的时间自动机重置问题的计算复杂性结论;（2）对部分规约的确定的时间自动机,研究得出,即使在输入字母表大小减至2的情况下,其复杂性仍是PSPACE-完全的;特别地,在单时钟情况下是NLOGSPACE-完全的;（3）对完全的非确定的时间自动机,研究得出其D_i-可重置问题（i=1,2,3）是不可判定的,其重置问题与非确定的寄存器自动机重置问题在指数时间可以相互归约,通过证明指数时间归约相对高复杂性类具有封闭性,利用非确定的寄存器自动机的结论得出单时钟的时间自动机的重置问题是Ackermann-完全的、限界的重置问题是NEXPTIME-完全的.这些复杂性结论,说明关于时间自动机的重置问题大都是难解的,一方面,为时间系统的可重置性的检测和求解奠定坚实的理论基础,另一方面,为以后寻找具有高效算法的特殊结构的时间系统（即具有高效算法的问题子类）给予理论指导.     

Computational Complexity of Certain Problems Related to Carefully Synchronizing Words for Partial Automata and Directing Words for Nondeterministic Automata

We show that the problem of checking careful synchronizability of partial finite automata is PSPACE-complete. Also the problems of checking D ₁-, D ₂-, and D ₃-directability of nondeterministic finite automata are PSPACE-complete; moreover, the restrictions of all these problems to automata with two input letters remain PSPACE-complete.     

The Quantum Complexity of Set Membership

   Abstract. We study the quantum complexity of the static set membership problem: given a subset S (|S| ≤ n ) of a universe of size m ( >> n ), store it as a table, T: {0,1} ^r --> {0,1} , of bits so that queries of the form ``Is x in S ?' can be answered. The goal is to use a small table and yet answer queries using few bit probes. This problem was considered recently by Buhrman et al. [BMRV], who showed lower and upper bounds for this problem in the classical deterministic and randomized models. In this paper we formulate this problem in the ``quantum bit probe model'. We assume that access to the table T is provided by means of a black box (oracle) unitary transform O _T that takes the basis state | y,b > to the basis state | y,b

On the Power of Las Vegas for One-Way Communication Complexity, OBDDs, and Finite Automata

The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main goal of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, ordered binary decision diagrams, and finite automata.(i) For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight.(ii) The result (i) is applied to show an at most polynomial gap between determinism and Las Vegas for ordered binary decision diagrams.(iii) For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the square root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound.     

On The Space Complexity Of Turn Bounded Pushdown Automata

A pushdown automaton is said to make a turn at a given instant if it changes at that instant from stack increasing to stack decreasing. Let \hbox{NPDA-TURN}(\,f(n)) and \hbox{DPDA-TURN}(\,f(n)) denote the classes of languages accepted by nondeterministic and deterministic pushdown automata respectively that make at most f v ( n ) turns for any input of length n . In this paper the following inclusions that express the space complexity of turn bounded pushdown automata are given: \hbox{DPDA-TURN}(\,f(n)) \subseteq {\bf DSPACE}(\log f(n)\log n) , and \hbox{NPDA-TURN}(\,f(n)) \subseteq {\bf NSPACE} (\log f(n)\log n) . In particular, it follows that finite-turn pushdown automata are logarithmic space bounded: \hbox{DPDA-TURN}(O(1))\subseteq {\bf DL} and \hbox{NPDA-TURN}(O(1))\subseteq {\bf NL} , from which two corollaries follow: one is that the class of metalinear context-free languages is complete for NL , and the other is that a more tight inclusion \hbox{NPDA-TURN}(\,f(n)) \subseteq {\bf DSPACE}(\log^2 f(n)\log n) cannot be derived unless {\bf DL}={\bf NL} , though \hbox{NPDA-TURN} (\,f(n)) \subseteq {\bf DSPACE}(\log^2 f(n)\log^2n) holds.     

The Complexity of Problems for Quantified Constraints

In this paper we are interested in quantified propositional formulas in conjunctive normal form with “clauses” of arbitrary shapes. i.e., consisting of applying arbitrary relations to variables. We study the complexity of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for such formulas, both with and without a bound on the number of quantifier alternations. For each of these computational goals we get full complexity classifications: We determine the complexity of each of these problems depending on the set of relations allowed in the input formulas. Thus, on the one hand we exhibit syntactic restrictions of the original problems that are still computationally hard, and on the other hand we identify non-trivial subcases that admit efficient algorithms.     

Automata over Finite T-Quasigroups

This paper investigates families of automata without outputs and also families of reversible Mealy and Moore automata specified by recurrence relations over finite T-quasigroups. Based on the decomposition of an Abelian group into the direct sum of primary cyclic groups, a unified approach is proposed to the hardware and software synthesis of such automata. Estimates are found for the time and space complexities of computations executed by these automata during one clock cycle.     

On the Computational Complexity of P Automata

We characterize the classes of languages over finite alphabets which may be described by P automata, i.e., accepting P systems with communication rules only. Motivated by properties of natural computing systems, and the actual behavior of P automata, we study computational complexity classes with a certain restriction on the use of the available workspace in the course of computations and relate these to the language classes described by P automata. We prove that if the rules of the P system are applied sequentially, then the accepted language class is strictly included in the class of languages accepted by one-way Turing machines with a logarithmically bounded workspace, and if the rules are applied in the maximally parallel manner, then the class of context-sensitive languages is obtained.     

Learning Fallible Deterministic Finite Automata

We consider the problem of learning from a fallible expert that answers all queries about a concept, but often gives incorrect answers. The expert can also be thought of as a truth table describing the concept which has been partially corrupted. In order to learn the underlying concept with arbitrarily high precision, we would like to use its structure in order to correct most of the incorrect answers. We assume that the expert's errors are uniformly and independently distributed, occur with any fixed probability strictly smaller than 1/2, and are persistent. In particular, we present a polynomial time algorithm using membership queries for correcting and learning fallible Deterministic Finite Automata under the uniform distribution.     

Multihead Two-Way Probabilistic Finite Automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata withlogspace constructible transition probabilities, and we describe a technique to simulate these automata by standard logspace probabilistic Turing machines. Next, we represent logspace probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and we show their (deterministic logspace) reducibility to the second levels of these hierarchies. We obtain a simple formula for the maximum inherent bandwidth of the configuration transition matrices associated with thek-head probabilistic finite automata processing a length-n input string. (The inherent bandwidth of the configuration transition matrices associated with an automaton processing a length-n input string is the smallest bandwidth we can get by changing the enumeration order of the automaton’s configurations.) Partially based on this relation, we find an apparently easier logspace complete problem forPL (the class of languages recognized by logspace unbounded-error probabilistic Turing machines), and we discuss possibilities for a space-efficient deterministic simulation of probabilistic automata. This research was supported by the National Science Foundation under Grant No. CDA 8822724 while the author was at the University of Rochester. An extended abstract of this paper appeared in Proceedings, Second Latin American Symposium, LATIN ’95: Theoretical Informatics, Valparaiso, Chile, April 1995.     

Automata Over Abstract Finite Quasigroups

This paper analyzes the structure of families of automata without output that are defined by recurrence relations on abstract finite quasigroups. The expediency of their use to design iterated hash functions with sufficiently high security is justified. It is shown how some families of reversible Mealy and Moore automata can be constructed based on such families of automata without output. The expediency of using the proposed families of Mealy and Moore automata as the basis to construct mathematical models for stream ciphers is substantiated.     

有限布尔代数上的自动机

本文对有限布尔代数上的自动机进行了研究，证明了有限布尔代数上的可逆内动机保持正交性；定出了有限布尔代数上的对称可逆内动机及一般ｎ（≤３）阶可逆内动机的图型。     

On the Complexity of 2-Monotone Restarting Automata

The -automaton is the weakest form of the nondeterministic version of the restarting automaton that was introduced by Jančar et al. to model the so-called analysis by reduction. Here it is shown that the class ℒ(R) of languages that are accepted by -automata is incomparable under set inclusion to the class of Church-Rosser languages and to the class of growing context-sensitive languages. In fact this already holds for the class of languages that are accepted by 2-monotone -automata. In addition, we prove that already the latter class contains -complete languages, showing that already the 2-monotone -automaton has a surprisingly large expressive power. The results of this paper have been announced at DLT 2004 in Auckland, New Zealand. This work was mainly carried out while T. Jurdziński was visiting the University of Kassel, supported by a grant from the Deutsche Forschungsgemeinschaft (DFG). F. Mráz and M. Plátek were partially supported by the Grant Agency of the Czech Republic under Grant-No. 201/04/2102 and by the program ‘Information Society’ under project 1ET100300517. F. Mráz was also supported by the Grant Agency of Charles University in Prague under Grant-No. 358/2006/A-INF/MFF.     

有限自动机研究的矩阵模型方法

1 引言随着现代科学技术的发展,有限自动机已成为许多学科的重要的理论和应用基础。然而近年来有限自动机理论发展缓慢,不能适应这些学科发展的需要。为了促进有限自动机研究的发展,本文中给出了矩阵模型方法。早在50年代,受图论研究的影响。人们就考虑过试图采用矩阵工具研究有限自动机,80年代主要以时     

Comparing Verboseness for Finite Automata and Turing Machines

A language is called (m,n)-verbose if there exists a Turing machine that enumerates for any n words at most m possibilities for their characteristic string. This notion is compared with (m,n)-fa-verboseness, where instead of a Turing machine a finite automaton is used. By use of a new diagonalisation method, where finite automata trick Turing machines, it is shown that all (m,n)-verbose languages are (h,k)-verbose iff all (m,n)-fa-verbose languages are (h,k)-fa-verbose. In other words, Turing machines and finite automata behave exactly the same way with respect to inclusion of verboseness classes. This identical behaviour implies that the nonspeedup theorem also holds for finite automata. As an application of the theoretical framework, a lower bound is derived on the number of bits that need to be communicated to finite automata protocol checkers for nonregular protocols.