Probabilistic and quantum finite automata with postselection

Two-way finite automata with quantum and classical states

We introduce 2-way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2-way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum state, but the tape head position is required to be classical. We show two languages for which 2qcfa's are better than classical 2-way automata. First, 2qcfa's can recognize palindromes, a language that cannot be recognized by 2-way deterministic or probabilistic finite automata. Second, in polynomial…

Language recognition power and succintness of affine automata

In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by Díaz-Caro and Yakaryılmaz \cite{DCY16A} referred as affine automata. First, we present efficient simulations of probabilistic and quantum automata by means of affine automata which allows us to characterize the class of exclusive stochastic languages. Then, we initiate a study on the succintness of affine automata. In particular, we show that an infinite family of unary regular languages can be recognized by 2-state affine automata but the state numbers o…

Unary probabilistic and quantum automata on promise problems

We continue the systematic investigation of probabilistic and quantum finite automata (PFAs and QFAs) on promise problems by focusing on unary languages. We show that bounded-error QFAs are more powerful than PFAs. But, in contrary to the binary problems, the computational powers of Las-Vegas QFAs and bounded-error PFAs are equivalent to deterministic finite automata (DFAs). Lastly, we present a new family of unary promise problems with two parameters such that when fixing one parameter QFAs can be exponentially more succinct than PFAs and when fixing the other parameter PFAs can be exponentia…

Succinctness of two-way probabilistic and quantum finite automata

We prove that two-way probabilistic and quantum finite automata (2PFA's and 2QFA's) can be considerably more concise than both their one-way versions (1PFA's and 1QFA's), and two-way nondeterministic finite automata (2NFA's). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than $ {1/2} $ by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFA's, 1QFA's and 2NFA's grow without bound. We also show that 2QFA…

经典与量子计数器自动机的新结果

New results on classical and quantum counter automata

本文证明零误差单向量子单计数器自动机在承诺问题上强于概率对应物，并推翻了关于盲计数器自动机识别Kleene闭包的猜想，获得多个分离结果。

Polynomially Ambiguous Probabilistic Automata on Restricted Languages

We consider the computability and complexity of decision questions for Probabilistic Finite Automata (PFA) with sub-exponential ambiguity. We show that the emptiness problem for strict and non-strict cut-points of polynomially ambiguous commutative PFA remains undecidable, implying that the problem is undecidable when inputs are from a letter monotonic language. We show that the problem remains undecidable over a binary input alphabet when the input word is over a bounded language, in the noncommutative case. In doing so, we introduce a new technique based upon the Turakainen construction of a…

有限代数结构上的非均匀确定性有限自动机

Nonuniform Deterministic Finite Automata over finite algebraic structures

本文刻画了恒等式检验和电路等价性可概率多项式时间求解的群及有限代数，并将非均匀确定性有限自动机推广到任意有限代数结构。

Language recognition by generalized quantum finite automata with unbounded error (abstract & poster)

In this note, we generalize the results of arXiv:0901.2703v1 We show that all one-way quantum finite automaton (QFA) models that are at least as general as Kondacs-Watrous QFA's are equivalent in power to classical probabilistic finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input does enlarge the class of languages recognized by such QFA's with unbounded error. (Note that, the proof of Theorem 1 in the abstract was presented in the previous version (arXiv:0901.2703v1).)

Sublinear-Time Probabilistic Cellular Automata

We propose and investigate a probabilistic model of sublinear-time one-dimensional cellular automata. In particular, we modify the model of ACA (which are cellular automata that accept if and only if all cells simultaneously accept) so that every cell changes its state not only dependent on the states it sees in its neighborhood but also on an unbiased coin toss of its own. The resulting model is dubbed probabilistic ACA (PACA). We consider one- and two-sided error versions of the model (in the same spirit as the classes $\mathsf{RP}$ and $\mathsf{BPP}$) and establish a separation between the …

Postselecting probabilistic finite state recognizers and verifiers

In this paper, we investigate the computational and verification power of bounded-error postselecting realtime probabilistic finite state automata (PostPFAs). We show that PostPFAs using rational-valued transitions can do different variants of equality checks and they can verify some nonregular unary languages. Then, we allow them to use real-valued transitions (magic-coins) and show that they can recognize uncountably many binary languages by help of a counter and verify uncountably many unary languages by help of a prover. We also present some corollaries on probabilistic counter automata.

Upper bounds for Newton's method on monotone polynomial systems, and P-time model checking of probabilistic one-counter automata

A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multi-type branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of {\em termination probabilities}, and computing these probabilities in turn boils down to computing the {\em least fixed point} (LFP) solution of a corresponding {\em monotone polynomial system} (MPS) of equations, denoted x=P(x). It was shown by Etessami & Yannakakis that a decom…

单向概率下推自动机的无结束标记定理

The No Endmarker Theorem for One-Way Probabilistic Pushdown Automata

本文证明单向概率下推自动机中的结束标记总是可以移除，通过概率变换以双指数栈状态复杂度为代价构造等价的无结束标记机器，且不损害错误概率，并统一涵盖确定性与非确定性模型。

作为概率有限状态自动机的循环神经语言模型

Recurrent Neural Language Models as Probabilistic Finite-state Automata

该研究从概率分布表征角度探究循环神经网络语言模型的能力，证明其等价于概率有限状态自动机子类，且表征确定性有限状态语言模型需Ω(N|Σ|)个神经元，为理解其能力局限提供基础。