Jacques Sakarovitch
This text is an extended version of the chapter 'Automata and rational expressions' in the AutoMathA Handbook that will appear soon, published by the European Science Foundation and edited by JeanEricPin.
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We present an algorithm to build an automaton from a rational expression. This approach introduces support for extended weighted expressions. Inspired by derived-term based algorithms, its core relies on a different construct, rational expansions. We introduce an inductive algorithm to compute the expansion of an expression from which the automaton follows. This algorithm is independent of the size of the alphabet, and actually even supports infinite alphabets. It can easily be accommodated to generate deterministic (weighted) automata. These constructs are implemented in Vcsn, a free-software…
The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature. Lifting this result to a weighted setting has been widely studied. Moreover, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalisin…
We introduce (weighted) rational expressions to denote series over Cartesian products of monoids. To this end, we propose the operator $|$ to build multitape expressions such as $(a^+|x + b^+|y)^*$. We define expansions, which generalize the concept of derivative of a rational expression, but relieved from the need of a free monoid. We propose an algorithm based on expansions to build multitape automata from multitape expressions.
Arden's Lemma is a classical result in language theory allowing the computation of a rational expression denoting the language recognized by a finite string automaton. In this paper we generalize this important lemma to the rational tree languages. Moreover, we propose also a construction of a rational tree expression which denotes the accepted tree language of a finite tree automaton.
To deal with system uncertainty, finite automata have been generalized into fuzzy automata. Stamenkovic and Ciric proposed an approach using the position automata for the construction of fuzzy automata from fuzzy regular expressions. There exist multifarious methodologies for the construction of finite automata from regular expressions known as Thompson construction, Antimirov partial derivatives, Glushkov automata and follow automata etc. In this paper, we propose an approach for the conversion of fuzzy regular expressions into fuzzy automata using the concept of follow automata. The number o…
Multiset automata are a class of automata for which the symbols can be read in any order and obtain the same result. We investigate weighted multiset automata and show how to construct them from weighted regular expressions. We present training methods to learn the weights for weighted regular expressions and for general multiset automata from data. Finally, we examine situations in which inside weights can be computed more efficiently.
Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata
本文研究了复制less成本寄存器自动机和多项式歧义加权自动机的表达能力，通过泵送序列族工具得出两者在不同域下的泵送样结果，并证明在有理数域下两者不可比较。
We extend Antimirov's partial derivatives from regular expressions to $μ$-regular expressions that describe context-free languages. We prove the correctness of partial derivatives as well as the finiteness of the set of iterated partial derivatives. The latter are used as pushdown symbols in our construction of a nondeterministic pushdown automaton, which generalizes Antimirov's NFA construction.
Coordination Sequences of Periodic Structures are Rational via Automata Theory
本文利用自动机理论的经典技术，简洁地证明了 Grosse-Kunstleve 等人关于 n 维欧几里得空间中周期结构配位序列为有理数的猜想。
Reversible Weighted Automata over Finite Rings and Monoids with Commuting Idempotents
本文研究了有限环和独异点上的可逆加权自动机，揭示了其与可交换幂等元有理语言的关系，并证明了相关问题的可判定性。
Determinisation and Unambiguisation of Polynomially-Ambiguous Rational Weighted Automata
本文研究有理数域上加权自动机的确定化与去歧义化问题，证明对多项式模糊加权自动机，这两个问题均属于 PSPACE，填补了 Bell 和 Smertnig 可判定性结果中复杂度界限的空白。
Higher-Dimensional Automata : Extension to Infinite Tracks
本文提出高维自动机扩展到无限轨道，并探讨其在不同语义下的语言识别能力，证明Muller语义比Büchi语义更强大，且两者不闭合于子sume，同时揭示ω-HDAs语言的有理性限制。
Constructible Words Characterize Rational Languages of Words Indexed by Scattered Linear Orderings
提出可构造词概念，通过有限次简单算子操作得到，并证明由散布线性序索引的有理语言完全由其可构造词刻画，证明依赖于Colcombet的半群理论定理。
We study the problem of constructing approximations to a weighted automaton. Weighted finite automata (WFA) are closely related to the theory of rational series. A rational series is a function from strings to real numbers that can be computed by a finite WFA. Among others, this includes probability distributions generated by hidden Markov models and probabilistic automata. The relationship between rational series and WFA is analogous to the relationship between regular languages and ordinary automata. Associated with such rational series are infinite matrices called Hankel matrices which play…
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the injectivity problem of determining if the acceptance probability function of a MO-QFA is injective over all input words, i.e., giving a distinct probability for each input word. We show that the injectivity problem is undecidable for 8 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial state vector is real algebraic. We also show undecidability of this problem when the initial vector is rat…
We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers.
Li and Pedrycz [Y. M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets and Systems 156 (2005) 68--92] have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice-ordered monoid, and generalize the well-known results of the classical theory of formal languages. In particular, they have shown that a fuzzy language over an integral lattice-ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a…
GrappaRE -- A Tool for Efficient Graph Recognition Based on Finite Automata and Regular Expressions
本文介绍工具GrappaRE，它利用有限自动机和正则表达式高效识别图语言，并验证了线性时间识别性能。
Note on Finite-Automata Bernoulli Factories for Rational Functions
本文指出 Mossel 和 Peres 关于多元有理函数有限自动机伯努利工厂的定理存在技术漏洞，并给出一个反例。
Kleene Theorems for Lasso Languages and $ω$-Languages
针对套索语言与$\omega$-语言缺乏克林定理的问题，研究引入有理套索语言与表达式，证明两类克林定理并给出套索自动机的Brzozowski构造方法。
The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of…
Active Learning of Symbolic Automata Over Rational Numbers
针对$L^*$算法仅能学习有限字母表上DFAs的局限，本文将其扩展到有理数域无限稠密字母表的符号自动机学习，使$L^*$可应用于新场景且查询数量与转移数和谓词表示大小成线性关系。
Weighted Automata and Regular Expressions for Financial Systems
本文引入加权有限金融自动机（WFFA）作为建模和分析金融系统定量属性的框架，通过股票价格、利率等不确定经济变量驱动。提出加权金融正则表达式用于描述定量金融属性，并建立WFFA与正则表达式之间的Kleene-Schützenberger型对应关系，研究WFFA的基本决策和优化问题。
We propose a novel automata model over the alphabet of rational numbers, which we call register automata over the rationals (RA-Q). It reads a sequence of rational numbers and outputs another rational number. RA-Q is an extension of the well-known register automata (RA) over infinite alphabets, which are finite automata equipped with a finite number of registers/variables for storing values. Like in the standard RA, the RA-Q model allows both equality and ordering tests between values. It, moreover, allows to perform linear arithmetic between certain variables. The model is quite expressive: i…
We consider linear cost-register automata (equivalent to weighted automata) over the semiring of nonnegative rationals, which generalise probabilistic automata. The two problems of boundedness and zero isolation ask whether there is a sequence of words that converge to infinity and to zero, respectively. In the general model both problems are undecidable so we focus on the copyless linear restriction. There, we show that the boundedness problem is decidable. As for the zero isolation problem we need to further restrict the class. We obtain a model, where zero isolation becomes equivalent to un…
Automata play important roles in wide area of computing and the growth of multicores calls for their efficient parallel implementation. Though it is known in theory that we can perform the computation of a finite automaton in parallel by simulating transitions, its implementation has a large overhead due to the simulation. In this paper we propose a new automaton called simultaneous finite automaton (SFA) for efficient parallel computation of an automaton. The key idea is to extend an automaton so that it involves the simulation of transitions. Since an SFA itself has a good property of parall…
We realize lamplighter groups $A\wr \mathbb Z$, with $A$ a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize $A\wr \mathbb Z$ as a bireversible automaton group if and only if the $2$-Sylow subgroup of $A$ has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of $2$.
$ω$-regular Expression Synthesis from Transition-Based Büchi Automata
本文提出直接从基于迁移的 Büchi 自动机合成 ω-正则表达式的新方法，证明其可靠完备，并实验表明所得表达式更紧凑，对 recurrence 型 LTL 公式平均缩减超 50%，且能处理更多 LTL 公式。
Regular expressions provide a flexible means for matching strings and they are often used in data-intensive applications. They are formally equivalent to either deterministic finite automata (DFAs) or nondeterministic finite automata (NFAs). Both DFAs and NFAs are affected by two problems known as amnesia and acalculia, and DFAs are also affected by a problem known as insomnia. Existing techniques require an automata conversion and compaction step that prevents the use of existing automaton databases and hinders the maintenance of the resulting compact automata. In this paper, we propose Paral…
Constrained Multi-Tildes: Derived Term and Position Automata
本文将 multi-tildes 从析取组合扩展到任意布尔组合，证明其可指数增强表达式分解能力，并给出转换为有限自动机的方法及 Haskell 实现。
We investigate commutative images of languages recognised by register automata and grammars. Semi-linear and rational sets can be naturally extended to this setting by allowing for orbit-finite unions instead of only finite ones. We prove that commutative images of languages of one-register automata are not always semi-linear, but they are always rational. We also lift the latter result to grammars: commutative images of one-register context-free languages are rational, and in consequence commutatively equivalent to register automata. We conjecture analogous results for automata and grammars w…
Several algorithms have been designed to convert a regular expression into an equivalent finite automaton. One of the most popular constructions, due to Glushkov and to McNaughton and Yamada, is based on the computation of the Null, First, Last and Follow sets (called Glushkov functions) associated with a linearized version of the expression. Recently Mignot considered a family of extended expressions called Extended to multi-tilde-bar Regular Expressions (EmtbREs) and he showed that, under some restrictions, Glushkov functions can be defined for an EmtbRE. In this paper we present an algorith…
The automata arising from the well known conversion of regular expression to non deterministic automata have rather particular transition graphs. We refer to them as the Glushkov graphs, to honour his nice expression-to-automaton algorithmic short cut (On a synthesis algorithm for abstract automata, Ukr. Matem. Zhurnal, 12(2):147-156, 1960, In Russian). The Glushkov graphs have been characterized (P. Caron and D. Ziadi, Characterization of Glushkov automata. Theoret. Comput. Sci., 233(1-2):75-90, 2000) in terms of simple graph theoretical properties and certain reduction rules. We show how to …
Computer science class enrollments have rapidly risen in the past decade. With current class sizes, standard approaches to grading and providing personalized feedback are no longer possible and new techniques become both feasible and necessary. In this paper, we present the third version of Automata Tutor, a tool for helping teachers and students in large courses on automata and formal languages. The second version of Automata Tutor supported automatic grading and feedback for finite-automata constructions and has already been used by thousands of users in dozens of countries. This new version…
This report is mostly written for educational purposes. It is meant as a self contained introduction to regular languages, regular expressions, and regular expression matching by using Brzozowski derivatives. As such it is mostly based on the work by Brzozowski[4] and Owens et al.[12] The language basics material have been inspired by books[2] and web material[16]. Chapter 1 introduces the fundamental concepts of formal languages, as well as the idea of string derivatives. In chapter 2 we define the class of regular languages, and further develops the theory of derivatives for that class. We u…
In this document, we propose a description, via a Haskell implementation, of a generalization of the notion of regular expression allowing us to group the definitions and the methods of (tree or word) automata constructions over one generic structure, based on enriched category theory tools. We first recall several methods of conversion from expressions to automata, enlightening the similarities between the words and trees cases. We then produce an original study of the power of enriched category theory applied 1) to automata and expressions implementation, and 2) to the study of associated al…
There exist several methods of computing an automaton recognizing the language denoted by a given regular expression: In the case of words, the position automaton P due to Glushkov, the c-continuation automaton C due to Champarnaud and Ziadi, the follow automaton F due to Ilie and Yu and the equation automaton E due to Antimirov. It has been shown that P and C are isomorphic and that E (resp. F) is a quotient of C (resp. of P). In this paper, we define from a given regular tree expression the k-position tree automaton P and the follow tree automaton F . Using the definition of the equation tre…
Regular languages are closed under a wealth of formal language operators. Incorporating such operators in regular expressions leads to concise language specifications, but the transformation of such enhanced regular expressions to finite automata becomes more involved. We present an approach that enables the direct construction of finite automata from regular expressions enhanced with further operators that preserve regularity. Our construction is based on an extension of the theory of derivatives for regular expressions. To retain the standard results about derivatives, we develop a derivabil…
From regular expressions to deterministic finite automata: $2^{\frac{n}{2}+\sqrt{n}(\log n)^{Θ(1)}}$ states are necessary and sufficient
本文研究了正则表达式到确定性有限自动机（DFA）的状态转换界限，证明了新的上下界均接近 $2^{\frac{n}{2}+\sqrt{n}(\log n)^{Θ(1)}}$，改进了现有结果并适用于特定 NFA 的确定化问题。
We Propose A Novel Automaton Model which uses Arithmetic Operations as the Evolving Rules, each cell has the states of the Natural Numbers k = (N), a radius of r = 1/2 and operates on an arbitrary input size. The Automaton reads an Arithmetic Expression as an input and outputs another Arithmetic Expression. In Addition, we simulate a variety of One Dimensional Cellular Automata Structures with different Dynamics including Elementary Cellular Automata.
Equivalence checking for weak bi-Kleene algebra
本文刻画了允许语言等价性判定的Pomset自动机片段，证明其与序列有理表达式对应，从而给出序列有理表达式等价性可判定的新证明。
We report on simulation, hierarchy, and decidability results for Practical Regular Expressions (PRE), which may include back references in addition to the standard operations union, concatenation, and star. The following results are obtained: PRE can be simulated by the classical model of nondeterministic finite automata with sensing one-way heads. The number of heads depends on the number of different variables in the expressions. A space bound O(n log m) for matching a text of length m with a PRE with n variables based on the previous simulation. This improves the bound O(nm) from (Câmpeanu …
We initiate the study of the verification power of AfAs as part of Arthur-Merlin (AM) proof systems. We show that every unary language is verified by a real-valued AfA verifier. Then, we focus on the verifiers restricted to have only integer-valued or rational-valued transitions. We observe that rational-valued verifiers can be simulated by integer-valued verifiers, and, their protocols can be simulated in nondeterministic polynomial time. We show that this bound tight by presenting an AfA verifier for NP-complete problem SUBSETSUM. We also show that AfAs can verify certain non-affine and non-…
Rational-Valued Affine Verifiers in Arthur--Merlin Proof Systems
本文研究了在限制仿射转移矩阵具有有理数条目的情况下，单向和双向仿射自动机作为Arthur-Merlin证明系统的验证器，展示了其在验证非正则语言和所有可识别语言中的优势。
In the last years, various extensions of ω-regular languages have been proposed in the literature, including ωB-regular (ω-regular languages extended with boundedness), ωS-regular (ω-regular languages extended with strict unboundedness), and ωBS-regular languages (the combination of ωB- and ωS-regular ones). While the first two classes satisfy a generalized closure property, namely, the complement of an ωB-regular (resp., ωS-regular) language is an ωS-regular (resp., ωB-regular) one, the last class is not closed under complementation. The existence of non-ωBS-regular languages that are the com…
Kleene Theorem for Higher-Dimensional Automata
本文证明了高维自动机的 Kleene 定理：其识别语言恰为有限区间偏序集的理性包含闭集，通过引入带接口的自动机及代数拓扑工具完成证明，为并发模型提供通用模板。
A featured transition system is a transition system in which the transitions are annotated with feature expressions: Boolean expressions on a finite number of given features. Depending on its feature expression, each individual transition can be enabled when some features are present, and disabled for other sets of features. The behavior of a featured transition system hence depends on a given set of features. There are algorithms for featured transition systems which can check their properties for all sets of features at once, for example for LTL or CTL properties. Here we introduce a model o…
Twinning automata and regular expressions for string static analysis
本文提出基于抽象解释理论的Tarsis抽象域，通过有限状态自动机近似字符串值，利用字符串字母表提升分析精度，通过实验验证其优于现有方法。
We provide syntactic derivative-like operations, defined by recursion on regular expressions, in the styles of both Brzozowski and Antimirov, for trace closures of regular languages. Just as the Brzozowski and Antimirov derivative operations for regular languages, these syntactic reordering derivative operations yield deterministic and nondeterministic automata respectively. But trace closures of regular languages are in general not regular, hence these automata cannot generally be finite. Still, as we show, for star-connected expressions, the Antimirov and Brzozowski automata, suitably quotie…
Brzozowski derivatives of a regular expression are developed for constructing deterministic automata from the given regular expression in the algebraic way. In this paper,some lemmas of the regular expressions are discussed and the regular languages of the derivatives are illustrated. Also the generalizations of the Brzozowski derivatives are proved as theorems with help of properties and known results.
Recently, the problem of obtaining a short regular expression equivalent to a given finite automaton has been intensively investigated. Algorithms for converting finite automata to regular expressions have an exponential blow-up in the worst-case. To overcome this, simple heuristic methods have been proposed. In this paper we analyse some of the heuristics presented in the literature and propose new ones. We also present some experimental comparative results based on uniform random generated deterministic finite automata.
The languages accepted by finite automata are precisely the languages denoted by regular expressions. In contrast, finite automata may exhibit behaviours that cannot be described by regular expressions up to bisimilarity. In this paper, we consider extensions of the theory of regular expressions with various forms of parallel composition and study the effect on expressiveness. First we prove that adding pure interleaving to the theory of regular expressions strictly increases its expressiveness up to bisimilarity. Then, we prove that replacing the operation for pure interleaving by ACP-style p…
We consider forkable regular expressions, which enrich regular expressions with a fork operator, to establish a formal basis for static and dynamic analysis of the communication behavior of concurrent programs. We define a novel compositional semantics for forkable expressions, establish their fundamental properties, and define derivatives for them as a basis for the generation of automata, for matching, and for language containment tests. Forkable expressions may give rise to non-regular languages, in general, but we identify sufficient conditions on expressions that guarantee finiteness of t…
Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices
本文针对扩张有理矩阵对应的数字系统，引入有限状态转换器自动机实现向量加法，用以刻画具有有限性性质的所有数字系统对及其吸引子。
We propose regular expressions to abstractly model and study properties of resource-aware computations. Inspired by nominal techniques -- as those popular in process calculi -- we extend classical regular expressions with names (to model computational resources) and suitable operators (for allocation, deallocation, scoping of, and freshness conditions on resources). We discuss classes of such nominal regular expressions, show how such expressions have natural interpretations in terms of languages over infinite alphabets, and give Kleene theorems to characterise their formal languages in terms …
Even the fastest SMT solvers have performance problems with regular expressions from real programs. Because these performance issues often arise from the problem representation (e.g. non-deterministic finite automata get determinized and regular expressions get unrolled), we revisit Boolean finite automata, which allow for the direct and natural representation of any Boolean combination of regular languages. By applying the IC3 model checking algorithm to Boolean finite automata, not only can we efficiently answer emptiness and universality problems, but through an extension, we can decide sat…
Regular expression matching is essential for many applications, such as finding patterns in text, exploring substrings in large DNA sequences, or lexical analysis. However, sequential regular expression matching may be time-prohibitive for large problem sizes. In this paper, we describe a novel algorithm for parallel regular expression matching via deterministic finite automata. Furthermore, we present our tool PaREM that accepts regular expressions and finite automata as input and automatically generates the corresponding code for our algorithm that is amenable for parallel execution on share…
Regular Expressions in a CS Formal Languages Course
本文提出在自动机理论与形式语言课程中用基于编程的方法讲授正则表达式，借助内置正则表达式类型的 FSM 语言，将正则表达式视为可编程的单词生成算法，以激发学生兴趣并促进相关算法学习。
Real-time Regular Expression Matching
针对自动机复杂度随正则表达式长度指数增长、制约网络入侵检测系统运行的问题，本文为几类复杂正则语言的指数爆炸问题提出理论加硬件的解决方案，并以正确性与复杂度定理提供支撑。
Graphoid Automata, Frobenius PROPs, and Semiring Evaluation
本文研究图状自动机的求值方程，证明经典图状方程等价于特殊交换Frobenius方程，并给出半环上矩阵求值相等的统一定义及骨律成立的条件。
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