Although regular expressions do not correspond univocally to regular languages, it is still worthwhile to study their properties and algorithms. For the average case analysis one often relies on the uniform random generation using a specific grammar for regular expressions, that can represent regular languages with more or less redundancy. Generators that are uniform on the set of expressions are not necessarily uniform on the set of regular languages. Nevertheless, it is not straightforward that asymptotic estimates obtained by considering the whole set of regular expressions are different from those obtained using a more refined set that avoids some large class of equivalent expressions. In this paper we study a set of expressions that avoid a given absorbing pattern. It is shown that, although this set is significantly smaller than the standard one, the asymptotic average estimates for the size of the Glushkov automaton for these expressions does not differ from the standard case.
Construction of fuzzy automata from fuzzy regular expressions
Li和Pedrycz [Y. M. Li，W. Pedrycz，Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoid，Fuzzy Sets and Systems 156（2005）68- 92]已经证明了基本结果，这些结果提供了在格序monoid中表示具有隶属值的模糊语言的不同等价方式，并且推广了形式语言的经典理论的众所周知的结果。特别是，他们已经证明，一个模糊语言的整体格序幺半群可以表示为模糊正则表达式，当且仅当它可以识别的模糊有限自动机。然而，他们没有给出任何有效的方法来构造一个等价的模糊有限自动机从一个给定的模糊正则表达式。在本文中，我们提供了这样一个有效的方法。通过将模糊正则表达式α中的标量变换为新的扩展字母表中的字母，将模糊正则表达式α转化为普通正则表达式α R。然后，从识别语言的任意非确定性有限自动机A开始，||α_R||用正则表达式α R表示，构造了模糊有限自动机A_α和A_α^r，它们的状态数与自动机A相同或更少，它们识别模糊语言||α||由模糊正则表达式α表示。起始的非确定性有限自动机A可以从α_R中使用任何一种众所周知的将正则表达式转换为非确定性有限自动机的构造来获得，例如Glushkov-McNaughton-Y…
A recent paper proposed an algorithm iSOIRE, which combines single-occurrence automaton (SOA) and maximum independent set (MIS) to learn a subclass single-occurrence regular expressions with interleaving (SOIREs) and claims the learnt expression is SOIRE, which has unrestricted usage for interleaving. However, in reality, the learnt expression still has many restrictions for using interleaving, even does for Kleene-star or interation, i.e, the learnt expression is not an SOIRE, we prove that by examples. In this paper, for the algorithm iSOIRE, we first give the basic notions, then provide ana…
FO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the long standing open problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star free expressions = aperiodic languages) result of Schützenberger. Our result also generalizes a lesser known characterization by Schützenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite wo…
We prove lower bounds on the length of regular expressions for finite languages by methods from arithmetic circuit complexity. First, we show a reduction: the length of a regular expression for a language $L\subseteq \{0,1\}^n$ is bounded from below by the minimum size of a monotone arithmetic formula computing a polynomial that has $L$ as its set of exponent vectors, viewing words as vectors. This result yields lower bounds for the binomial language of all words with exactly $k$ ones and $n-k$ zeros and for the language of all Dyck words of length $2n$. We also determine the blow-up of langua…
We show that the commutative closure combined with the iterated shuffle is a regularity-preserving operation on group languages. In particular, for commutative group languages, the iterated shuffle is a regularity-preserving operation. We also give bounds for the size of minimal recognizing automata. Then, we use these results to deduce that the commutative closure of any shuffle expression over group languages, i.e., expressions involving shuffle, iterated shuffle, concatenation, Kleene star and union in any order, starting with the group languages, always yields a regular language.
The class of regular transformations has several equivalent characterizations such as functional MSO transductions, deterministic two-way transducers, streaming string transducers, as well as regular transducer expressions (RTE). For algorithmic applications, it is very common and useful to transform a specification, here, an RTE, to a machine, here, a transducer. In this paper, we give an efficient construction of a two-way reversible transducer (2RFT) equivalent to a given RTE. 2RFTs are a well behaved class of transducers which are deterministic and co-deterministic (hence allows evaluation…
INESC-ID（INESC-ID（葡萄牙国家系统与计算机工程研究所））；Universidade de Lisboa（里斯本大学）；OutSystems（OutSystems公司）；Carnegie Mellon University（卡内基梅隆大学）
Form validators based on regular expressions are often used on digital forms to prevent users from inserting data in the wrong format. However, writing these validators can pose a challenge to some users. We present FOREST, a regular expression synthesizer for digital form validations. FOREST produces a regular expression that matches the desired pattern for the input values and a set of conditions over capturing groups that ensure the validity of integer values in the input. Our synthesis procedure is based on enumerative search and uses a Satisfiability Modulo Theories (SMT) solver to explor…
Milner (1984) introduced a process semantics for regular expressions as process graphs. Unlike for the language semantics, where every regular (that is, DFA-accepted) language is the interpretation of some regular expression, there are finite process graphs that are not bisimilar to the process interpretation of any regular expression. For reasoning about graphs that are expressible by regular expressions modulo bisimilarity it is desirable to have structural representations of process graphs in the image of the interpretation. For '1-free' regular expressions, their process interpretations sa…
Milner (1984) defined a process semantics for regular expressions. He formulated a sound proof system for bisimilarity of process interpretations of regular expressions, and asked whether this system is complete. We report conceptually on a proof that shows that Milner's system is complete, by motivating, illustrating, and describing all of its main steps. We substantially refine the completeness proof by Grabmayer and Fokkink (2020) for the restriction of Milner's system to `1-free' regular expressions. As a crucial complication we recognize that process graphs with empty-step transitions tha…
Key Lab. of System Software (CAS) Institute of Software, Chinese Academy of Sciences, \& University of Chinese Academy of Sciences Beijing, China；National Key Lab. of Space Integrated Information System Institute of Software, Chinese Academy of Sciences, \& University of Chinese Academy of Sciences Beijing, China；School of Computer Science \& MoE Key Lab. of High Confidence Software Technologies Peking University Beijing, China；School of Computer Science；Technology Tongji University Shanghai, China；Kyushu University Fukuoka, Japan.
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学力学与数学系）；National University of Science and Technology MISIS（国立研究型技术大学MISIS（莫斯科国立钢铁合金学院））；Neurosputnik LLC（Neurosputnik有限责任公司）；Rebis LLC（Rebis有限责任公司）
