自然语言是正则语言

自然语言由字母集、单词集、句子集、段落集和文章集5部分组成,而且,字母集包含于单词集,单词集包含于句子集,句子集包含于段落集,段落集包含于文章集。在此观点下,自然语言是正则语言。引入了字母空图和字母空图语言等10个概念。作为特例,英语由英语字母集、英语单词集、英语句子集、英语段落集和英语文章集5部分构成。在此观点下,英语是正则语言。引入了英语字母空图和英语字母空图语言等10个概念。汉语由汉字集、汉语词汇集、汉语句子集、汉语段落集和汉语文章集5部分构成。在此观点下,汉语是正则语言。引入了汉字空图和汉字空图语言等10个概念。这为计算机自然语言处理打开了一扇新的大门,开辟了语言学新的研究领域。     

One-Unambiguous Regular Languages

The ISO standard for the Standard Generalized Markup Language (SGML) provides a syntactic meta-language for the definition of textual markup systems. In the standard, the right-hand sides of productions are based on regular expressions, although only regular expressions that denote words unambiguously, in the sense of the ISO standard, are allowed. In general, a word that is denoted by a regular expression is witnessed by a sequence of occurrences of symbols in the regular expression that match the word. In an unambiguous regular expression as defined by Booket al.(1971,IEEE Trans. Comput.C-20(2), 149–153), each word has at most one witness. But the SGML standard also requires that a witness be computed incrementally from the word with a one-symbol lookahead; we call such regular expressions 1-unambiguous. A regular language is a 1-unambiguouslanguage if it is denoted by some 1-unambiguous regular expression. We give a Kleene theorem for 1-unambiguous languages and characterize 1-unambiguous regular languages in terms of structural properties of the minimal deterministic automata that recognize them. As a result we are able to prove the decidability of whether a given regular expression denotes a 1-unambiguous language; if it does, then we can construct an equivalent 1-unambiguous regular expression in worst-case optimal time.     

Odometers on Regular Languages

Odometers or "adding machines" are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A,<) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a one-to-one correspondence between ℕ and L. In this general setting the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer such as continuity, injectivity, surjectivity, minimality, .... We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integers whenever the set of greedy representations of all the integers is a regular language; we also consider substitution numeration systems as well as the connection with β-numerations.     

Regular Model Checking Using Inference of Regular Languages

Regular model checking is a method for verifying infinite-state systems based on coding their configurations as words over a finite alphabet, sets of configurations as finite automata, and transitions as finite transducers. We introduce a new general approach to regular model checking based on inference of regular languages. The method builds upon the observation that for infinite-state systems whose behaviour can be modelled using length-preserving transducers, there is a finite computation for obtaining all reachable configurations up to a certain length n. These configurations are a (positive) sample of the reachable configurations of the given system, whereas all other words up to length n are a negative sample. Then, methods of inference of regular languages can be used to generalize the sample to the full reachability set (or an overapproximation of it). We have implemented our method in a prototype tool which shows that our approach is competitive on a number of concrete examples. Furthermore, in contrast to all other existing regular model checking methods, termination is guaranteed in general for all systems with regular sets of reachable configurations. The method can be applied in a similar way to dealing with reachability relations instead of reachability sets too.     

Circuit Complexity of Regular Languages

We survey the current state of knowledge on the circuit complexity of regular languages and we prove that regular languages that are in AC⁰ and ACC⁰ are all computable by almost linear size circuits, extending the result of Chandra et al. (J. Comput. Syst. Sci. 30:222–234, 1985). As a consequence we obtain that in order to separate ACC⁰ from NC¹ it suffices to prove for some ε>0 an Ω(n ^1+ε ) lower bound on the size of ACC⁰ circuits computing certain NC¹-complete functions. Partially supported by grant GA ČR 201/07/P276, project No. 1M0021620808 of MŠMT ČR and Institutional Research Plan No. AV0Z10190503.     

Stochastic Inference of Regular Tree Languages

We generalize a former algorithm for regular language identification from stochastic samples to the case of tree languages. It can also be used to identify context-free languages when structural information about the strings is available. The procedure identifies equivalent subtrees in the sample and outputs the hypothesis in linear time with the number of examples. The results are evaluated with a method that computes efficiently the relative entropy between the target grammar and the inferred one.     

Property Testing of Regular Tree Languages

We consider the edit distance with moves on the class of words and the class of ordered trees. We first exhibit a simple tester for the class of regular languages on words and generalize it to the class of ranked and unranked regular trees. We also show that this distance problem is -complete on ordered trees. A preliminary version of this paper appeared in Proceedings of 31st International Colloquium on Automata, Languages and Programming, volume 3142 of Lecture Notes in Computer Science, pages 932–944, Springer, 2005. Work supported by ACI Sécurité Informatique: VERA of the French Ministry of research.     

Regular Languages Accepted by Quantum Automata

In this paper we analyze some features of the behaviour of quantum automata. In particular we prove that the class of languages recognized by quantum automata with isolated cut point is the class of reversible regular languages. As a more general result, we give a bound on the inverse error that implies the regularity of the language accepted by a quantum automaton.     

Regular Languages Defined with Generalized Quantifiers

We study an extension of first-order logic obtained by adjoining quantifiers that count with respect to an integer modulus. It is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups. We obtain an analogous result for regular ω-languages and establish some connections with complexity theory for fixed-depth families of circuits.     

Concatenation of Regular Languages and Descriptional Complexity

We investigate the deterministic and nondeterministic state complexity of languages resulting from the concatenation of two regular languages represented by deterministic and nondeterministic finite automata. We prove that the whole range of complexities up to the known upper bounds can be obtained in both cases.     

Learning Regular Languages from Simple Positive Examples

Learning from positive data constitutes an important topic in Grammatical Inference since it is believed that the acquisition of grammar by children only needs syntactically correct (i.e. positive) instances. However, classical learning models provide no way to avoid the problem of overgeneralization. In order to overcome this problem, we use here a learning model from simple examples, where the notion of simplicity is defined with the help of Kolmogorov complexity. We show that a general and natural heuristic which allows learning from simple positive examples can be developed in this model. Our main result is that the class of regular languages is probably exactly learnable from simple positive examples.     

Synthesis of Supervisors Using Learning Algorithm of Regular Languages

We propose an algorithm for thesynthesis of supervisors in discrete event systems. The algorithmis based on a learning algorithm of regular languages proposedby Angluin, and constructs a supervisor realizing an unknownspecification which is identified through the interaction betweenthe designer and the algorithm. We also consider the synthesisproblem for systems consisting of several processes which behaveconcurrently. One of serious problems in dealing with such aconcurrent system is that the number of states required for describingthe global behavior often grows exponentially in the size ofthe model. To improve this situation, we introduce the conceptof dependency defined on the set of events. It prevents the algorithmfrom considering all interleavings of independent occurrencesof events.     

Complete Classifications for the Communication Complexity of Regular Languages

We show that every regular language L has either constant, logarithmic or linear two-party communication complexity (in a worst-case partition sense). We prove similar classifications for the communication complexity of regular languages for the simultaneous, probabilistic, simultaneous probabilistic and Mod_p-counting models of communication.     

On the Representation of Real Numbers Using Regular Languages

Using a lexicographically ordered regular language, we show how to represent an interval of \R . We determine exactly the possible representations of any element in this interval and study the function which maps a representation onto its numerical value. We make explicit the relationship between the convergence of finite words to an infinite word and the convergence of the corresponding approximations to a real number. Received February 13, 2001, and in final form September 21, 2001. Online publication January 14, 2002.