Balanced timed regular expressions

Several classes of regular expressions for timed languages accepted by timed automata have been suggested in the literature. In this article we introduce balanced timed regular expressions with colored parentheses which are equivalent to timed automata, and, differently from existing definitions, do not refer to clock values, and do not use additional operations such as intersection and renaming.     

More concise representation of regular languages by automata and regular expressions

We consider two formalisms for representing regular languages: constant height pushdown automata and straight line programs for regular expressions. We constructively prove that their sizes are polynomially related. Comparing them with the sizes of finite state automata and regular expressions, we obtain optimal exponential and double exponential gaps, i.e., a more concise representation of regular languages.     

Algorithms for learning regular expressions from positive data

We describe algorithms that directly infer very simple forms of 1-unambiguous regular expressions from positive data. Thus, we characterize the regular language classes that can be learned this way, both in terms of regular expressions and in terms of (not necessarily minimal) deterministic finite automata.     

From regular expressions to smaller NFAs

Several methods have been developed to construct λ-free automata that represent a regular expression. Among the most widely known are the position automaton (Glushkov), the partial derivatives automaton (Antimirov) and the follow automaton (Ilie and Yu). All these automata can be obtained with quadratic time complexity, thus, the comparison criterion is usually the size of the resulting automaton. The methods that obtain the smallest automata (although, for general expressions, they are not comparable), are the follow and the partial derivatives methods. In this paper, we propose another method to obtain a λ-free automaton from a regular expression. The number of states of the automata we obtain is bounded above by the size of both the partial derivatives automaton and of the follow automaton. Our algorithm also runs with the same time complexity of these methods.     

Regular prefix relations

We define a class ofn-ary relations on strings called the regular prefix relations, and give four alternative characterizations of this class:

1. the relations recognized by a new type of automaton, the prefix automata,
2. the relations recognized by tree automata specialized to relations on strings,
3. the relations between strings definable in the second order theory ofk successors,
4. the smallest class containing the regular sets and the prefix relation, and closed under the Boolean operations, Cartesian product, projection, explicit transformation, and concatenation with Cartesian products of regular sets.

We give concrete examples of regular prefix relations, and a pumping argument for prefix automata. An application of these results to the study of inductive inference of regular sets is described.     

Duration-constrained regular expressions

This paper investigates the logic-automata-connection for Duration Calculus. It has been frequently observed that Duration Calculus with linear duration terms comes close to being a logic of linear hybrid automata. We attempt to make this relation precise by constructing Kleene-connection between duration-constrained regular expressions and a subclass of linear hybrid automata called loop-reset automata in which any variable tested in a loop is reset in the same loop. The formalism of duration-constrained regular expressions is an extension of regular expressions with duration constraints, which are essentially formulas of Duration Calculus without negation, yet extended by a Kleene-star operator. In this paper, we show that this formalism is equivalent in expressive power to loop-reset automata by providing a translation procedure from expressions to automata and vice verse.Received June 1999Accepted in revised form September 2003 by M. R. Hansen and C. B. Jones     

A general approach for building combinational P automata

In traditional automata theory, the closure with respect to certain operations is one of the most investigated properties, especially the methods of constructing automata for certain operations among the given sub-automata. However, up to now less effort has been addressed to this question for P automata. As an improvement on earlier results, we introduce P automata with communication and active membrane rules working in the initial mode (CAIP). We present methods for constructing automata that recognize the Union, the Concatenation, the Kleene Closure, or the ω Closure of the given languages which are represented by some P automata. We also show that for any language denoted by a regular expression, we can readily construct a CAIP automaton corresponding to it.     

Succinct representation of regular languages by boolean automata

Boolean automata are a generalization of finite automata in the sense that the ‘next state’, i.e. the result of the transition function given a state and a letter, is not just a single state (deterministic automata) or a union of states (nondeterministic automata) but a boolean function of states. Boolean automata accept precisely regular languages; furthermore they correspond in a natural way to certain language equations as well as to sequential networks. We investigate the succinctness of representing regular languages by boolean automata. In particular, we show that for every deterministic automaton A with m states there exists a boolean automaton with [log₂m] states which accepts the reverse of the language accepted by A (m≥1). We also show that for every n≥1 there exists a boolean automation with n states such that the smallest deterministic automaton accepting the same language has 2⁽²ⁿ⁾ states; moreover this holds for an alphabet with only two letters.     

On universality of concurrent expressions with synchronization primitives

Concurrent expressions are a class of extended regular expressions with a shuffle operator (6) and its closure (

). The class of concurrent expressions with synchronization primitives, called synchronized concurrent expressions, is introduced as an extended model of Shaw's flow expressions. This paper discusses some formal properties of synchronized concurrent expressions from a formal language theoretic point of view. It is shown that synchronized concurrent expressions with three signal/wait operations are universal in the sense that they can simulate any semaphore controlled concurrent expressions and they can describe the class of recursively enumerable sets. Some results on semaphore controlled regular expressions are also included to give a taste of more positive results.     

A multidimensional approach to syntactic pattern recognition

This paper describes a syntactic method for representing the primitive parts of a pattern as nodes of a type of directed graph. A linear representation of the digraph can then be presented to a regular unordered tree automaton for classification. Regular unordered tree automata can be simulated by deterministic pushdown automata, so this procedure can be implemented easily. Regular u-tree automata and the corresponding generative systems, regular u-tree grammars are formally defined. Several results are shown which are applicable to all syntactic pattern recognition schemes involving the use of primitives.     

Regular Language Representations in the Constructive Type Theory of Coq

We explore the theory of regular language representations in the constructive type theory of Coq. We cover various forms of automata (deterministic, nondeterministic, one-way, two-way), regular expressions, and the logic WS1S. We give translations between all representations, show decidability results, and provide operations for various closure properties. Our results include a constructive decidability proof for the logic WS1S, a constructive refinement of the Myhill-Nerode characterization of regularity, and translations from two-way automata to one-way automata with verified upper bounds for the increase in size. All results are verified with an accompanying Coq development of about 3000 lines.     

On the look-ahead problem in lexical analysis

Modern programming languages use regular expressions to define valid tokens. Traditional lexical analyzers based on minimum deterministic finite automata for regular expressions cannot handle the look-ahead problem. The scanner writer needs to explicitly identify the look-ahead states and code the buffering and re-scanning operations by hand. We identify the class of finite look-ahead finite automata, which is general enough to include all finite automata of practical lexical analyzers. Finite look-ahead finite automata are then transformed into suffix finite automata. A new lexical analyzer makes use of the suffix finite automata to identify tokens. The new lexical analyzer solves the look-ahead problem in a table-driven approach and it can detect lexical errors at an earlier time than traditional lexical analyzers. The extra cost of the new lexical analyzers is the larger state transition table and three additional 1-dimensional tables. Incremental lexical analysis is also discussed.This work was supported in part by National Science Council, Taiwan, R.O.C. under grants NSC 83-0111-S-009-001-CL and NSC 84-2213-E-009-043     

Compositional synthesis of asynchronous automata

Asynchronous automata are a model of communication processes with a control structure distributed on a set P of processes, global initializations and global accepting conditions. The well-known theorem of Zielonka states that they recognize exactly the class of regular Mazurkiewicz trace languages. The corresponding synthesis problem is, given a global specification A of any regular trace language L, to build an asynchronous automaton that recognizes L, automatically. Yet, all such existing constructions are quite involved and yield an explosion of the number of states in each process, which is exponential in both the sizes of A and P. In this paper, we introduce the particular case of distributed asynchronous automata, which require that the initializations and the accepting conditions are distributed as well. We present an original technique based on simple compositions/decompositions of these distributed asynchronous automata that results in the construction of smaller non-deterministic asynchronous automata: now, the number of states in each process is only polynomial in the size of A, but is still exponential in the size of P.     

Formal power series and regular operations on fuzzy languages

In this paper we study formal power series over a quantale with coefficients in the algebra of all languages over a given alphabet, and representation of fuzzy languages by these formal power series. This representation generalizes the well-known representation of fuzzy languages by their cut and kernel languages. We show that regular operations on fuzzy languages can be represented by regular operations on power series which are defined by means of operations on ordinary languages. We use power series in study of fuzzy languages which are recognized by fuzzy finite automata and deterministic finite automata, and we study closure properties of the set of polynomials and the set of polynomials with regular coefficients under regular operations on power series.     

Somma di serie numeriche mediante la trasformazione quadraticaT 2,2

In a previous paper the rational transformations of a sequence {S _n } have been studied and defined as operations which associate to the sequence {S _n } an other sequence whose terms are rational functions of the S _n ^' s. Such a subject is here continued for the parlicular rational transformation of order 2,T _2,2, which is the unique rational transformation giving «fast convergence». TheT _2,2 transformation is applied to the numerical series summation and, throught geometrical considerations, general methods to determine expressions which excede the rest of a given series are found. Besides, a «limit» for some classes of non-convergent series is defined.     

Non-uniform Cellular Automata based on Open-ended Rule Evolution

Cellular automata (CAs) are mathematical models of spatially and temporally discrete mathematical systems. Non-uniform CAs are the cellular automata in which each cell may contain a different transition rule and change it with time, while all cells share the same transition rule in regular CAs. Little is still known about the dynamics of open-ended evolution of rules in non-uniform CAs. The purpose of our study is to construct and investigate a model of non-uniform CAs capable of open-ended rule evolution exhibiting a wide variety of behavior across all Wolfram’s classes. For this purpose, we construct 1-dimensional 2-state 3 neighborhood non-uniform CAs with evolving transition rules. In the model, we found an interesting dynamics that Class II (periodical behavior) and III (chaotic behavior) patterns emerged alternately, between which Class IV patterns sometimes emerged.     

Automata theory based on lattice-ordered semirings

In this paper, definitions of K{\mathcal{K}} automata, K{\mathcal{K}} regular languages, K{\mathcal{K}} regular expressions and K{\mathcal{K}} regular grammars based on lattice-ordered semirings are given. It is shown that K{\mathcal{K}}NFA is equivalent to K{\mathcal{K}}DFA under some finite condition, the Pump Lemma holds if K{\mathcal{K}} is finite, and Ke{{\mathcal{K}}}\epsilonNFA is equivalent to K{\mathcal{K}}NFA. Further, it is verified that the concatenation of K{\mathcal{K}} regular languages remains a K{\mathcal{K}} regular language. Similar to classical cases and automata theory based on lattice-ordered monoids, it is also found that K{\mathcal{K}}NFA, K{\mathcal{K}} regular expressions and K{\mathcal{K}} regular grammars are equivalent to each other when K{\mathcal{K}} is a complete lattice.     

Fuzzy grammar theory based on lattices

In this paper, on the basis of breadth-first and depth-first ways, we establish a fundamental framework of fuzzy grammars based on lattices, which provides a necessary tool for the analysis of fuzzy automata. The relationship among finite automata with membership values in lattices (l-VFAs), lattice-valued regular grammars (l-RGs) and lattice-valued deterministic regular grammars (l-DRGs) is investigated. It is demonstrated that, based on each semantic way, l-VFAs and l-RGs are equivalent in the sense that they accept or generate the same classes of fuzzy languages. Furthermore, it is proved that l-VFAs,?l-valued deterministic finite automata, l-RGs and l-DRGs are equivalent based on depth-first way. For any l-RG,?the language based on breadth-first way coincides with the language based on depth-first way if and only if the truth-valued domain l is a distributive lattice.     

Expressiveness of propositional projection temporal logic with star

This paper investigates the expressiveness of Propositional Projection Temporal Logic with Star (PPTL*). To this end, Büchi automata and ω-regular expressions are first extended as Stutter Büchi Automata (SBA) and Extended Regular Expressions (ERE) to include both finite and infinite strings. Further, by equivalent transformations among PPTL* formulas, SBAs and EREs, PPTL* is proved to represent exactly the full regular language. Moreover, some fragments of PPTL* are characterized, and finally, PPTL* and its fragments are classified into five different language classes.