On the intersection of regex languages with regular languages

In this paper we revisit the semantics of extended regular expressions (regex), defined succinctly in the 90s [A.V. Aho, Algorithms for finding patterns in strings, in: Jan van Leeuwen (Ed.), Handbook of Theoretical Computer Science, in: Algorithms and Complexity, vol. A, Elsevier and MIT Press, 1990, pp. 255–300] and rigorously in 2003 by Câmpeanu, Salomaa and Yu [C. Câmpeanu, K. Salomaa, S. Yu, A formal study of practical regular expressions, IJFCS 14 (6) (2003) 1007–1018], when the authors reported an open problem, namely whether regex languages are closed under the intersection with regular languages. We give a positive answer; and for doing so, we propose a new class of machines — regex automata systems (RAS) — which are equivalent to regex. Among others, these machines provide a consistent and convenient method of implementing regex in practice. We also prove, as a consequence of this closure property, that several languages, such as the mirror language, the language of palindromes, and the language of balanced words are not regex languages.     

Succinctness of regular expressions with interleaving,intersection and counting

In this paper, we study the succinctness of regular expressions (REs) extended with interleaving, intersection and counting operators. We show that in a translation from REs with interleaving to standard regular expressions a double exponential size increase cannot be avoided. We also consider the complexity of translations to finite automata. We give a tight exponential lower bound on the translation of REs with intersection to NFAs, and, for each of the three classes of REs, we show that in a translation to a DFA a double exponential size increase cannot be avoided. Together with known results, this gives a complete picture of the complexity of translating REs extended with interleaving, intersection or counting into (standard) regular expressions, NFAs, and DFAs.     

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.     

On measuring nondeterminism in regular languages

It is well known that allowing nondeterminism in a finite automaton can produce in the most extreme case an exponential savings in the number of states required to recognize a regular language. This paper studies situations intermediate between forbidding nondeterminism and allowing it. The amount of nondeterminism used by a finite automaton is quantified, so that the decrease in the size of the state space that occurs as the amount of nondeterminism that is permitted increases in increments can be studied. These intermediate situations are shown always to lie between two extremes:(1) there are no savings as the amount of nondeterminism increases incrementally, so that savings occur only when the amount of nondeterminism becomes unlimited;(2) each increment of nondeterminism results in additional savings, the number s of states decreasing approximately as s^1/i, until exponential savings have been achieved after about i = logs/log log s increments.     

On regular drawn symbolic picture languages

Drawn symbolic pictures are an extension of drawn pictures obtained by associating a symbol from an alphabet to each point of the picture. In the paper we will address some new interesting issues derived from the introduction of the symbols and we will identify the conditions, which ensure the preservation of properties holding for drawn pictures in the setting of the proposed extension.     

On the state complexity of reversals of regular languages

We compare the number of states between minimal deterministic finite automata accepting a regular language and its reversal (mirror image). In the worst case the state complexity of the reversal is 2ⁿ for an n-state language. We present several classes of languages where this maximal blow-up is actually achieved and study the conditions for it. In the case of finite languages the maximal blow-up is not possible but still a surprising variety of different growth types can be exhibited.     

On regular realizability problems for context-free languages

We consider regular realizability problems, which consist in verifying whether the intersection of a regular language which is the problem input and a fixed language (filter) which is a parameter of the problem is nonempty. We study the algorithmic complexity of regular realizability problems for context-free filters. This characteristic is consistent with the rational dominance relation of CF languages. However, as we prove, it is more rough. We also give examples of both P-complete and NL-complete regular realizability problems for CF filters. Furthermore, we give an example of a subclass of CF languages for filters of which the regular realizability problems can have an intermediate complexity. These are languages with polynomially bounded rational indices.     

Adding regular expressions to graph reachability and pattern queries

It is increasingly common to find graphs in which edges are of different types, indicating a variety of relationships. For such graphs we propose a class of reachability queries and a class of graph patterns, in which an edge is specified with a regular expression of a certain form, expressing the connectivity of a data graph via edges of various types. In addition, we define graph pattern matching based on a revised notion of graph simulation. On graphs in emerging applications such as social networks, we show that these queries are capable of finding more sensible information than their traditional counterparts. Better still, their increased expressive power does not come with extra complexity. Indeed, (1) we investigate their containment and minimization problems, and show that these fundamental problems are in quadratic time for reachability queries and are in cubic time for pattern queries. (2) We develop an algorithm for answering reachability queries, in quadratic time as for their traditional counterpart. (3) We provide two cubic-time algorithms for evaluating graph pattern queries, as opposed to the NP-completeness of graph pattern matching via subgraph isomorphism. (4) The effectiveness and efficiency of these algorithms are experimentally verified using real-life data and synthetic data.     

Finite fuzzy automata, regular fuzzy languages, and pattern recognition

Fuzzy set theory is summarized, and finite fuzzy automata and languages are described. Some results in fuzzy languages based on the “max(min)” rule are discussed. An application of fuzzy languages in syntactic pattern analysis is described and is illustrated by a simple example.     

On the computation of quotients and factors of regular languages

Quotients and factors are important notions in the design of various computational procedures for regular languages and for the analysis of their logical properties. We propose a new representation of regular languages, by linear systems of language equations, which is suitable for the following computations: language reversal, left quotients and factors, right quotients and factors, and factor matrices. We present algorithms for the computation of all these notions, and indicate an application of the factor matrix to the computation of solutions of a particular language reconstruction problem. The advantage of these algorithms is that they all operate only on linear systems of language equations, while the design of the same algorithms for other representations often require translation to other representations.     

On regular tree languages and deterministic pushdown automata

The theory of formal string languages and of formal tree languages are both important parts of the theory of formal languages. Regular tree languages are recognized by finite tree automata. Trees in their postfix notation can be seen as strings. This paper presents a simple transformation from any given (bottom-up) finite tree automaton recognizing a regular tree language to a deterministic pushdown automaton accepting the same tree language in postfix notation. The resulting deterministic pushdown automaton can be implemented easily by an existing parser generator because it is constructed for an LR(0) grammar, and its size directly corresponds to the size of the deterministic finite tree automaton. The class of regular tree languages in postfix notation is a proper subclass of deterministic context-free string languages. Moreover, the class of tree languages which are in their postfix notation deterministic context-free string languages is a proper superclass of the class of regular tree languages.