Position Automaton Research Articles

Location automata for synchronised shuffle expressions

Several notions of synchronisation in concurrent systems can be modelled by regular shuffle operators. In this paper we consider regular expressions extended with three operators corresponding respectively to strong, arbitrary, and weak synchronisation. For these expressions, we define a location based position automaton. Furthermore, we show that the partial derivative automaton is still a quotient of the position automaton.

Location automata for regular expressions with shuffle and intersection

We define the notion of location for regular expressions with shuffle by extending the notion of position in standard regular expressions. Locations allow for the definition of the sets Follow, First, and Last with their usual semantics. From these, we construct an automaton for regular expressions with shuffle (APOS), which generalises the standard position/Glushkov automaton. The sets mentioned above are also the foundation for other constructions, such as the Follow automaton, and automata based on pointed expressions. As a consequence, all these constructions can be generalised to the shuffle operator. We show that the partial derivative automaton is a right-quotient of APOS. We relate APOS with another automaton construction based on positions that has been previously studied (A∂pos). The prefix automaton is extended to the shuffle operator and shown not to be a quotient of APOS. Locations are also used to define a position automaton for regular expressions with the intersection.

A deterministic parsing algorithm for ambiguous regular expressions

We introduce a new parser generator, called Berry–Sethi Parser (BSP), for ambiguous regular expressions (RE). The generator constructs a deterministic finite-state transducer that recognizes an input string, as the classical Berry–Sethi algorithm does, and additionally outputs a linear representation of all the syntax trees of the string; for infinitely ambiguous strings, a policy for selecting representative sets of trees is chosen. To construct the transducer, the RE symbols, including letters, parentheses and other metasymbols, are distinctly numbered, so that the corresponding language becomes locally testable. In this way a deterministic position automaton can be constructed, which recognizes and translates the input into a compact DAG representation of the syntax trees. The correctness of the construction is proved. The transducer operates in a linear time on the input. Its descriptive complexity is analyzed as a function of established RE parameters: the alphabetic width, the number of null string symbols and the height of the RE tree. A condition for checking RE ambiguity on the transducer graph is stated. Experimental results of running the parser generator and the parser on a large RE collection are presented. The POSIX RE disambiguation criterion has also been applied to the parser.

On Average Behaviour of Regular Expressions in Strong Star Normal Form

For regular expressions in (strong) star normal form a large set of efficient algorithms is known, from conversions into finite automata to characterisations of unambiguity. In this paper we study the average complexity of this class of expressions using analytic combinatorics. As it is not always feasible to obtain explicit expressions for the generating functions involved, here we show how to get the required information for the asymptotic estimates with an indirect use of the existence of Puiseux expansions at singularities. We study, asymptotically and on average, the alphabetic size, the size of the [Formula: see text]-follow automaton and of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions.

A mesh of automata

We contribute new relations to the taxonomy of different conversions from regular expressions to equivalent finite automata. In particular, we are interested in transformations that construct automata such as, the follow automaton, the partial derivative automaton, the prefix automaton, the automata based on pointed expressions recently introduced and studied, and last but not least the position, or Glushkov automaton (APOS), and their double reversed construction counterparts. We deepen the understanding of these constructions and show that with the artefacts used to construct the Glushkov automaton one is able to capture most of them. As a byproduct we define a dual version APOS← of the position automaton which plays a similar role as APOS but now for the reverse expression. Moreover, it turns out that the prefix automaton APre is central to reverse expressions, because the determinisation of the double reversal of APre (first reverse the expression, construct the automaton APre, and then reverse the automaton) can be represented as a quotient of any of the considered deterministic automata that we consider in this investigation. This shows that although the conversion of regular expressions and reversal of regular expressions to finite automata seems quite similar, there are significant differences.

Automata for regular expressions with shuffle

We generalize the partial derivative automaton and the position automaton to regular expressions with shuffle, and study their state complexity in the worst, as well as in the average case. The number of states of the partial derivative automaton (Apd) is, in the worst case, at most 2m, where m is the number of letters in the expression. The asymptotic average is bounded by (43)m. We define a position automaton (Apos) that is homogeneous, but in which several states can correspond to a same position, and we show that Apd is a quotient of Apos. The number of states of the position automaton is at most 1+m(2m−1), while the asymptotic average is no more than m(43)m.

Extended to multi-tilde–bar regular expressions and efficient finite automata constructions

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 algorithm which efficiently computes the Glushkov functions of an unrestricted EmtbRE. Our approach is based on a recursive definition of the language associated with an EmtbRE which enlightens the fact that the worst case time complexity of the conversion of an EmtbRE into an automaton is related to the worst case time complexity of the computation of the Null function. Finally we show how to extend the ZPC-structure to EmtbREs, which allows us to apply to this family of extended expressions the efficient constructions based on this structure (in particular the construction of the c-continuation automaton, the position automaton, the follow automaton and the equation automaton).

K-Position, Follow, Equation and K-C-Continuation Tree Automata Constructions

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 tree automaton E of Kuske and Meinecke and our previously defined k-C-continuation tree automaton C, we show that the previous morphic relations are still valid on tree expressions.

A Structural Construction of a Deterministic Position Automaton

Every regular expression can be transformed into a Nondeterministic Finite Automaton (NFA) with or without  transitions. A well known algorithm called subset construction technique is used for conversion of NFA into DFA. In this paper, initially, the construction of the position automaton is given for the same. Also, the algorithm to convert this Position Automaton into DFA using subset construction technique is discussed. AMS MSC2010 Certification: 68Q45, 68Q70

Construction of fuzzy automata from fuzzy regular expressions

Li and Pedrycz 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 fuzzy finite automaton. However, they did not give any efficient method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression. In this paper we provide such an efficient method. Transforming scalars appearing in a fuzzy regular expression αinto letters of the new extended alphabet, we convert the fuzzy regular expression αto an ordinary regular expression αR. Then, starting from an arbitrary nondeterministic finite automaton A that recognizes the language ‖αR‖ represented by the regular expression αR, we construct fuzzy finite automata Aα and Aαr with the same or even less number of states than the automaton A, which recognize the fuzzy language ‖α‖represented by the fuzzy regular expression α. The starting nondeterministic finite automaton A can be obtained from αR using any of the well-known constructions for converting regular expressions to nondeterministic finite automata, such as Glushkov–McNaughton–Yamada's position automaton, Brzozowski's derivative automaton, Antimirov's partial derivative automaton, or Ilie–Yu's follow automaton.

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.

An Efficient Computation of the Equation K-automaton of a Regular K-expression

The aim of this paper is to describe a quadratic algorithm for computing the equation K-automaton of a regular K-expression as defined by Lombardy and Sakarovitch. Our construction is based on an extension to regular K-expressions of the notion of c-continuation that we introduced to compute the equation automaton of a regular expression as a quotient of its position automaton.

Fast equation automaton computation

The most efficient known construction of equation automaton is that due to Ziadi and Champarnaud. For a regular expression E, it requires O ( | E | 2 ) time and space and is based on going from position automaton to equation automaton using c-continuations. This complexity is due to the sorting step that takes O ( | E | 2 ) time used to identify the identical sub-expressions of E. In this paper, we present a more efficient construction of the equation automaton which avoids the sorting step and replaces it by a minimization of an acyclic finite deterministic automaton. We show that this minimization allows the identification of identical sub-expressions as well as the sorting step used in Champarnaud and Ziadi's approach. Using the minimization we get O ( | E | + | E | ⋅ | E E | ) time and space complexity where | E E | is the number of states of the equation automaton.

Bit-Parallel Algorithms for Translating Regular Expressions into NFAs

The aim of the paper is to design efficient bit-parallel algorithms for translating regular expressions into nondeterministic finite automata (NFAs). Let r be a regular expression over an alphabet Σ, and let n be the length of r and let m be the number of symbols of Σ occurring in r. Then we present bit-parallel algorithms for translating regular expressions into Glushkov automata (position automata) and follow automata using Thompson automata. For Glushkov automata, we will give an algorithm which runs in O(n + m⌈m/W⌉)time and space. For follow automata, we will give a randomized algorithm which runs in O(n + m⌈m/W⌉) expected time and O(n + m⌈m/W⌉) space. We rely on a W-bit uniform RAM for estimating the complexities of algorithms. Since the best known algorithms for these automata runs in O(n + m2) time and space, our algorithms achieve an almost W-fold speed-up.

NORMALIZED EXPRESSIONS AND FINITE AUTOMATA

There exist two well-known quotients of the position automaton of a regular expression. The first one, called the equation automaton, was first introduced by Mirkin from the notion of prebase and has been redefined by Antimirov from the notion of partial derivative. The second one, due to Ilie and Yu and called the follow automaton, can be obtained by eliminating ε-transitions in an ε-NFA that is always smaller than the classical ε-NFAs (Thompson, Sippu and Soisalon–Soininen). Ilie and Yu discussed the difficulty of succeeding in a theoretical comparison between the size of the follow automaton and the size of the equation automaton and concluded that it is very likely necessary to realize experimental studies. In this paper we solve the theoretical question, by first defining a set of regular expressions, called normalized expressions, such that every regular expression can be normalized in linear time, and proving then that the equation automaton of a normalized expression is always smaller than its follow automaton.

FROM THE $\mathcal{ZPC}$ STRUCTURE OF A REGULAR EXPRESSION TO ITS FOLLOW AUTOMATON

The follow automaton of a regular expression, recently introduced by Ilie and Yu, is a quotient of the position automaton. The aim of this paper is to present an efficient computation of this quotient, based on specific properties of the [Formula: see text] structure of the expression.

FROM REGULAR WEIGHTED EXPRESSIONS TO FINITE AUTOMATA

In this article we generalize concepts of the position automaton and ZPC-structure to the regular [Formula: see text]-expressions. We show that the extended ZPC-structure can be built in linear time w.r.t. the size of the [Formula: see text]-expression and that the associated position automaton can be deduced from it in quadratic time.

Follow automata

We give two new algorithms for constructing small nondeterministic finite automata (NFA) from regular expressions. The first constructs NFAs with ε-transitions (εNFA) which are smaller than all the other εNFAs obtained by similar constructions. Their size is at most 32|α|+52, where α is the regular expression. This is very close to optimal since we prove also the lower bound 43|α|+52. The second constructs NFAs. It uses ε-elimination in the εNFAs we just introduced and builds a quotient of the well-known position automaton w.r.t. the equivalence given by the follow relation; therefore giving the name of follow automaton. The new automaton uses optimally the information from the positions of a regular expression. We compare the follow automaton with the best constructions to date and show that it has important advantages over those.

Canonical derivatives, partial derivatives and finite automaton constructions

Let E be a regular expression. Our aim is to establish a theoretical relation between two well-known automata recognizing the language of E, namely the position automaton P E constructed by Glushkov or McNaughton and Yamada, and the equation automaton E E constructed by Mirkin or Antimirov. We define the notion of c-derivative (for canonical derivative) of a regular expression E and show that if E is linear then two Brzozowski's derivatives of E are aci-similar if and only if the corresponding c-derivatives are identical. It allows us to represent the Berry–Sethi's set of continuations of a position by a unique c-derivative, called the c-continuation of the position. Hence the definition of C E , the c-continuation automaton of E, whose states are pairs made of a position of E and of the associated c-continuation. If states are viewed as positions, C E is isomorphic to P E . On the other hand, a partial derivative, as defined by Antimirov, is a class of c-derivatives for some equivalence relation, thus C E reduces to E E . Finally C E makes it possible to go from P E to E E , while this cannot be achieved directly (from the state graphs). These theoretical results lead to an O(|E| 2) space and time algorithm to compute the equation automaton, where |E| is the size of the expression. This is the complexity of the most efficient constructions yielding the position automaton, while the size of the equation automaton is not greater and generally much smaller than the size of the position automaton.

EVALUATION OF THREE IMPLICIT STRUCTURES TO IMPLEMENT NONDETERMINISTIC AUTOMATA FROM REGULAR EXPRESSIONS

The aim of this paper is to compare three efficient representations of the position automaton of a regular expression: the Thompson ∊-automaton, the [Formula: see text]-structure and the ℱ-structure, an optimization of the [Formula: see text]-structure. These representations are linear w.r.t. the size s of the expression, since their construction is in O(s) space and time, as well as the computation of the set δ(X,a) of the targets of the transitions by a of any subset X of states. The comparison is based on the evaluation of the number of edges of the underlying graphs respectively created by the construction step or visited by the computation of a set δ(X,a).