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Rational and recognisable power series
 DRAFT OF A CHAPTER FOR THE HANDBOOK OF WEIGHTED AUTOMATA
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From CContinuations to New Quadratic Algorithms For Automaton Synthesis
"... Two classical nondeterministic automata recognize the language denoted by a regular expression: the position automaton which deduces from the position sets defined by Glushkov and McNaughtonYamada, and the equation automaton which can be computed via Mirkin’s prebases or Antimirov’s partial deriva ..."
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Two classical nondeterministic automata recognize the language denoted by a regular expression: the position automaton which deduces from the position sets defined by Glushkov and McNaughtonYamada, and the equation automaton which can be computed via Mirkin’s prebases or Antimirov’s partial derivatives. Let E  be the size of the expression and �E � be its alphabetic width, i.e. the number of symbol occurrences. The number of states in the equation automaton is less than or equal to the number of states in the position automaton, which is equal to �E�+1. On the other hand, the worstcase time complexity of Antimirov algorithm is O(�E � 3 · E  2), while it is only O(�E � · E) for the most efficient implementations yielding the position automaton (BrüggemannKlein, Chang and Paige, Champarnaud et al.). We present an O(E  2) space and time algorithm to compute the equation automaton. It is based on the notion of canonical derivative which makes it possible to efficiently handle sets of word derivatives. By the way, canonical derivatives also lead to a new O(E  2) space and time algorithm to construct the position automaton. 1
On the average state complexity of partial derivative automata
 International Journal of Foundations of Computer Science
, 2011
"... The partial derivative automaton (Apd) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (Apos). By estimating the number of regular expressions that have ε as a partial derivative, we comp ..."
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Cited by 3 (3 self)
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The partial derivative automaton (Apd) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (Apos). By estimating the number of regular expressions that have ε as a partial derivative, we compute a lower bound of the average number of mergings of states in Apos and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k’s its limit approaches half the number of states in Apos. The lower bound corresponds to consider the Apd automaton for the marked version ofthe regularexpression, i.e.where allitsletters are made different. Experimental results suggest that the average number of states of this automaton, and of the Apd automaton for the unmarked regular expression, are very close to each other.
Derivatives of rational expressions and related theorems
 T.C.S
, 2004
"... Our aim is to study the set of Krational expressions describing rational series. More precisely we are concerned with the definition of quotients of this set by coarser and coarser congruences which lead to an extension – in the case of multiplicities – of some classical results stated in the Boole ..."
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Our aim is to study the set of Krational expressions describing rational series. More precisely we are concerned with the definition of quotients of this set by coarser and coarser congruences which lead to an extension – in the case of multiplicities – of some classical results stated in the Boolean case. In particular, multiplicity analogues of the well known theorems of Brzozowski and Antimirov are provided.
Inside Vaucanson
 In Proceedings of Implementation and Application of Automata, 10th International Conference (CIAA), Sophia Antipolis
, 2005
"... Abstract. This paper presents some features of the Vaucanson platform. We describe some original algorithms on weighted automata and transducers (computation of the quotient, conversion of a regular expression into a weighted automaton, and composition). We explain how complex declarations due to th ..."
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Abstract. This paper presents some features of the Vaucanson platform. We describe some original algorithms on weighted automata and transducers (computation of the quotient, conversion of a regular expression into a weighted automaton, and composition). We explain how complex declarations due to the generic programming are masked from the user and finally we present a proposal for an XML format that allows implicit descriptions for simple types of automata. 1
Description and analysis of a bottomup DFA minimization algorithm
, 2008
"... Abstract. We establish lineartime reductions between the minimization of a deterministic finite automaton (DFA) and the conjunction of 3 subproblems: the minimization of a strongly connected DFA, the isomorphism problem for a set of strongly connected minimized DFAs, and the minimization of a conne ..."
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Abstract. We establish lineartime reductions between the minimization of a deterministic finite automaton (DFA) and the conjunction of 3 subproblems: the minimization of a strongly connected DFA, the isomorphism problem for a set of strongly connected minimized DFAs, and the minimization of a connected DFA consisting in two strongly connected components, both of which are minimized. We apply this procedure to minimize, in linear time, automata whose nontrivial strongly connected components are cycles. 1.
Combinatorics Approach
, 2010
"... The partial derivative automaton (Apd) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (Apos). By estimating the number of regular expressions that have ε as a partial derivative, we com ..."
Abstract
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The partial derivative automaton (Apd) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (Apos). By estimating the number of regular expressions that have ε as a partial derivative, we compute a lower bound of the average number of mergings of states in Apos and describe its asymptotic behaviour. This depends on the alphabet size, k, and its limit, as k goes to infinity, is 1. The lower bound corresponds 2 exactly to consider the Apd automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the Apd automaton for the unmarked regular expression, are very close to each other. 1
Small NFAs from Regular Expressions: Some Experimental Results ⋆
"... Abstract. Regular expressions (REs), because of their succinctness and clear syntax, are the common choice to represent regular languages. However, efficient pattern matching or word recognition depend on the size of the equivalent nondeterministic finite automata (NFA). We present the implementatio ..."
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Abstract. Regular expressions (REs), because of their succinctness and clear syntax, are the common choice to represent regular languages. However, efficient pattern matching or word recognition depend on the size of the equivalent nondeterministic finite automata (NFA). We present the implementation of several algorithms for constructing small εfree NFAs from REs within the FAdo system, and a comparison of regular expression measures and NFA sizes based on experimental results obtained from uniform random generated REs. For this analysis, nonredundant REs and reduced REs in star normal form were considered. 1
On the Number of Broken . . .
, 2009
"... Bounds are given on the number of broken derived terms (a variant of Antimirov’s ’partial derivatives’) of a rational expression E. It is shown that this number is less than or equal to 2l(E)+1 in the general case, where l(E) is the literal length of the expression E, and that the classical bound l ..."
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Bounds are given on the number of broken derived terms (a variant of Antimirov’s ’partial derivatives’) of a rational expression E. It is shown that this number is less than or equal to 2l(E)+1 in the general case, where l(E) is the literal length of the expression E, and that the classical bound l(E) + 1 which holds for partial derivatives also holds for broken derived terms if E is in star normal form.
The Average Transition Complexity of Glushkov and Partial Derivative Automata ⋆
"... Abstract. In this paper, the relation between the Glushkov automaton (Apos) and the partial derivative automaton (Apd) of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of Apos was proved by Nicaud to be linear in the size of the correspo ..."
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Abstract. In this paper, the relation between the Glushkov automaton (Apos) and the partial derivative automaton (Apd) of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of Apos was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of Apos. Here we present a new quadratic construction of Apos that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2. Broda et al. computed an upper bound for the ratio of the number of states of Apd to the number of states of Apos, which is about 1 2 for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in Apd, which we then use to get an average case approximation. Some experimental results are presented that illustrate the quality of our estimate. 1