Abstract
In this paper, we extend the notion of (word) derivatives and partial derivatives due to (respectively) Brzozowski and Antimirov to tree derivatives using already known inductive formulae of quotients. We define a new family of extended regular tree expressions (using negation or intersection operators), and we show how to compute a Brzozowski-like inductive tree automaton; the fixed point of this construction, when it exists, is the derivative tree automaton. Such a deterministic tree automaton can be used to solve the membership test efficiently: the whole structure is not necessarily computed, and the derivative computations can be performed in parallel. We also show how to solve the membership test using our (Bottom-Up) partial derivatives, without computing an automaton.
Highlights
In 1956, Kleene [8] gave a fundamental theorem in automata theory
He showed that every regular expression E can be converted into a finite state machine that recognizes the same language as E, and vice versa
One of these approaches which appeared in 1964 was Brzozowski’s [3] construction; the idea is to use the notion of derivation to compute a deterministic automaton: the derivative of a regular expression E w.r.t. a word w is a regular expression that denotes the set of words w′ such that ww′ is denoted by E
Summary
In 1956, Kleene [8] gave a fundamental theorem in automata theory. He showed that every regular expression E can be converted into a finite state machine that recognizes the same language as E, and vice versa. A lot of methods have been proposed to provide the conversion of a given regular expression to a finite word automaton One of these approaches which appeared in 1964 was Brzozowski’s [3] construction; the idea is to use the notion of derivation to compute a deterministic automaton: the derivative of a regular expression E w.r.t. a word w is a regular expression that denotes the set of words w′ such that ww′ is denoted by E. Antimirov [1], in 1996 introduced the partial derivation which is a similar operation to the one defined by Brzozowski; a partial derivative of a regular expression is no longer a regular expression but a set of regular expressions, that leads to the construction of a non-deterministic automaton, with at most (n+1) states where n is the number of letters of the regular expression.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
