LEMMA 1.60If a language is regular, then it is deseribed by a regular expression.PROOF IDEA We need to show that if a language A is regular, a regularexpression describes it. Because A is regular, it is accepted by a DFA. We describea procedure for converting DFAS into equivalent regular expressions.We break this procedure into two parts, using a new type of finite automatoncalled a generalized nondeterministic finite automaton, GNFA. First we showhow to convert DFAS into GNFAS, and then GNFAS into regular expressions.Generalized nondeterministic finite automata are simply nondeterministic fi-nite automata wherein the transition arrows may have any regular expressions aslabels, instead of only members of the alphabet or e. The GNFA reads blocks ofsymbols from the input, not necessarily just one symbol at a time as in an ordi-nary NFA. The GNFA moves along a transition arrow connecting two states byreading a block of symbols from the input, which themselves constitute a stringdescribed by the regular expression on that arrow. A GNFA is nondeterministicand so may have several different ways to process the same input string. It ac-cepts its input if its processing can cause the GNFA to be in an accept state at theend of the input. The following figure presents an example of a GNFA.Fstartab U baa)*Tacceptab Use the procedure described in Lemma 1.60 to convert the following finite au-tomata to regular expressions.a,b2b3(a)(b)

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Use the procedure described in Lemma 1600 to convert the following finite au- tomata to regular expressions.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

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Question

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LEMMA 1.60
If a language is regular, then it is deseribed by a regular expression.
PROOF IDEA We need to show that if a language A is regular, a regular
expression describes it. Because A is regular, it is accepted by a DFA. We describe
a procedure for converting DFAS into equivalent regular expressions.
We break this procedure into two parts, using a new type of finite automaton
called a generalized nondeterministic finite automaton, GNFA. First we show
how to convert DFAS into GNFAS, and then GNFAS into regular expressions.
Generalized nondeterministic finite automata are simply nondeterministic fi-
nite automata wherein the transition arrows may have any regular expressions as
labels, instead of only members of the alphabet or e. The GNFA reads blocks of
symbols from the input, not necessarily just one symbol at a time as in an ordi-
nary NFA. The GNFA moves along a transition arrow connecting two states by
reading a block of symbols from the input, which themselves constitute a string
described by the regular expression on that arrow. A GNFA is nondeterministic
and so may have several different ways to process the same input string. It ac-
cepts its input if its processing can cause the GNFA to be in an accept state at the
end of the input. The following figure presents an example of a GNFA.
Fstart
ab U ba
a)*
Taccept
ab

Use the procedure described in Lemma 1.60 to convert the following finite au-
tomata to regular expressions.
a,b
2
b
3
(a)
(b)

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