Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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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
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Transcribed Image Text: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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Transcribed Image Text: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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