6 NFA and Regular Expressions
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1 Formal Language and Automata Theory: CS NFA and Regular Expressions 6.1 Nondeterministic Finite Automata A nondeterministic finite automata (NFA) is a 5-tuple where 1. is a finite set of states 2. is a finite alphabet 3. is the transition function where "!#$&% and ' is the power set of 4. ( is the start state and 5. *)+ is the set of accept or final states Example Consider a language -#/.0(1#2' 3%5467. has two 2 s separated by a string of length 89 9;:=< 9;( %?> Construct an NFA that accepts the language. The state transition diagram of the NFA is shown in Figure 1. Figure 1: A NFA
2 Page 2 of 6 NFA and Regular Expressions 6.2 Definition Now we formally define a language accepted by an NFA. Let.0 (" 4 where is the input alphabet. The string. is accepted by an NFA such that if there is a sequence of states & ( for all < 9 (. The language over decided by the NFA -#/.( 4 7. is accepted by % Example Construct an NFA accepting the language #/.( #2 3% is a substring of. %. The state transition diagram of the NFA is shown in Figure 2. Figure 2: A NFA.. A string is accepted by an NFA if there exists a computation of on the input starting from the start state to a final state. An accepting run of the above machine on the string is shown in Figure Equivalence of NFA and DFA Deterministic and nondeterministic finite automata recognize the same class of languages. Two machines are equivalent if they recognize the same language. We show that every NFA can be converted to an equivalent DFA. First we consider the case when an NFA does not have any $ -transition. Let!6 be an NFA. The constructed equivalent DFA " $# #0 # # where %# '& Rajat SethiSubhra Mazumdar Dept. of Computer Science & Engg IIT Kharagpur India
3 Chapter 6: NFA and Regular Expressions Page 3 of 6 Figure 3: Accepting run of NFA # -#5 6% # -# )+ % % # %# = %# is defined as #1 We claim that ". where ( %# '&. The equivalent DFA corresponding to NFA of Example is as shown in Figure 4. Figure 4: Equivalent DFA 6.4 Epsilon Closure of a state The epsilon closure & of a state of an NFA is defined inductively as follows: Basis: (?. Rajat SethiSubhra Mazumdar Dept. of Computer Science & Engg IIT Kharagpur India
4 Page 4 of 6 NFA and Regular Expressions Induction: if(? and ( $ then (?. We define the epsilon closure of set of states as?. Now we are ready to extend the equivalent DFA construction for NFAs with $ -transitions. The equivalent DFA is %# '& # 6 # -# )+ % % # 6 %# = %# is defined as # %# '&. We claim that ". The $ -closure of the start state of an NFA is shown in Figure 5.! where is an element of Figure 5: Epsilon closure of a state The number of states in an NFA is often smaller than the number of states in the equivalent DFA. In the worst case there may be exponential increase in the number of states. Also in some cases NFAs are simpler than DFAs to design. Designed NFA can be algorithmically converted to equivalent DFA. 6.5 Regular Expressions Regular expressions are expressions representing a specific language like mathematical expressions that represent a specific value. Thus each regular expression over an alphabet stands for a particular language over. It is another description (equivalent we shall see) of a class of languages. 6.6 Definition Regular expressions over an alphabet are defined inductively as follows: Rajat SethiSubhra Mazumdar Dept. of Computer Science & Engg IIT Kharagpur India
5 Chapter 6: NFA and Regular Expressions Page 5 of 6 Basis: < and are regular expressions (we assume that < ( ) for all1(0 is a regular expression. Induction: If and are regular expressions then / / 4 and are regular expressions Example Following are a few examples of regular expressions: 2 3 (23 ) (2 3 ) (2 3 ) 4. Some of the parenthesis in a regular expression can be avoided by introducing proper associativity and precedence of operators. Both the binary operators are left associative. The precedence order of the operators is as follows:. Thus stands for and not for 2 =3 4. Let be the collection of all regular expressions over. Let!"#? %. Then ) 4. Since 4 is a countable set is also countable. 6.7 Valuation We have already mentioned that each regular expression stands for a language. We define the valuation function from the set of regular expressions to the collection of languages over the alphabet. The definition of which maps a regular expression to the language expressed by it is on the inductive structure of the expression. Basis: < -#$?% -# % where1(. Induction: Let and be the languages of regular expressions and. Then we have! Example Consider the regular expression the language for it is ! % 2 %!1 4!1 4 #2'%!1 #53%'% #2'%!# ' # ' 3 22' 3 22 Rajat SethiSubhra Mazumdar Dept. of Computer Science & Engg IIT Kharagpur India
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