CS 461 Sept 2 Review NFA DFA Combining

CS 461 – Sept. 2 • Review NFA DFA • Combining FAs to create new languages – union, intersection, concatenation, star – We can basically understand how these work by drawing a picture. • Section 1. 3 – Regular expressions – A compact way to define a regular set, rather than drawing an FA or writing transition table.

Example #2 • NFA transition table given to the right. • DFA start state is A. • DFA accept state would be anything containing D. State A B C D inputs 0 1 A A, C ε - D B C - B D

continued Let’s begin. δ(A, 0) = A δ(A, 1) = AC We need new state AC. δ(AC, 0) = A δ(AC, 1) = ABC Continue from ABC… State A B C D inputs 0 1 A A, C ε - D B C - B D

answer State A B C D NFA inputs 0 1 A A, C ε - D B C - B D DFA inputs State A 0 A 1 AC AC A ABC AD ABC ACD ABCD ABCD

Moral • NFAs and DFAs have same computational power. • NFAs often have fewer states than corresponding DFA. • Typically, we want to design a DFA, but NFAs are good for combining 2+ DFAs. • After doing NFA DFA construction, we may see that some states can be combined. – Later in chapter, we’ll see how to simplify FAs.

U and ∩ • Suppose M 1 and M 2 are DFAs. We want to combine their languages. • Union: We create new start state. √ – Do you understand formalism p. 60 ? • How can we also do intersection? Hint: A ∩ B = (A’ U B’)’

Concatenation • Concat: For each happy state in M 1, turn it into a reject state and add ε-trans to M 2’s start. • Example L 1 = { does not contain 00 } L 2 = { has even # of 1’s } Let’s draw NFA for L 1 L 2. • Let’s decipher formal definition of δ on p. 61.

Star • We want to concat the language with itself 0+ times. • Create new start state, and make it happy. • Add ε-transitions from other happy states to the start state. • Example L = { begins with 1 and ends with 0 } Let’s draw NFA for L*. • Formal definition of δ on p. 63.

Regular expression • A concise way to describe a language – Text representation, straightforward to input into computer programs. • Use alphabet symbols along with operators + means “or” * means repetition Concatenation • There is no “and” or complement.

Examples • What do you think these regular expressions mean? 0* + 1* 0*1* 00*11* (a shorthand would be 0+1+ ) (0 + 1)* • What’s the difference between 10*1 and 1(0+1)*1 ? Does this mean “anything with 1 at beginning and end? ”

Practice • Words with 2 or more 0’s. What’s wrong with this answer: 1*01*01 ? • Words containing substring 110. • Every even numbered symbol is 0. – What’s wrong with: ((0 + 1)*0)* ? • Words of even length. • The last 2 symbols are the same. • What is the shortest word not in: 1*(01)*0* ? • True or false: (111*) = (11 + 111)*