CMPS 3223 Theory of Computation

CMPS 3223Theory of Computation Automata, Computability, & Complexity by Elaine Rich ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Slides provided by author Slides edited for use by MSU Department of Computer Science – R. Halverson

Chapter 6 Regular Expressions LOTS of problems at end of chapter for you to practice!

Regular Languages Regular Language L Regular Expression Accepts Finite State Machine

What is a regular expression? • A method to describe a regular language. • Different from that of a FSM • Consists of set of symbols + a syntax • Symbols • Special symbols: ∅  U ( ) * + • Alphabet ∑: from which strings in language are made

Regular Expressions The regular expressions over an alphabet  are all and only the strings that can be obtained as follows: 1.  is a regular expression. 2.  is a regular expression. 3. Every element of  is a regular expression. 4. If  ,  are regular expressions, then so is . 5. If  ,  are regular expressions, then so is . 6. If  is a regular expression, then so is *. 7. If  is a regular expression, then so is +. 8. If  is a regular expression, then so is (). Know this definition!

Regular Expression Examples If  = {a, b}, the following are regular expressions:   a (ab)* abba b*(aabb)+ b

Regular Expressions Define Languages Define L, a semantic interpretation function for regular expressions: 1. L() = . 2. L() = {}. 3. L(c), where c = {c}. 4. L() = L() L(). 5. L() = L() L(). 6. L(*) = (L())*. 7. L(+) = L(*) = L() (L())*. If L() is equal to , then L(+) is also equal to . Otherwise L(+) is language formed by concatenating together one or more strings drawn from L(). 8. L(()) = L().

The Role of the Rules • Rules 1, 3, 4, 5 & 6 give the language its power to define sets. • Rule 8 has as its only role grouping other operators. • Rules 2 & 7 appear to add functionality to regular expression language, but don’t. 2.  is a regular expression. * =  this is like 50 = 1, 0 =  7.  is a regular expression, then so is +. + = *

Analyzing a Regular ExpressionA formal view – but not what we will do L((ab)*b) = L((ab)*) L(b) = (L((ab)))* L(b) = (L(a) L(b))* L(b) = ({a}  {b})* {b} = {a, b}* {b}.

Examples Convention dictates that omit the L( ) portion and use the expression to represent a language. Give a description. L(a*b*) = a*b* = {a}*{b}* L((a  b)*) = (a  b)* = {a,b}* L((a  b)*a*b*)=(a  b)* a*b*={a,b}*{a}*{b}* L((a  b)*abba(a  b)*) = (ab)*abba(ab)*) = {a,b}*abba{a,b}*

Give a Regular Expression L = {w {a, b}*: |w| is even}

Solution L = {w {a, b}*: |w| is even} (a b) (a  b))* OR (aa ab ba  bb)* Explain how this guarantees an even number of characters in each string that fits the pattern of the regular expression.

Give a regular expression L = {w {a, b}*: w contains an odd number of a’s}

Solution L = {w {a, b}*: w contains an odd number of a’s} b* (ab*ab*)* ab* b* ab* (ab*ab*)*

More Regular Expression Examples L ( (aa*)  ) = L ( (a)* ) = L = {w {a, b}*: there is no more than one b in w} L = {w {a, b}* : no two consecutive letters in w are the same}

Common Idioms What do these mean? () (ab)* (ab)+

Operator Precedence in Regular Expressions Regular Arithmetic Expressions Expressions Highest Kleene star exponentiation concatenation multiplication Lowest union addition a b* c d* x y2 + i j2

The Details Matter Explain the differences! These will be components of MANY of your regular expressions a* b*  (ab)*  (ab)* (ab)* a*b*

The Details Matter L1 = {w {a, b}* : every a is immediately followed a b} A regular expression for L1: A FSM for L1: L2 = {w {a, b}* : every a has a matching b somewhere} A regular expression for L2: A FSM for L2:

Power of a Methodology In this course… • We will make claims that 2 methodologies are equivalent. i.e. Have the same power. • We will also claim that one methodology is more powerful than another What does that mean? • Descriptive, Define

6.2 Kleene’sTheorem Finite state machines &regular expressions define the same class of languages. i.e. They are equivalent. i.e. They are equally powerful. To prove this, we must show: Theorem: Any language that can be defined with a regular expression can be accepted by some FSM and so is regular. Theorem: Every regular language (i.e., every language that can be accepted by some DFSM) can be defined with a regular expression.

For Every Regular Expression There is a Corresponding FSM • We’ll show this by construction. • That is, for each of the components in the definition of a Regular Expression (page 128), we will develop a corresponding finite state machine. • The result will not necessarily be deterministic • The methods in the proof are not necessarily unique

For Every Regular Expression There is a Corresponding FSM For the first 3 components: : A single element c of : = (*):

Union of 2 Regular Expressionsexp1 U exp2 M1for Expression 1  M2 for Expression 2  Do any other states need to change? Minimal? S1 S S2

Concatenation of 2 Regular Expressionsexp1exp2 M1for Expression 1  is this still F?  M2 for Expression 2 Do any other states need to change? Finals? S1 F S2

Kleene Star of Regular Expressionexp1* M1for Expression 1   Do any other states need to change? Finals?  S1 F SF

Kleene Star of Regular Expressionexp1* - alternate way M1for Expression 1   Do any other states need to change? Finals?  SF F

Example 1 (b ab)* An FSM for b An FSM for a An FSM for b An FSM for ab: Note: This Example 6.5 page 136 is in error in text.

Example 1 (b ab)* An FSM for (bab): Can we reduce it?

Example 1 (b ab)* An FSM for (bab)*: Reduce?? Do Homework starting page 151.

Simplifying Regular Expressions Regex’s describe sets: ● Union is commutative:  = . ● Union is associative: ()  =  (). ●  is the identity for union:  =  = . ● Union is idempotent:  = . Concatenation: ● Concatenation is associative: () = (). ●  is the identity for concatenation:  =  = . ●  is a zero for concatenation:  =  = . Concatenation distributes over union: ● ()  = ()  (). ●  () = ()  (). Kleene star: ● * = . ● * = . ●(*)* = *. ● ** = *. ●()* = (**)*.

Chapter 6 Homework • End of Chapter – Page 161 + • Try all of the problems – Really!

CMPS 3223 Theory of Computation