Automata, Computability, & Complexity by Elaine Rich ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Slides provided by author Slides edited for use by MSU Department of Computer Science – R. Halverson 1

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

Regular Languages Regular Language Regular Expression Finite State Machine L Accepts

 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  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(  ).

 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 5 0 = 1,  0 =  7.  is a regular expression, then so is  +.  + =  *

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 *)* a b * b * a b * ( 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 RegularArithmeticExpressions HighestKleene starexponentiation concatenationmultiplication Lowestunionaddition a b *  c d *x y 2 + i j 2

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 L 1 = {w  { a, b }* : every a is immediately followed a b } A regular expression for L 1 : A FSM for L 1 : L 2 = {w  { a, b }* : every a has a matching b somewhere} A regular expression for L 2 : A FSM for L 2 :

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’s Theorem 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  :  = (  *):

M1for Expression 1  M2 for Expression 2  Do any other states need to change? Minimal? S1 S2 S

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

M1for Expression 1  Do any other states need to change? Finals? S1 SF F 

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: ●  * = . ●  * = . ●(  *)* =  *. ●  *  * =  *. ●(    )* = (  *  *)*.

 End of Chapter – Page  Try all of the problems – Really!