CMPS 3223Theory of

ComputationAutomata, Computability, & Complexity

by Elaine Rich~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Slides provided by authorSlides edited for use by MSU Department of

Computer Science – R. Halverson

1

Chapter 6

Regular ExpressionsLOTS of problems at end of chapter for

you to practice!

Regular Languages

Regular Language

Regular Expression

Finite State Machine

L

Accepts

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 LanguagesDefine 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.

The Role of the Rules

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)*) = (ab)*abba(ab)*)= {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 IdiomsWhat do these mean?

( )

(a b)*

(a b)+

Operator Precedence in Regular Expressions

Regular ArithmeticExpressions 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:

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

Power of a Methodology

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 deterministicThe 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?

Union of 2 Regular Expressionsexp1 U exp2

S1

S2

S

M1for Expression 1

is this still F?

M2 for Expression

2

Do any other states need to change? Finals?

Concatenation of 2 Regular Expressionsexp1exp2

S1

S2

F

M1for Expression 1

Do any other states need to change? Finals?

Kleene Star of Regular Expressionexp1*

S1SF F

M1for Expression 1

Do any other states need to change? Finals?

Kleene Star of Regular Expressionexp1* - alternate way

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 ExpressionsRegex’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 161 +

Try all of the problems – Really!

Chapter 6 Homework