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Complexity and Computability Theory I

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Complexity and Computability Theory I

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Complexity and Computability Theory I

Lecture #6

Instructor: Rina Zviel-Girshin

Lea Epstein

- Regular expressions
- Examples
- Properties of regular languages
- Transforming regular expressions into FA
- Transforming FA into regular expressions
- Examples

Rina Zviel-Girshin @ASC

- Regular languages are often described by means of algebraic expressions called regular expressions.
- In arithmetic we use the +, * operations to construct expressions:
(2+3)*5

- The value of the arithmetic expression is the number 25.

Rina Zviel-Girshin @ASC

- We can use regular operations to construct expressions describing regular languages:
(0+1)* 0*

where :

+ means OR

* Kleene star

concatenation

- The value of a regular expression is a regular language.

Rina Zviel-Girshin @ASC

- A set of regular expressions over an alphabet is defined inductively as follows:
Basis:

, , and (for all ) are regular expressions.

Induction:

If r and s are regular expressions then the following expressions are also regular:

(r) , r+s, r s and r*.

Rina Zviel-Girshin @ASC

, a, a+b, b*, (a+b)b, ab*, a*+b*

- To avoid using to many parentheses, we assume that the operations have the following priority hierarchy:
* - highest (do it first)

+ - lowest (do it last)

So:

(b+(a (b*)) = b + ab*

Rina Zviel-Girshin @ASC

L1 = { w | w has a single 1}

r1 = 0*10*

L2 = { w | w has at least one 1}

r2 = (0+1)*1 (0+1)*= *1*

L3 = { w | w contains the string 110}

r3 = (0+1)*110(0+1)* = *110*

Rina Zviel-Girshin @ASC

L4 = { w | |w| mod 2 =0}

r4 = ((0+1) (0+1))* = ()*

L5 = { w | w starts with 1}

r5 = 1(0+1)* = 1*

L6 = { w | w ends with 00}

r6 = (0+1)*00 = *00

Rina Zviel-Girshin @ASC

L7 = { w | w starts with 0 and ends with 10}

r7 = 0(0+1)*10 = 0*10

L8 = { w | w contains the string 010 or the string 101 }

r8 = (0+1)* 010 (0+1)* + (0+1)* 101 (0+1)*=

(0+1)*(010+101)(0+1)* = *(010+101)*

L9 = { w | w starts and ends with the same letter }

r9 = 0(0+1)*0 + 1(0+1)*1 + 0 +1 = 0*0 + 1*1 + 0 +1

Rina Zviel-Girshin @ASC

We associate each regular expression r with a regular language L(r) as follows:

- L()=,
- L()={},
- L()={} for each ,
- L(r+s)=L(r)L(s),
- L(r s)=L(r) L(s),
- L(r*)=(L(r))*.

Rina Zviel-Girshin @ASC

Some useful properties of regular expressions

r+s=s+r

r+=+r

r+r=r

r=r=

rr*=r+

r=r=r

Rina Zviel-Girshin @ASC

r(s+t)=rs+rt

r+(s+t)=(r+s)+t

r(st)=(rs)t

r+r*=(r*)*=r*r*=r*

r*+r+=r*

Rina Zviel-Girshin @ASC

- To prove that two regular expressions r and s are equivalent we have to prove that
L[r]L[s] and L[s]L[r].

- To show that two regular expressions are not equivalent we have to find a word that belongs to one expression and does not belong to the other.

Rina Zviel-Girshin @ASC

Are r and s equivalent?

r=+(0+1)*1

s=(0*1)*

Rina Zviel-Girshin @ASC

- Let wL[s] =(0*1)*.
- w= or w=x1x2..xn , n0 such that xiL[0*1], i=0,..n.
- If w= then wL[r].
- If w=x1x2..xn than we can represent w=w’1=x1x2..xn-100001 with zero or more 0.
But w’= x1x2..xn-10000L[(0+1)*].

So wL[r].

Rina Zviel-Girshin @ASC

- Let wL[r]=+(0+1)*1.
- If w= then wL[s] (by definition of *).
- If w then can be represented as w=w’1 where w’L[(0+1)*]. Assume that in w’ contains k instances of the letter 1. That means that w’ can be represented as w’= x11x21.. 1xk+1 where xi0*
But then w=w’1= (x11)(x21).. 1)(xk+1 1)

So wL[(0*1)*].

Rina Zviel-Girshin @ASC

Are r and s equivalent?

r=(0+1)*1+0*

s=(1+0)(0*1)*

No.

- We choose a word w = .
- wL[r]=(0+1)*1+0*, because w0*.
- But wL[s] =(1+0)(0*1)*, as all words in L[s] have at least one letter.

Rina Zviel-Girshin @ASC

- Regular expressions and finite automata are equivalent in terms of the languages they describe.
Theorem:

A language is regular if”f some regular expression describes it.

This theorem has two directions. We prove each direction separately.

Rina Zviel-Girshin @ASC

- If a language is described by a regular expression, then it is regular.
Proof idea:

Transformation of some regular expression r into a finite automaton N that accepts the same language L(r).

Rina Zviel-Girshin @ASC

- Given r we start the algorithm with N having a start state, a single accepting state and an edge labeled r:

Rina Zviel-Girshin @ASC

Now transform this machine into an NFA N by applying the following rules until all the edges are labeled with either a letter from or :

1.If an edge is labeled , then delete the edge.

Rina Zviel-Girshin @ASC

2.Transform any diagram of the type

into the diagram

Rina Zviel-Girshin @ASC

3.Transform any diagram of the type

into the diagram

Rina Zviel-Girshin @ASC

4.Transform any diagram of the type

into the diagram

Rina Zviel-Girshin @ASC

Construct an NFA for the regular expression b*+ ab.

We start by drawing the diagram:

Rina Zviel-Girshin @ASC

Next we apply rule 2 for b*+ab:

Rina Zviel-Girshin @ASC

Next we apply rule 3 for ab:

Rina Zviel-Girshin @ASC

Next we apply

rule 4 for b*:

Rina Zviel-Girshin @ASC

- If a language is regular , then it is described by a regular expression.
Proof idea:

transformation of some finite automaton N into a regular expression r that represents the regular language accepted by the finite automaton L(N).

Rina Zviel-Girshin @ASC

- The algorithm will perform a sequence of transformations into new machines with edges labeled with regular expressions.
- It stops when the machine has :
- two states:
- start
- finish

- one edge with regular expression on it.

- two states:

Rina Zviel-Girshin @ASC

Assume we have a DFA or an NFA N=(Q, , , q0, F).

Perform the following steps.

1. Create a new start state s and draw a new edge labeled with from s to the q0. (s,)= q0 .

Rina Zviel-Girshin @ASC

2. Create a new accepting state f and draw new edges labeled with from all the original accepting states to f. For each qF (q,)= f .

Rina Zviel-Girshin @ASC

3. For each pair of states i and j that have more than one edge from i to j, replace all the edges from i to j by a single edge labeled with the regular expression formed by the sum of the labels of all edges from i to j.

Rina Zviel-Girshin @ASC

4. Construct a sequence of new machines by eliminating one state at a time until the only two states remaining are s and f.

As each state is eliminated a new machine is constructed in the following way:

Rina Zviel-Girshin @ASC

Eliminating state k

- Let old(i,j) denote the label on edge <i,j> of the current machine.
- If there is no an edge <i,j> then old(i,j)=. Now for each pair of edges <i,k> and <k,j>, where ik and jk, calculate a new edge labeled new(i,j) as follows:
new(i,j)=old(i,j) + old(i,k)old(k,k)*old(k,j)

Rina Zviel-Girshin @ASC

new(i,j)=old(i,j) + old(i,k)old(k,k)*old(k,j)

Rina Zviel-Girshin @ASC

- For all other edges <i,j> where ik and jk,
new(i,j)=old(i,j).

- The states of the new machine are those of the current machine with state k eliminated. The edges of the new machine are those edges <i,j> for which new(i,j) has been calculated.

Rina Zviel-Girshin @ASC

- Now s and f are two remaining states. Regular expression new(s,f) represents the language of the original automaton.

Rina Zviel-Girshin @ASC

- Transform into regular expression the following DFA

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The result of first three steps of the algorithm:

- new start state
- new final state
- unify multiple edges

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Eliminate state 2

- No paths pass through 2. There are no states that connect through 2. So no need to change anything after deletion of state 2.
new(i,j)=old(i,j) for all i,j.

Rina Zviel-Girshin @ASC

Eliminate state 0

- The only path through it is s q1.
- We delete q0 and edges sq0 and q0q1.
- Instead we add an edge that is labeled by regular expression associated with the deleted edges.
new(s,q1)=old(s,q1)+old(s,q0)old(q0,q0)*old(q0,q1)=+*a=a

Rina Zviel-Girshin @ASC

Eliminate state q1

- The only path through it is s f.
new(s,f)=old(s,f) + old(s,q1)old(q1,q1)*old(q1,f) =+a(a+b)* = a(a+b)*

Rina Zviel-Girshin @ASC

Any Questions?

Rina Zviel-Girshin @ASC