INHERENT LIMITATIONS OF COMPUTER PROGRAMS

CSci 4011

THE REGULAR OPERATIONS

Union: A B = { w | w A or w B }

Intersection: A B = { w | w A and w B }

Negation: A = { w | w A }

Reverse: AR = { w1 …wk | wk …w1 A }

Concatenation: A B = { vw | v A and w B }

Star: A* = { w1 …wk | k ≥ 0 and each wi A }

1 0

1

0 1

0,1

0

NON-DETERMINISM1 0

1

0 1

0,1

0

An NFA accepts if there is a series of choices that take it to an accept state

DeterministicComputation

Non-DeterministicComputation

accept or reject accept

reject

Theorem: Every NFA has an equivalent DFA

Corollary: A language is regular iff it is recognized by an NFA

FROM NFA TO DFAInput: N = (Q, Σ, , Q0, F)

Output: M = (Q, Σ, , q0, F)

accept

reject

To learn if NFA accepts, we could do the computation in parallel, maintaining the

set of states where all threads are

Q = 𝒫(Q)

Idea:

(R,) = ε( (r,) )

Q = 𝒫(Q)

: Q Σ → Q

(R,) = ε( (r,) )rR

q0 = ε(Q0)

F = { R Q | f R for some f F }

FROM NFA TO DFAInput: N = (Q, Σ, , Q0, F)

Output: M = (Q, Σ, , q0, F)

(R,) = ε( (r,) )

EXAMPLES: NFA TO DFA

1

0,1

ε 02 31

a b

1

REGULAR LANGUAGES CLOSED UNDER CONCATENATION

Given DFAs M1 and M2, construct NFA by

connecting all accept states in M1 to start

states in M2

ε

ε

L(M1)=A L(M2)=B

REGULAR LANGUAGES ARE CLOSED UNDER REGULAR OPERATIONS

Union: A B = { w | w A or w B }

Intersection: A B = { w | w A and w B }

Negation: A = { w | w A }

Reverse: AR = { w1 …wk | wk …w1 A }

Concatenation: A B = { vw | v A and w B }

Star: A* = { w1 …wk | k ≥ 0 and each wi A }

REGULAR LANGUAGES CLOSED UNDER STAR

Let L be a regular language and M be a DFA for L

We construct an NFA N that recognizes L*

00,1

00

1

1

1

ε

ε

ε

Formally:

Input: M = (Q, Σ, , q1, F)

Output: N = (Q, Σ, , {q0}, F)

Q = Q {q0}

F = F {q0}

(q,a) =

{(q,a)}

{q1}

{q1}

if q Q and a ≠ ε

if q F and a = ε

if q = q0 and a = ε

if q = q0 and a ≠ ε

else

L(N) = L*Assume w = w1…wk is in L*, where w1,…,wk L

We show N accepts w by induction on k

Base Cases:

k = 0

k = 1

Inductive Step:

Assume N accepts all strings v = v1…vk L*, and let u = u1…ukuk+1 L*

Since N accepts u1…uk and M accepts uk+1, N must accept u

Assume w is accepted by N, we show w L*

If w = ε, then w L*

If w ≠ ε

accept

ε

ε

REGULAR LANGUAGES ARE CLOSED UNDER REGULAR OPERATIONS

Union: A B = { w | w A or w B }

Intersection: A B = { w | w A and w B }

Negation: A = { w | w A }

Reverse: AR = { w1 …wk | wk …w1 A }

Concatenation: A B = { vw | v A and w B }

Star: A* = { w1 …wk | k ≥ 0 and each wi A }

REGULAR EXPRESSIONS

REGULAR EXPRESSIONS

s is a regular expression representing {}

ε is a regular expression representing {ε}

is a regular expression representing

If R1 and R2 are regular expressions representing L1 and L2 then:

(R1R2) represents L1L2

(R1 R2) represents L1 L2

(R1)* represents L1*

PRECEDENCE

*

R2R1*(

EXAMPLE

R1*R2 R3 = ( ) ) R3

{ w | w has exactly a single 1 }

0*10*

{ w | w has length ≥ 3 and its 3rd symbol is 0 }

000(01)* 010(01)* 100(01)* 110(01)*

{ w | every odd position of w is a 1 }