D On input M2

1 Ron H on CM CM

2 Output the opposite

EE

air n

z r on cm

3

M w n 7

M Am Etten

LM w C Aim True

Eton my Em CM M EQm

total score lQ3Tg X 25 t Q 3 t XC

Rec Enumerator

Enumerator Reclong L

Rec on input

LBA

ALBA LM w M is an LBAand WE 4M

is decidable

ELBA MT M is an LBAand Lcm p

ALBA E ELBA

Elsa is dec

ATM E ELBA

M w C AtmM E ELBA

Accepting ComputationHistories

cACH

Co C Cm

fritter TTtransitioncontainsstart

config 9accept

Co Cm this is an

ACH for Mon w

Check

1 Co 90W

Esto's iii Eisasietetiathey must be the sane except aroundwhere tyre head is look in M's transition fr

Call this LBA Bif ELBA is decidable ther ca build B based

off of M W

if UB 70 say accept

if LCB D reject

remove write ontepe

only allow moving right

Deterministic Finite Automaton DFA

6291hi

Ex WE o B't oCw is odd and

w is ever

9euer.eu 9odd e qeYXe

Itf if f 90 e

9even.co 79oddoddOq Oe

A DEA is a 5 tuple Q Ei 8 go F

were fQ finite set of states

inputalphabet

9oEQ start state

F E Q set of final steles

8 Q XE Q is a hotelfunction

total functionhas a defined Value

every pair in E Eo is

I

W Wz Wr

A compilation for a DFA M

on input w w WB is a

FEI sequence off'stelesr rn

pwhere

ftp.ngcoop1 ro 9o if rn EF2 ri 8K i Wi

7The long recognized by DFA M

is WE M has on acceptingcomputation on w

Lang L is regulf if some DEA recognizes it

APFA M w M is a DEAand accepts w3

is decidable b c we can just simulate

Mon w 2 check if resulting slate C For not

F DFAM M is a DEA

Lcm 03is decidable

treat DFA as a directed graph

check if any path from 90to any state in F

If so rejectIf not accept

Regular Operations

Thu Regular tangs closed under complement

L reg I reg

Proof for any DFA for L swap final 2non final states

Thu Regular langs closed under oxenintersection

proof let 4 Lz be 1995gym diff

let D Q 81 F be DFAsXprodjoystwdion Dz Qa Ei 82 E

Q n Q2 70Make a DFA D Q I Io F

Q Q XQz 91,9 9 EQ q EQ90 19,0

skin 67 81 6 Sala 4

q q e Q a'EF a EE

F X Q U IQ Fzor but not

bothFixFzF xQz U Q xFz LIKE

o a

i

Piazza is this the smallest DFA

get Atm

ht F k 5

A B dee AUB des

A UB dec A B dee

let A AtmB Atm

E 0

E

E

E3L Ee3

B dec t AE B AdeeEton

Wz

Atm E ETM Atm E ELBA

ETm E F Qt m

f E o B

seatUMD Uma but fu f MD

Eton cm i unto

un IN I iE

i.Jf rejects everything Mi E ETm

Mz accepts Ma Eton

we

Aim E P

CM w

i coins tf a f I act

am

H vivzvz e.vn eiez emH