Question

Please do everything by hand.

and please draw the diagram by hand
Introduction to Automata and
Computability
Let Σ = {0, 1} and let L be the language over Σconsisting of those strings that consistting of those strings that consist of an even number of
1s and arbitrarily many 0s.
Give a regular expression for L.
Give a transition diagram for a finite state machine, M, such that L = L(M)

Answer 1

The language L can be represented by the regular expression (0*1*01*0*)* and visualized through a finite state machine with an accepting state for an even number of 1s and transitions that toggle between even and odd upon input of a 1.

Step-by-step explanation:

The question involves designing a finite state machine (FSM) for a specific language L over the alphabet Σ = {0, 1} such that L includes all strings with an even number of 1s, and any number of 0s. A regular expression that represents this language is (0*1*01*0*)*. It essentially means that you can have any number of 0s, followed by a pair of 1s, this pattern can repeat any number of times.

The transition diagram for this FSM includes two states: One that represents having an even number of 1s (which would be an accepting state), and another that represents having an odd number of 1s. From the even state, a '0' input would loop back to the same state, and a '1' input would transition to the odd state. From the odd state, again a '0' input would loop back, and a '1' input would transition back to the even (and accepting) state.

Even State (Accepting) <------ 0 ------ Even State (Accepting)
| |
1 1
| |
V V
Odd State ------ 0 ------ Odd State