Construct a dfa and nfa for a set of strings on{0,1} : the left most symbol differs from the right most one and odd length string

Construct dfa and nfa for a set of string 0,1 such that left most and right most element are different and odd length string
Can you plss draw it

How can I prove that the accepted language of a given DFA or NFA or REGEX is equivalent to a given language

I found this How do I verify that a DFA is equivalent to a NFA? but as it states it is not really a good question more of how can I check myself during an exam. Because as you might know to do this by hand for < 5 states is really hard to do without any mistake and I only get credit for a correct construction and not understanding that REGΣ = NEAΣ = DEAΣ

finite automata – NFA that ends with 0 and doesn’t have 11 after the first 0

So as the title says i m trying to find this NFA. So far i thought to make an NFA that “guesses” what comes after the first 0 and i got this: enter image description here

After some time trying to get rid of all the extra (7 !) states i put in i got to this beauty: enter image description here

But now i cant figure out how to get rid of the one extra state. Also if you have any tips on how to go to simpler NFA’s instead of what i did (like a thought process for this problem) i would appreciate it a lot. Thank you

finite automata – Is there a set of formal languages ​​such that any DFA for him has states $ Omega (c ^ k) $ and there is an NFA for him with states $ O (k) $?

Given an alphabet $ Sigma: | Sigma | = c $, can a set of languages $ {L_k } $ be created, so any DFA for $ L_k $ at $ Omega (c ^ k) $ States and an NFA for $ L_k $ exists with $ O (k) $ States?

I'm having trouble creating a $ L_k $ so that all DFA $ Omega (c ^ k) $ States. Is a string language with a suffix of $ s_k, | s_k | = k $ such a language? Here is a draft proof of that.

Proof by contradiction: leave a DFA $ langle Q, Sigma, delta, q_0, F rangle $ to have $ | Q | <c ^ {k-1} $. Let $ a, b $ to be chains of length $ k $ and $ a_k = (s_k) _1 not = b_k $

Let $ q_a $ and $ q_b $ denote $ delta (q_0, a) $ and $ delta (q_0, b) $, respectively.

There are two cases:

There is no $ a, b $ such as $ q_a = q_b $. Then each string corresponds to a different state, but there is $ c ^ {k-1} $ these chains so $ | Q | geq c ^ {k-1} $, which is not possible.

II. There is $ a, b $ such as $ q_a = q_b $. so $ delta (q_a, s_2s_3 ldots s_k) = delta (q_b, s_2s_3 ldots s_k) = q_c $. $ as_2s_3 ldots s_k $ must be accepted and $ bs_2s_3 ldots s_k $ therefore should not $ q_c $ is both an accepting state and not an accepting state, which is not possible.

This seems to prove that any DFA for $ L_k $ at least $ c ^ {k-1} $ knots, which is enough for $ Omega (c ^ k) $. If my proof is correct, the only remaining task is to prove that an NFA containing $ O (k) $ nodes exist for $ L_k $.

The easiest way to do it is to create such an NFA, but I don't know how to do it. $ O (k) $ suggests that $ i $-th node should match the state of the "prefix of $ s $ length $ i $ matches the suffix of the input string ", but I don't see how such an NFA can be created.

algorithms – McNaughton-Yamada Algotihm (1960) Regular expression to DFA without going through NFA

I have a homework problem here. It asks me to use the McNaughton-Yamada algorithm to convert a regular expression to DFA first, and then minimize this using a partition argument. I can do the latter. My problem is that I cannot access any real references on the algorithm. Their original document is behind a payment wall at the IEEE which my university does not have access to.

The algorithm went something like this:
1. For each symbol in the expression, give them a left-to-right index increasing by one for each occurrence of that symbol. For example, the expression, aa * would receive a_1 a_2 ^ *.

  1. We are building a diagram based on the possible word lengths.

If done correctly, it produces a DFA. I think labeling in (1) is to help label states.

Feel free to give your own example if you decide to give an answer. I will not provide any problem here as there is no guarantee that this is not really my homework exercise.

Why this NFA with an empty symbol

I'm trying to understand the difference between DFA and NFA
I have this example in my course book
enter description of image here
Are empty symbols at the start necessary?
or can we eliminate them to have something like that
enter description of image here

finite automata – Converting an NFA with 2 initial states into a regular expression using state elimination

The task is to determine the language accepted by the following NFA:enter description of image here

My approach to these problems is to convert the NFA into a regular expression and (for the algorithm):

If the initial state has incoming transitions: create a new starting state with an epsilon transition

If there are more than 1 final states OR the final state has outgoing edges: create a new final state with 1 (or more) epsilon transition (s).

In the end, only 2 states remain.

I can't find any solution related to the approach mentioned above in the case of 2 starting states. Any contribution on this situation would be useful.

Thank you!

Convert NFA to RE

I am fighting with these two. My final answers are

1. a*((a U b)b)*b

2.a*aab*(a U b)

NFA

What am i doing wrong?

Thank you
Rahman

automata – Help with converting NFA to regular expression

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automata – NFA DFA proof that for any normal language such as L in REG, this implies that s (L) IN REG

$$
varphi_ {1} (L) = left {w in Sigma ^ {*} | text {there is a} alpha in Sigma ^ {*} text {with} | alpha | = | w | text {and} alpha w in L right }
$$

$$ begin {array} {l} { text {Proof. We want to show that} varphi_ {1} (L) text {for all languages} L in operatorname {REG} text {is normal. Be there} L in operatorname {REG} text {random. Because} L text {is normal}} \ { text {there is a DFA} M text {with L} (M) = L text {We are building now} M text {a} lambda text {NFA} {} operatorname {with L} left (M ^ { prime} right) = varphi_ {1} (L). text {For this, we describe}} \ { text {the working method} M ^ { prime} text {formal the condition space of} M ^ { prime} text {is (of design) that of} M. text {The calculation}} \ { text {of} M ^ { prime} text {works (conceptually) as follows: instead of a stone at the beginning of the calculation , place}} q_ {0} text {,}} end {array}
$$

$$ begin {array} {l} { text {we place 3 stones (1 white, 1 red and 1 blue) on the states of} M. text {We place the blue}} \ { text { on} q_ {0} text {, and white on a step det. guessed} q_ {i} text {(both stones at the same condition). }} \ { text {The calculation of} M ^ { prime} text {at entry} w in Sigma ^ {*} text {functions (of design) as follows: <Describe here how}}} { text {the 3 stones per symbol read are moved (they can stay on a state). Define}} \ { text {extra when} M ^ { prime} text {a word is accepted? }} end {array} $$

Could someone help me with that? Searched for regular expressions DFA + NFA + Conversion, but still do not know how to solve this problem.