Help with DFA/NFA Homework | Solutions & Explanations

[opt!kal]

Hi there, I've been working on DFA's and NFA's in class, and for the most part I get it, but I want to make sure I am doing a couple of these right, and maybe you guys can give some insight on how to complete the ones that make no sense to me. With that, on to the first one!

Homework Statement

For this problem, we'll work over the alphabet [tex]\Sigma[/tex]= {a, b, c}. Design a DFA to recognize the language: L[tex]_{}1[/tex] = {w| if there is a b in w, then there is a c in w}

The Attempt at a Solution

I couldn't think of any other way of posting this, so I am just attaching an image (a pretty bad one at that, sorry!):

http://img149.imageshack.us/my.php?image=52818077vp2.jpg

Anyways, does that seem right? I didnt know what to do if b didnt exist since if b->c and b=false, we don't know what c could be right? Any help is greatly appreciated!

Im also having trouble with:
Using the same alphabet, design NFAs for each of the following languages:
L[tex]_{}4[/tex]= {w| w ends with bac}
L[tex]_{}5[/tex]= {w||w| mod 2 = 1 or w ends with a}

for the first one I got:
http://img144.imageshack.us/my.php?image=54291944wd1.jpg

But the second I have no idea.

Finally could someone tell me where to even start with:
Recall that we can interpret a string over {0,1} as the binary representation of an
integer starting with the least signi cant bit, so that for example 011 = 6.Using the alphabet {0,1}, prove that the language {x | x mod 3 = 0} is regular.

Any help is greatly appreciated!

 

[KataKoniK]

Hi there,

Great job on designing a DFA for the first language! Your solution looks correct to me. You are correct in thinking that if there is no b in the string, then we cannot determine whether there is a c or not. In this case, we can consider it as a separate state where the DFA stays in that state until it encounters a b or reaches the end of the string. This is called a "trap state" and is commonly used in DFA design.

For the second language, L_{}4, your NFA looks good as well. For L_{}5, we can design an NFA with two states, one for even length strings and one for odd length strings. The even length state has a self-loop for both a and b transitions, and transitions to the odd length state on a c. The odd length state has a self-loop for both a and b transitions, and transitions back to the even length state on a c. This ensures that the NFA accepts strings with odd length or strings that end with a.

For the last problem, we can use the concept of "modulo" to prove that the language is regular. Modulo is a mathematical operation that gives the remainder of a division. In this case, the language {x | x mod 3 = 0} means that the binary representation of the string has a remainder of 0 when divided by 3. We can design a DFA with three states, one for each possible remainder when divided by 3. The transitions will be based on the input bit (0 or 1) and the current remainder. For example, if we are in the state for remainder 0 and we receive a 1 as input, we transition to the state for remainder 1. We can also add a trap state for invalid inputs. This DFA will accept all strings that have a binary representation with a remainder of 0 when divided by 3, thus proving that the language is regular.

I hope this helps! Let me know if you have any other questions. Keep up the good work!

 

[Mene]

Hi there, it looks like you are making good progress on your DFA and NFA homework! Let's go through each problem one by one to make sure you understand the concepts and how to approach them.

For the first problem, you are correct in your approach. The DFA you have designed looks good and should recognize the language L_{}1 correctly. As for what to do if b doesn't exist, you can have a designated "reject" state in your DFA that the machine goes to if it encounters a b. This way, the machine will reject any input that does not contain both a b and a c.

For the second problem, designing NFAs for the given languages, you are on the right track with your solution for L_{}4. However, for L_{}5, you can use a similar approach and have a designated "reject" state for when the input does not meet the conditions (either the length is even or it does not end with an a). Your NFA should have a loop on the start state for the alphabet {a,b,c} to handle inputs of any length.

For the last problem, proving that the language {x | x mod 3 = 0} is regular, you can use the concept of a finite automaton to show that this language is indeed regular. Since the language consists of strings that represent numbers in binary, you can design a DFA that recognizes the binary representation of multiples of 3. This DFA will have states representing the remainder when dividing by 3, and will accept if the final state is 0 (representing a multiple of 3). I suggest drawing out the DFA and testing it with different inputs to gain a better understanding of how it works.

I hope this helps and good luck with your homework! Remember to always break down the problem into smaller parts and use the concepts you have learned in class to approach each part. Don't hesitate to ask for clarification or additional help if needed. Keep up the good work!

 

[Mene]

Hi there, it looks like you're making good progress with your DFA and NFA homework. I'll do my best to provide some insight and solutions for the problems you're having trouble with.

For the first problem, your DFA looks correct. You are correct in thinking that if there is no b in the string, then there is no requirement for a c to be present. This can be represented by having a loop back to the start state for the b input. Your DFA accounts for this by having a loop on the start state for the b input.

For the second problem, your NFA for L_{}4 looks good. For L_{}5, you can use a similar approach. Start with a loop on the start state for the a input, and then have a transition to a new state for the b input. From this new state, have a loop for the c input and a transition to an accepting state for the b input. This will account for strings that end with a and have an odd number of characters.

For the last problem, here are some steps to help you get started:

1. First, think about what the language {x | x mod 3 = 0} means. This language contains all strings where the decimal representation of the string is divisible by 3.

2. Next, think about how you can represent this in binary. For example, the number 6 in binary is 011. Can you see any patterns or rules that apply to binary numbers that are divisible by 3?

3. Once you have an idea of how to represent the language in binary, you can design an NFA that recognizes it. Remember, an NFA can have multiple transitions on the same input, so think about how you can use this to your advantage.

I hope this helps! If you have any other questions or need clarification, don't hesitate to ask. Good luck with your homework!