Solving a Finite Automaton Problem: A Troublesome Hint

[0rthodontist]

There is a problem on the latest homework that I am struggling with. From the way the problem is worded, I am pretty sure that the key to solving this problem is the same as the hint on a problem on the last homework. However when I did that other problem, I ignored the hint, and now I am having difficulty figuring this one out. Because I don't want to do anything that might be regarded as illicit, I'll only give the other problem:

The other problem is:
Present and justify an algorithm that decides whether a finite automaton M = (Q, sigma, delta, q0, F) recognizes the language sigma*, in time O(|Q| x |sigma|). Hint: Read the section of the notes that proves the regular languages are closed under the Boolean operations

The way I solved this was by disregarding the hint and simply searching M for any nonaccepting states reachable from q0, and I got full credit. But this current question I am working on seems to depend on this one. Does anyone have an idea about what the hint means?

 

[mighty2000]

Hi there,

It sounds like you are on the right track with recognizing that the key to solving this problem may be related to the hint given in the previous homework. It's always important to read and understand the hints given by your instructor, as they can often provide valuable insights and shortcuts to solving a problem.

In this case, the hint is directing you to the concept of regular languages and their closure properties under Boolean operations. This means that you can use the properties of regular languages, such as concatenation, union, and intersection, to build a new regular language from existing ones.

In order to solve this problem, you will need to use the closure properties to construct a regular language that recognizes sigma*. This can be done by combining the language sigma itself with the empty string, which is also a regular language. Then, you can use the algorithm you mentioned from the previous problem to determine if the finite automaton M recognizes this new language.

I hope this helps clarify the hint and gives you some ideas on how to approach the problem. Remember, it's always important to thoroughly understand the concepts and hints provided in your coursework, as they can lead to more efficient and accurate problem solving. Good luck!