Convert the following NFA into a DFA.

Convert the following NFA into a DFA. Implement the DFA using JFLAP and upload the DFA as a file named a2dfa.jff A3. Finite automata are useful components of computers and algorithms, thus it is important to be able to minimize the number of states of a given DFA. Convert the following DFA to a minimum-state equivalent DFA. Implement the minimized DFA using JFLAP and upload that DFA as a file named a3mindfa.jff A4. Consider the following regular grammar with start symbol S and terminals (x, y). S \rightarrow XS S \rightarrow yX S \rightarrow yY X \rightarrow XX X \rightarrow yS Y \rightarrow XY Y \rightarrow \lambda Using Online Text.. A4a. List four strings, each of length less than 4, accepted by this grammar. (Include \& if it is accepted.) A4b. List four strings, each of length less than 4, not accepted by this grammar. (Include \& If It is not accepted.) A4c. In English, describe the language represented by this grammar.

The Correct Answer and Explanation is:

A4a. Four strings (length < 4) accepted by the grammar:

1. y
2. yy
3. yyy
4. & (empty string, accepted via S→yY→yλS \rightarrow yY \rightarrow y\lambdaS→yY→yλ)

A4b. Four strings (length < 4) not accepted by the grammar:

1. x
2. yx
3. xy
4. xxy

These strings cannot be generated through the production rules starting from SSS.

A4c. English Description of the Language:

This grammar generates strings over the alphabet {x, y} that begin with one or more ys and may include xs only after at least one y. The grammar uses recursive rules and empty string transitions to ensure flexibility in generation. The empty string is allowed because Y→λY \rightarrow \lambdaY→λ. The structure of the rules shows that any accepted string must either:

• be composed entirely of ys,
• start with one or more ys and then have a mix of xs and ys in a recursive structure controlled by non-terminals XXX and YYY,
• or be empty due to S→yY→yλS \rightarrow yY \rightarrow y\lambdaS→yY→yλ.

Explanation

The regular grammar presented defines a language over the alphabet containing the symbols x and y. It uses non-terminal symbols S, X, and Y with the start symbol being S. The production rules show that a string generation must begin from S and follow specific paths through substitution.

The rule S→yYS \rightarrow yYS→yY or S→yXS \rightarrow yXS→yX ensures that any accepted string must start with at least one y. The rule S→XSS \rightarrow XSS→XS allows for the possibility of beginning with x indirectly if followed correctly by S expansions. However, since X itself cannot generate y directly without going through S again, strings that start with x and do not correctly resolve back into y cannot be accepted.

The X rules allow for repetition of x and possible transition back to S through X→ySX \rightarrow ySX→yS. This transition is key to generating additional y values after possibly encountering x, as it resets the expansion to S. The Y rules further allow for strings to terminate properly using Y→λY \rightarrow \lambdaY→λ, which means the string may end once the derivation moves to Y.

This grammar’s recursive nature allows for complex strings to be formed but always within the constraint that y appears before or between any x. The empty string is accepted due to Y deriving the empty string. Therefore, the language includes all strings that begin with y, possibly have x, and may end in any order conforming to the rules, including the empty string.