please ans it ASAP

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Consider the DFAs

$1$

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on the left

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and

$2$

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on the right

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:

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a

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If we use the general constructions discussed in class and in the book for building a DFA whose

language is

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$1$

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$($

$2$

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how many states would be in this DFA? Briefly justify your calculation.

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b

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Draw the state diagram that results from this construction and remove any unreachable states.

How many states are left?

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c

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Describe the language

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$1$

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$($

$2$

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in set builder notation or as a regular expression.

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d

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Is there any way to build a machine that recognizes the same language

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$1$

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$($

$2$

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but uses

fewer states?

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You can either draw the result or describe in words how to construct it

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student submitted image, transcription available

Your Submission

Rating

Sub$-$Subject

Discrete Structures

Topic

N$/$A

Step$-$by$-$step

Step $1$ of $3$

There are $3$ steps to solve this problem:

Step $1$

To answer your questions regarding DFAs $($ M$\_1)$ and $($ M$\_2),$ we will follow the typical constructions for building a DFA that recognizes the union of two DFA languages.

$($a$)$ Number of States in the Union DFA

To construct a DFA that recognizes the language $($ L$($M$\_1)\backslash$cup L$($M$\_2)),$ we use the cross$-$product construction method. If $($ M$\_1)$ has $($ n$\_1)$ states and $($ M$\_2)$ has $($ n$\_2)$ states, then the DFA for $($ L$($M$\_1)\backslash$cup L$($M$\_2))$ will have $($ n$\_1\backslash$times n$\_2)$ states.

Assuming $($ M$\_1)$ has $($ n$\_1)$ states and $($ M$\_2)$ has $($ n$\_2)$ states, the number of states in the new DFA would be $($ n$\_1\backslash$times n$\_2).$

Justification:

The cross$-$product construction combines each state of $($ M$\_1)$ with each state of $($ M$\_2)$ to form a new state in the union DFA. Therefore, the total number of states is the product of the number of states in each original DFA.

$($b$)$ State Diagram and Removing Unreachable States

$1.$ Construct the State Diagram:

Construct the state diagram using the states of $($ M$\_1)$ and $($ M$\_2).$ Each state in the new DFA is a pair $(($q$\_$i$,$ r$\_$j$))$ where $($ q$\_$i $)$ is a state from $($ M$\_1)$ and $($ r$\_$j $)$ is a state from $($ M$\_2).$

$2.$ Remove Unreachable States:

Identify and remove any states that cannot be reached from the initial state. This involves checking the transition function and tracing which states can actually be accessed from the starting state.

Example:

If $($ M$\_1)$ has states $(\{$A$,$ B$,$ C$\})$ and $($ M$\_2)$ has states $(\{$X$,$ Y$\}),$ the union DFA will initially have $(3\backslash$times $2=6)$ states: $(\{($A$,$X$),($A$,$Y$),($B$,$X$),($B$,$Y$),($C$,$X$),($C$,$Y$)\}).$

Explanation:

After removing unreachable states, you may find a reduced number of states, depending on the specific transitions and final states of $($ M$\_1)$ and $($ M$\_2).$

Step $2$ of $3$

Step $2$

$($c$)$ Language $($ L$($M$\_1)\backslash$cup L$($M$\_2))$

The language $($ L$($M$\_1)\backslash$cup L$($M$\_2))$ can be described in set builder notation or as a regular expression.

Set Builder Notation:

L$($M$\_1)\backslash$cup L$($M$\_2)=\{$ w $\backslash$mid w $\backslash$in L$($M$\_1)\backslash$text$\{$ or $\}$ w $\backslash$in L$($M$\_2)\}]$

Regular Expression:

If $($ R$\_1)$ is the regular expression for $($ L$($M$\_1))$ and $($ R$\_2)$ is the regular expression for $($ L$($M$\_2)),$ then $($ L$($M$\_1)\backslash$cup L$($M$\_2))$ can be represented by the regular expression $($ R$\_1+$ R$\_2).$

$($d$)$ Constructing a DFA with Fewer States

It is often possible to build a more optimized DFA that recognizes the same language but with fewer states. This typically involves state minimization techniques such as merging equivalent states.

$1.$ Identify Equivalent States:

States that behave identically for all possible input strings can be merged. Use methods like the partition refinement algorithm to find and merge these states.

$2.$ Construct the Minimized DFA:

Once equivalent states are identified and merged, the resulting minimized DFA will recognize the same language but with a potentially reduced number of states.

Explanation:

The theoretical explanation for the state diagram resulting from the product machine construction is that it is created by taking the Cartesian product of the state sets of the original DFAs

Step $3$ of $3$

Step $3$

Steps to Minimize:

Build the DFA for $($ L$($M$\_1)\backslash$cup L$($M$\_2))$ using the cross$-$product method.

Apply state minimization techniques to merge equivalent states.

Draw the minimized DFA or describe the construction steps.

Explanation:

The product machine construction uses the minimal number of states possible for this construction because it uses one state for every possible pair of states from the original DFAs. This is because every state in the product machine represents a unique combination of states from the original DFAs.

Final solution

Final Solution:

For a specific answer, knowing the exact states and transitions of $($ M$\_1)$ and $($ M$\_2)$ would allow a detailed minimized DFA to be drawn. Without that, the above steps provide the general approach.