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Problem 28

# Convert the following regular expressions to NFAs using the procedure given in Theorem 1.54. In all parts, $$\Sigma=\\{\mathrm{a}, \mathrm{b}\\}$$. a. $$\mathrm{a}(\mathrm{abb})^{*} \cup \mathrm{b}$$ b. $$a^{+} \cup(a b)^{+}$$ c. $$\left(\mathrm{a} \cup \mathrm{b}^{+}\right) \mathrm{a}^{+} \mathrm{b}^{+}$$

Expert verified
a. The NFA for $$\mathrm{a}(\mathrm{abb})^{*} \cup \mathrm{b}$$ is obtained by creating separate NFAs for $$\mathrm{a}$$, $$\mathrm{abb}$$, and $$\mathrm{b}$$, applying closure, concatenation, and union operations, then simplifying the result. b. The NFA for $$a^{+} \cup(a b)^{+}$$ is constructed by creating separate NFAs for $$a$$ and $$a b$$, applying the plus and union operations, and simplifying the result. c. The NFA for $$\left(\mathrm{a} \cup \mathrm{b}^{+}\right) \mathrm{a}^{+} \mathrm{b}^{+}$$ is derived by creating separate NFAs for $$a$$, $$b^{+}$$, $$a^{+}$$, and $$b^{+}$$, applying union and concatenation operations, and simplifying the final NFA.
See the step by step solution

## Step 1: a. Convert $$\mathrm{a}(\mathrm{abb})^{*} ∪ \mathrm{b}$$ to NFA

1. Create separate NFAs for $$\mathrm{a}$$, $$\mathrm{abb}$$, and $$\mathrm{b}$$. 2. Use the closure operation to create an NFA for $$(\mathrm{abb})^{*}$$. 3. Concatenate the NFAs for $$\mathrm{a}$$ and $$(\mathrm{abb})^{*}$$. 4. Apply the union operation to combine the NFAs for $$\mathrm{a}(\mathrm{abb})^{*}$$ and $$\mathrm{b}$$. 5. Simplify the resulting NFA, if possible.

## Step 2: b. Convert $$a^{+} \cup(a b)^{+}$$ to NFA

1. Create separate NFAs for $$a$$ and $$a b$$. 2. Use the plus operation to create NFAs for $$a^{+}$$ and $$(a b)^{+}$$. 3. Apply the union operation to combine the NFAs for $$a^{+}$$ and $$(a b)^{+}$$. 4. Simplify the resulting NFA, if possible.

## Step 3: c. Convert $$\left(\mathrm{a} \cup \mathrm{b}^{+}\right) \mathrm{a}^{+} \mathrm{b}^{+}$$ to NFA

1. Create separate NFAs for $$a$$, $$b^{+}$$, $$a^{+}$$, and $$b^{+}$$. 2. Apply the union operation to combine the NFAs for $$a$$ and $$b^{+}$$ and create an NFA for $$\left(\mathrm{a} \cup \mathrm{b}^{+}\right)$$. 3. Concatenate the NFAs for $$\left(\mathrm{a} \cup \mathrm{b}^{+}\right)$$, $$a^{+}$$, and $$b^{+}$$. 4. Simplify the resulting NFA, if possible.

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