Finding the Image of a Regular Language under Homomorphism

The problem asks us to find the regular language obtained by applying a given homomorphism \( h \) to a specified regular language \( L \). The language \( L \) is defined by the regular expression \( r = (w + x^*)(ww)^* \). The homomorphism \( h \) is defined over the alphabet \( \Sigma = \{w, x\} \) mapping to strings over the alphabet \( T = \{x, y, z\} \), with the following rules:

• \( h(x) = xzy \)
• \( h(w) = zxyy \)

Understanding String Homomorphism

A string homomorphism is a function from the set of strings over one alphabet to the set of strings over another alphabet that preserves the string concatenation operation. If \( h \) is a homomorphism from \( \Sigma^* \) to \( T^* \), then for any strings \( u, v \in \Sigma^* \), \( h(uv) = h(u)h(v) \). For an empty string \( \epsilon \), \( h(\epsilon) = \epsilon \).

When applying a homomorphism to a regular expression, we replace each symbol in the expression with its corresponding homomorphic image. The regular operations (union '+', concatenation '.', Kleene star '*') are preserved:

• \( h(r_1 + r_2) = h(r_1) + h(r_2) \)
• \( h(r_1 r_2) = h(r_1) h(r_2) \)
• \( h(r^*) = (h(r))^* \)

Applying Homomorphism to the Regular Expression

The given regular expression for \( L \) is \( r = (w + x^*)(ww)^* \). We apply the homomorphism \( h \) to this expression step-by-step:

\( h( (w + x^*)(ww)^* ) \)

Using the rule for concatenation \( h(r_1 r_2) = h(r_1) h(r_2) \):

\( = h(w + x^*) h((ww)^*) \)

Using the rule for union \( h(r_1 + r_2) = h(r_1) + h(r_2) \) on the first part, and the rule for Kleene star \( h(r^*) = (h(r))^* \) on the second part:

\( = (h(w) + h(x^*)) (h(ww))^* \)

Now, apply the Kleene star rule \( h(x^*) = (h(x))^* \) to the term \( h(x^*) \) and the concatenation rule \( h(ww) = h(w)h(w) \) to the term \( h(ww) \):

\( = (h(w) + (h(x))^* ) (h(w)h(w))^* \)

Finally, substitute the given definitions of \( h(w) \) and \( h(x) \):

• \( h(w) = zxyy \)
• \( h(x) = xzy \)

Substituting these values:

\( = ((zxyy) + (xzy)^* ) ((zxyy)(zxyy))^* \)

The resulting regular expression is \( (zxyy + (xzy)^* )(zxyy zxyy)^* \).

Comparing with Options

Let's compare our derived regular expression with the given options:

Our result: \( (zxyy + (xzy)^* )(zxyy zxyy)^* \)

Option 1: \( (z x yy + x z y) (z x yy) \)

Option 2: \( (zxyy + (xzy)*)(zxyy zxyy)* \)

Option 3: \( (zxyy + xzy)(zxyy)* \)

Option 4: \( (zxyy + (xzy)*)(zxyy zxyy) \)

Option 2 exactly matches our derived regular expression.

Conclusion

The regular language \( h(L) \) is denoted by the regular expression \( (zxyy + (xzy)^*)(zxyy zxyy)^* \), which corresponds to Option 2.

Revision Table: Key Concepts in Homomorphism

Additional Information: Homomorphisms and Regular Languages

Homomorphisms are fundamental tools in formal language theory. They represent a type of transformation on strings and languages. Applying a homomorphism to a language means applying the homomorphism to every string in that language.

For a regular language \( L \), the set of strings \( h(L) \) is defined as \( \{h(w) \mid w \in L\} \). The closure property of regular languages under homomorphism is very important. It confirms that regular languages remain within their class after such a transformation.

The process shown above demonstrates how to find the regular expression for the homomorphic image of a regular language given its regular expression. By applying the homomorphism rules to the structure of the regular expression, we obtain a new regular expression that denotes the transformed language.

Another related concept is the inverse homomorphism, denoted \( h^{-1}(L') \). This operation takes a language \( L' \) over the target alphabet \( T \) and finds the set of all strings \( w \) over the source alphabet \( \Sigma \) such that \( h(w) \in L' \). Regular languages are also closed under inverse homomorphism.

These closure properties are often proven using finite automata. If \( L \) is regular, there exists a finite automaton (FA) recognizing it. An FA for \( h(L) \) can be constructed based on the FA for \( L \) and the definition of \( h \). Similarly, an FA for \( h^{-1}(L') \) can be constructed from an FA for \( L' \).