Question

Which of the grammars below can generate the term: Foo(X, Bar(Y,Y, X), X)?

a) T | Foo(T, S, T)
| Bar(T, T,T)
| X
S Bar(S, S, T)
| Y
b) T Foo(T,S, T)
| X
S Bar(T, T, T)
Y
c) T Foo(T, T, T)
| Bar(T, T,T)
| X
d) T --> Foo(T, T, T)
| Bar(T, T, T)
| X
| Y

Answer 1

The correct grammar to generate the term Foo(X, Bar(Y,Y, X), X) is option (d), which provides the necessary flexibility to substitute variables X and Y independently to form the term.

Step-by-step explanation:

The given term Foo(X, Bar(Y,Y, X), X) can be generated by a grammar that is capable of recursively defining the functions Foo and Bar with the appropriate number of arguments that can themselves be terms.

Looking at the options, grammar (d) is the correct one. This is because it allows both terminals X and Y to be used independently. Given the grammar rules, we can break down our term using the grammar d as follows:

1. Start with T and replace using Foo(T, T, T) to get Foo(T, T, T)
2. Replace the first T in Foo with X based on grammar rule T --> X to get Foo(X, T, T)
3. For the second T in Foo, replace it with Bar(T, T, T) to get Foo(X, Bar(T, T, T), T)
4. Then replace every T inside Bar with Y based on grammar rule T --> Y to get Foo(X, Bar(Y, Y, T), T)
5. For the last T in Foo and the last T in Bar, replace them with X to get Foo(X, Bar(Y, Y, X), X), which matches the term we need to generate.

This process shows how grammar (d) generates the desired term. Grammars (a), (b), and (c) do not offer the flexibility of substituting separate variables like X and Y independently or the specific sequence needed for the term
Foo(X, Bar(Y,Y, X), X).