Find the grammar that generates inherently ambiguous context free language.

Question

Answer 1

IF a language is Ambiguous and that cannot be written in any other simplified way ( into unambiguous language) , then language is Inherently ambiguous.

A Language is ambiguous if a string of language can be derived by more than one way and having different derivation tree.

Such as L = aⁱb^jc^k     i=j or j= k , i,j,k>=0 

= aⁱbⁱc^k $\bigcup$ aⁱb^jc^j   

S$\rightarrow$ S1 | S2        

S1 $\rightarrow$ XC                 S2$\rightarrow$ AY

X$\rightarrow$ aXb|$\epsilon$               Y $\rightarrow$ bYc |$\epsilon$

C $\rightarrow$cC|$\epsilon$               A$\rightarrow$ aA|$\epsilon$

There is no other way to define this language , and there are strings abc, aabbcc, etc which can be derived from both  S$\rightarrow$S1 or  S $\rightarrow$s2 . So this language is inherent ambiguous.

In case of problem given

a.  L = { aⁿbⁿd^m n,m>=0 $\bigcup$  aⁱb^jd^j  i,j>=0 } which is inherently ambiguous . and strings abd,aabbdd, etc can be derived by 2 ways

b. L = { aⁿb^m m>=n m,n>=0   $\bigcup$   wbⁱ , w $\epsilon$ (a,b)* ,i>=0 } 

which can be simplified as 

L = w , w $\epsilon$(a,b)*   into an unambiguous language

Comments

In option B true because

 anbm m>=n m,n>=0   is. an  DCFL

And   wbi , w ϵϵ (a,b)* ,i>=0  is an Regular language

And DCFL union Regular language is DCFL and we know DCFL is inherently unambiguous

---

This is absolutely wrong, DCFL are not inherently ambiguous, {a^nb^n,n>0} is a DCFL and is not inherently ambiguous as there exists a unambiguous CFG. S->aSb|ab.