Whether $L(G)=L(R)$ is decidable for DCFL and CFL?

Let $$G_1$$ be the context free grammar and $$R$$ be regular language. Now I have to check whether $$L(G_1)=L(R)$$ is decidable or not? My approach $$\overline{L(G_1)}=\overline{L(R)}$$. Now $$L(G_1)$$ not closed under complement. Therefore $$L(G_1)=L(R)$$ is undecidable.

Now if $$G_1$$ be DCFG, then $$\overline{L(G_1)}=\overline{L(R)}$$. Now $$L(G_1)$$ closed under complement because of DCFL. Therefore $$L(G_1)=L(R)$$ is decidable. Don't know my approach is right or not, if it wrong correct me.

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Nov 8 '21 at 3:51