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Deterministic context-free languages are a proper subset of context-free languages. They are the context-free languages that can be accepted by a deterministic pushdown automaton. Deterministic context-free grammar and Deterministic context-free languages are always unambiguous grammar.

Here, a proof by contradiction is there, let’s assume that w are wz two unequal strings in L(G), where grammar is a deterministic context-free grammar.

Both are valid strings, so both have handles, and these handles must agree because we can write w = xhy and wz = xhyz = xhy' where h is the handle of w.

Hence, the first reduce steps of w and wz produce valid strings u and uz, respectively. Here, we can continue this process until we obtain S1 and S2 where S1z is the start variable. However, S1 does not appear on the right-hand side of any rule so we cannot reduce S1z. That gives a contradiction.

Every deterministic context-free grammar is an unambiguous context-free grammar the deterministic context-free grammar is formed only one parse tree and no other parse tree is formed by the context-free grammar; either the string is able to form a left-hand derivation tree or a right hand derivation tree, so from a grammar made by a string of a language are formed only single parse tree is called unambiguous context-free grammar.

There always exists a unambiguous context-free grammar corresponding to unambiguous context-free language.

Deterministic context-free grammars are always unambiguous and are parsed by LR parsers. the first reduce steps of w and wz produce valid strings u and uz, respectively. And here we can continue this process until we obtain S1 and S2 z where S1 is the start variable, However, S1 does not appear on the right-hand side of any rule. so we cannot reduce S1z.

So, it is proved that every deterministic context-free grammar generates a prefix-free language.