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The $DK-$Test is the procedure that determines whether the context-free grammar. For any CFG the DFA is constructed that identify he handles.

Consider in contradiction, that C is not a deterministic CFG. The valid ahb is considered, that has unforced handle h . It has a possibility that some string has another handle. Thus, ahb' can be rewritten as $\stackrel{铃�}{a}\stackrel{铃�}{h}\stackrel{铃�}{b}$ where, b' is a terminal.

While, ah=$\stackrel{铃�}{a}\stackrel{铃�}{h}$ , the reduce rule will be changed, because h and $\stackrel{铃�}{h}$are different handles. Thus, it does not satisfy the DK-test since it has two completed rules.

While ah$鈮�$$\stackrel{铃�}{a}\stackrel{铃�}{h}$ , assume that the proper prefix of $\stackrel{铃�}{a}\stackrel{铃�}{h}$is ah . Assume that s is an accepting state of the input in DK. The s accepts the states because h is the handle.

Hence, the transition must label b' because, $b\text{'}鈭�\underset{}{鈭�}\text{'}$ doesnot has null rule. It does not satisfy the DK-test.

Therefore, the DK-test fails, In contradiction to the assumption, CFG is a deterministic context-free grammar.