Prefix property and DPDA

Question

Construct the DPDA with empty stack and final state method for the language

L = ${ a^n b^n / n>= 0}$.

Comments

@ Shubhanshu

A language has prefix property means if w∈L, then no proper prefix of w∈L. 

see this

 https://gateoverflow.in/35419/identify-language

https://gateoverflow.in/377/gate1999_1-6

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Yes sir, the above language also do not satisfy the prefix property. as L contain

L = { epsilon, ab, aabb, aaabbb, ....................}

so if we take any string from this language lets say ab, then prefix of ab will be:

epsilon, //proper prefix

a, //proper prefix

ab. //prefix

since {epsilon} belongs to the language L, that;s why the language does not have prefix property,

But why we can't draw its DPDA with empty stack.???

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@Shubhanshu

If a language L is accepted by the empty-stack DPDA M then it must be having prefix property.

Suppose language L, that not following prefix property, is accepted by DPDA M using empty stack and uv ∈ L and also u ∈ L. Since u is accepted by M, there is computation path from (q₀,u,S₀) to (q_k,ϵ,ϵ). Since M is deterministic, the only possible computation for input uv is  (q₀,u,S₀) ⇒∗ (q_k,v,ϵ). But M has already had empty stack and so it cannot move further from (q_k,v,ϵ) and hence does not accept uv.