Given a context-free grammar G, there exists a Nondeterministic Pushdown Automaton N that accepts exactly the language G accepts. (and visa versa)

There may also exist a Deterministic Pushdown Automaton D that accepts exactly the language G accepts too. It depends on the grammar.

By what algorithm on the productions of G can we determine if D exists?

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    $\begingroup$ Even if you know beforehand that there is a DPDA for G, there is no algorithm to find it. See here. $\endgroup$ – sdcvvc May 21 '12 at 23:55

There is no algorithm that given a context-free grammar, decide if a DPDA recognizes the same language and computes it if it exists.

Because if such an algorithm existed, we would be able to decide the undecidable problem of the universality of a context-free grammar i.e. whether a given context-free grammar $G$ on $Σ$ recognizes the whole language $Σ^*$.

Suppose there is such an algorithm $A_{DPDA}$. Let $G$ be some context-free grammar. Let $L$ be $\mathcal L(G)$. Then the algorithm $A_{DPDA}$ will decide if there is a DPDA $A$ recognizing $L$.

  • If there is no such DPDA, then $L$ is not recognizable by a DPDA, in particular it is not regular, so it can't be $Σ^*$.

  • If a DPDA $A$ exists then we can decide if $L$ is equal to $Σ^*$ because universality is decidable for DPDAs. Why? Because:

    • DPDA languages are closed under complementation (because DPDAs are deterministic)
    • emptiness is decidable for DPDAs (because it is for PDAs)

Using $A_{DPDA}$ we have built an algorithm deciding whether $L(G)=Σ^*$ for any context-free grammar $G$, which has been proven impossible. Therefore $A_{DPDA}$ does not exist.

  • $\begingroup$ I don't understand this sorry. Are you using Σ to refer to the alphabet used by G? And what do you mean "is L the full language $Σ^∗$" ? Are you saying that we will execute the algorithm on a grammar G that generates $Σ^∗$, the language of any string over Σ? We would clearly find not only a DPDA, but a simple DFA for this language. The complement of $Σ^∗$ is the empty language, this is also easily recognized by both a DPDA and a DFA - so I am clearly missing something in your explanation. $\endgroup$ – Andrew Tomazos May 21 '12 at 16:34
  • $\begingroup$ I hope this is clearer now. Note that I answer a slightly different question: I would have sworn you asked if we could compute $D$, not just decide whether or not it exists. $\endgroup$ – jmad May 21 '12 at 17:42

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