How does part two of the proof in the equivalence between CFGs and PDAs work?
Part two of the proof in the equivalence between Context-Free Grammars (CFGs) and Pushdown Automata (PDAs) builds upon the foundation laid in part one, which establishes that every CFG can be simulated by a PDA. In this part, we aim to show that every PDA can be simulated by a CFG, thus establishing the equivalence between the two formalisms.
To begin, let us recall the definition of a PDA. A PDA is a 7-tuple (Q, Σ, Γ, δ, q0, Z0, F), where:
– Q is a finite set of states,
– Σ is the input alphabet,
– Γ is the stack alphabet,
– δ is the transition function,
– q0 is the initial state,
– Z0 is the initial stack symbol, and
– F is the set of accepting states.
The key idea in this proof is to construct a CFG that generates the same language as the given PDA. We will achieve this by simulating the behavior of the PDA using nonterminals and productions of the CFG.
Let's outline the steps involved in this construction:
1. Start by introducing a nonterminal S as the start symbol of the CFG.
2. For each state q in Q, introduce a nonterminal Aq in the CFG.
3. For each terminal symbol a in Σ and each stack symbol X in Γ, introduce a nonterminal Aq,a,X in the CFG. This nonterminal represents the configuration of the PDA when it is in state q, reading input symbol a, and having X on top of the stack.
4. Add productions to the CFG to simulate the transitions of the PDA. For each transition (q, a, X) → (p, γ) in the PDA, where γ is a string of stack symbols, add the production Aq,a,X → Apgamma to the CFG.
5. Add productions to the CFG to simulate the PDA's acceptance. For each state q in F, add the production Aq,ε,Z0 → ε, where ε represents the empty string.
6. Finally, add productions to the CFG to ensure that the generated language matches the language accepted by the PDA. For each terminal symbol a in Σ, add the production S → Aq,a,Z0 for each state q in Q.
By constructing the CFG as described above, we have effectively simulated the behavior of the given PDA. The start symbol S generates strings that correspond to accepting computations of the PDA, thus generating the same language.
To illustrate this construction, let's consider a simple example. Suppose we have a PDA with the following transitions:
δ(q0, a, Z0) = {(q1, XZ0)}
δ(q1, b, X) = {(q2, ε)}
Using the steps outlined above, we can construct the following CFG:
S → Aq0,a,Z0
Aq0,a,Z0 → Aq1,X,Z0
Aq1,X,Z0 → Aq2,ε
This CFG generates the same language as the given PDA.
Part two of the proof in the equivalence between CFGs and PDAs demonstrates that every PDA can be simulated by a CFG. By constructing a CFG that mirrors the behavior of the PDA, we establish the equivalence between the two formalisms.
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