How do we construct a context-free grammar (CFG) from a given PDA to recognize the same set of strings?
To construct a context-free grammar (CFG) from a given pushdown automaton (PDA) to recognize the same set of strings, we need to follow a systematic approach. This process involves converting the PDA's transition function into production rules for the CFG. By doing so, we establish an equivalence between the PDA and the CFG, ensuring that both recognize the same language.
Let's consider a PDA defined by 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.
To construct a CFG from this PDA, we need to define a set of production rules that mimic the behavior of the PDA's transition function. Each production rule consists of a non-terminal symbol on the left-hand side and a string of terminals and/or non-terminals on the right-hand side.
The non-terminal symbols in the CFG represent the states of the PDA, while the terminals represent the input symbols. Additionally, we introduce a new non-terminal symbol, S, which serves as the start symbol of the CFG.
The production rules are constructed as follows:
1. For each transition (q, a, X) → (p, γ) in the PDA's transition function:
– If γ is not empty, add a production rule q → aXp.
– If γ is empty, add a production rule q → aXp for each a in Σ.
2. For each transition (q, ε, X) → (p, γ) in the PDA's transition function:
– If γ is not empty, add a production rule q → Xp.
– If γ is empty, add a production rule q → Xp for each a in Σ.
3. For each accepting state f in F, add a production rule f → ε.
4. Add a production rule S → q0Z0, where q0 is the initial state and Z0 is the initial stack symbol.
By constructing the CFG using these production rules, we ensure that it recognizes the same language as the original PDA. The CFG can then be used to generate valid strings in the language or to determine if a given string belongs to the language.
Let's illustrate this process with an example:
Consider a PDA with the following transition function:
δ(q0, a, Z0) = {(q1, AZ0)}
δ(q1, a, A) = {(q1, AA)}
δ(q1, b, A) = {(q1, ε)}
δ(q1, ε, Z0) = {(qf, Z0)}
To construct the CFG, we apply the steps outlined above:
1. From δ(q0, a, Z0) = {(q1, AZ0)}, we add the production rule q0 → aq1Z0.
2. From δ(q1, a, A) = {(q1, AA)}, we add the production rule q1 → aq1A.
3. From δ(q1, b, A) = {(q1, ε)}, we add the production rule q1 → bq1.
4. From δ(q1, ε, Z0) = {(qf, Z0)}, we add the production rule q1 → εqf.
5. Finally, we add the production rule S → q0Z0.
The resulting CFG is:
S → aq1Z0
q0 → aq1Z0
q1 → aq1A
q1 → bq1
q1 → εqf
This CFG recognizes the same set of strings as the original PDA.
To construct a CFG from a given PDA to recognize the same set of strings, we follow a systematic approach of converting the PDA's transition function into production rules for the CFG. By constructing the CFG using these rules, we establish an equivalence between the PDA and the CFG, ensuring that both recognize the same language.
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