Explain the equivalence between PDAs and DCFLs.

1. Definition: Deterministic Context-Free Languages (DCFLs) are those languages that can be accepted by a deterministic pushdown automaton (DPA). PDAs are automata with a stack used for additional memory.

2. Deterministic PDAs: Every DCFL can be accepted by a deterministic PDA, meaning a deterministic PDA can parse and recognize exactly the languages classified as DCFLs.

3. Context-Free Languages: All DCFLs are a subset of context-free languages, but not all context-free languages are deterministic. DCFLs are characterized by the lack of ambiguity in parsing.

4. Closure Properties: DCFLs are closed under operations such as intersection with regular languages and complementation, which highlights their structured and well-defined nature.

5. Non-Deterministic vs. Deterministic: Non-deterministic PDAs can accept a broader class of context-free languages, including those that are not deterministic, but deterministic PDAs are restricted to DCFLs.

In essence, the equivalence shows that deterministic PDAs precisely characterize DCFLs, highlighting the relationship between deterministic parsing and context-free language theory.