Which ONE of the following languages is accepted by a deterministic pushdown automaton (DPDA)?

The correct option is Any regular language.

To understand why this is correct, let us analyze the capabilities and formal relationships between deterministic pushdown automata (DPDAs), regular languages, and other language families in the Chomsky hierarchy.

1. What is a Deterministic Pushdown Automaton (DPDA)?
A pushdown automaton (PDA) is a finite automaton that is equipped with an auxiliary memory structure called a stack. A deterministic pushdown automaton (DPDA) is a restricted version of a PDA where at most one transition is possible for any given state, input symbol, and stack top symbol. In other words, its behavior is completely deterministic.

2. Relationship between Regular Languages and DPDAs:
Every regular language is recognized by a Deterministic Finite Automaton (DFA). A DFA can be easily converted into a DPDA by simply ignoring the stack (i.e., the DPDA never pushes or pops anything, or performs dummy operations that do not affect its state control). Thus, the class of regular languages is a proper subset of the class of languages accepted by DPDAs (deterministic context-free languages, or DCFLs). That is:
$\mathrm{Regular\; Languages}\subset \mathrm{DCFL}$
Therefore, any regular language is guaranteed to be accepted by a DPDA.

3. Why the other options are incorrect:

• Any context-free language: Not all context-free languages (CFLs) can be accepted by a deterministic pushdown automaton. For example, the language $L=\left\{w{w}^R\right|w\in {\{a,b\}}^{*}\}$ (even-length palindromes) is a context-free language but requires non-determinism to guess the middle of the string. Thus, it cannot be recognized by any DPDA.
• Any language accepted by an NPDA: Non-deterministic Pushdown Automata (NPDAs) recognize the exact class of all context-free languages. As established above, there are context-free languages (like even-length palindromes) that cannot be accepted by a DPDA. Hence, not every language accepted by an NPDA can be accepted by a DPDA.
• Any decidable language: Decidable languages (recursive languages) are those recognized by Turing machines that always halt. This is a much larger class of languages than context-free or deterministic context-free languages. For example, the language $L_{\mathrm{anbncn}}=\left\{a^n{b}^n{c}^n\right|n\ge 0\}$ is decidable but not context-free, and therefore cannot be accepted by any DPDA.