A deterministic finite automaton (DFA) $D$ with alphabet $\Sigma = \{a, b\}$ is given below. Which of the following finite state machines is a valid minimal DFA which accepts the same language as $D$?

To minimize the given DFA $D$ with states $\{p, q, r, s, t\}$ where $s$ is the only final state:

1. Equivalence of Trap/Dead States:

   • States $r$ and $t$ are both non-accepting dead/trap states (self-loop on both $a$ and $b$, no transition leading to an accepting state).

   • Thus, states $r$ and $t$ can be merged into a single non-accepting state.

2. Evaluating Options:

   • Option (B) accepts $b$ (since $p \xrightarrow{a,b} q \xrightarrow{b} s$ or reaches an accepting state on input $b$), which is not accepted by the original DFA $D$.

   • Option (C) is non-deterministic or invalid as state $q$ has multiple transitions on $b$ and accepts invalid strings.

   • Option (D) incorrectly accepts strings not in the language of $D$.

   • Option (A) correctly represents the language accepted by $D$, where non-equivalent states are preserved and equivalent dead states are appropriately merged.

Therefore, (A) is the correct option.