Agent-Alternation-Free Epistemic Metric Temporal Logic with Past: Model Checking and Complexity

We study model checking for an epistemic metric temporal logic with past, interpreted over finite B\"uchi automata under synchronous perfect recall. The logic is motivated by observation-based verification problems such as diagnosis and opacity, where an observer sees only a projection of an execution and reasons about events that may have occurred earlier. These requirements use no alternation between different agents' knowledge. We therefore consider the agent-alternation-free fragment, in which nested knowledge operators must refer to the same agent. We show that model checking for this fragment is EXPSPACE-complete. The lower bound already holds with one agent, one occurrence of the knowledge operator, and no non-trivial metric bounds. For the upper bound, we combine temporal test automata with perfect-recall observers. Because past formulas may have different truth values on indistinguishable histories ending in the same system state, the observer must track temporal automaton states in addition to system states.