This paper points out that despite significant efforts to extend the AGM belief-changing paradigm beyond finite logic, the computational aspects of AGMs have remained largely unexplored. We investigate the computability of AGM reductions in infinite logics and reveal an intriguing negative consequence: there exist infinitely many incomputable AGM reduction functions in these logics. More dramatically, we show that the current de facto standard computability control strategy, which relies on restricting the knowledge state space, fails, maintaining incomputability in all infinite cases. Given these devastating results, we propose a novel approach to controlling computability beyond finite domains. Using linear temporal logic (LTL) as a case study, we identify an infinite class of perfectly rational AGM reduction functions that are computable by design. We construct these functions using Büchi automata to represent and reason about LTL beliefs.