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Model-checking is a popular technique for verifying finitestate concurrent systems, whose behaviour can be modeled using Labeled Transition Systems (Ltss). In this paper, we study the model-checking problem for the modal 碌-calculus on acyclic LTSS. This has various applications of practical interest such as trace analysis, log information auditing, run-time monitoring, etc. We show that on acyclic LTSS, the full 碌-calculus has the same expressive power as its alternation-free fragment. We also present two new local model-checking algorithms based upon a translation to boolean equation systems. The first algorithm handles 碌-calculus formulas 驴 with alternation depth ad(驴) 驴 2 and has time complexity O(|驴|2 驴 (|S|+|T|)) and space complexity O(|驴|2 驴 |S|), where |S| and |T| are the number of states and transitions of the acyclic LTS and |驴| is the number of operators in 驴. The second algorithm handles formulas 驴 with alternation depth ad(驴) = 1 and has time complexity O(|驴| 驴 (|S| + |T|)) and space complexity O(|驴| 驴 |S|).