The complexity bounds of the computed uniform interpolants using syntactic approaches are generally lower than those obtained through semantic methods. Syntactic approaches, such as the proof-theoretic method used in the paper, provide more precise complexity bounds and guarantee termination of the algorithms. This can be advantageous in practical applications where efficiency and predictability are crucial. Semantic approaches, on the other hand, may involve more complex computations based on model theory or bisimulation relations, leading to potentially higher computational complexity.
The practical implications of these differences lie in the efficiency and scalability of the algorithms. Syntactic approaches are more suitable for automated reasoning systems and formal verification tools where fast computation and termination are essential. Semantic approaches may offer more insights into the semantics of the logic but could be computationally more demanding and harder to implement in automated systems. Ultimately, the choice between syntactic and semantic approaches depends on the specific requirements of the application and the trade-offs between complexity and precision.