On Strategies in Synthesis Over Finite Traces

The innovations in reactive synthesis from {\em Linear Temporal Logics over finite traces} ($\ltlf$) will be amplified by the ability to verify the correctness of the strategies generated by $\ltlf$ synthesis tools. This motivates our work on {\em $\ltlf$ model checking}. $\ltlf$ model checking, however, is not straightforward. The strategies generated by $\ltlf$ synthesis may be represented using {\em terminating} transducers or {\em non-terminating} transducers where executions are of finite-but-unbounded length or infinite length, respectively. For synthesis, there is no evidence that one type of transducer is better than the other since they both demonstrate the same complexity and similar algorithms. In this work, we show that for model checking, the two types of transducers are fundamentally different. Our central result is that $\ltlf$ model checking of non-terminating transducers is \emph{exponentially harder} than that of terminating transducers. We show that the problems are \expspace-complete and $\pspace$-complete, respectively. Hence, considering the feasibility of verification, $\ltlf$ synthesis tools should synthesize terminating transducers. This is, to the best of our knowledge, the \emph{first} evidence to use one transducer over the other in $\ltlf$ synthesis.