Synthesis of Succinct Systems

Synthesis of correct by design systems from specification has recently attracted much attention. The theoretical results imply that this problem is highly intractable, e.g., synthesizing a system is 2EXPTIMEcomplete for an LTL specification and EXPTIME-complete for CTL. An argument in favor of synthesis is that the temporal specification is highly compact, and the complexity reflects the large size of the system constructed. A careful observation reveals that the size of the system is presented in such arguments as the size of its state space. This view is a bit biased, in the sense that the state space can be exponentially larger than the size of a reasonable implementation such as a circuit or a program. Although this alternative measure of the size of the synthesized system is more intuitive (e.g., this is the standard way model checking problems are measured), research on synthesis has so far stayed with measuring the system in terms of the explicit state space. This raises the question of whether or not there exists a small bound on the circuits or programs. In this paper, we show that this is the case if, and only if, PSPACE = EXPTIME.