Symbolic Reactive Synthesis for the Safety and EL-fragment of LTL

We suggest an expressive fragment of LTL for which reactive synthesis can be performed by symbolically analyzing games. For general LTL, this kind of analysis is impossible due to the complexity of determinization. Bypasses are either by enumerative handling of determinization or by restricting attention to fragments of the language. Here, we take the second approach and suggest a fragment combining a safety specification and a liveness part. The safety part is unrestricted but allows symbolic treatment due to the simplicity of determinization in the case of safety languages. The liveness part is very general, allowing to define Emerson-Lei conditions on occurrences of letters. We elaborate the construction of an Emerson-Lei game that captures the synthesis problem. We also show how Emerson-Lei games can be analyzed symbolically by providing a fixpoint-based characterization of the winning region, which is obtained from an analysis of the Zielonka tree of the winning condition. Our algorithm generalizes the solutions of games with known winning conditions such as B\"uchi, GR[1], parity, Streett, Rabin and Muller objectives, and in the case of these conditions reproduces previously known algorithms and complexity results; the algorithm solves unrestricted Emerson-Lei games with $n$ nodes, $m$ edges and $k$ colors in time $\mathcal{O}(k!\cdot m\cdot n^k)$ and yields winning strategies with memory $\mathcal{O}(k!)$. The runtime of the resulting overall synthesis algorithm is single-exponential in the size of the liveness part and doubly-exponential in the size of the safety part, as it is for (safety) LTL. However, the trade-off between enumerative and symbolic aspects is maximized by enumeratively analyzing the liveness condition and generating from it a symbolic game analysis algorithm.