Distributed Reactive Systems Are Hard to Synthesize

This paper considers the problem of synthesizing a finite-state distributed reactive system. Given a distributed architecture A, identifying several processors 9, . . . , 9, and their interconnection scheme, and a propositional temporal specification 'P, a solution to the synthesis problem consists of finite-state programs II1 , . . . , I I k (one for each processor), whose joint (synchronous) behavior maintains 'P against all possible inputs from the environment. We refer to such a solution as the realization of the specification 'P over the architecture A. The work reported here extends the finite-state reactive synthesis studies reported in [PR88, PR89aI and also in [ALW89], that did not impose a given architecture, and hence standardly yielded a solution based on the easiest architecture, that of a single processor. Specifically, we show that the problem of realizing a given propositional specification over a given architecture is undecidable, and it is nonelementarily decidable for the very restricted class of hierarchical architectures. These results are based on Peterson and Reif's [PR79] work on games of incomplete information. We further give an extensive characterization of architecture classes for which the realizability problem is elementarily decidable, and of classes for which it is undecidable. 'This research was supported in part by the European Community ESPRIT Basic Research Action project 3096 (SPEC). tThis author's research was partially supported by a grant from the Israel Ministry of Science and Development, the National Council for Research and Development. tAddress: Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel. Email: ami&wisdom.weizmann.ac.IL, [email protected] . Another approach to the implementation of 'P over A is to synthesize first (using the technique of [PR89a]) a single processor strategy that satisfies 'P, and then to decompose this specific strategy into a set of programs . . . , n k for A. The problem of decomposing a given finite-state program for a single processor into a set of programs over a given architecture A is shown to be decidable for all acyclic architectures.