Distributed reactive systems are hard to synthesize

Abstract

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. 'hovot 76100, Israel. Email: ami&wisdom.weizmann.ac.IL, roni@wisdom.weizmann.ac.IL .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.

The Planning Spectrum - One, Two, Three, Infinity

“…The proposed algorithm has a time complexity that is doubly exponential in the size of the goal formula. It is known that the planning problem is 2EXPTIME-complete for goals of the form A.ϕ (Pnueli & Rosner, 1990), and hence the complexity of our algorithm is optimal.…”

“…Indeed, the realizability problem for an LTL formula ϕ, which is known to be 2EXPTIME-complete (Pnueli & Rosner, 1990), can be reduced to a planning problem for the goal A.ϕ. In a realizability problem one assumes that a program and the environment alternate in the control of the evolution of the system.…”

Distributed Control Synthesis

“…An even more appealing idea is to automatically synthesize correct-by-construction systems from their formal specifications. This goal is hard to achieve due to complexity issues: the synthesis of distributed algorithms is shown to be undecidable [23]. Synthesis algorithms for distributed systems would take as input a formal specification for the behavior of the system and an architecture that reflects the communication allowed between the components of the system.…”

“…Synthesis algorithms for distributed systems would take as input a formal specification for the behavior of the system and an architecture that reflects the communication allowed between the components of the system. Limiting the architecture of the system [23,14,15,5,29,30], one can achieve decidability for synthesis for some architectures, like chains or rings.…”

“…Observing carefully the assumptions in the undecidability results of [23,33,32], one requirement that can be relaxed is the ability of the processes to exchange messages. We still require the basic algorithm to have a particular set of processes, variables shared between them in a predefined way, including the prescribed communication network.…”