Synthesis in Distributed Environments

TL;DR

This paper addresses the synthesis of reactive systems in distributed environments by modeling them as Petri games, providing complexity results for different numbers of environment tokens and enabling synthesis with multiple information sources.

Contribution

It introduces a Petri game-based model for distributed environment synthesis and analyzes its computational complexity for various numbers of environment tokens.

Findings

Polynomial-time solvability for up to two environment tokens

NP-completeness for three or more environment tokens

EXPTIME-completeness when environment tokens grow with net size

Abstract

Most approaches to the synthesis of reactive systems study the problem in terms of a two-player game with complete observation. In many applications, however, the system's environment consists of several distinct entities, and the system must actively communicate with these entities in order to obtain information available in the environment. In this paper, we model such environments as a team of players and keep track of the information known to each individual player. This allows us to synthesize programs that interact with a distributed environment and leverage multiple interacting sources of information. The synthesis problem in distributed environments corresponds to solving a special class of Petri games, i.e., multi-player games played over Petri nets, where the net has a distinguished token representing the system and an arbitrary number of tokens representing the environment. While, in general, even the decidability of Petri games is an open question, we show that the synthesis problem in distributed environments can be solved in polynomial time for nets with up to two environment tokens. For an arbitrary but fixed number of three or more environment tokens, the problem is NP-complete. If the number of environment tokens grows with the size of the net, the problem is EXPTIME-complete.