Parameterized Synthesis for Fragments of First-Order Logic over Data Words

TL;DR

This paper investigates the parameterized synthesis problem for systems with an unknown number of processes, showing undecidability in general but decidability with bounded environment access, and introduces a cutoff for analysis.

Contribution

It establishes decidability results for parameterized synthesis in first-order logic fragments and identifies conditions under which synthesis is feasible.

Findings

Undecidability when both system and environment access all processes

Decidability when environment access is bounded

Existence of a cutoff for process architecture analysis

Abstract

We study the synthesis problem for systems with a parameterized number of processes. As in the classical case due to Church, the system selects actions depending on the program run so far, with the aim of fulfilling a given specification. The difficulty is that, at the same time, the environment executes actions that the system cannot control. In contrast to the case of fixed, finite alphabets, here we consider the case of parameterized alphabets. An alphabet reflects the number of processes that are static but unknown. The synthesis problem then asks whether there is a finite number of processes for which the system can satisfy the specification. This variant is already undecidable for very limited logics. Therefore, we consider a first-order logic without the order on word positions. We show that even in this restricted case synthesis is undecidable if both the system and the environment have access to all processes. On the other hand, we prove that the problem is decidable if the environment only has access to a bounded number of processes. In that case, there is even a cutoff meaning that it is enough to examine a bounded number of process architectures to solve the synthesis problem.