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

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, which is 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. Partly supported by ANR FREDDA (ANR-17-CE40-0013).

Realizability problem for constraint LTL

“…The parametrized alphabet reflects the number of processes. Similar to the results in our paper, the synthesis problem in [5] is undecidable in the general case but turns decidable when the number of processes that are controllable by the environment is bounded, while the number of system processes remains unbounded. In [26], a parametrized extension of the Church Synthesis Problem of MSO Logic over (N, <) is considered.…”

“…Church Synthesis problem for a restriction of First Order Logic over parametrized alphabets has been studied in [5]. The parametrized alphabet reflects the number of processes.…”