Predictable and Performant Reactive Synthesis Modulo Theories via Functional Synthesis

TL;DR

The paper addresses static synthesis for $LTL^{\mathcal{T}}$, enabling controlled generation of deterministic programs that reason over rich theories rather than just propositional atoms. It combines a Boolean controller for the equi-realizable abstraction $\varphi^{\mathbb{B}}$ with a $<\partitioner>$ that maps environment valuations to Boolean inputs and a $<\provider>$ that outputs theory-valued system variables, avoiding on-the-fly SMT solving. A key advance is the use of Skolem functions to realize a static, predictable $<\provider>$, with optional adaptivity to optimize outputs under user-defined criteria; the framework also supports adaptive extensions and quantifier-elimination to handle broader theories. Empirical evaluation demonstrates that the static approach yields significant speedups and improved predictability over dynamic, solver-based methods, making it suitable for embedded and real-time contexts. Overall, the work establishes a first decidable, full static reactive synthesis method for $LTL^{\mathcal{T}}$ and opens avenues for adaptive, neurosymbolic extensions and broader theory integration.

Abstract

Reactive synthesis is the process of generating correct controllers from temporal logic specifications. Classical LTL reactive synthesis handles (propositional) LTL as a specification language. Boolean abstractions allow reducing LTLt specifications (i.e., LTL with propositions replaced by literals from a theory calT), into equi-realizable LTL specifications. In this paper we extend these results into a full static synthesis procedure. The synthesized system receives from the environment valuations of variables from a rich theory calT and outputs valuations of system variables from calT. We use the abstraction method to synthesize a reactive Boolean controller from the LTL specification, and we combine it with functional synthesis to obtain a static controller for the original LTLt specification. We also show that our method allows responses in the sense that the controller can optimize its outputs in order to e.g., always provide the smallest safe values. This is the first full static synthesis method for LTLt, which is a deterministic program (hence predictable and efficient).