Realizability modulo theories

In this paper we study the problem of realizability of reactive specifications written in ${\mathrm{LTL}}_T$, which is the extension of LTL where atomic propositions can be literals from a first-order theory, including arithmetic theories. We present a solution based on transforming ${\mathrm{LTL}}_T$ specifications into purely Boolean specifications by (1) substituting theory literals by Boolean variables, and (2) computing an additional Boolean formula that captures the dependencies between the new variables imposed by the literals. We prove that the resulting specification is realizable if and only if the original specification is realizable. Moreover, the resulting specification can be passed to existing Boolean off-the-shelf synthesis and realizability tools, which can handle only Boolean LTL specifications.

A second contribution is to prove that ${\mathrm{LTL}}_T$ realizability of theories with a decidable ${\exists }^{⁎}{\forall }^{⁎}$ fragment is decidable for all combinations of LTL temporal modalities. We present a simple version of our method, which relies on SMT solving, and performs a brute-force search to construct the “extra requirement”. A third contribution is an algorithm that checks whether a candidate is a correct Booleanization in non-Boolean LTL realizability.