Translating formulas of Linear Temporal Logic (LTL) over finite traces, or LTLf, to symbolic Deterministic Finite Automata (DFA) plays an important role not only in LTLf synthesis, but also in synthesis for Safety LTL formulas. The translation is enabled by using MONA, a sophisticated tool for symbolic DFA construction from logic specifications. Recent works used a first-order encoding of LTLf formulas to translate LTLf to First Order Logic (FOL), which is then fed to MONA to construct the symbolic DFA. This encoding was shown to perform well, but other encodings have not been studied. Specifically, the natural question of whether second-order (MSO) encoding, which has significantly simpler quantificational structure, can outperform first-order (FOL) encoding remained open.

In this paper we meet this challenge and study MSO encoding for LTLf formulas. We introduce a specific MSO encoding, and show that this encoding and its simplicity indeed allow more potential than FOL for optimization, thus benefiting symbolic DFA construction. We then show by empirical evaluations that, surprisingly, the FOL encoding outperforms in practice the MSO encodings. We analyze the results and explain how to improve MONA in order to allow the MSO encoding to outperform the FOL encoding.