A Symbolic Approach to Safety ltl Synthesis

Abstract

Temporal synthesis is the automated design of a system that interacts with an environment, using the declarative specification of the system's behavior. A popular language for providing such a specification is Linear Temporal Logic, or LTL. LTL synthesis in the general case has remained, however, a hard problem to solve in practice. Because of this, many works have focused on developing synthesis procedures for specific fragments of LTL, with an easier synthesis problem. In this work, we focus on Safety LTL, defined here to be the Until-free fragment of LTL in Negation Normal Form (NNF), and shown to express a fragment of safe LTL formulas. The intrinsic motivation for this fragment is the observation that in many cases it is not enough to say that something "good" will eventually happen, we need to say by when it will happen. We show here that Safety LTL synthesis is significantly simpler algorithmically than LTL synthesis. We exploit this simplicity in two ways, first by describing an explicit approach based on a reduction to Horn-SAT, which can be solved in linear time in the size of the game graph, and then through an efficient symbolic construction, allowing a BDD-based symbolic approach which significantly outperforms extant LTL-synthesis tools.

Incremental Determinization for Quantifier Elimination and Functional Synthesis

“…More recent works revisited the use of BDDs, e.g. the tools SSyft [25] and RSynth [26,27]. This motivated the search for alternatives to BDDs [8][9][10].…”

First-Order vs. Second-Order Encodings for $$\textsc {ltl}_f$$-to-Automata Translation

“…First-order encoding was shown to perform well in the context of LTL f -to-automata translation [30], but other encodings have not been studied. Specifically, the natural question of whether second-order (MSO) outperforms first-order in the same context remained open.…”

“…The reason is that those algorithms require determinization of automata on finite words (rather than infinite words), and solving reachability games (rather than parity games) [8]. Another application, as shown in [30], is that temporal synthesis of Safety LTL formulas, a syntactic fragment of LTL expressing safety properties, can be reduced to reasoning about finite words (see also [18,19]). This approach has been implemented in [31] for LTL f synthesis and in [30] for synthesis of Safety LTL formulas, and has been shown to outperform existing temporal-synthesis tools such as Acacia+ [2].…”

“…The key algorithmic building block in these approaches is a translation of LTL f to symbolic Deterministic Finite Automata (DFA) [30,31]. In fact, translating LTL f formula to DFA has other algorithmic applications as well.…”

Eventually Safe Languages

“…Considering restricted classes of specifications is a classical way to try to tackle the difficulty of the synthesis problem. The classes of safety and liveness properties [1] have gathered particular interest [19,21], as they simplify algorithms while expressing typical requirements on reactive systems. We introduce a class of properties called "eventually safe" and noted ESafe, for which we give a specification language E νTL and an algorithm systematically producing GFG automata from this language.…”