LTL Reactive Synthesis with a Few Hints

Abstract

We study a variant of the problem of synthesizing Mealy machines that enforce LTL specifications against all possible behaviours of the environment including hostile ones. In the variant studied here, the user provides the high level LTL specification ϕ of the system to design, and a set E of examples of executions that the solution must produce. Our synthesis algorithm works in two phases. First, it generalizes the decisions taken along the examples E using tailored extensions of automata learning algorithms. This phase generalizes the user-provided examples in E while preserving realizability of ϕ. Second, the algorithm turns the (usually) incomplete Mealy machine obtained by the learning phase into a complete Mealy machine that realizes ϕ. The examples are used to guide the synthesis procedure. We provide a completness result that shows that our procedure can learn any Mealy machine M that realizes ϕ with a small (polynomial) set of examples. We also show that our problem, that generalizes the classical LTL synthesis problem (i.e. when E = ∅), matches its worst-case complexity. The additional cost of learning from E is even polynomial in the size of E and in the size of a symbolic representation of solutions that realize ϕ. This symbolic representation is computed by the synthesis algorithm implemented in Acacia-Bonzai when solving the plain LTL synthesis problem. We illustrate the practical interest of our approach on a set of examples.

LTL Reactive Synthesis with a Few Hints

“…As a consequence, L(P 1 ) ⊆ L(P 2 ) and L ω (P 1 ) ⊆ L ω (P 2 ). Given a preMealy machine P, we say that a specification S is P-realizable if there exists a Mealy machine M such that P M and M realizes S. Note that if P is a (complete) Mealy machine, S is P-realizable iff P realizes S. The next result is proved in [5]: Theorem 2. Given a universal co-Büchi automaton A with n states defining a specification S = L ∀ (A) and a preMealy machine P with m states and n h holes, deciding whether S is P-realizable is ExpTime-hard and in ExpTime (in n and polynomial in m).…”

“…State merging is a standard operation in automata learning algorithms which intuitively means that merging the ∼-class of e and the ∼-class of e , and propagating this merge to the descendants of e and e , does not result any conflict. The formal definition is in [5]. At line 5, it filters the previous set by keeping only the states which, when merged with e, produce a preMealy machine which can be completed into a Mealy machine realizing S (again by Thm.…”

“…Amongst those candidates, a single one is selected and the transition on (m, i) is added to M accordingly at line 10. given as a deterministic safety automaton, a completion strategy σC Output: A (complete) Mealy machine M such that S is M0-realizable, otherwise UNREAL. The pseudo-code of SynthSafe can be found in [5]. The completion procedure may not terminate for some completion strategies.…”

“…The proof of the latter theorem is a consequence of several results proved on the generalization and completion phases, and is given in [5].…”

“…i∈{1,2} (r i R¬g i ). Strix, on the specification ϕ ME CORE ∧ ϕ ME LOW , provides us with a better solution, but it is more complex than needed (it has 9 states: refer [5]) and clearly does not look like an optimal solution to our mutual exclusion problem. E.g., the model of Fig.…”