Weighted safety

Safety properties, which assert that the system always stays within some allowed region, have been extensively studied and used. In the last years, we see more and more research on quantitative formal methods, where systems and specifications are weighted. We introduce and study safety in the weighted setting. For a value v ∈ ℚ, we say that a weighted language L : ∑* → ℚ is v-safe if every word with cost at least v has a prefix all whose extensions have cost at least v. The language L is then weighted safe if L is v-safe for some v. Given a regular weighted language L, we study the set of values v ∈ ℚ for which L is v-safe. We show that this set need not be closed upwards or downwards and we relate the v-safety of L with the safety of the (Boolean) language of words whose cost in L is at most v. We show that the latter need not be regular but is always context free. Given a deterministic weighted automaton A, we relate the safety of L(A) with the structure of A, and we study the problem of deciding whether L(A) is v-safe for a given v. We also study the weighted safety of L(A) and provide bounds on the minimal value |v| for which a weighted language L(A) is v-safe.