Probabilistic Automata for Safety LTL Specifications

Automata constructions for logical properties play an important role in the formal analysis of the system both statically and dynamically. In this paper, we present constructions of finite-state probabilistic monitors (FPM) for safety properties expressed in

LTL

. FPMs are probabilistic automata on infinite words that have a special, absorbing reject state, and given a cut-point

λ

 ∈ [0,1], accept all words whose probability of reaching the reject state is at most 1 − 

λ

. We consider

Safe-LTL

, the collection of

LTL

formulas built using conjunction, disjunction, next, and release operators, and show that (a) for any formula

ϕ

, there is an FPM with cut-point 1 of exponential size that recognizes the models of

ϕ

, and (b) there is a family of

Safe-LTL

formulas, such that the smallest FPM with cut-point 0 for this family is of doubly exponential size. Next, we consider the fragment

LTL

(

G

) of

Safe-LTL

wherein always operator is used instead of release operator and show that for any formula

ϕ

 ∈

LTL

(

G

) (c) there is an FPM with cut-point 0 of exponential size for

ϕ

, and (d) there is a robust FPM of exponential size for

ϕ

, where a robust FPM is one in which the acceptance probability of any word is bounded away from the cut-point. We also show that these constructions are optimal.