Lecture Notes in Computer Science 3472

538 Séverine Colin and Leonardo Mariani
and the gate set on closing; is flashing is evaluated to false, is off is evaluated
true, is closing is evaluated to true and the final result is false, therefore the
formula has been violated.
Exercise 18.2 (Algorithm Application). Apply the BTT-FMS of Fig 18.4 on the
following traces and determine the resulting truth value:
• flashing, closing, closed, opening, open, closing, off
• flashing, off, closing, closed, opening, flashing, open
When a new event is received, a BTT-FSM only need to evaluate at most all
the propositions to compute the next state, so at worst the run-time overhead
is linear with respect to the number of distinct variables. The size of the BTT-
FSMs can become a problem when storage is a scarce resource, hence particular
attention is given to the generation of optimal BTT-FSMs. However, the number
of propositions to be evaluated tends to decrease with the number of states, so
the overall monitoring overhead is also reduced. This algorithm is recommended
in situations where the monitored LTL formulas are relatively small in size.
Future time propositional LTL may not be the most appropriate formalism
for logic based monitoring due to the conceptual contradiction between the finite
past traces and the logic expressiveness referring to infinite future. Past time LTL
can be a more natural logic for run-time monitoring, in fact safety requirements
are usually expressed by means of past events.
18.5.2 PathExplorer and Past Time LTL
Before presenting the algorithm used by PathExplorer to verify requirements
specified by past time LTL [HR02], we briefly remind the reader of the basic
notions of finite trace linear past time temporal logic, and introduce some
operators used by PathExplorer.
Past Time LTL Syntactically, PathExplorer allows the following formulas,
where A is a set of “atomic propositions”:
F ::= true | false | A |¬F | FopF Propositional operators
⊙F | ✸· F | ⊡F | F SS F | F SW F Standard past time operators
↑ F |↓F | [F , F )S | [F , F )W
Monitoring operators
The propositional binary operators op are the standard ones, and the other
operators should be read :
•⊙F as “previously F ”,
• ✸· F as “eventually in the past F ”,
• ⊡F as “always in the past F ”,
• F1 SS F2 as “F1 strong since F2”,
• F1 SW F2 as “F1 weak since F2”,
•↑F as “start F ”,
•↓F as “end F ”,
• [F1, F2) as“intervalF1, F2”.