Lecture Notes in Computer Science 3472

562 Therese Berg and Harald Raffelt
insert coin refuse coin
insertcoffee spendcoffee
insert coin
spendcoffee
Fig. 19.2. Example Labeled Transition System
a certain atomic proposition holds, starting from the initial state?”. Temporal
logics are logical formalisms designed for expressing such properties. There are
two kinds of temporal logics, linear-time and branching-time. Linear-time logics
are concerned with paths and treat each possible execution-path independently,
branching-time logics, on the other hand, describe properties that depend on the
branching structure of the model. The pros and cons of both logics are compared
by Moshe Y. Vardi [Var01]. Both temporal logics have different expressiveness
and therefore the kind of properties a model checker can prove depends on the
choice of the underlying temporal logic. As an example, consider the two rooted
labeled transitions systems in Figure 19.3, showing two different vending machines
offering coffee and tea. Both machines serve coffee or tea after a coin
has been inserted, but the right machine decides internally whether to serve
coffee or tea, in contrast to the left machine which leaves the decision to the
customer. Both machines have the same set of computations (maximal paths):
{(coin, coffee) , (coin, tea)}. Unfortunately they can not be distinguished in
linear-time logics, since in linear-time logics each path is examined separately.
Branching-time logic, in contrast, can distinguish these two machines, since it
is possible to express properties like “a coffee action is possible after any coin
action”.
The choice of using linear-time or branching-time logic depends on the properties
to be analyzed. Linear-time logics are preferred when only path properties
are of interest, as when analyzing data-flow properties, like dead-locks.
Branching-time logics are often better for analyzing reactive systems, due to
their greater selectivity.
Linear Temporal Logic (LTL)[Pnu77] can be seen as the “standard” lineartime
logic. It is often presented in a form to be interpreted over Kripke structures.
Its formulas are constructed as follows:
φ ::= true | p |¬φ | φ ∧ φ | X (φ) | φUφ