Probabilistic Computation Tree Logic (PCTL) is an extension of computation tree logic (CTL) that allows for probabilistic quantification of described properties. It has been defined in the paper by Hansson and Jonsson.[1]
PCTL is a useful logic for stating soft deadline properties, e.g. "after a request for a service, there is at least a 98% probability that the service will be carried out within 2 seconds". Akin CTL suitability for model-checking PCTL extension is widely used as a property specification language for probabilistic model checkers.
PCTL syntax
A possible syntax of PCTL can be defined as follows:
Therein,
is a comparison operator and
is a probability threshold.
Formulas of PCTL are interpreted over discrete Markov chains. An interpretation structure is a quadruple
, where
is a finite set of states,
is an initial state,
is a transition probability function,
, such that for all
we have
, and
is a labeling function,
, assigning atomic propositions to states.
A path
from a state
is an infinite sequence of states
. The n-th state of the path is denoted as
and the prefix of
of length
is denoted as
.
Probability measure
A probability measure
on the set of paths with a common prefix of length
is given by the product of transition probabilities along the prefix of the path:
![{\displaystyle \mu _{m}(\{\sigma \in X:\sigma \uparrow n=s_{0}\to \dots \to s_{n}\})={\mathcal {T}}(s_{0},s_{1})\times \dots \times {\mathcal {T}}(s_{n-1},s_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e939811ebfdad1d2fcd35004df5b3fc7129f9d63)
For
the probability measure is equal to
.
Satisfaction relation
The satisfaction relation
is inductively defined as follows:
if and only if
,
if and only if not
,
if and only if
or
,
if and only if
and
,
if and only if
, and
if and only if
.
See also
- Computation tree logic
- Temporal logic
References
- ^ Hansson, Hans, and Bengt Jonsson. "A logic for reasoning about time and reliability." Formal aspects of computing 6.5 (1994): 512-535.