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 edit

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 edit

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:

 

For   the probability measure is equal to  .

Satisfaction relation edit

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 edit

References edit

  1. ^ Hansson, Hans, and Bengt Jonsson. "A logic for reasoning about time and reliability." Formal aspects of computing 6.5 (1994): 512-535.