Probabilistic CTL

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Probabilistic Computation Tree Logic (PCTL) is an extension of computation tree logic (CTL) which 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]

One of the possible syntax of PCTL is defined as follows:

Therein, is 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 of the set of path with the common prefix of length is equal to the product of transitions probabilitites 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]


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