Hennessy–Milner logic


In computer science, Hennessy–Milner logic is a dynamic logic used to specify properties of a labeled transition system, a structure similar to an automaton. It was introduced in 1980 by Matthew Hennessy and Robin Milner in their paper "On observing nondeterminism and concurrency".
Another variant of the HML involves the use of recursion to extend the expressibility of the logic, and is commonly referred to as 'Hennessy-Milner Logic with recursion'. Recursion is enabled with the use of maximum and minimum fixed points.

Syntax

A formula is defined by the following BNF grammar for Act some set of actions:
That is, a formula can be
; constant truth : always true
; constant false : always false
; formula conjunction
; formula disjunction
; formula : for all Act-derivatives, Φ must hold
; formula : for some Act-derivative, Φ must hold

Formal semantics

Let be a labeled transition system, and let
be the set of HML formulae. The satisfiability
relation relates states of the LTS
to the formulae they satisfy, and is defined as the smallest relation such that, for all states
and formulae,
  • ,
  • there is no state for which,
  • if there exists a state such that and, then,
  • if for all such that it holds that, then,
  • if, then,
  • if, then,
  • if and, then.