Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

122 CHAPTER 6. HML WITH RECURSION
F U s G min
= G ∨ (F ∧ 〈Act〉tt ∧ [Act](F U s G)) , and
F U w G max
= G ∨ (F ∧ [Act](F U w G)) .
It should be clear that, as the names indicate, strong until is a stronger condition
than weak until. We can use the ‘until’ operators to express Even(F ) and Inv(F ).
In fact, Even(G) ≡ tt U s G and Inv(F ) ≡ F U w ff.
Properties like ‘some time in the future’ and ‘until’ are examples of what we
call temporal properties. Tempora is Latin—it is plural for tempus, which means
‘time’—, and a logic that expresses properties that depend on time is called temporal
logic. The study of temporal logics is very old and can be traced back to Aristotle.
Within the last 30 years, researchers in computer science have started showing
interest in temporal logic as, within this framework, it is possible to express properties
of the behaviour of programs that change over time (Clarke, Emerson and
A.P. Sistla, 1986; Manna and Pnueli, 1992; Pnueli, 1977).
The modal µ-calculus (Kozen, 1983) is a generalization of Hennessy-Milner
logic with recursion that allows for largest and least fixed point definitions to be
mixed freely. It has been shown that the modal µ-calculus is expressive enough
to describe any of the standard operators that occur in the framework of temporal
logic. In this sense by extending Hennessy-Milner logic with recursion we obtain
a temporal logic.
From the examples in this section we can see that least fixed points are used to
express that something will happen sooner or later, whereas the largest fixed points
are used to express invariance of some state of affairs during computations, or that
something does not happen as a system evolves.
6.2 Syntax and semantics of HML with recursion
The first step towards introducing recursion in HML is to add variables to the syntax.
To start with we only consider one recursively defined property. We will study
the more general case of properties defined by mutual recursion later.
The syntax for Hennessy-Milner-logic with one variable X, denoted by M {X},
is given by the following grammar:
F ::= X | tt | ff | F1 ∧ F2 | F1 ∨ F2 | 〈a〉F | [a]F .
Semantically a formula F (that may contain a variable X) is interpreted as a function
OF : 2 Proc → 2 Proc that, given a set of processes that are assumed to satisfy
X, gives us the set of processes that satisfy F .