2.2  Well-formed CTL formula

Let us see some examples of well-formed CTL formulas and some examples which are not well-formed, in order to understand the syntax. Suppose that p , q and r are atomic formulas. The following are well-formed CTL formulas:

The following are not well-formed formulas:

i.  Gp : G must be preceded by a path quantifier ( A or E ).

ii.  EFG r: Same reason as for formula (i)

iii.  F[ r U q ]: Until operator must be preceded by a path quantifier ( A or E ).

iv.  AEF r: Path quantifier A should be followed by a temporal connective G,F or X . For example if we write AFEFr or AGEFr, then the formula is well-formed.

v. A[( r U q ) ( p U r )]: Boolean connectives (like or ) cannot be directly inside path quantifier A or E . However we can write A[(r q)U(p r)].

3.  Semantics of CTL

The semantics of CTL is defined over a model ‘ M' , which is defined as 3-tuple , where

•  S : set of states of M .

•  :set of transitions T S X S such that for every s S there exists a state s' such that s s' . In other words, there is at-least one transition emanating from each state.

•  L: labeling for each state and defined by L:Sρ(P), P is set of atomic propositions and ρ(P) is the power set of P.

This model is also known as Kripke structure. A Kripke structure is similar to a state transition diagram, which has states and transitions from one state to another, but includes two extra features, given as follows:

•  A ll states must have at least one outgoing edge.

•  Each state is labeled with one of the element of the power set of atomic propositions. These atomic propositions will be assigned to either true (T) or false (F) values.

Figure 14 shows two state transition diagrams (the labeling of the states is not shown, but assumes that all states have labels with a subset of atomic propositions).Figure 14a is not a Kripke structure because state s6 does not have an outgoing transition. The state transition diagram of Figure 14b is a Kripke structure because it satisfies all the requirements of the model discussed above.