IIST and UNU - UNU-IIST - United Nations University

Linking Duration Calculus and TLA 12
Law 6 (1) ⌈p⌉ = 〈 ∞ ↑ p ↓ [0, 0] 〉
(2) 〈 ∞ ↑ p ↓ ∞ 〉 = ⌈p⌉ ∨ ⌈¬p⌉
(3) ⌈p⌉ ⌈¬p⌉ = 〈 [0, ∞] ↑ p ↓ ∞ 〉 ∧ ¬〈 ∞ ↑ p ↓ ∞ 〉 .
We use an abbreviation P ⊳ p ⊲ Q of choice to denote that two durational specifications P and
Q are controlled by a boolean p : during any interval in which p is always true, P must hold;
during any interval in which ¬p is always true, Q must hold. There is no restriction for those
intervals in which p is sometimes true and sometimes not true.
Def 8 P ⊳ p ⊲ Q ̂= ⌈p⌉ ⇒ P ∧ ⌈¬p⌉ ⇒ Q .
The following law shows that the chop composition of any two durational specifications can
be implemented as a choice between the specifications, and the change of the choice is made
sometime during an interval. If p is independent of P and Q and becomes hidden, the two
sides will be equal.
Law 7 P Q ⊇ P ⊳ p ⊲ Q ∧ 〈 [0, ∞] ↑ p ↓ ∞ 〉 ∧ ¬〈 ∞ ↑ p ↓ ∞ 〉.
General patterns of specifications and their refinement
A durational specification is a predicate P (i, x) on the interval i and the real-time function
x : [0, ∞] → S . A common durational specification is a predicate P (l, ∫ p) on the length l ̂= |i|
of the interval and the integral of a boolean function p during the interval. This reflects the fact
that the controlling of a system is normally independent of the starting time and insensitive to
state changes at isolated time points.
A durational specification D
∫
p f(l) requires the total time of the state satisfying p in
any interval with length l to be bounded by a characteristic function f(l). For example, the
specification D (l 30 ⇒ ∫ leak 4) is a special case of this pattern with the characteristic
function f(l) ̂= 4 ⊳ l ∈ [4, 30] ⊲ l where we use a ⊳ b ⊲ c to denote the value a if b is true, or
the value c otherwise. The dual pattern D
∫
p g(l) is equal to D
∫
¬p (l−f(l)).
The following law shows that two specifications with different characteristic functions may describe
the same requirement.
Law 8 D
∫
p f(l) = D
∫
p f ′ (l) , if any of the following condition is satisfied ( l, l 0 , l 1 0 ):
1. f ′ (l) = min(l, f(l)) ,
2. f ′ (l) = min {f(l ′ ) | l ′ l} ,
3. or f ′ (l) = min {f(l 0 ) + f(l 1 ) | l 0 + l 1 = l} .
Report No. 301,
UNU-IIST, P.O. Box 3058, Macau