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Temporal logic of resource cumulation 9
another state satisfying ¬p in some time t ∈ U and then moves back to a state satisfying p in
some time t ∈ V where U, V ⊆ [0, ∞] . The two-state automaton can be formalised as follows:
(2)
Automaton(p, U, V ) ̂=
K [(p(s) ∧ ¬p(s ′ ) ∧ t ′ −t ∈ U) ∨ (¬p(s) ∧ p(s ′ ) ∧ t ′ −t ∈ V )] (s,t) .
Figure 2: Automaton with two states
Duration calculus (DC) is a special interval logic. A durational specification is a predicate on a
variable i ∈ I that denotes the interval and an auxiliary variable x : [0, ∞] → S that denotes a
real-time Boolean function. We use a boolean function p : S → {0, 1} to denote whether a state
x(t) at the time t satisfies the predicate p(·) . The space of durational specifications is denoted
by D.
Again, we may introduce some dependent variables. For example, instead of specifying the
relation (i.e. a predicate) between the interval and the real-time function, we may specify the
relation between the length of the interval and the integral of the real function in the interval.
Although not all computation can be specified in such a restricted way, it has been expressive
enough for most applications and covers most common design patterns [13]. The following table
lists the primitives of DC:
P (l, ∫ p) general pattern
⌈p⌉ lift
D P modality of sub-interval closure
P Q chop operation
P ∨ Q logical disjunction
¬P negation
For example, the Gas Burner problem [13] includes a requirement that gas leak is bounded by
4 for any interval no longer than 30. This can be formalised as a specification in DC:
(3)
D (|i| 30 ⇒ ∫ i
Leak(x(t)) dt 4)
Report No. 301,
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