IIST and UNU - UNU-IIST - United Nations University

Temporal logic of resource cumulation 8
Examples of temporal logics
The amount of time that a computation consumes corresponds to the cumulator RTime. A
real-time specification is a predicate on a typed modal variable t ∈ [0, ∞] that denotes time and
an untyped auxiliary variable s that denotes the system’s state at the time. We let T denote
the space of such specifications. Since addition is commutative i.e. a + b = b + a, it makes no
difference whether time is extended from the left-hand side or the right-hand side. For example,
e t = x described a system’s temporature growing exponentially over time.
Intervals within a time domain form the cumulator Interval . A specification on intervals is
a predicate on a variable i ∈ I that denotes the interval and an auxiliary variable x that denotes
some system feature related to the interval. We let I denote the space of all temporal
specifications on intervals. An interval can be extended from either left-hand side or right-hand
side.
Traces of elements of X form a cumulator Trace(X) . A trace specification is a predicate on
a single variable tr ∈ X † . We let S denote the space of trace specifications. For example, the
specification |S|