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Temporal logic of resource cumulation 6
3 Temporal logic of resource cumulation
Resource cumulation
Many aspects of computing can be modelled as the cumulation of resources. In real-time computing,
time is a kind of resource. A process “consumes” a non-negative amount of time. A
computation may also produce resources. For example, a reactive process generates an increasingly
longer sequence of intermediate states called a trace. Resource cumulation can be
formalized as a quintuple called a cumulator: (X, ; 0, ⌢ ; | · |) , which consists of three
parts: a well-founded partial order (X, ) in which each element is called a cumulation and
the greatest lower bound exists for any non-empty subset, a monoid (0, ⌢ ) in which 0 , or
zero cumulation is the least cumulation, and a monotonic and associative binary operation concatenation
⌢ corresponds to the addition of cumulations, and a monotonic and strict volume
function | · | : X → [0, ∞] : We assume that the partial order and the monoid are consistent:
a b ⇔ ∃c∈X · a ⌢ c = b . The unusual part of a cumulator is the volume function. A volume
function measures the amount of resource cumulated. With such additional information we can
then reason about the dynamics of resource cumulation. For example, a resource is exhausted
when its volume reaches infinity ∞ . The use of volume functions can substantially simplify the
reasoning of limit points, continuity, and other topological properties. Such modelling is aimed
at avoiding complicated domain construction and has reflected our pragmatic view on resources.
For a more complete account of resource cumulation, please refer to [3].
Example: The amount of time that a computation consumes can be modelled as a cumulator:
RTime ̂= ([0, ∞], ; 0, + ; id) where + is addition. id is the identity function.
Example: In some applications, we are interested in temporal properties over a period of time
and thus need to reason about temporal intervals. Intervals form a cumulator Interval ̂= (I,
; ∅, ⌢ ; | · |) where I denotes the set of intervals, each of which is a convex subset i of the
real domain [0, ∞] (such that for any t 1 , t 2 ∈ i and t 3 ∈ T , t 1 t 3 t 2 implies t 3 ∈ i ). For
example, [1, 2] , [1, 2) , (1, 2] , (1, 2) and the empty set ∅ are intervals. Let I denote the set of
all intervals. a ⌢ b ̂= a ∪ b if a ∩ b = ∅ , ⊔a = ⊓b and a ∪ b ∈ I . The volume of a non-empty
interval is its length: |a| ̂= ⊔a − ⊓a where ⊔a and ⊓a denote the lub and glb of the interval
a respectively. The volume of the empty set is zero |∅| = 0 . The orders a b means that b is
a right-hand extension of a , i.e. ∃c ∈ I · a ⌢ c = b .
Example: Finite and infinite traces form a typical cumulator: Trace(X) ̂= (X † , ; 〈〉, ∧ ; |·
|) where X is the type of each element, and X † the set of all sequences of elements (including
the infinite ones). For two sequences a, b ∈ X † , a ∧ b denotes their concatenation. If a is an
infinite sequence, then for any b, a ∧ b = a . a b iff a is a prefix (i.e. pre-cumulation) of b . |a|
denotes the length of a . For exampe, the length of the empty sequence 〈〉 is 0. a i denotes the
i -th element of the sequence where 1 i |a| .
Report No. 301,
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