Dynamic Dataflow Modeling in Ptolemy II - Ptolemy Project ...

A partially-ordered set (poset) is a set S where an ordering relation (denoted ) is
defined that is reflexive ( ∀s∈S, s s),
transitive ( ∀ss , ′ , s′′ ∈Ss , s′ , s′ s′′ ⇒s
s′′
)
and anti-symmetric ( ∀ss , ′ ∈Ss , s′ , s′ s⇒ s= s′
). The bottom element of a poset S, if
it exists, is an element s∈ S that satisfies ∀s′ ∈ S, s s′
. An upper bound of a subset
W ⊆ S is an element s∈S that satisfies ∀w∈W, w s.
A least upper bound (LUB) of a
subset W ⊆ S is an upper bound s such that for any other upper bound s′ , s s′ . A
chain in S is a totally ordered subset of S, i.e., any two elements in the subset are ordered.
A complete partial order (CPO) is a poset with a bottom element where every chain has a
LUB.
There is a natural ordering relation in the set S consisting of all finite and infinite
sequences. It is the prefix order, defined as ∀ss , ′ ∈ Ss , s′
if s is a prefix of s′ . This
definition easily generalizes to S n element-wise. This turns out to be a very useful
ordering relation in defining the denotational semantics of Kahn process network because
under this ordering S n is a CPO and there is strong theorem for continuous functions
defined on a CPO.
A function F: S m S n m
is monotonic if ∀ss , ′ ∈S, ss′ ⇒Fs
( ) Fs ( ′ ) . This expresses
an untimed notion of causality since according to the definition, giving additional inputs
can only result in additional outputs. A function F: S m S n is continuous if for every
m
chain W ⊆ S , F( W)
has a LUB and FLUBW ( ( )) = LUBFW ( ( )) . Here F(W) denotes a
set obtained by applying the function F to each element of W. Intuitively continuity
means the response of the function to an infinite input sequence is the limit of its
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