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

functional, therefore formally establishing dataflow process network as a special case of
Kahn process network. We give a brief review of that paper for completeness.
A dataflow actor with m inputs and n outputs is a pair { f , R } , where:
1. f: S m S n is a function called the firing function,
2.
m
R ⊆ S is a set of finite sequences called the firing rules,
3. f(r) is finite for all r∈ R,
and
4. no two distinct rr , ′∈ Rare
joinable (i.e. they don’t have a LUB).
A Kahn process F based on the dataflow actor { f , R } can be defined as follows:
⎧ f ( r). F( s′ ) if there exists r∈ R such that s = rs . ′
Fs () = ⎨
⎩Λ
otherwise
where . represents concatenation. Since F is self-referential in this definition, we can not
guarantee that F exists, nor that F is unique. To show its existence and uniqueness, it has
to be interpreted as the least-fixed-point function of functional
m n m n
φ :( S →S ) →( S → S ) defined as
⎧ f ( r). F( s′ ) if there exists r∈ R such that s= rs . ′
( φ(
F))( s)
= ⎨
⎩Λ
otherwise
m n
where ( S → S ) represents the set of functions mapping m
S to
18
n
S . It can be proved
m n
that ( S → S ) is a CPO under pointwise prefix order. Another similar notation
m n
[ S → S ] represents the set of continuous functions mapping m
S to
n
S , and it can also
m n m n
be proved to be a CPO under pointwise prefix order. Obviously [ S →S ] ⊂( S → S ) .
It can be proved that φ is both monotonic and continuous (see [8]). Again using the same
theorem, φ has a least fixed point (which is a function in this case) and there is a