and 
it could be that 
.
TERMS DEFINED (glossary)
Ordered signal process,actor,token,dataflow process,firing signal,firing,order preserving,order embedding,order isomorphism,partially ordered set,prefix,prefix order,poset.
The notation for a tuple of signals in the html version is: ss.
ORDERED SIGNAL PROCESS NETWORKS
Let T(s) denote the tags in the signal s and T(ss) the union of the tags in the signals in the tuple ss.
In an ordered signal process network, T(s) is totally ordered for each signal s but the set of all tags T(ss) is typically partially ordered.
In OSP networks there is no notion of time, but only the notion of ordered sequence of events.
For any two distinct signals and it could be that .
We are interested in maps between ordered sets 
.
f can be:
in P 
in Q
in P 
in Q
f is an order embedding and onto
If we have a signal and there is an order preserving bijection with a subset of the natural numbers 
, we assume that the tags are the natural numbers themselves.
Given two or more signals there is an ordering relation between events in a signal but not between events of different signals.
In OSP networks events are called tokens and a signal is a sequence of tokens with no notion of time.
In the class we went through the following example. It was given an ordered signal process with an input and an output signal, respectively 
and 
. This process consumes one token from the input and produces one token at the output.
From the ordering of tokens in each signal follows that
for 
, 
for k=1,2.
For one token consumed at the input exactly one token is produced at the output, therefore the following inequality holds
for all i.
From the above inequalities 
, but we cannot say anything about the ordering between 
and 
.
Intuitively, in the specification nothing says that the process produces a token before the next token arrives. A total order of the tokens of all the signals of the process would cause an overspecification of the system.
Hasse diagrams (see the textbook) are useful to describe the partial order relations between tokens. In the Hasse diagrams two events are comparable only if there is a path from one to the other.
In the class further examples were described to motivate the introduction of the OSP networks model.
DATAFLOW
A dataflow process, also called an actor, is an OSP with a firing signal.
A firing signal is both an input and an output signal and contains only events that are comparable with all events in other input and output signals.
Firings happen after a set of tokens has been consumed at the input and before a set of tokens produced at the output. Dataflow is a way of implement ing process networks where each process is constructed as a sequence of firings.
Given two successive firings 
:
-an output event e' where 
is said to be produced by firing 
.
-an input event e' where is said to be consumed by firing 
.
Let's consider the merge process.
The output signal preserves the ordering of the tokens of the input signals, but there is no ordering between tokens of different input signals. Therefore there are many possible output signals, each corresponding to a linearization of the partial order of the tokens of all the input signals. The merge process is therefore non determinate.
If we have two processes, the firing signals are partially ordered. We can come up with any sequence of firings which satisfies the partial order constraints in order to implement the system. The partial ordering relation between firings gives the range of possible schedules.
PARTIAL ORDER BETWEEN SIGNALS
if every event in is also in .
if every event in is also in and if there is an event in not in it must occur after all the events in .
Given a set of signals S and a partially order relation E, (S,E) is a partially ordered set or POSET.
If 
is a tuple of signals we have the pointwise extension (,
): given two tuples and , if each component of is prefix of the corresponding component of .
Given a subset of possible signals 
.
is a maximal element of Q if 
, 
is a maximum if 
has a minimum (
), a maximal but not a maximum.