The
Universality of the NULL Function
Gates
vary in their behavior only for the data function.
The NULL function is identical for all gates.
Any arbitrary combination of gates will behave
identically with the NULL wavefront. They will
all transition to NULL. So the NULL wavefront
behavior can be ignored in the specification of
a combinational expression. Only the behavior
of the data wavefront need be considered when
specifying a combinational expression.
Completeness
of Participation
If
there is a don't care signal anywhere in the expression,
one which may or may not transition, it would
not be possible to logically determine when a
resolution is complete. So every gate and every
signal path of the combinational expression effectively
participates in every wavefront resolution. This
will be referred to as completeness of participation.
Summary
of 4NCL Completenesses
There
are a number of completeness relationships that
contribute to the logical determination of 4NCL.
Completeness of transition between disjoint
logical value domains.
Completeness
criterion.
Completeness
of participation.
Singleform
completeness of expression
Relationship
of 4NCL to Boolean Logic
Since
4NCL uses TRUE and FALSE to differentiate data
values, 4NCL can be directly mapped gate for gate
with a Boolean logic expression to form a 4NCL
combinational expression as shown in Figure
6.
Figure
6. Boolean logic full adder and the corresponding
4NCL full adder.
4NCL
as a Standard of Reference
4NCL
can be used as a standard of logical completeness
for combinational expression. As an expression
deviates from 4NCL logical completeness, the deviations
can be characterized as compromises of the completeness
relationships. In this way it is possible to see
more clearly the inherent nature of other forms
of expression and to relate them to each other.
In
this instance the consequences of reducing the
number of logic values will be considered.
Can
a 3 value logic be logically complete?
The
four value logic can be reduced to a three value
logic that expresses the completeness criterion
by replacing the expression of the INTERMEDIATE
value with a hysteresis behavior of the gate as
shown in Figure 7. This forms
a three value NULL Convention Logic or 3NCL.
Figure
7. Correspondence of 4NCL and 3NCL.
4NCL 
3NCL 
TRUE 
TRUE 
FALSE 
FALSE 
INTERMEDIATE 
Hysteresis
Behavior 
NULL 
NULL 
In
4NCL, the INTERMEDIATE value indicates a partial
completion state to be ignored by the watcher.
The partial completion states can be effectively
ignored with a hysteresis behavior or state holding
behavior of each gate. When the input of a gate
is completely data, the gate outputs a data value
and maintains the data value until the input of
the gate is completely NULL, whereupon, the gate
outputs a NULL value and maintains the NULL value
until the input of the gate is completely data,
and so on. Instead of being explicitly expressed
as a logic value at the gate output, the partially
complete states are ignored by the gate output.
Only wavefronts of NULL values and data values
flow through the gates and through the expression.
This
hysteresis behavior can be expressed logically
as a feedback relationship making the gate a state
machine. The truth tables expressing the hysteresis
behavior for the 3NCL AND gate, OR gate, XOR gate
and NOT gate are shown in Figure
8. The NOT gate is a single input gate that
cannot posses any intermediate input states so
it does not require hysteresis to explicitly express
completeness for both data and NULL. The input
R value is the output result value fed back to
the input. The gate data function is expressed
by the values inside the bold boxes. This feedback
path includes a time relationship that must be
expressed to completely characterize the behavior
of a combinational expression.
Figure
8. 3NCL gates.
