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In the previous section, we presented the TAS systems of AOP. In this section we will present all theorems that are related to the STAS and ITAS systems only because these two TASes exist in all radices systems. A few will be presented for some OTAS systems. To generate function tables (Truth-Tables) for all the theorems listed below go to the download page and download AOP program which will do the job for you. Notations and Terminology Intrinsic theorems are theorems that describe each prioritor as single entity. Such theorems tell us the properties of each prioritor. Extrinsic theorems describe the functional behavior resulted from the interactions between two prioritors. They describe a pair of two prioritor as a single entity. For example, the commutation property of a prioritor is an intrinsic property but the distribution property of a prioritor is an extrinsic property because not all prioritors are distributable with each other. Global Theorems are theorems that hold true for all TAS systems and for all radii. Local Theorems are theorems that do not hold true for all TAS systems and do not hold true for all radii. For example, the prioritors of the binary system have properties that do not exist in other systems such as A+A─=1. Thus, such a property is called a local intrinsic property. On the other hand Aa a─L=a─L is a global intrinsic property because it holds true for all prioritors and in all An empirical theorem is a theorem that does not have analytical proof but have an experimental proof (tested for specific radii and for specific TAS systems). AOP has a large number of theorems, thus the process of naming each theorem is difficult. Therefore, AOP categorizes theorems into types to simplify the naming process by using an index scheme. So, each theorem in AOP has a formula, name and type. The formula spells out the action of the theorem. The name identifies the theorem and type classifies the theorem. When it is impossible to derive a name for a property we use the type and add an index to it. For example, we say "absorption-II theorem". The "absorption' is type and "II or 2" is the index The basic types used so far by AOP are Static, absorption, and virtual. A theorem is said to be a
static theorem if
one of its sides is constant. For example, A+A─=1 is a static theorem.
A theorem is said to be an absorption theorem
if there is at least one variable in one side that does not appear on the
other side. For example, Aa (Aa*B)=A
In Boolean and Post algebras use the
"S"
symbol to stand for their consecutive operations by OR and MAX operators.
For example, X1+X2+X3+X4
is written as Unlike Boolean
and Post algebras, the number of binary operations in AOP increases very rapidly
in the order of z! as we increase the system radix.
For example, the decimal system has 3.6 million prioritors.
Thus, AOP cannot provide a symbol for each operation.
Instead, AOP uses the `symbol as a global symbol to stand for its
consecutive operations. For example, X1aX2aX3aX4
is written as The intrinsic theorems describe all the characteristics that a prioritor may posses. That is they describe the intrinsic properties of prioritors. They are listed in Table-1 and their proofs is in the appendix.
Local intrinsic theorems are special theorems that hold true for some radices and do not hold true for the others. For example, A+A¯=1 is true for the binary system and not true for other systems.
Extrinsic theorems are properties that exist between two prioritors. This section lists all the extrinsic properties of the STAS system. These properties are global because the STAS system is a global system. A global TAS system may have local extrinsic properties. Such properties are not listed in this section. They are listed under Local extrinsic properties. The STAS system has the form (a, a*). Its base is equal to up-del (D) operator, its mate is star a!=a*, and its comate is costar a?=a#.
Table-1 lists the a and a* of the (a, a*) STAS
systems in the quaternary, ternary and binary digital systems. The first pair in
Table-1
each system corresponds to the (MIN, MAX) in Post algebras
which are (MIN,MAX)=(Q1,Q2), (MIN,MAX)=(T1,T6) and (AND,OR)= (B1,B2).
This sections lists some of the local extrinsic theorems of OTAS systems in AOP for the first three radices: binary, ternary, and quaternary. Table-4 list the theorem type and formula.
For the binary system, all the virtual theorems are satisfied and reduce to the A+(A¾ *B)=A+B or A*(A¾ +B)=A*B in Boolean algebra. For the ternary system, the virtual-I theorem exists in TAST4 and TAST5. Virtual-II theorem exists in TAST5. The Virtual-III theorem exists in TAST2 for s=T5. For
the quaternary system the virtual-I theorem exists in TASTQH, TASTQI, TASTQM,
and TASTQN. virtual-II theorem exists in TASQN. The virtual-III
theorem exists in TASQ4 for s=QH,
TASQ6 for s=QN,
TASQC for s=QM,
TASQE for s=QH,
and TASQK for s=QN.
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