To describe the morphology of tables it is necessary to explain how 1's and 0's are distributed in rows and columns. Notice that the values under the main column p, referring to the table laid out in an earlier section, changes from 1 to 0 half way through. If this point is considered the folding point of the table, observe that all of p is reflected at a reduced scale in the upper and lower portion of the q column and q is reflected in r. A similar quality appears in resulting columns of a 2-trace operation with the difference that the pattern now merges aspects of the truth table constructed for atomic sentences. This is expected since each value in a row is the product of the 2/3 operation. Notice that truth tables identify the objects which are traces and also, because of the ordering, the order in which the partitioning is performed. L-systems, as a systematic approach, can highlight the partitioning mechanism. To determine the role of partitioning in the identification of traces may provide clues in predicting what kind of object trace algorithms can produce. This may also help in identifying what is intrinsic to traces or if there is such a thing.
The first task is to map a geometry to the principal column. Figure 5 shows the end of a segment (the generator) placed perpendicular (the initiator) to the middle of a line . The generator splits a segment to symbolize where values change from 1's to 0's in the p column. We associate this action to a 2-partitioning. Each production uses a similar geometry. The production of r rewrites on the production of q and the same rules apply to q on r. Figure 6 shows the mapping of columns to lines as described above. Each perpendicular segment highlight a transition between 1's and 0's or black and blank spaces, referring to the visualization of truth tables. So for main column p there is 1 segment for 1 transition; for main column q there are 3 transitions and so on. Notice that each segment has a corresponding length to events of sequence of 1's and 0's such that a sequence of four 0's will yield a longer line than a sequence of two 0's, this equally applies to 1's. The only column which is truly represented is p as a single line that traverses the whole system, so as to give a certain orientation and a meaning to the building of perpendicular lines. This is how an L-system for trivial traces using three atoms is build in this experiment.
Non-trivial traces are produced by performing a trace operation on trivial traces. The resulting morphology is one which encompasses pieces of the initial truth table and additional blank and black spaces. It is known that the process of partitioning is done systematically, independently of the objects which are operated on. However looking at truth tables for traces it not obvious how to extract the partitioning mechanism from the resulting object which are traces. Since the basic structure of the l-system which has been built for trivial traces refers to transitions between 1's and 0's, it is consistent to pursue this approach in defining subsequent rules to express the rest of the table. There are obvious self similarities, which have been highlighted in figures 1 to 4. The challenge is to establish what those similarities are and how are they distributed in the system.
The systematic way in which L-systems rewrite their geometry seems to parallel the development of truth tables for traces. The kind of apical growth that has been defined in an L-system for trivial traces follows the natural evolution of the representation of truth-tables that we already have. The remaining task is to find a geometry more appropriate to representing the remainder of the table. Since its been observed that resulting traces are made of trivial traces plus something or other, it follows that the geometry which is to be fed would be the structure for trivial traces and that the something part will be taken care of by the natural evolution of the system.
Rules are given such that a black box is created as the generator and is later filled by the trivial traces l-system' structure. The new system cycles through three productions. The result is then fed as a new generator, and so on. Figure 7shows a production which is equivalent to 1 iteration of a 2-trace on three atoms. The mapping of transitions from 1's to 0's, for the non-trivial construction of the truth table, to the L-system is highlighted and so is the recipe for the related L-system. It is possible to state the approach in more formal terms:
and
all 
-partition of ,
It is possible to say that this recipe applies to any truth table for traces.
The analysis of the resulting picture shows that the mapping of segments to the 2-trace on 3 atoms' truth table is not entirely successful. Although the mapping for main columns p and q is satisfactory however the generation of segments for main column r produces 2 segments too many comparing to transition points between 1's and 0's on the related truth table.
The generation for the rest of the table follows instructions given by the table. However a 2-trace on 3 atoms yields only one result. It is therefore difficult to see if the repeated pattern given by a second production is relevant in answering questions about traces. The L-system might be showing isomorphisms that are overlapped in the truth table representation. The reason being that the order of the atomic sentences never changes in the partitioning process such that p will always represent the first main column, q the second and so on. Subsequently, for example, the representation of the p,q set will merge with the representation of the q,p set. This kind of ordering is absent in L-systems because the productions are simultaneously calculated.
The experiment is repeated for a 2-trace on 4 atoms. The L-system is produced under similar premises as for the L-system for a 2-trace on 3 atoms. Examination of the results show that the production for main column r yields 2 segments too many, as in the 3 atoms case, and that the production for main column s yields 12 segments to many. This kind of discrepancy is expected since the 2 extra segments in the production for r will further extra segments exponentially. Figure 8 highlights the mapping for a 2-trace on 4 atoms as in figure 7.
A 2-trace on 4 atoms generates 4 new results, represented as 4 new columns in the truth table. The morphology of the first result shows a repetition in pattern of a peace of main column q, at a reduced scale, showing 3 transition form 1's to 0's. The second result shows 7 transitions, as in main column r, and so do the subsequent results, even though the particular arrangement of the blank and black spaces do vary.
The strategy to represent the non-trivial results of the truth table for the 4-atom case, is to produce a representation for the first result and then add a production to represent the subsequent results. Figure 9 indicates where this happens in the recipe as well as the resulting L-system. A mapping from truth table to L-system is also highlighted.
Notice that only one production is generated for results that yield 7 transitions of 1's to 0's. However this production is repeated 3 times on the production of the result representing 4 transitions. This is consistent with the representation given by the table. Again the extra production can be interpreted as representations of isomorphisms.
The difference between truth table and L-system might not be in contradiction with the theoretical objects which are traces. After all Truth tables are a particular representation of traces and should not be taken as an ultimate point of reference.
