8. Crown Prince Shotoku Said that Peace Is the Most Supreme

How far have we progressed?

This essay is composed of merely a part of results of our study by the present of March 1, 1996 and it is not only that the study is not completed but also that there remains plenty of pages not described except chapter titles. Of course we are not in the stage to write a postscript then providing a simple summary at the present and a prospect from now on, let lay down the pen for the time.

The figure below shows rough conceptual scheme of our Hamiltonian experimentation at the present stage.

• Terminal graph division is a method dividing a graph into right subgraph and left subgraph and proceeding experiments recursively, and it brings drastic effects to reduce experimentation costs. For the reason of searching cost and others, it may be implemented in such restriction that only 2 or 3 terminal should be selected. Terminal graph division is executed on the top layer of Hamiltonian experiments.
  
• Three modes experiment includes three kinds of experiment of IO-division experiment, mesh-diagram experiment and series-parallel experiment and it is implemented principally to support short-circuit graph method. (1)Due to the three modes experiment, some kinds of graphs are separated as non-Hamiltonian graphs by judgment with non-retrial. (2)Making three modes experiment for the original graph or short-circuit graphs, it can detect early contradictions which cannot be detected by factor experiment in the stage. (3)Further, it promotes auto-determination process in operations of arbitrary selection with constraint conditions on three modes experiment.
  
• IO-division experiment is a method dividing a graph into left and right subgraph and examining interfaces of passage points or branches of these two subgraphs and also it includes terminal division experiment and cut-set division experiment. As either of them cannot dispose all divisions on the graph for the reason of costs, it experiments divisions of such branches or points as just determined or removed by arbitrary selection.
  
• Mesh-diagram experiment, giving each points of a graph point value and calculating points score and branches score, can regulate distribution of branches of a graph conforming to the constraint conditions. It is adoptive to experiment a graph of terminal division difficult.
  
• Series-parallel experiment detects topological contradictions of a graph by experimenting a reduced graph obtained by Hamiltonial series-parallel conversion.
  
• Short-circuit graph method is a fundamental experiment for Hamiltonian experimentation based upon factor experiment and owing to the short-circuit method, since each stage of Hamiltonian experiment can be executed recursively as a Hamiltonian experiment of short-circuited and reduced graph, it makes possible to implement three modes experiment into experimental process. Short-circuit method makes thorough examination in all numbers in the worst cases.
  
• Non-temporal experimentation, using retrieval transformation on revival trees, breaks through the limitation of short-circuit method and realizes the experiments never retrograding. In non-temporal experimentation, it is able to generate a retrieval map constituted as a semi-point graph from deadlocked determinative paths of short-circuit graph, to derive revival trees from this map to resolve the contradictions and further to obtain retrieval paths without deadlock. In the case of such experiment as deviates from the range of deadlock map, non-temporal experimentation is executed recursively.
  
• In short-circuit graph method, when it lockouted at the stage where just obtained a Hamiltonian path, after taking out maximum Hamiltonian ties, the switch panel game introduced in the 1st chapter is played out to inquire a Hamiltonian tie from these materials. It is not clear whether switch panel game is able to be realized or not at the present. Switch panel game should be formalized as one small problem of Hamiltonian experiments. It is to be considered that the Hamiltonian experiment problem has it's most peculiar characteristic at the point where dividing the problem into sub-problems does not necessarily make the problem small. In this meaning, we cannot stop regarding that it is a completely non-deterministic problem. However, we suppose that the switch panel game is obviously transferred to other dimension than Hamiltonian experiment problem, and so we confirm that it is hopeful enough to resolve this small problem.
  
• If it is possible to carry out the re-experiments never retrograding by non-temporal experimentation on revival trees, there exists a possibility that we can constitute Hamiltonian experiments with the combination of only short-circuit graph method and non-temporal experimentation (without using three modes experiment). At the present, there remains some parts not formalized yet in the ending phases of non-temporal experiment.
  
• Possibly, such a suggestion might be considered that can't we play the switch panel game itself on the retrieval map? (In other words, to implement the game as non-temporal experiment). We somewhat worry that the composition of the small problem of switch panel game seems resemble to an non-decidable problem known as Post's correspondence problem.
  

What shall we do from now on?

There exist so many matters left behind. Problems not yet resolved (switch panel game, complete judgment with non-retrial), problems with unfinished parts (experiment deviated from the range of deadlock map), problems not formalized yet (re-experimental graph method including non-temporal experiment, cross IO contradiction), problems not obtained their proofs (Hamiltonian series-parallel conversion, etc., problems belong to such category as unsolved, uncompleted, not formalized are of course problems unproved), problems not disposed (experimentation for undirected graphs), problems without formalized algorithms (search algorithms of terminal division, cut-set division), etc. To resolve all of these problems is the present subject.

In these problems, there are such two classes as is able to reach a settlement in time and needs something than time. According the arrangement of the chapter titles of this essay, we anticipate that non-determinability problem of chapter 3, algorithms of chapter 4 and calculation costs of chapter 5 are able to be resolved basically only if times are given.

In parallel with formalizing algorithms, we should compose programs to demonstrate them, and both of such works must be done simultaneously, as actual proofs of rightness of algorithms and measurements of practical abilities. Experiments for such graphs that rare to occur in hand-writing scale sample like cross IO contradiction may be easily realized on a computer. Simulations on computer might be useful to extract peculiar problems not found yet. To optimize the algorithms, especially to inquire the resolutions of graphs with Hamiltonian ties as rapid as possible, some kind of optimization or improvement would come to be necessary.

Dealing with those subjects, how long time should we estimate to solve completely Hamiltonian Circuit Problem? What we understand just now is only that at least the time must be in the order of polynomial time algorithms. On the stage when this essay is completed, we must compose a formally publishable revel paper and publish it. The greatest difficulty at the present is that we are in such status that we lack all of primary elements indispensable to accomplish them and yet we cannot abandon these things.

Assuming that the Hamiltonian Circuit Problem is a problem equals to the method to control one's fortune and a Hamiltonian circuit represents a ring = peace, what we have to prove from now on are enough with the following two propositions.

• Only one method to control the fortune is to select the best way at every moment.
• If there is a problem to be solved for bringing peace, it is the problem of resource distribution.

If it is so, to solve the Hamiltonian Circuit Problem is indeed a matter of time for us.