Duality

In response to the previous post, I may have some insight on the structure of the oracle spaces. This is not rigorous and the definitions will continue to evolve.

Lets say that we have constructed our oracle space and SAT space. We can now transform the SAT space into an oracle space for itself, but witness satisfiability of those expressions from our previous oracle space.
This would in essence, take each conjunction and expand to be itself a combinatorial space from this set of k, generating 2^k combinations. We could then consider if one of these strands would be satisfiable by witnessing through the terms of our previous oracle space.

I still favor the previous method, as it lessens confusion, and for the instance that one of these new SAT spaces, our old oracle space, could overlap, thus providing false witnessing. That is not good! Although with rigor, I do believe this is the same arena that our duality will occur; with refinement and time, determined to be finite by our finite space bounds...

Fragments of themes

You see, when I began developing my axioms, I was not sure that they would work in every instance; the language was created to be simple ensuring flexibility and strong syntactical witnessing features.

One particular instance deals with querying a combinatorial space oracle, for the extent we will refer to this as an EXPSPACE oracle. This space contains k disjunctive strings of all permutations of particular set of n objects, with the relationship k = 2^n.

Now say with a complement of this set of n objects we construct an expression, it may take on many forms, for the time being we will concentrate on nSAT, this expression can be represented in CNF, just as our full EXPSPACE. Although by the complement, we have that binding of sites will not occur until execution.

This is where the problem gets interesting. Suppose that we construct both an EXPSPACE and nSAT, composed of L and complement(L), respectfully; and these two will decide on themselves by the syntax of the construction of the objects and ultimately by the physical description of the complement function. We are posed with the question: If one of these strings from nSAT meets with its corresponding satisfying member, what would stop the small bit length from going to where a larger bit length string could have obtained a stronger satisfiability assignment, (of course we assume that a small bit length has lesser degree of conjunctions than a string with a larger bit length, this may not be true in an application!).

So I consider the cases:
(Case 1): It does not make a difference, you are querying an oracle, it should have the same output regardless.
(Case 2): It does make a difference, the longer strings should have priority, allowing for shorter assignments to take places at a later iteration in the runtime.

Both cases are promising, and (Case 1) is favorable, but (Case 2) remains practical; as O(case1)=1 and O(case2)=k.
I remain to contemplate the consequences...

Working on school

The partitions of my time must be divided up accordingly. So, after I finish my homework, and read for school, I continue to work on logic. It is very interesting, although in a branch away from my studies, it seems that the more I focus on my individual research the more I understand those other topics.
I discussed some of my work with a few professors this week, and they were excited; however Dr. Su, my computer science professor, said that I need to focus on the text. Thus to satisfy CMPSC 360 (a discrete mathematics course), and at large my GPA, I will put my foremost effort into school.
At the same time, I am reading Handbook of Proof Theory by Samuel Buss. Enriching the coursework from CMPSC 360, but at large the development of an axiomatic language for a propositional logic.

Begining to compile work

I have just begun on a draft for JSTOR's journal: The Journal of Symbolic Logic, as a compilation of my work in the axioms of propositional logic. This work bounds the Entscheidungsproblem to a discrete amount of terms to compute problems belonging to EXPSPACE in a finite time bound.

The details must be omitted, but at the same time I have started to realize relativistic physics in a sense of time independence among a confined system. We describe this change over the body of a space time to be one entity, thus allowing for calculations of this space time to be determined with certainty, at the present this is done only in some probabilistic sense. I do not yet fully understand the underlying mechanisms that are operating this function; however I do believe that it deals with the constant relationship between mass and energy along with some phase transition element.

Relativistically speaking, I do believe that this time independence can be measured from a perspective with time dependence; say your watch at the current position. Although this constant force guiding the space time is not known, insight allows me to believe that it, perhaps is some the limit as this function approaches from an infinite degree to some lower degree of zero. And the time dependence would act very fast relative to the perspective in someone in the future, but for someone in the past would be observable as very slow.