The existential predicate and the acquisition of information.

You are reading this; or you may be looking outside, listening to your music, or doing any other activity as we can go on for hours and days... But, you are processing the events of the moment. This is the simple activity that we are interested in on today's post.

As we learn new information from our various senses, information is never forgotten (perhaps misplaced), that information is stored in a non-volatile memory location in our CNS. This process, continues throughout the span of our lives. It is not the memory that we are concerned with however, rather we must be interested in what it is that our CNS does with the information gathered.

Consider each synaptic moment to be a write and read cycle, thought will not differentiate itself, for if thought were to then no new thoughts would ever exist, and we would be limited to a distinct potential set by only our input. Thought does, on the contrary, allow us to consider implications of situations or possibility's that would be absurd in the known realm of the universe that we operate. Hence, there is no bound on the imagination.

Although this may be precisely that same story that our mind told us that can be true. It is not, and the following will help convince you that it may not be.

Suppose that each synaptic moment we gather a well defined set of information, this may be restricted to our senses for a human, but beyond that we can think of other sensor's which may capture events in a situation. This array of information must be immediately stored in acted upon with those other past information that was collected. To do this, we must apply combinatorics so that every possibility of the situation may be represented. Note: this is purely a conservative case supplying the ideal situation, in a mind one may need to ponder an idea to generate the same effect.

Thus, with this operation acting on every object of our input array, we build a stronger complete view of that same input data set. And, from this we can then see the possibilitilities for a given situation (SAT).

We will continue, and I will continue to unleash this journey. Although, at the moment this is much of the writing that I get done. I am using the majority of my time to focus on school work and cannot diverge far from the topics...

The truth of the matter.

I moved back to school this past Wednesday and have been relaxing before courses start for the fall semester.

In the mean time I am continuing to work through Gödel's work on the incompleteness theorem, along with the ZF axiom system.

Truth is the main premise to this thesis, and how this truth can be interpreted through a machine; as for you and almost any being (perhaps there are some that do not perceive the world as I do) realize our surrounding as what we perceive it to be. Where if it is not the case and we rather perceive some false premise, this is also fine, as you have first have perceived those fallacies as truths, but have then (from more sensory input, or cognitive consideration) realized the inconsistency of the situation and have adapted your reality accordingly.

Given a statement of logic, we would like to see what variable combinations give truth to this statement. This is first done by accepting all variables and possible negations as truths, contrary to the Boolean algebraic approach of neg(1)=0, and assigning a specific mapping to these variables (to integers, letters, symbols, etc...) such that each variable along with any necessary negations of variables of the same class to be defined independently. It is now that we can define the statement of logic in a variable independent sense, allowing us to understand completly under the basis of truth.

Next, on to bounded combinatorical spaces and the existential prediacte axioms!

1/3 of Break

It has been a few weeks... tied down with earning money, hence work has not been at full throttle. During this time I began to construct a formal definition of a ``combinatorial bounded space machine'', which allows a bounded vocabulary to be reduced to two sets based on the satisfiability constraints of some bounded expression.

I picked up, once again, Gödel's collected work from 1929 to 1936; it is a great book composing his groundbreaking results in both native German and opposing page translations to English. This is a great inspiration, along as a reference; Gödel was only 23 when he received his Ph.D for his Completeness Theorem; I can feel my time is not far away!

Beginning of summer

All finals are over with and I am back home.  

I am reading "Introduction to Metamathematics" by Steven Kleene, a classic book on computation and formal languages.  It was first published in 1952 and is said to cover the breadth of the field of theoretical computer science before many subtopics were created.  

Nearing the end of Spring 2008 semester

I am now finishing off the semester with finals for my courses. Perspectives on PH have not changed dramatically, but once in the summer I will focus time on this as a priority. Myodesopsia, observation of floaters, is a common occurrence, but not yet a practiced scientific method. I do not want to distort the mathematical constructs of PH; but the findings may have an isomorphic consequence to this phenomenon. As a scientist I am not obliged to these theories, but merely view them as an alternate perspective to reality. Alternate perspectives may merge with models (must remain skeptical to situation).

Emphesis on the current semester

Over the past several weeks I have started to compile some of the work, namely definitions and some algorithms that can be used to construct a combinatorial space oracle. I have hit a point though where proof techniques will be implemented to demonstrate the witnessing of some expression directly with syntax. This is required and must be rigorous; thus much care to detail must be accounted for; and the outcome is to demonstrate the collapse of PH. Thus, I am in a phase of learning, specifically the details of formally communicating proofs.

In the mean time with the current semester over halfway done, my grades must remain a priority. And homework remains to supersede work.

Resolving independent negation

The handling of a logical statement, that we would like to see if there exist some validity, is currently done with the method of expressing variables with a boolean assignment and thus there is a maximum of 2^n truth combinations. This is attainable to brute force attack all of the combinations in feasible time bounds for low values of n, however when n increases to even maximal bounds as low as 50 this procedure becomes inefficient; (this is a far approximation, and in actuality, 50 will seem quite high). This would take over 156 computing hours at 2 GHz, feasible. But suppose that we take n = 100, then it would take over 2.0098*10^11 computing centuries at 2 GHz, infeasible. This is where the problem begins to be overwhelming, and why EXPSPACE computation is not practiced commonly, only for small input sizes.

Lets now compare to another mode of computation. Assume that we handle the negation using independent terms, thus for our n variables we can have a maximum of t=2n terms. This is assuming that we use both the original variable and its negation, perhaps the expression will not require the maximal negations, we are then at some value n <= l <= 2n. Now comes to the part of the expression. When the combinatorical space is constructed it takes O(n) synthesis steps to generate a space containing 2^n strings. These represent every truth possibility for the bound of n terms.

But say that we realize that from the expression there are instances of negations of variables in the same conjunction. In the current logic, we would have these destroy themselves. I think now, as direction removes commutativity with the AND operator, we are left with the instance of removal of crucial information about the expression. Thus, for the maximal bounds of the cardinality of terms, should be somewhere in the range from the amount of terms to the amount of terms in the product statement prior to existential operations on the expression. (This maximum may be reduced to somewhere near half of the original amount of instances, but more care will be given to take care of all instances.) This is an approximate neighborhood of how many negation terms will be included in the combinatorical set, and if all were in then satisfiability will always be attained, giving false witnessing; (very bad!).

So even if we are to execute this maximum bound for a combinatorical space construction we would only extend our efficiency by some constant multiplier, and not increase the complexity of the generation: O(kn) -> 2^(kn), but dramatically increase the complexity of the size of our combinatorical space, precisely increasing our base by a multiplier of k.

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.