త్రుథ్ అబౌట్ రోబోత్నింజ.కం

2009-05-06T05:22:00.000-07:00

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1. Error Line 91, Column 5: end tag for element "li" which is not open

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3. Error Line 187, Column 276: document type does not allow element "p" here; missing one of "object", "applet", "map", "iframe", "button", "ins", "del" start-tag

…t="300" />

Extending the Model over Zero

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4. Error Line 187, Column 315: end tag for "dt" omitted, but OMITTAG NO was specified

…tending the Model over Zero

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5. Info Line 187: start tag was here

v

6. Error Line 187, Column 315: end tag for "dl" omitted, but OMITTAG NO was specified

…tending the Model over Zero

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7. Info Line 186: start tag was here

>

8. Error Line 187, Column 319: end tag for element "p" which is not open

…ing the Model over Zero

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10. Error Line 189, Column 5: end tag for element "dl" which is not open

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12. Error Line 219, Column 26: end tag for element "p" which is not open

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14. Error Line 526, Column 75: required attribute "alt" not specified

….net/mini/RobotPiratNinja.png" border="0">

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The attribute given above is required for an element that you've used, but you have omitted it. For instance, in most HTML and XHTML document types the "type" attribute is required on the "script" element and the "alt" attribute is required for the "img" element.

Typical values for type are type="text/css" for
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Robot Pirate Ninja

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- The Single Best Source of Robot Pirate Ninja
- Pages
  - 0 Robot
  - Would You (yes, YOU) like to subscribe to a Scribe?
  - 1 Pirate
  - 2 Ninja
  - 3 Being
  - 1. The Introduction
  - 1. The Chapter 1: The Voice
  - 1. The Chapter 2: God Math (Logik, LOGOS, P=NP)
  - 1. The Chapter 3: Me and the God Path (Pi is a Relationship)
  - 1. The Rest of the Book
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1. The Chapter 2: God Math (Logik, LOGOS, P=NP)

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Chapter 2: God Math (LOGOS, Logik, P=NP)

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God Math starts with two opposite limits and the midpoint between them. Those two opposites are known to humans now as “infinity” and “zero.” Fascinating concepts in and of themselves, it took a global culture to bring the two together and one writer (ed. Howdy!) to figure out how they can be used to explain a whole lot more than everything and nothing.

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I should be very clear to note here, this is my approach to “math” (with a bit of help). I made this stuff up because the math I was being taught in college didn’t apply to the real world as I saw it. I needed a better way to calculate anything easily. Calculus didn’t cut it for me.

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Part of my problem with Calculus was a professor who went by the name Dr. H at Trinity University in San Antonio, Texas, which is where I was formally edumacated[1]. I’ve got the $100K piece of paper to prove it, and that piece of paper has already paid for itself many times over, and I’m yet a young man as of this typing. Dr. H hailed from India, and had the accent to prove it. It was my unfortunate inability to decipher his lectures, along with various social concerns, which ultimately ended my relationship with formal mathematical education.

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Hence, I hope you can understand why I ask forgiveness of the various folks in the reading audience who will find my treatment of infinity and zero to be damn near heretical. Check that, what follows is absolutely and completely mathematically heretical. This is my P=NP proof, after all. One would have to be a bit outside the norm to put forth such a concept in public, and so I’m applying for that position. I don’t think it leads to tenure.

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I needed a system of math that could be universally applied, easily calculated, be massively adaptable, easy to teach and learn, and you needed to be able to do it in your head. Calculus just wasn’t it for me. So I had to make up my own thing. It ultimately took nearly ten years of study and practice and thinking and searching and asking and praying and hoping and hurting to get to where I ended up.

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I ended up at a place where the voice finally said, “Yes, that’s right. Now go tell them about it.”[2]

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ZERO and The Model™ [NP]

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A great many great mathematicians hail from India, much like Dr. H-the-Un-Understandable. One of the more recent (in math years) was Subrahmanyan Chandrasekhar, who figured out how much matter, in how much space, would lead to a collapse of that section of the universe. This is now known as the Chandrasekhar Limit. A quick public knowledge definition follows…

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Via the Wikipedia: “Stars with higher mass [than the C-Limit] will develop a degenerate core whose mass will grow until it exceeds the limit. At this point the star will explode in a core-collapse supernova, leaving behind either a neutron star or a black hole”

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In my humble opinion, it is the black hole that defines what one could consider the “edges” of the Universe. A true edge to something is a place from which you cannot return. Once you cross what is called the “event horizon” of a black hole there is no turning back to the Universe you knew before. The universe as we know it is flat, and Newton’s Laws apply is this very real sense, an object once in motion will travel in a straight and continuous line until affected by another force forever. When it finally is affected it will react in exact proportion to that force. We later learned from Einstein that the existence of matter and energy made a lot of these “forces” universal, and indeed there exists a universal constant that turns straight, infinite lines into a fictional concept. There is no escape from the Force.[3] This is why straight lines, perfectly straight lines that go on forever, are so rare as to be nearly unique in nature. The only one I’ve ever seen has been in my mind.

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However, there is a limit to those forces mentioned above, and we also know thanks to Chandrasekhar, and Darth Vader, that the force can eat itself. The ego can consume the soul. The black hole lets nothing escape, and as such is about the only stark “edge” one will ever find in the vast reaches of space.

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Zero is NP in my proof. It is the “anti-prpposition” or the “wrong answer” or the “limit to the set of all wrong answers.” Zero is the Dark Tower. Death.

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I’ll be coming back to this black hole metaphor later as I demonstrate one of the ways that The Model ™ can be applied. By The Model™ I am referring to the mathematical system I created over the roughly ten year period mentioned earlier. Understanding TM begins with an understanding of zero and infinity and their intimate relationship to the rest of rest of reality. I found it logical that a system which could be universally applied needed to have both the largest and smallest possible limits. Zero and Infinity qualified nicely.

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The “zero” of which I speak was the one defined by another Indian by the name of Brahmagupta in the year 628. His work “Brahmasputha Siddhanta” (The Opening of the Universe) defined the rules for using zero as follows…

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The sum of zero and a negative number is negative.

The sum of zero and a positive number is positive.

The sum of zero and zero is zero.

The sum of a positive and a negative is their difference; or, if they are equal, zero.

A positive or negative number when divided by zero is a fraction with the zero as denominator.

Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.

Zero divided by zero is zero.

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All but one of these rules is still in effect, as it were, in this modern age. The only one we have “improved” upon is the one about dividing by zero. We have since decided that this mathematical operation was “undefined”.

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This was one of the things about higher math that had bothered me for a long time. How could one of the most important and absolute limits of numbers not be used to divide?

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The solution that Brahmagupta missed had been present in Indian mathematical thought for nearly a thousand years. Sadly, infinity and zero weren’t really compared until it had been absolutely established that one could not divide by zero.

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We settled on rules that said if one tried to divide by zero, one was left with a non-answer, wandering in the mathematical wasteland forever. This disturbed me greatly, as I had been instilled as a younger man with a great desire to know the truth of all things and a great confidence that I could re-solve any problem. After all, the Voice had never steered me wrong, as it so quietly reminded me, shining brightly from above.

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Then one day, about midway through my twenty-second trip around the Sun, as my formal education neared an end, the answer came to me. I described it then like a bolt of lightning, and it set my soul afire. I had figured it out. I knew what was halfway between infinity and zero. What was precisely and provably halfway between everything and nothing. I had learned to divide by zero.

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INFINITY and The Model ™ [P]

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What followed after that day was a flurry of activity. I had resolved one of the great mysteries left to figure out, and the answer fit perfectly. It was applicable to any number of things, and I saw the great potential that could come from my discovery. I wrote up a hasty explanation, breathlessly mailed out copies to every major news outlet and waited for the responses to come. I was careful to be what I thought was engagingly cryptic and looking back now, I laugh heartily at what I thought was good work then. When I look back at this work sometime in the future, I hope to receive a similar jolt of mirth.

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It wasn’t until years later that I realized what I thought was a new and novel concept had been discovered, analyzed and formalized in the 19^th century by Bernhard Riemann, in what is affectionately known as Riemann’s Sphere.

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It took quite a few more years before I realized I might have just solved one of the most difficult equations in math. Or, more accurately, “math” itself.

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Infinity is P in my proof. It is the “proposition” or the “right answer” or the “limit to the set of all right answers.” Infinity is the Light Tower. God.

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I think I mention somewhere in this book exactly how it is that P = NP and how P ≠ NP [Appendix P=NP]. It really depends on how you define terms, and “math”. The missing element to finalize the equation is that One Unanswered Question, which is always swirling around, that extra dimension of analysis that often makes the question moot. Is P equal to NP? The correct clarifying question is, “When and Where?” or simply “In time/space?” If one is limited to answering the questions on our plane of existence, then one must have at least four dimensions to describe it, accurately. A proof then is somewhat trivial, as time becomes a finite property. “Pragmatic Infinity” replaces “Fictional Infinity” and solutions abound.

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The concept of Infinity itself had been around since being recorded in India (again) around the 4^th century B.C. If was formalized and given the familiar symbol ∞ in 1655 by John Wallis in England. One of the many plunders of the English from the Indians (not quite as bad as Americans taking the whole concept of “Indian” and misapplying it), infinity has been in common if not problematic usage in a number of fields since it became more formalized [citation needed][4].

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We’ll get back to the whole sphere thing in moment, but right now I want to get back to that dividing-by-zero concept. What I had learned was that when you asked the question “what is infinity divided by zero”, the answer is in fact, “one.”

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This follows quite logically from the well accepted equation (one divided by infinity equals zero. This is generally applied to any finite number, X, in the equation )[5]. However, the concept that no number could be divided by zero was so ingrained into the minds of so many that it wasn’t understood that accepting that (one divided by infinity equals zero) automatically implies the two following equations be true, (zero multiplied by infinity equals one) and the kicker (infinity divided by zero equals one [P/NP = 1, P=NP]).

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I know it might seem strange to many that figuring out the answer to a math problem could bring someone such joy, and somehow be related to the subject of the first chapter of this book, but the profoundness of this simple logic and beauty, a beauty that I felt was revealed to me for a larger purpose, overwhelmed my senses. Truly and absolutely. I ended up quitting a good college job (bartender at the best, closest bar to campus), to try and spread the simple message. The one that had hit me like a thunderbolt and it sent me to my knees in tears.

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It’s not undefined. I figured it out. I know what God is. And I can prove it.[6]

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The Model™ and God (aka. Allah, Y__H, etc.)

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The model applies when one has to, as they say, show their work. When taken literally and compared to the religious concept of creation, I think you’ll find the following argument amusing. I most certainly did.

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Graphing the equations , , and can all be done with the following simple diagram….

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The logic of the diagram is as follows. It is only zero, infinity, and one that are the stable “values” on the diagram. Any other value, when left to act upon itself, will trend toward either infinity or zero, never actually reaching their destination. This simple fact has been known since the time of Zeno and is well accepted. For example, 1.1 * 1.1 = 1.21, 1.21*1.21 = 1.4641, 1.4641 * 1.4641 = 2.14-something and were off to the races, headed toward the largest conceivable number + 1. On the other side of the dividing live the same and opposite trend emerges, 0.9 * 0.9 = 0.81, 0.81 * 0.81 = .06561 and the race is now headed the other direction, hurtling towards oblivion.

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What we are left with is a graph of the limits of all possible numbers, the exact midpoint of all possible numbers, and all possible numbers themselves. A good place to start when building a universal model, one would hope.

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The next step in understanding how math can lead to God is to loosen the reigns on the model and let it wander a bit. Normally we like to have graphs all nice and neat, flat and simple. Reality is far more chaotic than that. Reality tosses things every direction, exploring all possible angles. Our nice and balanced simplified model is now without direction, a ship without anchor. We have limits, sure, but we don’t know what they mean. And we don’t know where they are. As we shall see as we explore the implications of The Model™, how one defines the axes defines the context of the question/proposition. An answer without a question and a context is useless[7]. A proposition with a without a fully defined context, including a temporal one, is not a well-formed proposition.

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To “illustrate” this concept, we first add the other side of the “0” line (the X axis) and add another “1”. Note how we now have a “negative” 0. This is possible because when we add the fourth dimension to the model we now have a way to discuss propositions in both the past and the future in the same manner [discussed further in Chapter 4].

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Extending the Model over Infinity

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We must also extend the graph over each of the “0” lines and extend the “∞” line, giving a full quadrant view. This gives us a “negative” ∞ to play with, which becomes very useful when discussing a timeline that is infinite in both directions.

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Extending the Model over Zero

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This is then extended to all quadrants and a “Z” axis (another set of “0”s) is added to give the full three dimensional form. The Blue lines in the following graph define the plane created by the addition of the “Z” axis.

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Adding the Z Axis

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Now we are beginning to see an ideal form begin to take shape, all built from a simple concept, and the applied to multiple dimensions. The ability to transfer relationships over each axis is allowed because of the symmetry of mathematics and reality. My reading of Emmy Noether’s work (perhaps the greatest female mathematician in modern history) says this is possible, accepted, and natural (which is to say, correct).

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Hence we are left with a three dimensional model that includes limits and relationships in all possible directions.

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What we don’t have, necessarily, is an exact direction in which each axis should point. As we factor in the chaos inherent in reality, and allow everything to spin around the origin, we are left with a “derivative” of the simple model. There is a part of reality that the model cannot touch.

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Translating pictures of three- dimensional objects on two-dimensional surfaces such as a page in a book requires a bit of work on your part, but I think you can manage.

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As we then recombine all the possible scenarios into a single image, and we get the following:

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The Full Drawing (with a big error)

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This step is the final one to create a three-dimensional model (outside of time) and we finally see the fully realized (simple) model in its ideal form. Where once there were endless possibilities and numbers beyond imagining, now we have a simple something left over; a small piece of reality sitting in the middle untouched by the chaos of creation and possibility. What we have left is a sphere. A perfect, untouched sphere: The Gyroscope of Logic.

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Perhaps I should say what we have created through all the possible interactions and endless possibilities is a simple core that is, in fact, outside those possibilities. Or in this case, inside of them.

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It is this process of being both essential to the system yet outside of it that makes this model so wonderfully explanatory. It is both this aspect of the model, the simple logic of its creation, which adds to the massive amount of evidence that a sphere is a perfectly efficient shape. And in this case, I am using “massive” literally, as the vast majority of matter we have observed in the Universe is spherical in shape.

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So now we have a sphere, and an origin, and yet no defining line. We need only one. Either infinity or zero will do.

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This is why we consider God to be the infinite line, the definer of reality, and a million other names spoken with reverence since time immemorial. Zero is death, the end of the line, and the ceasing of action. It too, defines reality for the living. Absolute zero, in the scientific sense, is the ceasing of all action of energy. Our species has not yet achieved this goal, and I consider it to be unattainable, much like traveling at the speed of light. We can get close to this lower limit of energy, and have done so and created something called the “Bose-Einstein Condensate” which is very useful for some cool science, but if there was no energy left in a system, there would be no matter, and no system. Hence, we now have the ability to brush against the lower limit of reality, in a very real sense. Yet we cannot cross it, for there is nothing on the other side to greet us.

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The metaphor between your personal reality (that’s you, the one reading this book) and this model is that the infinite line represents the perfect action. That line represents the perfect response to any given stimulus or situation. What Would God Do? That action is represented by the God Path, the infinite line.

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The zero line, on the other hand, is usually representative of the worst possible action you could take, and one that would immediately make it impossible to reach your goal (The God Path). We can do this with the simplified version of the model and simply ascribe the words “Good” and “Evil” to actions on either side of the dividing line. It is for this reason that all cultures find purposeless suicide to be an immoral and selfish act.

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The Simple Moral Graph

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It would seem at first glance that this model would indicate there to be an equal amount of good and evil actions. However, as we will explore further in Chapter 6, once The Model is applied in four dimensions, the number of “good” actions becomes quite a bit smaller, as they become more similar and converge on the God Path, which is a single Infinite Line in a sea of Infinite Zeros (the plane created by the by the X and Z axes). Which is to say, there are many more wrong actions that right ones, as the right ones all begin to look like the same action.

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I would also like to point out that the sphere in the middle, defining the model, is you.

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Vitruvian Man

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I would include a picture of a woman in the same sphere, but I don’t want to make this book too titillating for the younger (male) readers.

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The line with slope 1 in the model then becomes a neutral action, the procrastinator’s paradise, which would bring you no closer toward, yet no further from, your goal. “Purgatory”, which has some nice slopes in the real world [‽]. The Place of Purging. I will discuss this later as in my model of morality, it is always possible to respond to a negative action (defined as somewhere between 0 and 1) with an action of slope 1. What one is purging, in this sense, is the evil of the action of another.

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Understanding the metaphorical implications and abilities of the model requires intertwining the two concepts, stimulus (action) and response(reaction), with the three-dimensional model we’ve just created (the sphere, cognition, a “function” in math terms) to produce a functioning four-dimensional whole (The Model).

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This will be discussed further in Chapter 4: Adding the Fourth Dimension and Generalizing the Model.

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If you want to skip the emotional baggage I’ve carried on this journey of turning a simple equation into a worldview, and some of the reasons I’ve taken it this far, you can turn to Chapter 4 now, it begins on page [[xx]]. Chapter 3 deals with what happened to me to set me on this path. You will miss a great deal of exposition if you skip ahead, so choose your own adventure carefully.

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I most certainly have. Or perhaps I should say it has chosen me.

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[1] Names have been shortened, occasionally, to protect the innocent.

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[2] While writing this, I just remembered when this happened and it brings me to my knees in overwhelming joy. I’ll get into the brain and feelings and emotion and memory and perceived reality and feedback loops in a later chapter…but all I need to do, no matter how dark the current day might be, is remember the day I heard that, and I smile.

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[3] Yes, this is a direct and obvious reference to the mythology of Star Wars. Great stuff. Empire was the best, followed by the last one, where we see the greatest good turned into the greatest evil. We’ll have more on Joseph Campbell’s understanding of the human condition later. It should become apparent through further reading why this story appealed to me so very much.

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[4] See Wikipedia, I do a lot. And if there’s something I know is wrong in the ‘Pedia. I fix it. I hope you do the same, and argue about it. It’s an AMAZING resource for our species, try and keep it tidy and running. I do. I’m a big fan of Jimmy Wales’ goal, “Imagine a world in which every single person on the planet is given free access to the sum of all human knowledge. That’s our commitment” Sounds like a good world to me. If there’s something I don’t explain or footnote, trust me, it’s in the ‘Pedia. Look it up. I told you to bring an internet to this reading in the introduction, and I wasn’t kidding. That’s how I get to keep this book shorter and sweeter.

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[5] Word has been screwing with my equations as I have transferred this file from computer to computer due to some unforeseen consequences of writing it [Chapter 11]. I’ll fix it in a later version. What can I say, I ain’t perfect.

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[6] I probably should have said “prove” it. The emotional aspects of the equations will be dealt with in further chapters. Ultimately the proof can only be provided by one to their self, as we will see that each of us has a rather astounding power to decide what reality means to us. What I can and will offer is what I consider a logical argument that, when applied to situations where “God” is invoked, a very familiar pattern emerges. And it has happened in every surviving culture on the planet.

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255. Comment by Martin M. Musatov on March 12, 2009 2:20 am
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Great post.

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265. Comment by m on March 18, 2009 5:49 pm
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Luke 18:43

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275. Comment by Marty on March 19, 2009 2:18 am
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What was earlier posted by me was an experiment. The fact is, you are exploiting my knowledge and me and you will pay for it or you will pay me for it. My name is Martin M. Musatov and this website is exploting my work and I am requesting a lawyer to aid me in a suit against this exploitation scheme. I am not a “troll” or a “couch boy”. I am a human being, and what you are doing is illegal, immoral, and sickening. You make me sad, and want to pray to God to ask him, how could you let such sick nonsense and exploitation occur at this level. I have faith in my God and His will be done.

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285. Comment by Martin Musatov on April 9, 2009 9:11 pm
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EZZ = Musatov

288. Proof:

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The Classes P and NP

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We now shift gears slightly and move to the examination of two families of problems, which are very important to computer scientists. Two families which constitute the bulk of our practical computational problems and have been central to the theory of computation for many years.

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The first is a class which contains all of the problems we solve using computers. If we think about the problems we actually present to the computer we note that not too many computations require more than O(n3) or O(n4) time. In fact, most of the important algorithms we compute are somewhere in the O(log n) to O(n3) range. Thus we shall state that practical computation resides within polynomial time bounds. There is a name for this class of problems.

292.
Definition. The class of polynomially solvable problems, P contains all sets in which membership may be decided by an algorithm whose running time is bounded by a polynomial.

293.
Besides containing all of what we have decided to consider practical computational tasks, the class P has another attractive attribute. Its use allows us to not worry about our machine model since all reasonable models of computation (including programs and Turing machines) have time complexities, which are polynomially related.

294.
That was the class of problems we actually compute. But there is another important class. This one is the class of problems that we would love to solve but are unable to do so exactly. Since that sounds strange, let’s look at an example. Consider final examination scheduling. A school has n courses and five days in which to schedule examinations. An optimal schedule would be one where no student has to take two examinations on the same day. This seems like an easy problem. But there are O(5n) possible different schedules. If we looked at them all with a computer which could check a million schedules every second, the time spent checking for a value of n = 50 would be about

295.
200,000,000,000,000,000,000 years!

296.
Yes, that’s right. Obviously this will not be done between registration and the end of the semester.

297.
One might wonder if the above analysis was needed, because after all, who would look at all of the schedules? You only need to check a few of the obvious ones. Or do you? Think back over all of the examination schedules you have seen. Were there any, which were optimal? No! So, there must be a small problem somewhere. We shall see more on this problem later.

298.
Let us think a little more about examination schedules. While it might be very difficult to find a good one, it is easy to check a schedule to see how near perfect it is. This process is called verification and allows us to know quickly if we stumble upon a good schedule.

299.
Consider another problem that of finding a minimal length tour of n cities where we begin and end at the same place. (This is called the closed tour problem.) Again, there are many solutions, in fact n factorial different tours are possible. And, once more, if we have a tour, we can easily check to see how long it is. Thus if we want a tour of less than some fixed length, we can quickly check candidates to see if they qualify.

300.
This is interesting and provides some hope of solving problems of this kind. If we can determine the worth of an answer, then maybe we can investigate promising solutions and keep the best one.

301.
Let us consider a class of problems, which all seem very complex, but have solutions, which are easily checked. Here is a class, which contains the problems for which solutions can be verified in polynomial time.

302.
Definition. The class of nondeterministic polynomially acceptable problems, NP, contains all sets in which membership can be verified in polynomial time.

303.
This may seem to be quite a bizarre collection of problems. But think for a moment. The examination scheduling problem does fit here. If we were to find a solution, it could be checked out very quickly. Lots of other problems fall into this category. Another instance is closed tours of groups of cities. Many graph problems used in CAD algorithms for computer chip design fit in here also. Also, most scheduling problems. This is a very interesting collection of problems.

304.
One might wonder about the time actually involved in solving membership in this class. The only known relationship between NP and deterministic time is the following result.

305.
Theorem 1. For every set A in NP there is a polynomial p(n) such that the problem of determining whether a data item of size n is a member of A can be solved in 2p(n) time.

306.
A useful tool in studying the relationships between members of a class is the translation or mapping of one to another. If we can translate one set into another, we can often deduce properties of one by the properties that we know the other possesses. This is called reducibility, is pictured in Figure 1, and defined below.

307.
Definition. The set A is many-one polynomial-time reducible to the set B (this is written as A £p B) if and only if there is a recursive function g(x) which can be computed in polynomial time such that for all x: x Î A if and only if g(x) Î B.

308.
Figure 1 - A many to one mapping between sets

309.
Note that all of the members of A map into a portion of B and all elements not in A map into a part of B’s complement. This gives us a way to solve membership in A if we know how to solve membership in B. If A is reducible to B via the function g(x), then all we need do to determine if x is in A is to check to see if g(x) is in B.

310.
One of the properties preserved by reducibility is complexity. Recall that to decide whether x was in A, we had to:

311.
1. Compute g(x), and

312.
2. Check to see if g(x) was in B.

313.
Thus the complexity of deciding membership in A is the sum of the complexities of computing g(x) and deciding membership in B. If computing g(x) does not take very long then we can say that B is no more complex than A. From this discussion we can state the following theorem.

314.
Theorem 2. If A £p B and B is in P, then A is in P also.

315.
And of course if A £pB and B is in NP, then A is in NP for exactly the same reasons. This brings up another concept.

316.
Definition. The set A is hard for a class of sets if and only if every set in the class is many-one reducible to A.

317.
If the reducibility function is not very complex, this means that the set A is at least as complex as any of the members of the class it is hard for. Thus an NP-hard set would be as difficult to decide membership in as any set in NP. If it were itself in NP, it would be the most complex set in the class. We have a name for this.

318.
Definition. A set is complete for a class if and only if it is a member of the class and hard for the class.

319.
Here is another fact about NP-complete sets and polynomial reducibilities, which will be our major tool in proving sets NP-complete.

320.
Theorem 3. If A £p B for a set B in NP, and A is NP-complete, then B is NP-complete also.

321.
Polynomial reducibilities also may be used to place upper bounds upon sets in P. For example, the following result is based on this.

322.
Theorem 4. If A is NP-complete then A is a member of P if and only if P = NP.

323.
Proof. Almost obvious. If A is a member of P then every set polynomially reducible to A is also in P. Thus the NP-completeness of A forces every single one of the sets in NP to be members of P.

324.
On the other hand, if P = NP then of course A is a member of P as well.

325.
This is very interesting. If we know that membership in one NP-complete set can be decided in polynomial time then we know that every set in NP can be decided using some polynomial algorithm! This means that we would get all of their recognition algorithms for the price of one. But, it is felt that this is highly unlikely since we know of no sub-exponential algorithms for membership in any of these sets and the problem has been around for a while.

326.
In closing, here is a small list of some of the many problems that are members of NP, and are in fact, NP-complete.

327.
0-1 Integer Programming (0-1 INT). Given a matrix A and a vector b, is there a vector x with values from {0, 1} such that Ax ³ b?

328.
CLIQUE. Given a graph and an integer k, are there k vertices in the graph which are all adjacent to each other?

329.
Vertex Cover (VC). Given a graph and an integer k, is there a collection of k vertices such that each edge is connected to one of the vertices in the collection?

330.
Chromatic Number (COLOR). Given a graph and an integer k, is there a way to color the vertices with k colors such that adjacent vertices are colored differently?

331.
Examination Scheduling (EXAM). Given a list of courses, a list of conflicts between them, and an integer k; is there an exam schedule consisting of k dates such that there are no conflicts between courses which have examinations on the same date?

332.
Closed Tour (TOUR). Given n cities and an integer k, is there a tour, of length less than k, of the cities which begins and ends at the same city?

333.
Rectilinear Steiner Spanning Tree (STEINER). Given n points in Euclidean space and an integer k, is there a collection of vertical and horizontal lines of total length less than k, which spans the points?

334.
Knapsack. Given n items, each with a weight and a value, and two integers k and m, is there a collection of items with total weight less than k, which has a total value greater than m?

335.

336. Reply

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342. Comment by pversusnp on May 5, 2009 4:07 am
343.

344.
Comment by Martin Musatov on April 9, 2009 9:11 pm. EZZ = Musatov Proof: The Classes P and NP. We now shift gears slightly and move to the examination of …Martin Musatov P=NP

345.

346. Reply

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351.

352. Comment by pversusnp on May 5, 2009 6:28 am
353.

354.
Here is the original drawing:

355.
The Original Drawing (with no errors)

356.
Thanks for playing,

357.
Martin Musatov

358.

359. Reply

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364.

365. Comment by pversusnp on May 5, 2009 6:29 am
366.

367.
http://mail.google.com/mail/?ui=2&ik=3e15f913b6&view=att&th=12110bdc95b4c480&attid=0.1&disp=inline&zw

368.

369. Reply

370.

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375. Comment by pversusnp on May 5, 2009 6:30 am
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377.
how do I put it in the post? hmm…

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385. Comment by pversusnp on May 5, 2009 6:36 am
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<!– Hide from old browsers

388. // Cache-busting and pageid values

389. if (!pageNum) var pageNum = Math.round(Math.random() * 10000000000);

390. document.write(”);

391. document.write(”);

392. // End Hide –>

393.

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Advertisement

396.
<!– Hide from old browsers

397. // Cache-busting and pageid values

398. if (!pageNum) var pageNum = Math.round(Math.random() * 10000000000);

399. document.write(”);

400. document.write(”);

401. // End Hide –>

402.

403.

404.
Customize

405.
Advertisement

406.
Customize

407.
<!– Hide from old browsers

408. // Cache-busting and pageid values

409. if (!pageNum) var pageNum = Math.round(Math.random() * 10000000000);

410. document.write(”);

411. document.write(”);

412. // End Hide –>

413.

414.

415.
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  431. Comment by pversusnp on May 5, 2009 6:46 am
  432.
  
  433.
  # Calculus
  
  434. #
  
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452. Comment by pversusnp on May 5, 2009 6:43 am
453.

454.
umm… little help here ropini ‘robot ninja’ whatever the hell your name is… I’m going to just keep trying until I turn your site into the bulletin board from MySpace…

455.
how do I embed by image you ugly *douche*…

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wah

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