P=NP Proved, Disproved, Varies Based on Problem or issue, and Indeterminate Based on Similarity between Checked and Find

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1 From the SelectedWorks of James T Struck Winter March 2, 2018 P=NP Proved, Disproved, Varies Based on Problem or issue, and Indeterminate Based on Similarity between Checked and Find James T Struck Available at:

2 P=NP Proved, Disproved, Varies Based on Problem or issue, and Indeterminate Based on Similarity between Checked and Find By James T. Struck BA, BS, AA, MLIS Abstract We consider Dr. Stephen A. Cook s P=NP Relation with regard to P=NP being an algebraic relationship, a relationship between being able to check a solution and find a solution and its other possible relationships. We find the relation to be provable, disprovable and varying based on problem or issue and in some cases, in which for example the value of N is not clear, to be indeterminate or uncertain. Method We first consider algebraic relationships of P and NP as algebraic symbols, then case studies on the issue of checking and finding where NP is for checking and P is for finding. We consider abstract cases where a computer is restricted from doing finding or checking so that the other process is faster and then concrete examples like we can find a computer, but cannot easily check its operation, check for a book but cannot easily find the book, can find a mailbox but cannot check for the letter in the box and cases in which we cannot easily find or check for something such as in response to the query What? What operation is easier is not clear? Are we to check for or find something or how do the words differ in function? Discussion 1

3 Dr. Stephen A. Cook argued about the P=NP relation in 1971, but if we consider the equation itself on a mathematical algebraic basis the relation can be proved, disproved and found indeterminate. Disproved P Does Not Equal NP is based on the idea that a P is not equal to an NP. There are different algebraic symbols or quantities on both sides of the equation. With different variables or algebraic symbols on the sides of the equation, there would not be equality. Proved P=NP is based on the value of N=1, in which case P=NP Indeterminate As we do not know the value of N, it cannot be determined if P=NP or P does not equal NP This is an algebraic symbol consideration of the equation. Based strictly on seeing the NP as algebraic symbols of something, we can see the equation as provable, disprovable and indeterminate. Relationship Varies on Case Studies or Problem Being Considered Solutions vary based on problem being studied Case study Proof of P=NP based on identical definitions of finding and checking solutions Checking and finding a solution to a problem can be seen as identical. Checking for something can be seen as the same as finding something, in which case the relation P=NP could be seen as 2

4 proved in that checking can be seen as the same as finding. When I check for God or a bird or myself, I can be seen as doing the same thing as finding God or a bird or myself. Case study Disproof where P Does not equal NP I can set up a computer in which checking a solution can be done, but finding a solution cannot be done. The machine s algorithm is that checking is doable but finding cannot be done. Therefore NP can be seen as done before P as P is not permitted. Case study where NP or checking can be done faster than P I can be sitting at a library circulation desk and be asked Can you check if that book is held by the library? I can check the catalog and find that the book is held by the library. Finding the library book on the bookshelf or somewhere would be harder as I have to check the shelf or the patrons (possibly millions of them) for the book. Checking for a book can be easier than finding a particular book. Case Study where P or finding Can be Easier than Checking for something or NP I put a piece of mail in a mailbox in front of someone, but it is unlawful for a non-mail carrier to go into the box and retrieve my letter. I then ask Can you find the mail box? which the person could as I put the mail in the box in front of him. I then ask Can you check for the letter? which he cannot do as he is not supposed to go into the mailbox. He can find the letter, but he is not permitted to check for the letter by the jurisdiction s laws. Case Study where P is possible but NP is not I set up an algorithm on a computer and tell the computer to find P but not to check for NP. Therefore on that computer, P is possible but NP is not possible. 3

5 Case study where the P or NP being faster is not able to be determined? The relation can be indeterminate in a specific case for P=NP. I ask someone What? What is there to check or to find? That is unclear so the relation between P and NP cannot be determined. I ask someone Is there a God? God can be checked for and found in dictionary and as a word in many books, but what is the difference between checking for God and finding God? The difference in the words is not clear or cannot easily be determined. Does God mean Paul Tillich s ultimate concern, does God mean all things as in Spinoza, does God mean Being as in Martin Heidegger s work, does God mean first mover, creator as in the work of Thomas Aquinas. Word meanings are unclear so the relation N=NP can be seen as indeterminate at times as someone s meaning of problems, checking for and finding can be unclear. Case study of Checking or NP faster than finding or P I can teach someone the word checking or NP in which case they know how to check for something, but they do not know the word find so they are not able to find something. Checking for would be faster than finding. Inverse finding easier than Checking- We can teach someone the word find in which they can find, but not the word check in which case they can find but cannot check for. 4

6 Case Study of finding easier than Checking I ask someone to find a computer, which they see in front of them. Does the computer operate? The checking for the operation of the computer would be more difficult than the finding of the computer. Case study of Checking on a solution more simple to find than finding a solution I am asked I need you to say thank you to Stephen A. Cook for his P=NP article. It is easy for me to check that if I had tried to take the Edmund Fitzgerald iron ore carrier in Thanksgiving 1975, I would have probably died on the ship with the other sailors. Finding Stephen Cook to say thank you to him would involve more work, more difficulty than imagining that sailing on the Edmund Fitzgerald would result in me sinking to the bottom of the Great Lakes in Checking for something like what would have happened if I would have taken the Edmund Fitzgerald to say thank you to Dr. Cook? Can be easier than finding something like actually saying Thank you, Dr. Cook in some cases. Conclusion The P=NP relation then can be seen as proved, disproved, and uncertain varying based on the issue or problem presented to the relationship. There is not one clear answer to the relation P=NP, but rather the relation can be proved, disproved and found to vary based on problem presented to a computer or problem solver. The way the P=NP relation is understood is also shown here to vary as well. Treated as a strictly algebraic relationship, we are also able to show provable, disprovable and uncertain relationship between P and NP. Using case studies, we are also able to show P=NP to be provable, disprovable and uncertain based on issue studied. 5

7 Recall that the whole discussion of HAL by Arthur C. Clarke revolves around the issue of the computer HAL resolving the issue independently of humans; it creates a new star near Jupiter and the moon Europa around Jupiter to settle or bring peace to a Central American war. A computer or human can actually be designed to think for itself, like humans, in which it does not have to solve or verify following the programs or algorithms we give the computer or human. P can be seen as unrelated to NP, P=NP, or P does not equal NP, or one can see the issue as mystery a different kind of relationship. James T. Struck BA, BS, AA, MLIS 6

8 An earlier discussion by me Disproof and Proof of P versus NP Problem Postulated by Stephen Cook By James T. Struck BA, BS, AA, MLIS Quoted from THE P VERSUS NP PROBLEM STEPHEN COOK accessed on 2/21/2017 Problem Statement. Does P = NP? Different Verified versus Solved Computer Invention Disproof James T. Struck argued from that one can invent a different computer which will always verify a solution differently than the solved computer solution. As a disproof, solved solutions can always be worked out differently than verified solutions. I can develop a computer so that any solution that someone has will give a different verified solution. That is a disproof would be that one can develop a 7

9 computer so that every solution is not what is verified, every solution is different than what gets verified. Different computers would have different answers to P=NP. P does not NP with the case study of the different verification method computer invention. Same Computer Proof of P=NP Further, one can develop a computer so that when there is a computer solution it is the same as the verified solution. A solution can be both always different than a verified solution and always the same as a verified solution. Stephen Cook's conjecture Same Computer Disproof of P=NP Does P=NP can be easily disproved by developing a computer where solutions do not equal what is verified. But one can also develop a computer where every computer is the same as what is verified. No Objective or single Solution to P=NP Dr. Cook and solvers forgot that there is no objective answer to the query. P a solution can be the same as what is verified and can be different than what is verified. The P=NP conjecture is both provable and deniable based on the computer that is used to test his theory. P can equal NP but P does not have to equal NP. Method or Path Disproof of P=NP In addition the routes through which we gain solutions can be different than verified solutions, but they do not have to be different routes. P can be achieved the 8

10 same way as NP, but P does not have to equal NP. I can have solution A=B but then verify it using a different method. The Method disproof is a legitimate type of disproof. Paths to solutions do not have to equal paths to verification. Method or Path proof of P=NP On the other hand I can set up a path to a solution to be the same as the path to verification. I can have solution A=B and verify on the same computer that A=B. Easily Provable and Deniable based on computer, verification method Computer selection based proof and disproof- every computer would be a different type of finding and checking. Verification method proof and disproof; every computer would be a different verification method as they are different computers. There is no objective solution to Dr. Cook's query, in some cases P=NP and in other cases P does not equal NP. There is not one proof, but rather the conjecture is Provable and deniable. Answers vary on method, computer, verification, question, issue. Proof and disproof based on subject, issue or question or topic? Here we are solving God while that would have a different P=NP answer than adding. P=NP varies in being proved or denied based on issue studied. So not only does computer, method, verification, question vary but the whole Conjecture is subjective based on what surface, geometry, question is being queried. P =NP or P could =NP for 2+2 but P not equal to NP for queries about God, for example. 9

11 P=NP can be proven and disproven for subject matter issues Is there a God? I or you design a computer to say yes. Then it verifies the Yes. So P does =NP Is there are a God? I or you design and computer to say maybe. Then it verifies no. Then it verifies yes. Cook s conjecture is deniable as what a computer solves is not what it verifies. Is 2+2=4 I or you design a computer to say no. That within 2 there are an infinite numbers and 2 is an Arabic numeral not necessarily needing to equal 4. The 2 rocks break up and become an infinite number of rocks. The computer provides different verification and solution. Is 2+2=4 I or you design a computer to not give a solution. Then the computer verifies that 2+2=4. My solution is not what is verified. Is 2+2=4? My computer solves that 2+2=4 but then verification is programmed to show that 2+2 does not equal 4 due to an infinite number of pieces or decimal numbers within 2 and 2 and 4. Solved is not what is verified. Does P=NP? Stephen Cook forgot that computers, methods, issues, subjects, paths vary. P can equal NP, but P does not have to equal NP. 10

12 If I were to walk directly North of where I am now I would end up in Ontario Canada to thank you Dr. Cook for his conjecture. I am trying to solve the Problem can I say thank you to Dr. Cook? But if I took a car my method would get me to Ontario at a different time. If I took a boat, I might sink like the Edmund Fitzgerald in If I take a plane I would say thanks to Dr. Cook earlier if my plane did not crash like about 2 crashes recently and 6.8 crashes in the US in 1972 each day. My solution will vary by the path taken. Solution can be same as verification method, but solution does not have to be the same as verification. P=NP and P does not equal NP. My plane could crash therefore my verification method would differ from my solution. I can assert that I can send Dr. Cook an , but to verify that I could use different computers to send the message, or different systems from gmail to yahoo to Hotmail to twitter to facebook etc. If I ever get to Toronto to discuss the issue with Dr. Cook my solution is different than verification method. Thank you Is different than Thank you very much And both those are verified differently Thank you is understood differently than Thank you very much. P=NP but P does not have to equal NP P=NP and 11

13 P does not equal NP The conjecture is both provable and deniable. Linguistically I can say for example P=NP Linguistically I can say P does not equal NP With color I can say P=NP with the same color P does not equal NP with different colors P does =NP with the same or different colors Solution and verification can be the same method, but solution and verification do not have to be the same. Some problems cannot Be Solved or Verified-Neither P nor NP By the way a different solution is that certain computers cannot verify or solve a question? For example for the question IS a Dog a cat? Or the problem How or Where? or When? or What? A computer can choose not to verify or solve. Computers and human beings can make choices too to not solve or not check. There is not one answer to Cook s P=NP as algorithms vary based on questions and problems so that P s relationship with NP also varies. How something is solved varies in its relationship with how something or a problem is checked, so the 12

14 solution to Cook s P=NP query is that the relationship would actually vary based on problem asked. James T. Struck BA,BS, AA, MLIS PO BOX 61 Evanston IL