{"paper_meta":{"paper_id":"arxiv:0709.1207","title":"0709.1207","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"The P versus NP Brief \n \n \nType of paper: \nBrief study \n \n \nDate: \n*September 2007 \n \n \n \n \nCorrespondence: \nMikael Franzén \nWillgood Institute \nVegagatan 58, V5 Unit 43 \n413 11 Göteborg \nSweden \n \nmikael.franzen@willgood.org \n \n \n \n \n \nAbstract: \nThis paper discusses why P and NP are likely to be different. It analyses the essence of the concepts \nand points out that P and NP might be diverse by sheer definition. It also speculates that P and NP \nmay be unequal due to natural laws. \n \n \n \n \n \n \n* The paper was written and completed in Sept 2007 (RV01PNP01). It was then revised twice between the date of completion \nand April 2009, and hence two updated versions (RV02PNP01, RV03PNP01) were produced. As with RV02PNP01, changes \nin RV03PNP01 are minor and consist of additional content such as, notational declarations, formatting changes, and the latest \ncorrespondence details.\n\nWillgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n1\n \n \n1. Introduction \n \nIt may be defined as a matter of polynomial time versus non-deterministic polynomial \ntime; and hence the question whether P equals NP, or P does not equal NP. It is hailed \nthe most significant problem within theoretical computer science and serves as a \ntestimony of the complexity of seemingly simple logic and “everyday” axioms. But, it \nruns far deeper than that. In a sense it addresses the very nature of logic and physical \ntruth; because the way we ask ourselves this question is well beyond the realm of \nmathematics, and therefore, it can be viewed as being metamathematical , as Scott \nAronson1 referred to it. \n \nSo, maybe the way of approach is far more important than the actual problem. Perhaps \nthe manner in which P vs. NP is assessed and analyzed makes a significant difference in \nterms of importance. That is not to say that the essence of the problem is a variable \ndepending on approach, but rather that the approach may hold the answer within itself. \n \nLet us take a look at what the following entails: \n \nP means Polynomial time and is defined as the computation runtime when no greater \nthan the function of the problem. It would almost be foolish to assume that NP is \nanything else than the opposite of P. So NP means Non-deterministic Polynomial time, \nand is defined as the computation runtime when greater than the function of the problem. \nHowever, this definition is not exactly correct; the reason being indeterminism. In other \nwords, problems in NP (besides those that are also in P) always rely on some sort of \nbrute-force analysis or probabilistic measure. Therefore NP should be defined as the \nnumber of steps that may exponentially evolve in order to fit the relevant polynomial \nfunction. Thus, computation runtime is likely to be exponentially as long as a potential \nsolution can be verified in polynomial time.2 \n \nSimply put, P problems are relatively fast to compute and hence also fast to solve and to \nverify, while NP problems are complicated to compute and therefore time-consuming to \nsolve, but easy to verify once the answer is known. So, what is the differentiating factor \nin all this? Well, the variables of course. In order for P to equal NP, we need to eliminate \nthe variables – Not literally speaking, but algorithmically speaking. However, is this at all \npossible? \n \n2. Input, processor and output \n \nDefinition of (a), a polynomial time problem, and (b), a non deterministic polynomial \ntime problem, where k is an independent constant; n the input string and m the instance of \nprocess work: \n(a), \n)\nn\n(\n)\nn\n(\nm\nn\nk\n(n)\nk\nf\nO\n \n \n))\n(\nO(\n \n \n=\n→\n=\n. \n(b), \n)\nO(\n \n \n \n))\nO(n(\n \n \n \n )\nO(\n \n \n \n))\n(\nO(\n \n \nn\nk\nk\n)\nn\n(\nm\nk\nn\nm\nn\n)\nn\n(\nm\nn\nk\n(n)\nf\n=\n→\n→\n≠\n→\n=\n→\n=\n. \nIn essence this tells us that a problem applied to the function f(n) characterizes that function \nby definition. Thus, a question in P or NP will render the function accordingly. So in \ngeneral terms, all input with variables exceeding the number of functionalities of the \nprocessor, will most likely be NP type of problems. Let us examine this a little bit closer:\n\nWillgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n2\n \nFor example, providing an input of 1 and 1 to a processor, which performs addition, will \nresult in an input of 2 units with an equal 2-unit processor functionality. The operation of \naddition will count as 1 unit and the production of a result, or output, may be considered \nas another 1 unit. Hence, in a straightforward \nmanner, we could say that the plus (+) and \nequals (=) signs are the 2 units of \nfunctionality. This type of processor-analogy \nworks very well for P problems, but it would \nnot work for any type of NP problem. NP \nproblems would not even fit the input intake; \nin a way it would be like feeding skittles through your inkjet, or putting stones in your \njuice blender. So, if we consider the Traveling salesman problem and the Subset sum \nproblem: the variables are individually, or in some combination, processable by the kind \nof processor pictured above. However, all together it will render the processor in terms of \nINTIME. So, concretely described, NP complexities are then solved by different, non-\ndeterministic kinds of processors, as opposed to deterministic ones, like in Figure 1. \nHowever, non-deterministic processors are intrinsically dependent on deterministic \nprocessors (i.e. P is a subset of NP). Therefore, as implied earlier, a complexity function \nis characterized by its input, thus suggesting that P is not the same as NP; strongly \nimplying, (perhaps naively), that in general, P is not equal to NP. \n \n \n3. Characterizations \n \nP complexities are processed at their very essence, instantly, even if time-wise it may \ntake very long: \n \n \n)\nO(\n \n \n \n \n))\n(\nO(\n \n \n \n \nk\nn\nn\n.\nm\nn\nk\n(n)\nm\n.n\nf\n≤\n∀\n∝\n=\n∝\n∀\n. \n \n-Therefore, n may approach infinity, and consequently, the function and its output \napproaches equal bounds. Although it possibly could be impractical and lingering, the \nruntime, on the other hand, does not approach infinity – Well, the total runtime may \napproach infinity to the extent that the input does, but, total runtime would normally not \nbe counted until an end is reached, and subsequently, in this context, total runtime may be \nan invalid notion:\n 3 \n \nII\n∑\n∑\n∞\n=\n∞\n=\n∞\n→\n↔\n→\n→\n>\n1\n1\nn\nn\n)\nn\n(\nm\n)\nn\n(\n)\nn\n(\n)\nn\n(\nm\n)\nn\n(\nm\n(n)\nf\nf\nf\n \n \n \n \n \n \n \n \n0\n \n ,\n \n \nn lim\n. \n \nNP complexities may or may not be processed at their essence. That is to say, a brute-\nforce analysis, or some other type of non-deterministic activity may yield computational \nresults that are relevant or irrelevant to the core of the question. Consequently: \n \nIII\n. )\nO(\n \nΩ\n \n))\n(\nO(\n \n \n \n \n \n \nn\nk\nn\nk\n(n)\nm\n.n\n f\n≥\n∝\n=\n∝\n∃\n \n \n \nI O(f(n)) = NP = NTIME, \n)\n(\nΝ\nNTIME\nNP\n \nk\nn\nkU\n∈\n=\n | O(2f(n) ) =\n)\n(2\nΝ\nNTIME\nNEXPTIME\n \nk\nn\nkU\n∈\n=\n \n II Let the sequential terms {an } and {bn} be the instances of functionality and input, f(n) and m(n) \nIII Let Ω denote all possible outcomes, {x1, x2, ...} \nFigure 1\n\nWillgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n3\n-So, as with P complexities, n may approach infinity, but with one very significant \ndifference, namely, the relationship between runtime and function. Here we see a \ndislocation between input, function and runtime. Firstly in terms of components and \nfunctionality; input may be non-deterministic in a way that yields no immediate function, \nor, it may be deterministic, but ‘incorrect’, thus yielding an irrelevant function. Secondly in \nterms of totality; input may produce disproportionate exponential or factorial runtimes, and \nruntime may depend on variables that are outside immediate and closely related sets. In \nother words, the scope of Ω may be considered unbounded by its own limits: \n \n \n0\n \n \n→\n>\n)\nn\n(\nm\n(n),\nf\n Ω ⊢\n \n \n \n \n \n \n \n \nΩ\nΩ \n \nn \n \n \n \n \n∑\n∞\n∑\n∞\n \n \n \n \n \n \n=\n=\n↔\n→\n∴\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nf\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nm\nlim\n. \nSo: \n \nIf we create an algorithm that renders P to equal NP it would have to remodel all \ndependencies and variables into a nice polynomial package. In addition it needs a real-\ntime performance by the square root of all possible answers: \n \n \nΩ\n \n \n0\n \n , \n=\n>\n)\nn\n(\nm\n(n)\n f\n –This would yield the same expression as the one for P \ncomplexities on previous page, but the problem with this is that it first entails the \nfollowing: \n \nΩ\n \n \n \n \n \n \n \n \nf\n \n \nΩ\nΩ \n \nn \n∝\n∑\n∞\n∑\n∞\n \n \n \n \n \n \n \n \n \n \n=\n=\n↔\n→\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\n)\nn\n(\nm\nlim\n –And thus we are back where \nwe started: \n \n0\n \n \n→\n>\n)\nn\n(\nm\n(n),\nf\n Ω ⊢\n \n \n \n \n \n \n \n \n \nΩ\nΩ \n \nn \n \n \n \n \n∑\n∞\n∑\n∞\n \n \n \n \n \n \n=\n=\n↔\n→\n∴\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nf\n)\nn\n(\nm\n)\nn\n(f\n)\nn\n(\nm\nlim\n. \nSo in this context: \nΩ\nΩ\n \n \nΩ\n \nΩ\n \n∈\n=\nQ\n. \n \nHence, trying to reduce size and limit the scope, while at the same time taking all the variables into \naccount, results in an extrapolation of all possible outcomes (Ω). Therefore all steps that would \nnormally be taken, when not using the algorithm, are reproduced as algorithmic processes that are a \npolynomial function of the input size. \n \n \n5. Quantum connection? \n \nQuantum mechanics deals with things at the very essence of reality. And in a \nmathematical sense it would not be farfetched to say the same thing about the P vs. NP \nproblem. Consider for a moment the Heisenberg uncertainty principle... Probability \nrules; things are dealt with in a non-deterministic manner. \n \nWhat about Quantum computers and the equality of P and NP? \nTo answer this question, let us go back to the latter: Nature at its core, as far as we know, \nis probabilistic; this indeterminism leads to the deterministic phenomena observable in \nour everyday macroscopic world. So does this not in itself answer the question? Does this \nmean that quantum computers, as a concept and a tool to solve the P vs. NP problem, are \nno better than “normal” computers? It probably does.\n\nWillgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n4\n \n6. Conclusion \n \nThis brief has shown how NP type problems are intrinsically tied into themselves, almost \nlike a subset concurrently outside and within the boundaries of a greater set. P problems, \non the other hand, are free from such ties, and tend not to become bogged down within. \nSo, it is tempting to speculate that if P equaled NP, this fact would be evident and clear. \nBut of course, there may exist obscure conditions such that P equals NP; and the reason \nwe cannot identify them could lie in possible short-comings of current mathematics. \nHence we may need to discover new mathematics and/or a novel type of reasoning. \nHowever, given what has been presented, it must be said that the likelihood of P being \nequal to NP is very slim. It might just be that the laws of nature prohibit this equality: \nThink quantum mechanics, think indeterminism...–But, if drawing parallels to nature is \nvalid, does not that mean that P is just as probabilistic as NP? Well, that is a fair point. \nHowever, remembering that quantum mechanical phenomena leads to classical \nphenomena, then NP complexities would be the mathematical counterpart of quantum \nmechanical processes or systems; subsequently, P complexities would be the \nmathematical counterpart of processes or systems in the classical world. Then, \ndisregarding potential fallacies, P is to Newton as NP is to Heisenberg. \n \nSo, to end on a positive note: whatever our convictions; let us keep an open mind. \n \n \nReferences \n \n1 Aaronson S, (2003), ‘Is P Versus NP Formally Independent’, Bulletin of the EATCS 81, p 1. \n \n2 Howe D (1995) ‘Foldoc - NTIME’ [on-line], London, UK, Imperial College. Available from: \nhttp://burks.bton.ac.uk/burks/foldoc/82/80.htm \n \n3 Weisstein, E W. ‘MathWorld - Limit Comparison Test’ [on-line] Champaign, USA, Wolfram Research \nInc, Available from: http://mathworld.wolfram.com/LimitComparisonTest.html [accessed Aug 2007]. \n \n \n# Bibliography \n \nAaronson S, (2003), ‘Is P Versus NP Formally Independent’, Bulletin of the EATCS 81. Also available \nonline from: http://www.scottaaronson.com/papers/ [accessed Aug 2007]. \n \nBroadhurst D -et al., (2002), The Quantum World, Milton Keynes, UK, The Open University, pp 26-41. \n \nHowe D (1985-2007) ‘Free On-line Dictionary of Computing (Foldoc)’ [on-line], London, UK, Imperial \nCollege. Available from: http://foldoc.org/ [accessed July 2007]. \n \nGmH -et al., ‘P versus NP’ [on-line], New York, USA. Available from QEden: \nhttp://www.qeden.com/wiki/P_versus_NP:Math_Intense_Intro [accessed Aug 2007]. \n \nNeill H (2003), Calculus, 3rd edtn, London, UK, Hodder Headline ltd - tech yourself, pp 113-123. \n \nNielsen D -et al., (2002), Dictionary of Mathematics, 3rd edtn, London, UK, Penguin Books, pp 254, 301. \n \nWeisstein, E W. ‘Complexity Theory’ [on-line] Champaign, USA. From MathWorld –A Wolfram Web \nResource: http://mathworld.wolfram.com/ComplexityTheory.html [accessed July 2007].","paragraphs":[{"paragraph_id":"p1","order":1,"text":"The P versus NP Brief"},{"paragraph_id":"p2","order":2,"text":"Type of paper: \nBrief study"},{"paragraph_id":"p3","order":3,"text":"Date: \n*September 2007"},{"paragraph_id":"p4","order":4,"text":"Correspondence: \nMikael Franzén \nWillgood Institute \nVegagatan 58, V5 Unit 43 \n413 11 Göteborg \nSweden"},{"paragraph_id":"p5","order":5,"text":"mikael.franzen@willgood.org"},{"paragraph_id":"p6","order":6,"text":"Abstract: \nThis paper discusses why P and NP are likely to be different. It analyses the essence of the concepts \nand points out that P and NP might be diverse by sheer definition. It also speculates that P and NP \nmay be unequal due to natural laws."},{"paragraph_id":"p7","order":7,"text":"* The paper was written and completed in Sept 2007 (RV01PNP01). It was then revised twice between the date of completion \nand April 2009, and hence two updated versions (RV02PNP01, RV03PNP01) were produced. As with RV02PNP01, changes \nin RV03PNP01 are minor and consist of additional content such as, notational declarations, formatting changes, and the latest \ncorrespondence details."},{"paragraph_id":"p8","order":8,"text":"Willgood Institute"},{"paragraph_id":"p9","order":9,"text":"Hamilton Institute \nAuthor: Mikael Franzén"},{"paragraph_id":"p10","order":10,"text":"Ref: RV03PNP01"},{"paragraph_id":"p11","order":11,"text":"1"},{"paragraph_id":"p12","order":12,"text":"1. Introduction"},{"paragraph_id":"p13","order":13,"text":"It may be defined as a matter of polynomial time versus non-deterministic polynomial \ntime; and hence the question whether P equals NP, or P does not equal NP. It is hailed \nthe most significant problem within theoretical computer science and serves as a \ntestimony of the complexity of seemingly simple logic and “everyday” axioms. But, it \nruns far deeper than that. In a sense it addresses the very nature of logic and physical \ntruth; because the way we ask ourselves this question is well beyond the realm of \nmathematics, and therefore, it can be viewed as being metamathematical , as Scott \nAronson1 referred to it."},{"paragraph_id":"p14","order":14,"text":"So, maybe the way of approach is far more important than the actual problem. Perhaps \nthe manner in which P vs. NP is assessed and analyzed makes a significant difference in \nterms of importance. That is not to say that the essence of the problem is a variable \ndepending on approach, but rather that the approach may hold the answer within itself."},{"paragraph_id":"p15","order":15,"text":"Let us take a look at what the following entails:"},{"paragraph_id":"p16","order":16,"text":"P means Polynomial time and is defined as the computation runtime when no greater \nthan the function of the problem. It would almost be foolish to assume that NP is \nanything else than the opposite of P. So NP means Non-deterministic Polynomial time, \nand is defined as the computation runtime when greater than the function of the problem. \nHowever, this definition is not exactly correct; the reason being indeterminism. In other \nwords, problems in NP (besides those that are also in P) always rely on some sort of \nbrute-force analysis or probabilistic measure. Therefore NP should be defined as the \nnumber of steps that may exponentially evolve in order to fit the relevant polynomial \nfunction. Thus, computation runtime is likely to be exponentially as long as a potential \nsolution can be verified in polynomial time.2"},{"paragraph_id":"p17","order":17,"text":"Simply put, P problems are relatively fast to compute and hence also fast to solve and to \nverify, while NP problems are complicated to compute and therefore time-consuming to \nsolve, but easy to verify once the answer is known. So, what is the differentiating factor \nin all this? Well, the variables of course. In order for P to equal NP, we need to eliminate \nthe variables – Not literally speaking, but algorithmically speaking. However, is this at all \npossible?"},{"paragraph_id":"p18","order":18,"text":"2. Input, processor and output"},{"paragraph_id":"p19","order":19,"text":"Definition of (a), a polynomial time problem, and (b), a non deterministic polynomial \ntime problem, where k is an independent constant; n the input string and m the instance of \nprocess work: \n(a), \n)\nn\n(\n)\nn\n(\nm\nn\nk\n(n)\nk\nf\nO"},{"paragraph_id":"p20","order":20,"text":"))\n(\nO("},{"paragraph_id":"p21","order":21,"text":"=\n→\n=\n. \n(b), \n)\nO("},{"paragraph_id":"p22","order":22,"text":"))\nO(n("},{"paragraph_id":"p23","order":23,"text":")\nO("},{"paragraph_id":"p24","order":24,"text":"))\n(\nO("},{"paragraph_id":"p25","order":25,"text":"n\nk\nk\n)\nn\n(\nm\nk\nn\nm\nn\n)\nn\n(\nm\nn\nk\n(n)\nf\n=\n→\n→\n≠\n→\n=\n→\n=\n. \nIn essence this tells us that a problem applied to the function f(n) characterizes that function \nby definition. Thus, a question in P or NP will render the function accordingly. So in \ngeneral terms, all input with variables exceeding the number of functionalities of the \nprocessor, will most likely be NP type of problems. Let us examine this a little bit closer:"},{"paragraph_id":"p26","order":26,"text":"Willgood Institute"},{"paragraph_id":"p27","order":27,"text":"Hamilton Institute \nAuthor: Mikael Franzén"},{"paragraph_id":"p28","order":28,"text":"Ref: RV03PNP01"},{"paragraph_id":"p29","order":29,"text":"2"},{"paragraph_id":"p30","order":30,"text":"For example, providing an input of 1 and 1 to a processor, which performs addition, will \nresult in an input of 2 units with an equal 2-unit processor functionality. The operation of \naddition will count as 1 unit and the production of a result, or output, may be considered \nas another 1 unit. Hence, in a straightforward \nmanner, we could say that the plus (+) and \nequals (=) signs are the 2 units of \nfunctionality. This type of processor-analogy \nworks very well for P problems, but it would \nnot work for any type of NP problem. NP \nproblems would not even fit the input intake; \nin a way it would be like feeding skittles through your inkjet, or putting stones in your \njuice blender. So, if we consider the Traveling salesman problem and the Subset sum \nproblem: the variables are individually, or in some combination, processable by the kind \nof processor pictured above. However, all together it will render the processor in terms of \nINTIME. So, concretely described, NP complexities are then solved by different, non-\ndeterministic kinds of processors, as opposed to deterministic ones, like in Figure 1. \nHowever, non-deterministic processors are intrinsically dependent on deterministic \nprocessors (i.e. P is a subset of NP). Therefore, as implied earlier, a complexity function \nis characterized by its input, thus suggesting that P is not the same as NP; strongly \nimplying, (perhaps naively), that in general, P is not equal to NP."},{"paragraph_id":"p31","order":31,"text":"3. Characterizations"},{"paragraph_id":"p32","order":32,"text":"P complexities are processed at their very essence, instantly, even if time-wise it may \ntake very long:"},{"paragraph_id":"p33","order":33,"text":")\nO("},{"paragraph_id":"p34","order":34,"text":"))\n(\nO("},{"paragraph_id":"p35","order":35,"text":"k\nn\nn\n.\nm\nn\nk\n(n)\nm\n.n\nf\n≤\n∀\n∝\n=\n∝\n∀\n."},{"paragraph_id":"p36","order":36,"text":"-Therefore, n may approach infinity, and consequently, the function and its output \napproaches equal bounds. Although it possibly could be impractical and lingering, the \nruntime, on the other hand, does not approach infinity – Well, the total runtime may \napproach infinity to the extent that the input does, but, total runtime would normally not \nbe counted until an end is reached, and subsequently, in this context, total runtime may be \nan invalid notion:\n 3"},{"paragraph_id":"p37","order":37,"text":"II\n∑\n∑\n∞\n=\n∞\n=\n∞\n→\n↔\n→\n→\n>\n1\n1\nn\nn\n)\nn\n(\nm\n)\nn\n(\n)\nn\n(\n)\nn\n(\nm\n)\nn\n(\nm\n(n)\nf\nf\nf"},{"paragraph_id":"p38","order":38,"text":"0"},{"paragraph_id":"p39","order":39,"text":","},{"paragraph_id":"p40","order":40,"text":"n lim\n."},{"paragraph_id":"p41","order":41,"text":"NP complexities may or may not be processed at their essence. That is to say, a brute-\nforce analysis, or some other type of non-deterministic activity may yield computational \nresults that are relevant or irrelevant to the core of the question. Consequently:"},{"paragraph_id":"p42","order":42,"text":"III\n. )\nO("},{"paragraph_id":"p43","order":43,"text":"Ω"},{"paragraph_id":"p44","order":44,"text":"))\n(\nO("},{"paragraph_id":"p45","order":45,"text":"n\nk\nn\nk\n(n)\nm\n.n\n f\n≥\n∝\n=\n∝\n∃"},{"paragraph_id":"p46","order":46,"text":"I O(f(n)) = NP = NTIME, \n)\n(\nΝ\nNTIME\nNP"},{"paragraph_id":"p47","order":47,"text":"k\nn\nkU\n∈\n=\n | O(2f(n) ) =\n)\n(2\nΝ\nNTIME\nNEXPTIME"},{"paragraph_id":"p48","order":48,"text":"k\nn\nkU\n∈\n="},{"paragraph_id":"p49","order":49,"text":"II Let the sequential terms {an } and {bn} be the instances of functionality and input, f(n) and m(n) \nIII Let Ω denote all possible outcomes, {x1, x2, ...} \nFigure 1"},{"paragraph_id":"p50","order":50,"text":"Willgood Institute"},{"paragraph_id":"p51","order":51,"text":"Hamilton Institute \nAuthor: Mikael Franzén"},{"paragraph_id":"p52","order":52,"text":"Ref: RV03PNP01"},{"paragraph_id":"p53","order":53,"text":"3\n-So, as with P complexities, n may approach infinity, but with one very significant \ndifference, namely, the relationship between runtime and function. Here we see a \ndislocation between input, function and runtime. Firstly in terms of components and \nfunctionality; input may be non-deterministic in a way that yields no immediate function, \nor, it may be deterministic, but ‘incorrect’, thus yielding an irrelevant function. Secondly in \nterms of totality; input may produce disproportionate exponential or factorial runtimes, and \nruntime may depend on variables that are outside immediate and closely related sets. In \nother words, the scope of Ω may be considered unbounded by its own limits:"},{"paragraph_id":"p54","order":54,"text":"0"},{"paragraph_id":"p55","order":55,"text":"→\n>\n)\nn\n(\nm\n(n),\nf\n Ω ⊢"},{"paragraph_id":"p56","order":56,"text":"Ω\nΩ"},{"paragraph_id":"p57","order":57,"text":"n"},{"paragraph_id":"p58","order":58,"text":"∑\n∞\n∑\n∞"},{"paragraph_id":"p59","order":59,"text":"=\n=\n↔\n→\n∴\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nf\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nm\nlim\n. \nSo:"},{"paragraph_id":"p60","order":60,"text":"If we create an algorithm that renders P to equal NP it would have to remodel all \ndependencies and variables into a nice polynomial package. In addition it needs a real-\ntime performance by the square root of all possible answers:"},{"paragraph_id":"p61","order":61,"text":"Ω"},{"paragraph_id":"p62","order":62,"text":"0"},{"paragraph_id":"p63","order":63,"text":", \n=\n>\n)\nn\n(\nm\n(n)\n f\n –This would yield the same expression as the one for P \ncomplexities on previous page, but the problem with this is that it first entails the \nfollowing:"},{"paragraph_id":"p64","order":64,"text":"Ω"},{"paragraph_id":"p65","order":65,"text":"f"},{"paragraph_id":"p66","order":66,"text":"Ω\nΩ"},{"paragraph_id":"p67","order":67,"text":"n \n∝\n∑\n∞\n∑\n∞"},{"paragraph_id":"p68","order":68,"text":"=\n=\n↔\n→\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\n)\nn\n(\nm\nlim\n –And thus we are back where \nwe started:"},{"paragraph_id":"p69","order":69,"text":"0"},{"paragraph_id":"p70","order":70,"text":"→\n>\n)\nn\n(\nm\n(n),\nf\n Ω ⊢"},{"paragraph_id":"p71","order":71,"text":"Ω\nΩ"},{"paragraph_id":"p72","order":72,"text":"n"},{"paragraph_id":"p73","order":73,"text":"∑\n∞\n∑\n∞"},{"paragraph_id":"p74","order":74,"text":"=\n=\n↔\n→\n∴\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nf\n)\nn\n(\nm\n)\nn\n(f\n)\nn\n(\nm\nlim\n. \nSo in this context: \nΩ\nΩ"},{"paragraph_id":"p75","order":75,"text":"Ω"},{"paragraph_id":"p76","order":76,"text":"Ω"},{"paragraph_id":"p77","order":77,"text":"∈\n=\nQ\n."},{"paragraph_id":"p78","order":78,"text":"Hence, trying to reduce size and limit the scope, while at the same time taking all the variables into \naccount, results in an extrapolation of all possible outcomes (Ω). Therefore all steps that would \nnormally be taken, when not using the algorithm, are reproduced as algorithmic processes that are a \npolynomial function of the input size."},{"paragraph_id":"p79","order":79,"text":"5. Quantum connection?"},{"paragraph_id":"p80","order":80,"text":"Quantum mechanics deals with things at the very essence of reality. And in a \nmathematical sense it would not be farfetched to say the same thing about the P vs. NP \nproblem. Consider for a moment the Heisenberg uncertainty principle... Probability \nrules; things are dealt with in a non-deterministic manner."},{"paragraph_id":"p81","order":81,"text":"What about Quantum computers and the equality of P and NP? \nTo answer this question, let us go back to the latter: Nature at its core, as far as we know, \nis probabilistic; this indeterminism leads to the deterministic phenomena observable in \nour everyday macroscopic world. So does this not in itself answer the question? Does this \nmean that quantum computers, as a concept and a tool to solve the P vs. NP problem, are \nno better than “normal” computers? It probably does."},{"paragraph_id":"p82","order":82,"text":"Willgood Institute"},{"paragraph_id":"p83","order":83,"text":"Hamilton Institute \nAuthor: Mikael Franzén"},{"paragraph_id":"p84","order":84,"text":"Ref: RV03PNP01"},{"paragraph_id":"p85","order":85,"text":"4"},{"paragraph_id":"p86","order":86,"text":"6. Conclusion"},{"paragraph_id":"p87","order":87,"text":"This brief has shown how NP type problems are intrinsically tied into themselves, almost \nlike a subset concurrently outside and within the boundaries of a greater set. P problems, \non the other hand, are free from such ties, and tend not to become bogged down within. \nSo, it is tempting to speculate that if P equaled NP, this fact would be evident and clear. \nBut of course, there may exist obscure conditions such that P equals NP; and the reason \nwe cannot identify them could lie in possible short-comings of current mathematics. \nHence we may need to discover new mathematics and/or a novel type of reasoning. \nHowever, given what has been presented, it must be said that the likelihood of P being \nequal to NP is very slim. It might just be that the laws of nature prohibit this equality: \nThink quantum mechanics, think indeterminism...–But, if drawing parallels to nature is \nvalid, does not that mean that P is just as probabilistic as NP? Well, that is a fair point. \nHowever, remembering that quantum mechanical phenomena leads to classical \nphenomena, then NP complexities would be the mathematical counterpart of quantum \nmechanical processes or systems; subsequently, P complexities would be the \nmathematical counterpart of processes or systems in the classical world. Then, \ndisregarding potential fallacies, P is to Newton as NP is to Heisenberg."},{"paragraph_id":"p88","order":88,"text":"So, to end on a positive note: whatever our convictions; let us keep an open mind."},{"paragraph_id":"p89","order":89,"text":"References"},{"paragraph_id":"p90","order":90,"text":"1 Aaronson S, (2003), ‘Is P Versus NP Formally Independent’, Bulletin of the EATCS 81, p 1."},{"paragraph_id":"p91","order":91,"text":"2 Howe D (1995) ‘Foldoc - NTIME’ [on-line], London, UK, Imperial College. Available from: \nhttp://burks.bton.ac.uk/burks/foldoc/82/80.htm"},{"paragraph_id":"p92","order":92,"text":"3 Weisstein, E W. ‘MathWorld - Limit Comparison Test’ [on-line] Champaign, USA, Wolfram Research \nInc, Available from: http://mathworld.wolfram.com/LimitComparisonTest.html [accessed Aug 2007]."},{"paragraph_id":"p93","order":93,"text":"# Bibliography"},{"paragraph_id":"p94","order":94,"text":"Aaronson S, (2003), ‘Is P Versus NP Formally Independent’, Bulletin of the EATCS 81. Also available \nonline from: http://www.scottaaronson.com/papers/ [accessed Aug 2007]."},{"paragraph_id":"p95","order":95,"text":"Broadhurst D -et al., (2002), The Quantum World, Milton Keynes, UK, The Open University, pp 26-41."},{"paragraph_id":"p96","order":96,"text":"Howe D (1985-2007) ‘Free On-line Dictionary of Computing (Foldoc)’ [on-line], London, UK, Imperial \nCollege. Available from: http://foldoc.org/ [accessed July 2007]."},{"paragraph_id":"p97","order":97,"text":"GmH -et al., ‘P versus NP’ [on-line], New York, USA. Available from QEden: \nhttp://www.qeden.com/wiki/P_versus_NP:Math_Intense_Intro [accessed Aug 2007]."},{"paragraph_id":"p98","order":98,"text":"Neill H (2003), Calculus, 3rd edtn, London, UK, Hodder Headline ltd - tech yourself, pp 113-123."},{"paragraph_id":"p99","order":99,"text":"Nielsen D -et al., (2002), Dictionary of Mathematics, 3rd edtn, London, UK, Penguin Books, pp 254, 301."},{"paragraph_id":"p100","order":100,"text":"Weisstein, E W. ‘Complexity Theory’ [on-line] Champaign, USA. From MathWorld –A Wolfram Web \nResource: http://mathworld.wolfram.com/ComplexityTheory.html [accessed July 2007]."}],"pages":[{"page":1,"text":"The P versus NP Brief \n \n \nType of paper: \nBrief study \n \n \nDate: \n*September 2007 \n \n \n \n \nCorrespondence: \nMikael Franzén \nWillgood Institute \nVegagatan 58, V5 Unit 43 \n413 11 Göteborg \nSweden \n \nmikael.franzen@willgood.org \n \n \n \n \n \nAbstract: \nThis paper discusses why P and NP are likely to be different. It analyses the essence of the concepts \nand points out that P and NP might be diverse by sheer definition. It also speculates that P and NP \nmay be unequal due to natural laws. \n \n \n \n \n \n \n* The paper was written and completed in Sept 2007 (RV01PNP01). It was then revised twice between the date of completion \nand April 2009, and hence two updated versions (RV02PNP01, RV03PNP01) were produced. As with RV02PNP01, changes \nin RV03PNP01 are minor and consist of additional content such as, notational declarations, formatting changes, and the latest \ncorrespondence details."},{"page":2,"text":"Willgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n1\n \n \n1. Introduction \n \nIt may be defined as a matter of polynomial time versus non-deterministic polynomial \ntime; and hence the question whether P equals NP, or P does not equal NP. It is hailed \nthe most significant problem within theoretical computer science and serves as a \ntestimony of the complexity of seemingly simple logic and “everyday” axioms. But, it \nruns far deeper than that. In a sense it addresses the very nature of logic and physical \ntruth; because the way we ask ourselves this question is well beyond the realm of \nmathematics, and therefore, it can be viewed as being metamathematical , as Scott \nAronson1 referred to it. \n \nSo, maybe the way of approach is far more important than the actual problem. Perhaps \nthe manner in which P vs. NP is assessed and analyzed makes a significant difference in \nterms of importance. That is not to say that the essence of the problem is a variable \ndepending on approach, but rather that the approach may hold the answer within itself. \n \nLet us take a look at what the following entails: \n \nP means Polynomial time and is defined as the computation runtime when no greater \nthan the function of the problem. It would almost be foolish to assume that NP is \nanything else than the opposite of P. So NP means Non-deterministic Polynomial time, \nand is defined as the computation runtime when greater than the function of the problem. \nHowever, this definition is not exactly correct; the reason being indeterminism. In other \nwords, problems in NP (besides those that are also in P) always rely on some sort of \nbrute-force analysis or probabilistic measure. Therefore NP should be defined as the \nnumber of steps that may exponentially evolve in order to fit the relevant polynomial \nfunction. Thus, computation runtime is likely to be exponentially as long as a potential \nsolution can be verified in polynomial time.2 \n \nSimply put, P problems are relatively fast to compute and hence also fast to solve and to \nverify, while NP problems are complicated to compute and therefore time-consuming to \nsolve, but easy to verify once the answer is known. So, what is the differentiating factor \nin all this? Well, the variables of course. In order for P to equal NP, we need to eliminate \nthe variables – Not literally speaking, but algorithmically speaking. However, is this at all \npossible? \n \n2. Input, processor and output \n \nDefinition of (a), a polynomial time problem, and (b), a non deterministic polynomial \ntime problem, where k is an independent constant; n the input string and m the instance of \nprocess work: \n(a), \n)\nn\n(\n)\nn\n(\nm\nn\nk\n(n)\nk\nf\nO\n \n \n))\n(\nO(\n \n \n=\n→\n=\n. \n(b), \n)\nO(\n \n \n \n))\nO(n(\n \n \n \n )\nO(\n \n \n \n))\n(\nO(\n \n \nn\nk\nk\n)\nn\n(\nm\nk\nn\nm\nn\n)\nn\n(\nm\nn\nk\n(n)\nf\n=\n→\n→\n≠\n→\n=\n→\n=\n. \nIn essence this tells us that a problem applied to the function f(n) characterizes that function \nby definition. Thus, a question in P or NP will render the function accordingly. So in \ngeneral terms, all input with variables exceeding the number of functionalities of the \nprocessor, will most likely be NP type of problems. Let us examine this a little bit closer:"},{"page":3,"text":"Willgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n2\n \nFor example, providing an input of 1 and 1 to a processor, which performs addition, will \nresult in an input of 2 units with an equal 2-unit processor functionality. The operation of \naddition will count as 1 unit and the production of a result, or output, may be considered \nas another 1 unit. Hence, in a straightforward \nmanner, we could say that the plus (+) and \nequals (=) signs are the 2 units of \nfunctionality. This type of processor-analogy \nworks very well for P problems, but it would \nnot work for any type of NP problem. NP \nproblems would not even fit the input intake; \nin a way it would be like feeding skittles through your inkjet, or putting stones in your \njuice blender. So, if we consider the Traveling salesman problem and the Subset sum \nproblem: the variables are individually, or in some combination, processable by the kind \nof processor pictured above. However, all together it will render the processor in terms of \nINTIME. So, concretely described, NP complexities are then solved by different, non-\ndeterministic kinds of processors, as opposed to deterministic ones, like in Figure 1. \nHowever, non-deterministic processors are intrinsically dependent on deterministic \nprocessors (i.e. P is a subset of NP). Therefore, as implied earlier, a complexity function \nis characterized by its input, thus suggesting that P is not the same as NP; strongly \nimplying, (perhaps naively), that in general, P is not equal to NP. \n \n \n3. Characterizations \n \nP complexities are processed at their very essence, instantly, even if time-wise it may \ntake very long: \n \n \n)\nO(\n \n \n \n \n))\n(\nO(\n \n \n \n \nk\nn\nn\n.\nm\nn\nk\n(n)\nm\n.n\nf\n≤\n∀\n∝\n=\n∝\n∀\n. \n \n-Therefore, n may approach infinity, and consequently, the function and its output \napproaches equal bounds. Although it possibly could be impractical and lingering, the \nruntime, on the other hand, does not approach infinity – Well, the total runtime may \napproach infinity to the extent that the input does, but, total runtime would normally not \nbe counted until an end is reached, and subsequently, in this context, total runtime may be \nan invalid notion:\n 3 \n \nII\n∑\n∑\n∞\n=\n∞\n=\n∞\n→\n↔\n→\n→\n>\n1\n1\nn\nn\n)\nn\n(\nm\n)\nn\n(\n)\nn\n(\n)\nn\n(\nm\n)\nn\n(\nm\n(n)\nf\nf\nf\n \n \n \n \n \n \n \n \n0\n \n ,\n \n \nn lim\n. \n \nNP complexities may or may not be processed at their essence. That is to say, a brute-\nforce analysis, or some other type of non-deterministic activity may yield computational \nresults that are relevant or irrelevant to the core of the question. Consequently: \n \nIII\n. )\nO(\n \nΩ\n \n))\n(\nO(\n \n \n \n \n \n \nn\nk\nn\nk\n(n)\nm\n.n\n f\n≥\n∝\n=\n∝\n∃\n \n \n \nI O(f(n)) = NP = NTIME, \n)\n(\nΝ\nNTIME\nNP\n \nk\nn\nkU\n∈\n=\n | O(2f(n) ) =\n)\n(2\nΝ\nNTIME\nNEXPTIME\n \nk\nn\nkU\n∈\n=\n \n II Let the sequential terms {an } and {bn} be the instances of functionality and input, f(n) and m(n) \nIII Let Ω denote all possible outcomes, {x1, x2, ...} \nFigure 1"},{"page":4,"text":"Willgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n3\n-So, as with P complexities, n may approach infinity, but with one very significant \ndifference, namely, the relationship between runtime and function. Here we see a \ndislocation between input, function and runtime. Firstly in terms of components and \nfunctionality; input may be non-deterministic in a way that yields no immediate function, \nor, it may be deterministic, but ‘incorrect’, thus yielding an irrelevant function. Secondly in \nterms of totality; input may produce disproportionate exponential or factorial runtimes, and \nruntime may depend on variables that are outside immediate and closely related sets. In \nother words, the scope of Ω may be considered unbounded by its own limits: \n \n \n0\n \n \n→\n>\n)\nn\n(\nm\n(n),\nf\n Ω ⊢\n \n \n \n \n \n \n \n \nΩ\nΩ \n \nn \n \n \n \n \n∑\n∞\n∑\n∞\n \n \n \n \n \n \n=\n=\n↔\n→\n∴\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nf\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nm\nlim\n. \nSo: \n \nIf we create an algorithm that renders P to equal NP it would have to remodel all \ndependencies and variables into a nice polynomial package. In addition it needs a real-\ntime performance by the square root of all possible answers: \n \n \nΩ\n \n \n0\n \n , \n=\n>\n)\nn\n(\nm\n(n)\n f\n –This would yield the same expression as the one for P \ncomplexities on previous page, but the problem with this is that it first entails the \nfollowing: \n \nΩ\n \n \n \n \n \n \n \n \nf\n \n \nΩ\nΩ \n \nn \n∝\n∑\n∞\n∑\n∞\n \n \n \n \n \n \n \n \n \n \n=\n=\n↔\n→\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\n)\nn\n(\nm\nlim\n –And thus we are back where \nwe started: \n \n0\n \n \n→\n>\n)\nn\n(\nm\n(n),\nf\n Ω ⊢\n \n \n \n \n \n \n \n \n \nΩ\nΩ \n \nn \n \n \n \n \n∑\n∞\n∑\n∞\n \n \n \n \n \n \n=\n=\n↔\n→\n∴\n∞\n→\n1\nn\n1\nn\n)\nn\n(\nm\n)\nn\n(\nf\n)\nn\n(\nf\n)\nn\n(\nm\n)\nn\n(f\n)\nn\n(\nm\nlim\n. \nSo in this context: \nΩ\nΩ\n \n \nΩ\n \nΩ\n \n∈\n=\nQ\n. \n \nHence, trying to reduce size and limit the scope, while at the same time taking all the variables into \naccount, results in an extrapolation of all possible outcomes (Ω). Therefore all steps that would \nnormally be taken, when not using the algorithm, are reproduced as algorithmic processes that are a \npolynomial function of the input size. \n \n \n5. Quantum connection? \n \nQuantum mechanics deals with things at the very essence of reality. And in a \nmathematical sense it would not be farfetched to say the same thing about the P vs. NP \nproblem. Consider for a moment the Heisenberg uncertainty principle... Probability \nrules; things are dealt with in a non-deterministic manner. \n \nWhat about Quantum computers and the equality of P and NP? \nTo answer this question, let us go back to the latter: Nature at its core, as far as we know, \nis probabilistic; this indeterminism leads to the deterministic phenomena observable in \nour everyday macroscopic world. So does this not in itself answer the question? Does this \nmean that quantum computers, as a concept and a tool to solve the P vs. NP problem, are \nno better than “normal” computers? It probably does."},{"page":5,"text":"Willgood Institute \n \nHamilton Institute \nAuthor: Mikael Franzén \n \n Ref: RV03PNP01 \n \n4\n \n6. Conclusion \n \nThis brief has shown how NP type problems are intrinsically tied into themselves, almost \nlike a subset concurrently outside and within the boundaries of a greater set. P problems, \non the other hand, are free from such ties, and tend not to become bogged down within. \nSo, it is tempting to speculate that if P equaled NP, this fact would be evident and clear. \nBut of course, there may exist obscure conditions such that P equals NP; and the reason \nwe cannot identify them could lie in possible short-comings of current mathematics. \nHence we may need to discover new mathematics and/or a novel type of reasoning. \nHowever, given what has been presented, it must be said that the likelihood of P being \nequal to NP is very slim. It might just be that the laws of nature prohibit this equality: \nThink quantum mechanics, think indeterminism...–But, if drawing parallels to nature is \nvalid, does not that mean that P is just as probabilistic as NP? Well, that is a fair point. \nHowever, remembering that quantum mechanical phenomena leads to classical \nphenomena, then NP complexities would be the mathematical counterpart of quantum \nmechanical processes or systems; subsequently, P complexities would be the \nmathematical counterpart of processes or systems in the classical world. Then, \ndisregarding potential fallacies, P is to Newton as NP is to Heisenberg. \n \nSo, to end on a positive note: whatever our convictions; let us keep an open mind. \n \n \nReferences \n \n1 Aaronson S, (2003), ‘Is P Versus NP Formally Independent’, Bulletin of the EATCS 81, p 1. \n \n2 Howe D (1995) ‘Foldoc - NTIME’ [on-line], London, UK, Imperial College. Available from: \nhttp://burks.bton.ac.uk/burks/foldoc/82/80.htm \n \n3 Weisstein, E W. ‘MathWorld - Limit Comparison Test’ [on-line] Champaign, USA, Wolfram Research \nInc, Available from: http://mathworld.wolfram.com/LimitComparisonTest.html [accessed Aug 2007]. \n \n \n# Bibliography \n \nAaronson S, (2003), ‘Is P Versus NP Formally Independent’, Bulletin of the EATCS 81. Also available \nonline from: http://www.scottaaronson.com/papers/ [accessed Aug 2007]. \n \nBroadhurst D -et al., (2002), The Quantum World, Milton Keynes, UK, The Open University, pp 26-41. \n \nHowe D (1985-2007) ‘Free On-line Dictionary of Computing (Foldoc)’ [on-line], London, UK, Imperial \nCollege. Available from: http://foldoc.org/ [accessed July 2007]. \n \nGmH -et al., ‘P versus NP’ [on-line], New York, USA. Available from QEden: \nhttp://www.qeden.com/wiki/P_versus_NP:Math_Intense_Intro [accessed Aug 2007]. \n \nNeill H (2003), Calculus, 3rd edtn, London, UK, Hodder Headline ltd - tech yourself, pp 113-123. \n \nNielsen D -et al., (2002), Dictionary of Mathematics, 3rd edtn, London, UK, Penguin Books, pp 254, 301. \n \nWeisstein, E W. ‘Complexity Theory’ [on-line] Champaign, USA. From MathWorld –A Wolfram Web \nResource: http://mathworld.wolfram.com/ComplexityTheory.html [accessed July 2007]."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"equals (=) signs are the 2 units of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"I O(f(n)) = NP = NTIME,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"| O(2f(n) ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":12992,"parse_confidence":0.5,"equation_parse_rate_proxy":0.15,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}