structured/0710.3961.structured.json

{"paper_meta":{"paper_id":"arxiv:0710.3961","title":"0710.3961","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":"arXiv:0710.3961v1 [cs.CC] 22 Oct 2007\nOn a New Type of Information Processing\nfor Efficient Management of Complex Systems\nVictor Korotkikh∗and Galina Korotkikh†\nSchool of Computing Sciences\nCentral Queensland University\nMackay, Queensland, 4740\nAustralia\nIt is a challenge to manage complex systems efficiently without confronting NP-hard problems.\nTo address the situation we suggest to use self-organization processes of prime integer relations\nfor information processing. Self-organization processes of prime integer relations define correlation\nstructures of a complex system and can be equivalently represented by transformations of two-\ndimensional geometrical patterns determining the dynamics of the system and revealing its structural\ncomplexity. Computational experiments raise the possibility of an optimality condition of complex\nsystems presenting the structural complexity of a system as a key to its optimization. From this\nperspective the optimization of a system could be all about the control of the structural complexity\nof the system to make it consistent with the structural complexity of the problem. The experiments\nalso indicate that the performance of a complex system may behave as a concave function of the\nstructural complexity. Therefore, once the structural complexity could be controlled as a single\nentity, the optimization of a complex system would be potentially reduced to a one-dimensional\nconcave optimization irrespective of the number of variables involved its description. This might\nopen a way to a new type of information processing for efficient management of complex systems.\nPACS numbers: 89.75.-k, 89.75.Fb\nI.\nINTRODUCTION\nIt is a challenge to manage complex systems efficiently\nwithout confronting NP-hard problems. To address the\nsituation we consider the description of complex systems\nin terms of self-organization processes of prime integer\nrelations [1] and suggest to use the processes for infor-\nmation processing.\nII.\nTHE HIERARCHICAL NETWORK OF\nPRIME INTEGER RELATIONS\nThe description is realized through the unity of two\nequivalent forms, i.e., arithmetical and geometrical. We\nbriefly present the forms in order to introduce the hier-\narchical network of prime integer relations. More details\nmay be found in [1], [2].\nA.\nTHE ARITHMETICAL FORM\nIn the arithmetical form a complex system is char-\nacterized by hierarchical correlation structures built in\naccordance with self-organization processes of prime in-\nteger relations. As each of the correlation structures is\nready to exercise its own scenario and there is no mecha-\nnism specifying which of them is going to take place, an\n∗Electronic address: v.korotkikh@cqu.edu.au\n†Electronic address: g.korotkikh@cqu.edu.au\nintrinsic uncertainty about the complex system exists.\nAt the same time, the information about the correlation\nstructures can be used to evaluate the probability of an\nobservable to take each of the measurement outcomes.\nTherefore, the arithmetical form of the description pro-\nvides the statistical information about a complex system.\nThe form reveals nonlocal correlations without refer-\nence to signals as well as the distances and local times\nof the parts. Thus, the arithmetical form suggests that\nparts of a complex systems may be far apart in space and\ntime and yet remain interconnected with instantaneous\neffect on each other, but no signalling. Namely, if a cor-\nrelation structure of a system is selected and some parts\nare specified, then through the prime integer relations in\ncontrol of the correlation structure the other parts are\nimmediately defined.\nThrough the arithmetical form of the description we\nbecome aware of the hierarchical network, i.e., a set\nof mutually self-consistent elements built by all self-\norganization processes of prime integer relations. Arith-\nmetic ensures that not even a minor change can be made\nto any element of the network (Figure 1). It is helpful\nto think that self-organization processes of prime integer\nrelations take place in the hierarchical network as they\ncompose more and more complex structures.\nThe hierarchical network of prime integer relations is\na causal structure. As parts of a system change and the\nprime integer relations in control of the system cause the\nother parts to change accordingly, an event takes place.\nOnce the changes have been realized, the event is fixed\nin space and time with respect to the reference frames\nof the parts. For the parts the effect of the event has\nnot necessarily be the same, but for each part it is ap-\n\n2\n16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1\n+16 - 15 - 14 + 13 - 12 + 11 + 10 - 9 - 8 + 7 + 6 - 5 + 4 - 3 - 2 + 1 = 0\n+16 - 15 - 14 + 13 - 12 + 11 + 10 - 9 = 0\n- 8 + 7 + 6 - 5 + 4 - 3 - 2 + 1 = 0\n+16 - 15 - 14 + 13 = 0\n- 12 + 11 + 10 - 9 = 0\n- 8 + 7 + 6 - 5 = 0\n+ 4 - 3 - 2 + 1 = 0\n+16 - 15 = 0\n-14 + 13 = 0\n+10 - 9 = 0\n-12 + 11 = 0\n-8 + 7 = 0\n+6 - 5 = 0\n+4 - 3 = 0\n-2 + 1 = 0\n+16\n- 15\n- 14\n+ 13\n- 12\n + 11\n+ 10\n- 9\n- 8\n+ 7\n+ 6\n- 5\n+ 4\n- 3\n- 2\n+ 1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\nFIG. 1: The figure shows a hierarchical structure of prime\ninteger relations defined as a result of a self-organization pro-\ncess starting at the zero level with integers 16, 13, 11, 10, 7, 6,\n4, 1 in the ”positive state” and integers 15, 14, 12, 9, 8, 5, 3, 2\nin the ”negative”. The process is controlled by arithmetic and\ncannot progress to level 5, because arithmetic determines that\n+164−154−144+134−124+114+104−94−84+74+64−54+\n44 −34 −24 +14 ̸= 0. This hierarchical structure can specify a\ncorrelation structure of a system made of 16 elementary parts\nPi, i = 1, ..., 16 whose changes ∆xi, i = 1, ..., 16 of the observ-\nables xi, i = 1, ..., 16 relative to their reference frames can be\ngiven by the sequence ∆x1...∆x16 = +1 −1 −1 + 1 −1 + 1 +\n1 −1 −1 + 1 + 1 −1 + 1 −1 −1 + 1.\npropriately determined by the prime integer relations.\nHowever, the prime integer relations at work for a sys-\ntem have no causal power to effect systems controlled by\nseparate prime integer relations. As a result, informa-\ntion about the systems is blocked for the observers of the\nsystem.\nB.\nTHE GEOMETRICAL FORM\nSpecified by two parameters ε > 0 and δ > 0 the ge-\nometrical form arises as the self-organization processes\nof prime integer relations find isomorphic realization in\nterms of transformations of two-dimensional geometrical\npatterns [1]. As a result, hierarchical structures of prime\ninteger relations defining the correlation structures of a\ncomplex system become equivalently represented by hi-\nerarchical structures of geometrical patterns determining\nthe dynamics of the system and revealing its complexity.\nThe quantitative description of the system turns out to\nbe given by the description of the geometrical patterns\n[1], [2].\nFigure 2 shows a hierarchical structure of geometrical\npatterns, which for given ε > 0 and δ > 0 is isomorphic\nto the hierarchical structure of prime integer relations\ndepicted in Figure 1. A scale invariant property and a\n2\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n2\n1\n1\n[1]\n1\n[2]\n1\n[3]\n1\n[4]\n2\n[1]\n[0]\n1\n2\n3\n4\n4\n8\n4\nFIG. 2: The figure shows a hierarchical structure of geomet-\nrical patterns, which for given ε and δ is isomorphic to the\nhierarchical structure of prime integer relations shown in Fig-\nure 1. Under the integration of the function Ψ[l]\n1 , Ψ[l+1]\n1\n(t0) =\n0, l = 0, 1, 2, 3 the geometrical patterns of the level l form\nthe geometrical patterns of the higher level l + 1. The width\nWl of a geometrical pattern at level l, l = 1, ..., 4 equals the\nsum Wl = Wl−1,left + Wl−1,right = 2Wl−1 of the widths of\nthe two geometrical patterns it is made up of, where Wl−1,left\nand Wl−1,right are the widths of the left and right geometrical\npatterns and Wl−1,left = Wl−1,right = Wl−1 as each geomet-\nrical pattern at level l −1 has the same width Wl−1 = 2l−1ε\nand ti = εi, i = 0, 1, ..., 16. The height Hl of the geometrical\npattern equals Hl = Sl−1, where Sl−1 is the area of each of the\ngeometrical patterns it is composed of. The width Wl and the\nheight Hl can specify length scales of the level l, l = 0, 1, ..., 4.\nrenormalization group transformation come to our atten-\ntion, as we consider the connection between a geometrical\npattern and the geometrical patterns it is made of.\nAlthough the geometrical patterns are not triangles at\nlevel l, l = 2, 3, 4, yet the boundary curve, as the graph of\nthe function Ψ[l]\n1 , is such that the area Sl of a geometrical\npattern can be simply given, as if it were a triangle, by\nSl = WlHl\n2\n,\nwhere Wl and Hl are its width and height.\nThe renormalization group transformation at level 4\ndefines\nε′ = 23ε,\nδ′ = ε3δ\n\n3\nand uses a coarse-grained procedure replacing the geo-\nmetrical pattern made up of 8 geometrical patterns at\nlevel 1 by their enlarged version with the boundary curve\nas the graph of the function Ψ[1]\n2 . Each of the geometrical\npatterns at level 1 is elementary in the sense that it is\nfully specified by a prime integer relation made up of two\nintegers without further internal structure. The width\n2ε′ and the height δ′ε′ of the renormalized geometrical\npattern at level 4 are given in terms of ε′ and δ′ in the\nsame way as the width 2ε and the height δε of a geomet-\nrical pattern at level 1 are given in terms of ε and δ. The\ntwo geometrical patterns at level 4 have the same width,\nheight and area\nZ t16\nt0\nΨ[4]\n1 (t)dt =\nZ t16\nt0\nΨ[1]\n2 (t)dt,\nbut the lengths of their boundary curves are different\n(Figure 2).\nThe arithmetical and geometrical forms unite the dy-\nnamics and the structure of a complex system as two\nsides of the same entity in the preservation of the system\nas a whole. In particular, at one side the dynamics of\nthe parts is determined to produce spacetime patterns of\nthe parts to fit precisely into the geometrical patterns of\nthe complex system. Under this condition at the side the\ncorresponding prime integer relations can provide the re-\nlationships between the parts for the complex system to\nexist. If the spacetime patterns do not fit even slightly,\nthen one or more of the relationships are not in place and\nthe complex system collapses.\nTo measure the complexity of a system in terms of self-\norganization processes of prime integer relations a con-\ncept of structural complexity is introduced [1]. Starting\nwith the integers at the zero level, the self-organization\nprocesses of prime integer relations progress to different\nlevels and thus produce a hierarchical complexity order.\nIn particular, the higher the level self-organization pro-\ncesses progress to, the greater is the structural complex-\nity of a corresponding system. In our description sys-\ntems can be compared in terms of complexity by using\ntwo equivalent forms: in structure - by the hierarchical\nstructures of prime integer relations and in dynamics -\nby the hierarchical structures of geometrical patterns.\nRemarkably, based on integers and controlled by arith-\nmetic only self-organization processes of prime integer re-\nlations can describe complex systems by information not\nrequiring further simplification.\nIII.\nTESTING A NEW MEDIUM FOR\nINFORMATION PROCESSING\nThe correlation structures of a complex system contain\ninformation about the parts. By changing some parts the\ninformation can be processed as the other parts change in\naccordance with the correlation structures. This shows\nthe importance of self-organization processes of prime in-\nteger relations for information processing. Namely, for a\ngiven problem they could be used to build the correlation\nstructures of a system in processing information demon-\nstrating the optimal performance for the problem.\nAs a result, we suggest the hierarchical network of\nprime integer relations as a new medium for information\nprocessing and are interested in its navigating proper-\nties. It would be important if for any problem the per-\nformance of a system could behave as a concave function\nof its structural complexity. Guided by this property the\nperformance global maximum for a problem could be ef-\nficiently found. It would be also beneficial if at the global\nmaximum the structural complexities of the system and\nthe problem could be related through an optimality con-\ndition.\nThe optimality condition might be interpreted as fol-\nlows: if through arithmetical interdependencies emerging\nin the hierarchical network between a computing system\nand a problem a new building block is formed at the high-\nest possible level, then the optimal performance takes\nplace. Or, in other terms, a computing system finds the\nsolution to a problem, once arithmetic interdependencies\nemerging between the system and the problem provide a\nchannel to obtain the desired information.\nIt is worth to note that since the correlation structures\nof a system are completely determined by prime integer\nrelations, which are equivalent to two-dimensional geo-\nmetrical patterns, the entropy of the system, measuring\nits information content, can be connected with the areas\nof the two-dimensional patterns. Thus, in our approach\nthere is a general connection between entropy and area.\nComputational experiments have been conducted [3]\nto test the navigating properties. In particular, an opti-\nmization algorithm A, as a complex system, of N com-\nputational agents minimizing the average distance in the\ntravelling salesman problem (TSP) is developed.\nThe\nagents work in parallel and start in the same city by\nchoosing the next city at random. Then an agent at each\nstep visits the next city by using one of the two strategies:\na random strategy or the greedy strategy.\nIn the solution of a problem with n cities the state of\nthe agents at step j, j = 1, ..., n−1 can be described by a\nbinary sequence sj = s1j...sNj, where sij = +1, if agent\ni, i = 1, ..., N uses the random strategy and sij = −1,\nif the agent i uses the greedy strategy. The dynamics\nof the complex system is realized as the agents step by\nstep choose their strategies and can be encoded by an\nN × (n −1) binary strategy matrix\nS = {sij, i = 1, ..., N, j = 1, ..., n −1}.\nWe try to change the structural complexity of the al-\ngorithm A monotonically by forcing the system to make\nthe transition from regular behaviour to chaos by period-\ndoubling. To control the system in this transition a pa-\nrameter v, 0 ≤v ≤1 is introduced. It specifies a thresh-\nold point dividing the interval of current distances passed\nby the agents into two parts, i.e., successful and unsuc-\ncessful. This information is provided for an optimal if-\nthen rule [1] that each agent uses to choose the next strat-\n\n4\negy. The rule relies on the Prouhet-Thue-Morse (PTM)\nsequence\n+1 −1 −1 + 1 −1 + 1 + 1 −1 ...\nand has the following description:\n1. if the last strategy is successful, continue with the\nsame strategy.\n2.\nif the last strategy is unsuccessful, consult PTM\ngenerator which strategy to use next.\nRemarkably, for each problem p tested from a class\nP it has been found that the performance of the algo-\nrithm A indeed behaves as a concave function of the\ncontrol parameter with the only global maximum at a\nvalue v∗(p). The global maximums {v∗(p), p ∈P} are of\ninterest to probe whether the structural complexities of\nthe algorithm A and the problem are related through an\noptimality condition. For this purpose strategy matrices\n{S(v∗(p)), p ∈P}\ncorresponding to the global maximums {v∗(p), p ∈P}\nare used and the structural complexities of the algorithm\nA and a problem p are approximated as follows. The\nstructural complexity C(A(p)) of the algorithm A is ap-\nproximated by the quadratic trace\nC(A(p)) =\n1\nN 2 tr(V 2(v∗(p))) =\n1\nN 2\nN\nX\ni=1\nλ2\ni\nof the variance-covariance matrix V (v∗(p)) obtained from\nthe strategy matrix S(v∗(p)), where λi, i = 1, ..., N are\nthe eigenvalues of V (v∗(p)).\nThe structural complexity C(p) of the problem p is\napproximated by the quadratic trace\nC(p) = 1\nn2 tr(M 2(p)) = 1\nn2\nn\nX\ni=1\n(λ′\ni)2\nof the normalized distance matrix\nM(p) = {dij/dmax, i, j = 1, ..., n},\nwhere λ′\ni, i = 1, ..., n are the eigenvalues of M(p), dij\nis the distance between cities i and j and dmax is the\nmaximum of the distances.\nTo reveal the optimality condition the points with the\ncoordinates\n{x = C(p), y = C(A(p)), p ∈P}\nare considered. The result has been indicative of a linear\nrelationship between the structural complexities and thus\nsuggests an optimality condition of the algorithm A [3]:\nIf the algorithm A demonstrates the optimal perfor-\nmance for a problem p, then the structural complexity\nC(A(p)) of the algorithm A is in the linear relationship\nC(A(p)) = 0.67C(p) + 0.33\nwith the structural complexity C(p) of the problem p.\nAccording to the optimality condition if the optimal\nperformance takes place, then in terms of the structural\ncomplexity the dynamics of the algorithm A is in a cer-\ntain relation with the structure of the problem p, i.e., the\ndistance network with the vertices as the cities and the\nedges specifying the pairwise distances.\nThe optimality condition is a practical tool.\nFor a\ngiven problem p by using the distance matrix we can\ncalculate the structural complexity C(p) of the problem\np and from the optimality condition find the structural\ncomplexity\nC(A(p)) = 0.67C(p) + 0.33.\nThen to obtain the optimal performance of the algorithm\nA for the problem p we need to tune the control param-\neter for the algorithm A to function with the structural\ncomplexity C(A(p)).\nIV.\nTHE ORDER OF THE PROCESSES AND\nEFFICIENT QUANTUM ALGORITHMS\nThe computational results point that by using self-\norganization processes of prime integer relations it may\nbe possible to design classical algorithms comparable to\nefficient quantum algorithms.\nQuantum algorithms rely on the practical use of entan-\nglement, whose sensitivity challenges the development of\nrelevant technologies. In a TSP quantum algorithm the\nwave function would be evolved to maximize through the\namplitudes the probability of the shortest routes to be\nmeasured. However, there is no general direction known\nfor the evolution to take in order to make the quantum\nalgorithm efficient. In this regard the majorization prin-\nciple seems to play an important role [4].\nIt provides\na local navigation, but without information about the\nwhole landscape.\nWhile the nature of quantum entanglement is yet to be\nunderstood [5] in our approach the nonlocal correlations\nare known from their origin in the self-organization pro-\ncesses of prime integer relations. The question is whether\nthis knowledge could be used to provide computational\nresources comparable to quantum computation.\nIn this paper we focus on whether such a resource could\nbe obtained from the nonlocal correlations although used\nin classical computation, but with the order of the pro-\ncesses preserved. If that could be possible, then the or-\nder of the self-organization processes would establish a\ngeneral direction for efficient computation. Remarkably,\nthe experiments raise the possibility that if the evolution\ngoes in this direction, then the performance landscape\nbecomes concave.\nIn the experiments we use the parameter to control\nthe correlation structures of the computing system with\ntheir consequences observed in the routes taken by the\nagents. To help in associations with the quantum case\n\n5\nthe average distance for a value v of the parameter can\nbe written as\n ̄D = 1\nN (γ1,...,n−1(v)d([1, ..., n −1 >) + ...\n+γn−1,...,1(v)d([n −1, ..., 1 >))\nwhere γi1,...,in−1(v) is the number of the agents followed\nthe route [i1, ..., in−1 >, d([i1, ..., in−1 >) is its distance\nand the n cities are labelled by 0, 1, ..., n −1 with the\ninitial city by 0. The interpretation of the coefficient\nγi1,...,in−1(v)\nN\nas the probability of the route [i1, ..., in−1 > suggests that\nthe minimization of the average distance considered in\nthe algorithm A can be connected with the maximization\nof the probability of the shortest route considered in TSP\nquantum algorithms.\nSignificantly, the experiments have shown that in the\ncase of the algorithm A the maximization of the prob-\nability turns out to be a one-dimensional concave opti-\nmization. In its course the computing system adapting\nto the problem got more complex or simpler until the\nglobal maximum is reached and the structural complex-\nities of the computing system and the problem become\nrelated through the optimality condition. More connec-\ntions might arise if the wave function could be involved\nin the description of the correlation structures.\nWe note that the algorithm A shares a common fea-\nture with Shor’s algorithm, which also relies on the PTM\nsequence [6].\nV.\nCONCLUSIONS\nWe have suggested that self-organization processes of\nprime integer relations could be used for information pro-\ncessing.\nIn particular, for a given problem self-organization pro-\ncesses of prime integer relations could be used to build\nthe correlation structures of a system in processing infor-\nmation demonstrating the optimal performance for the\nproblem. Remarkably, the processes can be equivalently\nrepresented by transformations of two-dimensional geo-\nmetrical patterns determining the dynamics of the sys-\ntem and revealing in the information processing its struc-\ntural complexity.\nThe information processing would be distinctive, be-\ncause self-organization processes of prime integer rela-\ntions can define the correlation structures of a system\nwithout reference to the distances, local times and sig-\nnals between the parts.\nComputational experiments testing competitive ad-\nvantages of the information processing have been pre-\nsented. They raise the possibility of an optimality condi-\ntion of complex systems: if the structural complexity of\na system is in a certain relationship with the structural\ncomplexity of a problem, then the system demonstrates\nthe optimal performance for the problem.\nThe optimality condition presents the structural com-\nplexity of a system as a key to its optimization. From\nits perspective the optimization of a system could be all\nabout the control of the structural complexity of the sys-\ntem to make it consistent with the structural complexity\nof the problem.\nImportantly, the experiments also indicate that the\nperformance of a complex system may behave as a con-\ncave function of the structural complexity.\nTherefore,\nonce the structural complexity could be controlled as a\nsingle entity, the optimization of a complex system would\nbe potentially reduced to a one-dimensional concave opti-\nmization irrespective of the number of variables involved\nits description. This might open a way to a new type of\ninformation processing for efficient management of com-\nplex systems.\n[1] V.\nKorotkikh,\nA Mathematical\nStructure\nfor Emer-\ngent Computation, Kluwer Academic Publishers, Dor-\ndrecht/Boston/London, 1999.\n[2] V. Korotkikh and G. Korotkikh, Description of Complex\nSystems in terms of Self-Organization Processes of Prime\nInteger Relations, in Complexus Mundi: Emergent Pat-\nterns in Nature, M. M. Novak (ed.), World Scientific, New\nJersey/London, 2006, pp. 63-72, arXiv:nlin.AO/0509008;\nV. Korotkikh and G. Korotkikh, On an Irreducible Theory\nof Complex Systems, InterJournal of Complex Systems,\n1751, 2006, arXiv:nlin.AO/0606023.\n[3] V. Korotkikh, G. Korotkikh and D. Bond, On Optimality\nCondition of Complex Systems: Computational Evidence,\narXiv:cs/0504092.\n[4] R. Orus, J. Latorre and M. A. Martin-Delgado, System-\natic Analysis of Majorization in Quantum Algorithms,\narXiv:quant-ph/0212094.\n[5] N. Gisin, Can Relativity be Considered Complete? From\nNewtonian Nonlocality to Quantum Nonlocality and Be-\nyond, arXiv:quant-ph/0512168.\n[6] K. Maity and A. Lakshminarayan, Quantum Chaos in\nthe Spectrum of Operators Used in Shor’s Algorithm,\narXiv:quant-ph/0604111.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0710.3961v1 [cs.CC] 22 Oct 2007\nOn a New Type of Information Processing\nfor Efficient Management of Complex Systems\nVictor Korotkikh∗and Galina Korotkikh†\nSchool of Computing Sciences\nCentral Queensland University\nMackay, Queensland, 4740\nAustralia\nIt is a challenge to manage complex systems efficiently without confronting NP-hard problems.\nTo address the situation we suggest to use self-organization processes of prime integer relations\nfor information processing. Self-organization processes of prime integer relations define correlation\nstructures of a complex system and can be equivalently represented by transformations of two-\ndimensional geometrical patterns determining the dynamics of the system and revealing its structural\ncomplexity. Computational experiments raise the possibility of an optimality condition of complex\nsystems presenting the structural complexity of a system as a key to its optimization. From this\nperspective the optimization of a system could be all about the control of the structural complexity\nof the system to make it consistent with the structural complexity of the problem. The experiments\nalso indicate that the performance of a complex system may behave as a concave function of the\nstructural complexity. Therefore, once the structural complexity could be controlled as a single\nentity, the optimization of a complex system would be potentially reduced to a one-dimensional\nconcave optimization irrespective of the number of variables involved its description. This might\nopen a way to a new type of information processing for efficient management of complex systems.\nPACS numbers: 89.75.-k, 89.75.Fb\nI.\nINTRODUCTION\nIt is a challenge to manage complex systems efficiently\nwithout confronting NP-hard problems. To address the\nsituation we consider the description of complex systems\nin terms of self-organization processes of prime integer\nrelations [1] and suggest to use the processes for infor-\nmation processing.\nII.\nTHE HIERARCHICAL NETWORK OF\nPRIME INTEGER RELATIONS\nThe description is realized through the unity of two\nequivalent forms, i.e., arithmetical and geometrical. We\nbriefly present the forms in order to introduce the hier-\narchical network of prime integer relations. More details\nmay be found in [1], [2].\nA.\nTHE ARITHMETICAL FORM\nIn the arithmetical form a complex system is char-\nacterized by hierarchical correlation structures built in\naccordance with self-organization processes of prime in-\nteger relations. As each of the correlation structures is\nready to exercise its own scenario and there is no mecha-\nnism specifying which of them is going to take place, an\n∗Electronic address: v.korotkikh@cqu.edu.au\n†Electronic address: g.korotkikh@cqu.edu.au\nintrinsic uncertainty about the complex system exists.\nAt the same time, the information about the correlation\nstructures can be used to evaluate the probability of an\nobservable to take each of the measurement outcomes.\nTherefore, the arithmetical form of the description pro-\nvides the statistical information about a complex system.\nThe form reveals nonlocal correlations without refer-\nence to signals as well as the distances and local times\nof the parts. Thus, the arithmetical form suggests that\nparts of a complex systems may be far apart in space and\ntime and yet remain interconnected with instantaneous\neffect on each other, but no signalling. Namely, if a cor-\nrelation structure of a system is selected and some parts\nare specified, then through the prime integer relations in\ncontrol of the correlation structure the other parts are\nimmediately defined.\nThrough the arithmetical form of the description we\nbecome aware of the hierarchical network, i.e., a set\nof mutually self-consistent elements built by all self-\norganization processes of prime integer relations. Arith-\nmetic ensures that not even a minor change can be made\nto any element of the network (Figure 1). It is helpful\nto think that self-organization processes of prime integer\nrelations take place in the hierarchical network as they\ncompose more and more complex structures.\nThe hierarchical network of prime integer relations is\na causal structure. As parts of a system change and the\nprime integer relations in control of the system cause the\nother parts to change accordingly, an event takes place.\nOnce the changes have been realized, the event is fixed\nin space and time with respect to the reference frames\nof the parts. For the parts the effect of the event has\nnot necessarily be the same, but for each part it is ap-"},{"paragraph_id":"p2","order":2,"text":"2\n16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1\n+16 - 15 - 14 + 13 - 12 + 11 + 10 - 9 - 8 + 7 + 6 - 5 + 4 - 3 - 2 + 1 = 0\n+16 - 15 - 14 + 13 - 12 + 11 + 10 - 9 = 0\n- 8 + 7 + 6 - 5 + 4 - 3 - 2 + 1 = 0\n+16 - 15 - 14 + 13 = 0\n- 12 + 11 + 10 - 9 = 0\n- 8 + 7 + 6 - 5 = 0\n+ 4 - 3 - 2 + 1 = 0\n+16 - 15 = 0\n-14 + 13 = 0\n+10 - 9 = 0\n-12 + 11 = 0\n-8 + 7 = 0\n+6 - 5 = 0\n+4 - 3 = 0\n-2 + 1 = 0\n+16\n- 15\n- 14\n+ 13\n- 12\n + 11\n+ 10\n- 9\n- 8\n+ 7\n+ 6\n- 5\n+ 4\n- 3\n- 2\n+ 1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\nFIG. 1: The figure shows a hierarchical structure of prime\ninteger relations defined as a result of a self-organization pro-\ncess starting at the zero level with integers 16, 13, 11, 10, 7, 6,\n4, 1 in the ”positive state” and integers 15, 14, 12, 9, 8, 5, 3, 2\nin the ”negative”. The process is controlled by arithmetic and\ncannot progress to level 5, because arithmetic determines that\n+164−154−144+134−124+114+104−94−84+74+64−54+\n44 −34 −24 +14 ̸= 0. This hierarchical structure can specify a\ncorrelation structure of a system made of 16 elementary parts\nPi, i = 1, ..., 16 whose changes ∆xi, i = 1, ..., 16 of the observ-\nables xi, i = 1, ..., 16 relative to their reference frames can be\ngiven by the sequence ∆x1...∆x16 = +1 −1 −1 + 1 −1 + 1 +\n1 −1 −1 + 1 + 1 −1 + 1 −1 −1 + 1.\npropriately determined by the prime integer relations.\nHowever, the prime integer relations at work for a sys-\ntem have no causal power to effect systems controlled by\nseparate prime integer relations. As a result, informa-\ntion about the systems is blocked for the observers of the\nsystem.\nB.\nTHE GEOMETRICAL FORM\nSpecified by two parameters ε > 0 and δ > 0 the ge-\nometrical form arises as the self-organization processes\nof prime integer relations find isomorphic realization in\nterms of transformations of two-dimensional geometrical\npatterns [1]. As a result, hierarchical structures of prime\ninteger relations defining the correlation structures of a\ncomplex system become equivalently represented by hi-\nerarchical structures of geometrical patterns determining\nthe dynamics of the system and revealing its complexity.\nThe quantitative description of the system turns out to\nbe given by the description of the geometrical patterns\n[1], [2].\nFigure 2 shows a hierarchical structure of geometrical\npatterns, which for given ε > 0 and δ > 0 is isomorphic\nto the hierarchical structure of prime integer relations\ndepicted in Figure 1. A scale invariant property and a\n2\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n2\n1\n1\n[1]\n1\n[2]\n1\n[3]\n1\n[4]\n2\n[1]\n[0]\n1\n2\n3\n4\n4\n8\n4\nFIG. 2: The figure shows a hierarchical structure of geomet-\nrical patterns, which for given ε and δ is isomorphic to the\nhierarchical structure of prime integer relations shown in Fig-\nure 1. Under the integration of the function Ψ[l]\n1 , Ψ[l+1]\n1\n(t0) =\n0, l = 0, 1, 2, 3 the geometrical patterns of the level l form\nthe geometrical patterns of the higher level l + 1. The width\nWl of a geometrical pattern at level l, l = 1, ..., 4 equals the\nsum Wl = Wl−1,left + Wl−1,right = 2Wl−1 of the widths of\nthe two geometrical patterns it is made up of, where Wl−1,left\nand Wl−1,right are the widths of the left and right geometrical\npatterns and Wl−1,left = Wl−1,right = Wl−1 as each geomet-\nrical pattern at level l −1 has the same width Wl−1 = 2l−1ε\nand ti = εi, i = 0, 1, ..., 16. The height Hl of the geometrical\npattern equals Hl = Sl−1, where Sl−1 is the area of each of the\ngeometrical patterns it is composed of. The width Wl and the\nheight Hl can specify length scales of the level l, l = 0, 1, ..., 4.\nrenormalization group transformation come to our atten-\ntion, as we consider the connection between a geometrical\npattern and the geometrical patterns it is made of.\nAlthough the geometrical patterns are not triangles at\nlevel l, l = 2, 3, 4, yet the boundary curve, as the graph of\nthe function Ψ[l]\n1 , is such that the area Sl of a geometrical\npattern can be simply given, as if it were a triangle, by\nSl = WlHl\n2\n,\nwhere Wl and Hl are its width and height.\nThe renormalization group transformation at level 4\ndefines\nε′ = 23ε,\nδ′ = ε3δ"},{"paragraph_id":"p3","order":3,"text":"3\nand uses a coarse-grained procedure replacing the geo-\nmetrical pattern made up of 8 geometrical patterns at\nlevel 1 by their enlarged version with the boundary curve\nas the graph of the function Ψ[1]\n2 . Each of the geometrical\npatterns at level 1 is elementary in the sense that it is\nfully specified by a prime integer relation made up of two\nintegers without further internal structure. The width\n2ε′ and the height δ′ε′ of the renormalized geometrical\npattern at level 4 are given in terms of ε′ and δ′ in the\nsame way as the width 2ε and the height δε of a geomet-\nrical pattern at level 1 are given in terms of ε and δ. The\ntwo geometrical patterns at level 4 have the same width,\nheight and area\nZ t16\nt0\nΨ[4]\n1 (t)dt =\nZ t16\nt0\nΨ[1]\n2 (t)dt,\nbut the lengths of their boundary curves are different\n(Figure 2).\nThe arithmetical and geometrical forms unite the dy-\nnamics and the structure of a complex system as two\nsides of the same entity in the preservation of the system\nas a whole. In particular, at one side the dynamics of\nthe parts is determined to produce spacetime patterns of\nthe parts to fit precisely into the geometrical patterns of\nthe complex system. Under this condition at the side the\ncorresponding prime integer relations can provide the re-\nlationships between the parts for the complex system to\nexist. If the spacetime patterns do not fit even slightly,\nthen one or more of the relationships are not in place and\nthe complex system collapses.\nTo measure the complexity of a system in terms of self-\norganization processes of prime integer relations a con-\ncept of structural complexity is introduced [1]. Starting\nwith the integers at the zero level, the self-organization\nprocesses of prime integer relations progress to different\nlevels and thus produce a hierarchical complexity order.\nIn particular, the higher the level self-organization pro-\ncesses progress to, the greater is the structural complex-\nity of a corresponding system. In our description sys-\ntems can be compared in terms of complexity by using\ntwo equivalent forms: in structure - by the hierarchical\nstructures of prime integer relations and in dynamics -\nby the hierarchical structures of geometrical patterns.\nRemarkably, based on integers and controlled by arith-\nmetic only self-organization processes of prime integer re-\nlations can describe complex systems by information not\nrequiring further simplification.\nIII.\nTESTING A NEW MEDIUM FOR\nINFORMATION PROCESSING\nThe correlation structures of a complex system contain\ninformation about the parts. By changing some parts the\ninformation can be processed as the other parts change in\naccordance with the correlation structures. This shows\nthe importance of self-organization processes of prime in-\nteger relations for information processing. Namely, for a\ngiven problem they could be used to build the correlation\nstructures of a system in processing information demon-\nstrating the optimal performance for the problem.\nAs a result, we suggest the hierarchical network of\nprime integer relations as a new medium for information\nprocessing and are interested in its navigating proper-\nties. It would be important if for any problem the per-\nformance of a system could behave as a concave function\nof its structural complexity. Guided by this property the\nperformance global maximum for a problem could be ef-\nficiently found. It would be also beneficial if at the global\nmaximum the structural complexities of the system and\nthe problem could be related through an optimality con-\ndition.\nThe optimality condition might be interpreted as fol-\nlows: if through arithmetical interdependencies emerging\nin the hierarchical network between a computing system\nand a problem a new building block is formed at the high-\nest possible level, then the optimal performance takes\nplace. Or, in other terms, a computing system finds the\nsolution to a problem, once arithmetic interdependencies\nemerging between the system and the problem provide a\nchannel to obtain the desired information.\nIt is worth to note that since the correlation structures\nof a system are completely determined by prime integer\nrelations, which are equivalent to two-dimensional geo-\nmetrical patterns, the entropy of the system, measuring\nits information content, can be connected with the areas\nof the two-dimensional patterns. Thus, in our approach\nthere is a general connection between entropy and area.\nComputational experiments have been conducted [3]\nto test the navigating properties. In particular, an opti-\nmization algorithm A, as a complex system, of N com-\nputational agents minimizing the average distance in the\ntravelling salesman problem (TSP) is developed.\nThe\nagents work in parallel and start in the same city by\nchoosing the next city at random. Then an agent at each\nstep visits the next city by using one of the two strategies:\na random strategy or the greedy strategy.\nIn the solution of a problem with n cities the state of\nthe agents at step j, j = 1, ..., n−1 can be described by a\nbinary sequence sj = s1j...sNj, where sij = +1, if agent\ni, i = 1, ..., N uses the random strategy and sij = −1,\nif the agent i uses the greedy strategy. The dynamics\nof the complex system is realized as the agents step by\nstep choose their strategies and can be encoded by an\nN × (n −1) binary strategy matrix\nS = {sij, i = 1, ..., N, j = 1, ..., n −1}.\nWe try to change the structural complexity of the al-\ngorithm A monotonically by forcing the system to make\nthe transition from regular behaviour to chaos by period-\ndoubling. To control the system in this transition a pa-\nrameter v, 0 ≤v ≤1 is introduced. It specifies a thresh-\nold point dividing the interval of current distances passed\nby the agents into two parts, i.e., successful and unsuc-\ncessful. This information is provided for an optimal if-\nthen rule [1] that each agent uses to choose the next strat-"},{"paragraph_id":"p4","order":4,"text":"4\negy. The rule relies on the Prouhet-Thue-Morse (PTM)\nsequence\n+1 −1 −1 + 1 −1 + 1 + 1 −1 ...\nand has the following description:\n1. if the last strategy is successful, continue with the\nsame strategy.\n2.\nif the last strategy is unsuccessful, consult PTM\ngenerator which strategy to use next.\nRemarkably, for each problem p tested from a class\nP it has been found that the performance of the algo-\nrithm A indeed behaves as a concave function of the\ncontrol parameter with the only global maximum at a\nvalue v∗(p). The global maximums {v∗(p), p ∈P} are of\ninterest to probe whether the structural complexities of\nthe algorithm A and the problem are related through an\noptimality condition. For this purpose strategy matrices\n{S(v∗(p)), p ∈P}\ncorresponding to the global maximums {v∗(p), p ∈P}\nare used and the structural complexities of the algorithm\nA and a problem p are approximated as follows. The\nstructural complexity C(A(p)) of the algorithm A is ap-\nproximated by the quadratic trace\nC(A(p)) =\n1\nN 2 tr(V 2(v∗(p))) =\n1\nN 2\nN\nX\ni=1\nλ2\ni\nof the variance-covariance matrix V (v∗(p)) obtained from\nthe strategy matrix S(v∗(p)), where λi, i = 1, ..., N are\nthe eigenvalues of V (v∗(p)).\nThe structural complexity C(p) of the problem p is\napproximated by the quadratic trace\nC(p) = 1\nn2 tr(M 2(p)) = 1\nn2\nn\nX\ni=1\n(λ′\ni)2\nof the normalized distance matrix\nM(p) = {dij/dmax, i, j = 1, ..., n},\nwhere λ′\ni, i = 1, ..., n are the eigenvalues of M(p), dij\nis the distance between cities i and j and dmax is the\nmaximum of the distances.\nTo reveal the optimality condition the points with the\ncoordinates\n{x = C(p), y = C(A(p)), p ∈P}\nare considered. The result has been indicative of a linear\nrelationship between the structural complexities and thus\nsuggests an optimality condition of the algorithm A [3]:\nIf the algorithm A demonstrates the optimal perfor-\nmance for a problem p, then the structural complexity\nC(A(p)) of the algorithm A is in the linear relationship\nC(A(p)) = 0.67C(p) + 0.33\nwith the structural complexity C(p) of the problem p.\nAccording to the optimality condition if the optimal\nperformance takes place, then in terms of the structural\ncomplexity the dynamics of the algorithm A is in a cer-\ntain relation with the structure of the problem p, i.e., the\ndistance network with the vertices as the cities and the\nedges specifying the pairwise distances.\nThe optimality condition is a practical tool.\nFor a\ngiven problem p by using the distance matrix we can\ncalculate the structural complexity C(p) of the problem\np and from the optimality condition find the structural\ncomplexity\nC(A(p)) = 0.67C(p) + 0.33.\nThen to obtain the optimal performance of the algorithm\nA for the problem p we need to tune the control param-\neter for the algorithm A to function with the structural\ncomplexity C(A(p)).\nIV.\nTHE ORDER OF THE PROCESSES AND\nEFFICIENT QUANTUM ALGORITHMS\nThe computational results point that by using self-\norganization processes of prime integer relations it may\nbe possible to design classical algorithms comparable to\nefficient quantum algorithms.\nQuantum algorithms rely on the practical use of entan-\nglement, whose sensitivity challenges the development of\nrelevant technologies. In a TSP quantum algorithm the\nwave function would be evolved to maximize through the\namplitudes the probability of the shortest routes to be\nmeasured. However, there is no general direction known\nfor the evolution to take in order to make the quantum\nalgorithm efficient. In this regard the majorization prin-\nciple seems to play an important role [4].\nIt provides\na local navigation, but without information about the\nwhole landscape.\nWhile the nature of quantum entanglement is yet to be\nunderstood [5] in our approach the nonlocal correlations\nare known from their origin in the self-organization pro-\ncesses of prime integer relations. The question is whether\nthis knowledge could be used to provide computational\nresources comparable to quantum computation.\nIn this paper we focus on whether such a resource could\nbe obtained from the nonlocal correlations although used\nin classical computation, but with the order of the pro-\ncesses preserved. If that could be possible, then the or-\nder of the self-organization processes would establish a\ngeneral direction for efficient computation. Remarkably,\nthe experiments raise the possibility that if the evolution\ngoes in this direction, then the performance landscape\nbecomes concave.\nIn the experiments we use the parameter to control\nthe correlation structures of the computing system with\ntheir consequences observed in the routes taken by the\nagents. To help in associations with the quantum case"},{"paragraph_id":"p5","order":5,"text":"5\nthe average distance for a value v of the parameter can\nbe written as\n ̄D = 1\nN (γ1,...,n−1(v)d([1, ..., n −1 >) + ...\n+γn−1,...,1(v)d([n −1, ..., 1 >))\nwhere γi1,...,in−1(v) is the number of the agents followed\nthe route [i1, ..., in−1 >, d([i1, ..., in−1 >) is its distance\nand the n cities are labelled by 0, 1, ..., n −1 with the\ninitial city by 0. The interpretation of the coefficient\nγi1,...,in−1(v)\nN\nas the probability of the route [i1, ..., in−1 > suggests that\nthe minimization of the average distance considered in\nthe algorithm A can be connected with the maximization\nof the probability of the shortest route considered in TSP\nquantum algorithms.\nSignificantly, the experiments have shown that in the\ncase of the algorithm A the maximization of the prob-\nability turns out to be a one-dimensional concave opti-\nmization. In its course the computing system adapting\nto the problem got more complex or simpler until the\nglobal maximum is reached and the structural complex-\nities of the computing system and the problem become\nrelated through the optimality condition. More connec-\ntions might arise if the wave function could be involved\nin the description of the correlation structures.\nWe note that the algorithm A shares a common fea-\nture with Shor’s algorithm, which also relies on the PTM\nsequence [6].\nV.\nCONCLUSIONS\nWe have suggested that self-organization processes of\nprime integer relations could be used for information pro-\ncessing.\nIn particular, for a given problem self-organization pro-\ncesses of prime integer relations could be used to build\nthe correlation structures of a system in processing infor-\nmation demonstrating the optimal performance for the\nproblem. Remarkably, the processes can be equivalently\nrepresented by transformations of two-dimensional geo-\nmetrical patterns determining the dynamics of the sys-\ntem and revealing in the information processing its struc-\ntural complexity.\nThe information processing would be distinctive, be-\ncause self-organization processes of prime integer rela-\ntions can define the correlation structures of a system\nwithout reference to the distances, local times and sig-\nnals between the parts.\nComputational experiments testing competitive ad-\nvantages of the information processing have been pre-\nsented. They raise the possibility of an optimality condi-\ntion of complex systems: if the structural complexity of\na system is in a certain relationship with the structural\ncomplexity of a problem, then the system demonstrates\nthe optimal performance for the problem.\nThe optimality condition presents the structural com-\nplexity of a system as a key to its optimization. From\nits perspective the optimization of a system could be all\nabout the control of the structural complexity of the sys-\ntem to make it consistent with the structural complexity\nof the problem.\nImportantly, the experiments also indicate that the\nperformance of a complex system may behave as a con-\ncave function of the structural complexity.\nTherefore,\nonce the structural complexity could be controlled as a\nsingle entity, the optimization of a complex system would\nbe potentially reduced to a one-dimensional concave opti-\nmization irrespective of the number of variables involved\nits description. This might open a way to a new type of\ninformation processing for efficient management of com-\nplex systems.\n[1] V.\nKorotkikh,\nA Mathematical\nStructure\nfor Emer-\ngent Computation, Kluwer Academic Publishers, Dor-\ndrecht/Boston/London, 1999.\n[2] V. Korotkikh and G. Korotkikh, Description of Complex\nSystems in terms of Self-Organization Processes of Prime\nInteger Relations, in Complexus Mundi: Emergent Pat-\nterns in Nature, M. M. Novak (ed.), World Scientific, New\nJersey/London, 2006, pp. 63-72, arXiv:nlin.AO/0509008;\nV. Korotkikh and G. Korotkikh, On an Irreducible Theory\nof Complex Systems, InterJournal of Complex Systems,\n1751, 2006, arXiv:nlin.AO/0606023.\n[3] V. Korotkikh, G. Korotkikh and D. Bond, On Optimality\nCondition of Complex Systems: Computational Evidence,\narXiv:cs/0504092.\n[4] R. Orus, J. Latorre and M. A. Martin-Delgado, System-\natic Analysis of Majorization in Quantum Algorithms,\narXiv:quant-ph/0212094.\n[5] N. Gisin, Can Relativity be Considered Complete? From\nNewtonian Nonlocality to Quantum Nonlocality and Be-\nyond, arXiv:quant-ph/0512168.\n[6] K. Maity and A. Lakshminarayan, Quantum Chaos in\nthe Spectrum of Operators Used in Shor’s Algorithm,\narXiv:quant-ph/0604111."}],"pages":[{"page":1,"text":"arXiv:0710.3961v1 [cs.CC] 22 Oct 2007\nOn a New Type of Information Processing\nfor Efficient Management of Complex Systems\nVictor Korotkikh∗and Galina Korotkikh†\nSchool of Computing Sciences\nCentral Queensland University\nMackay, Queensland, 4740\nAustralia\nIt is a challenge to manage complex systems efficiently without confronting NP-hard problems.\nTo address the situation we suggest to use self-organization processes of prime integer relations\nfor information processing. Self-organization processes of prime integer relations define correlation\nstructures of a complex system and can be equivalently represented by transformations of two-\ndimensional geometrical patterns determining the dynamics of the system and revealing its structural\ncomplexity. Computational experiments raise the possibility of an optimality condition of complex\nsystems presenting the structural complexity of a system as a key to its optimization. From this\nperspective the optimization of a system could be all about the control of the structural complexity\nof the system to make it consistent with the structural complexity of the problem. The experiments\nalso indicate that the performance of a complex system may behave as a concave function of the\nstructural complexity. Therefore, once the structural complexity could be controlled as a single\nentity, the optimization of a complex system would be potentially reduced to a one-dimensional\nconcave optimization irrespective of the number of variables involved its description. This might\nopen a way to a new type of information processing for efficient management of complex systems.\nPACS numbers: 89.75.-k, 89.75.Fb\nI.\nINTRODUCTION\nIt is a challenge to manage complex systems efficiently\nwithout confronting NP-hard problems. To address the\nsituation we consider the description of complex systems\nin terms of self-organization processes of prime integer\nrelations [1] and suggest to use the processes for infor-\nmation processing.\nII.\nTHE HIERARCHICAL NETWORK OF\nPRIME INTEGER RELATIONS\nThe description is realized through the unity of two\nequivalent forms, i.e., arithmetical and geometrical. We\nbriefly present the forms in order to introduce the hier-\narchical network of prime integer relations. More details\nmay be found in [1], [2].\nA.\nTHE ARITHMETICAL FORM\nIn the arithmetical form a complex system is char-\nacterized by hierarchical correlation structures built in\naccordance with self-organization processes of prime in-\nteger relations. As each of the correlation structures is\nready to exercise its own scenario and there is no mecha-\nnism specifying which of them is going to take place, an\n∗Electronic address: v.korotkikh@cqu.edu.au\n†Electronic address: g.korotkikh@cqu.edu.au\nintrinsic uncertainty about the complex system exists.\nAt the same time, the information about the correlation\nstructures can be used to evaluate the probability of an\nobservable to take each of the measurement outcomes.\nTherefore, the arithmetical form of the description pro-\nvides the statistical information about a complex system.\nThe form reveals nonlocal correlations without refer-\nence to signals as well as the distances and local times\nof the parts. Thus, the arithmetical form suggests that\nparts of a complex systems may be far apart in space and\ntime and yet remain interconnected with instantaneous\neffect on each other, but no signalling. Namely, if a cor-\nrelation structure of a system is selected and some parts\nare specified, then through the prime integer relations in\ncontrol of the correlation structure the other parts are\nimmediately defined.\nThrough the arithmetical form of the description we\nbecome aware of the hierarchical network, i.e., a set\nof mutually self-consistent elements built by all self-\norganization processes of prime integer relations. Arith-\nmetic ensures that not even a minor change can be made\nto any element of the network (Figure 1). It is helpful\nto think that self-organization processes of prime integer\nrelations take place in the hierarchical network as they\ncompose more and more complex structures.\nThe hierarchical network of prime integer relations is\na causal structure. As parts of a system change and the\nprime integer relations in control of the system cause the\nother parts to change accordingly, an event takes place.\nOnce the changes have been realized, the event is fixed\nin space and time with respect to the reference frames\nof the parts. For the parts the effect of the event has\nnot necessarily be the same, but for each part it is ap-"},{"page":2,"text":"2\n16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1\n+16 - 15 - 14 + 13 - 12 + 11 + 10 - 9 - 8 + 7 + 6 - 5 + 4 - 3 - 2 + 1 = 0\n+16 - 15 - 14 + 13 - 12 + 11 + 10 - 9 = 0\n- 8 + 7 + 6 - 5 + 4 - 3 - 2 + 1 = 0\n+16 - 15 - 14 + 13 = 0\n- 12 + 11 + 10 - 9 = 0\n- 8 + 7 + 6 - 5 = 0\n+ 4 - 3 - 2 + 1 = 0\n+16 - 15 = 0\n-14 + 13 = 0\n+10 - 9 = 0\n-12 + 11 = 0\n-8 + 7 = 0\n+6 - 5 = 0\n+4 - 3 = 0\n-2 + 1 = 0\n+16\n- 15\n- 14\n+ 13\n- 12\n + 11\n+ 10\n- 9\n- 8\n+ 7\n+ 6\n- 5\n+ 4\n- 3\n- 2\n+ 1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\n3\nFIG. 1: The figure shows a hierarchical structure of prime\ninteger relations defined as a result of a self-organization pro-\ncess starting at the zero level with integers 16, 13, 11, 10, 7, 6,\n4, 1 in the ”positive state” and integers 15, 14, 12, 9, 8, 5, 3, 2\nin the ”negative”. The process is controlled by arithmetic and\ncannot progress to level 5, because arithmetic determines that\n+164−154−144+134−124+114+104−94−84+74+64−54+\n44 −34 −24 +14 ̸= 0. This hierarchical structure can specify a\ncorrelation structure of a system made of 16 elementary parts\nPi, i = 1, ..., 16 whose changes ∆xi, i = 1, ..., 16 of the observ-\nables xi, i = 1, ..., 16 relative to their reference frames can be\ngiven by the sequence ∆x1...∆x16 = +1 −1 −1 + 1 −1 + 1 +\n1 −1 −1 + 1 + 1 −1 + 1 −1 −1 + 1.\npropriately determined by the prime integer relations.\nHowever, the prime integer relations at work for a sys-\ntem have no causal power to effect systems controlled by\nseparate prime integer relations. As a result, informa-\ntion about the systems is blocked for the observers of the\nsystem.\nB.\nTHE GEOMETRICAL FORM\nSpecified by two parameters ε > 0 and δ > 0 the ge-\nometrical form arises as the self-organization processes\nof prime integer relations find isomorphic realization in\nterms of transformations of two-dimensional geometrical\npatterns [1]. As a result, hierarchical structures of prime\ninteger relations defining the correlation structures of a\ncomplex system become equivalently represented by hi-\nerarchical structures of geometrical patterns determining\nthe dynamics of the system and revealing its complexity.\nThe quantitative description of the system turns out to\nbe given by the description of the geometrical patterns\n[1], [2].\nFigure 2 shows a hierarchical structure of geometrical\npatterns, which for given ε > 0 and δ > 0 is isomorphic\nto the hierarchical structure of prime integer relations\ndepicted in Figure 1. A scale invariant property and a\n2\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n4\n0\n8\n12\n16\n2\n1\n1\n[1]\n1\n[2]\n1\n[3]\n1\n[4]\n2\n[1]\n[0]\n1\n2\n3\n4\n4\n8\n4\nFIG. 2: The figure shows a hierarchical structure of geomet-\nrical patterns, which for given ε and δ is isomorphic to the\nhierarchical structure of prime integer relations shown in Fig-\nure 1. Under the integration of the function Ψ[l]\n1 , Ψ[l+1]\n1\n(t0) =\n0, l = 0, 1, 2, 3 the geometrical patterns of the level l form\nthe geometrical patterns of the higher level l + 1. The width\nWl of a geometrical pattern at level l, l = 1, ..., 4 equals the\nsum Wl = Wl−1,left + Wl−1,right = 2Wl−1 of the widths of\nthe two geometrical patterns it is made up of, where Wl−1,left\nand Wl−1,right are the widths of the left and right geometrical\npatterns and Wl−1,left = Wl−1,right = Wl−1 as each geomet-\nrical pattern at level l −1 has the same width Wl−1 = 2l−1ε\nand ti = εi, i = 0, 1, ..., 16. The height Hl of the geometrical\npattern equals Hl = Sl−1, where Sl−1 is the area of each of the\ngeometrical patterns it is composed of. The width Wl and the\nheight Hl can specify length scales of the level l, l = 0, 1, ..., 4.\nrenormalization group transformation come to our atten-\ntion, as we consider the connection between a geometrical\npattern and the geometrical patterns it is made of.\nAlthough the geometrical patterns are not triangles at\nlevel l, l = 2, 3, 4, yet the boundary curve, as the graph of\nthe function Ψ[l]\n1 , is such that the area Sl of a geometrical\npattern can be simply given, as if it were a triangle, by\nSl = WlHl\n2\n,\nwhere Wl and Hl are its width and height.\nThe renormalization group transformation at level 4\ndefines\nε′ = 23ε,\nδ′ = ε3δ"},{"page":3,"text":"3\nand uses a coarse-grained procedure replacing the geo-\nmetrical pattern made up of 8 geometrical patterns at\nlevel 1 by their enlarged version with the boundary curve\nas the graph of the function Ψ[1]\n2 . Each of the geometrical\npatterns at level 1 is elementary in the sense that it is\nfully specified by a prime integer relation made up of two\nintegers without further internal structure. The width\n2ε′ and the height δ′ε′ of the renormalized geometrical\npattern at level 4 are given in terms of ε′ and δ′ in the\nsame way as the width 2ε and the height δε of a geomet-\nrical pattern at level 1 are given in terms of ε and δ. The\ntwo geometrical patterns at level 4 have the same width,\nheight and area\nZ t16\nt0\nΨ[4]\n1 (t)dt =\nZ t16\nt0\nΨ[1]\n2 (t)dt,\nbut the lengths of their boundary curves are different\n(Figure 2).\nThe arithmetical and geometrical forms unite the dy-\nnamics and the structure of a complex system as two\nsides of the same entity in the preservation of the system\nas a whole. In particular, at one side the dynamics of\nthe parts is determined to produce spacetime patterns of\nthe parts to fit precisely into the geometrical patterns of\nthe complex system. Under this condition at the side the\ncorresponding prime integer relations can provide the re-\nlationships between the parts for the complex system to\nexist. If the spacetime patterns do not fit even slightly,\nthen one or more of the relationships are not in place and\nthe complex system collapses.\nTo measure the complexity of a system in terms of self-\norganization processes of prime integer relations a con-\ncept of structural complexity is introduced [1]. Starting\nwith the integers at the zero level, the self-organization\nprocesses of prime integer relations progress to different\nlevels and thus produce a hierarchical complexity order.\nIn particular, the higher the level self-organization pro-\ncesses progress to, the greater is the structural complex-\nity of a corresponding system. In our description sys-\ntems can be compared in terms of complexity by using\ntwo equivalent forms: in structure - by the hierarchical\nstructures of prime integer relations and in dynamics -\nby the hierarchical structures of geometrical patterns.\nRemarkably, based on integers and controlled by arith-\nmetic only self-organization processes of prime integer re-\nlations can describe complex systems by information not\nrequiring further simplification.\nIII.\nTESTING A NEW MEDIUM FOR\nINFORMATION PROCESSING\nThe correlation structures of a complex system contain\ninformation about the parts. By changing some parts the\ninformation can be processed as the other parts change in\naccordance with the correlation structures. This shows\nthe importance of self-organization processes of prime in-\nteger relations for information processing. Namely, for a\ngiven problem they could be used to build the correlation\nstructures of a system in processing information demon-\nstrating the optimal performance for the problem.\nAs a result, we suggest the hierarchical network of\nprime integer relations as a new medium for information\nprocessing and are interested in its navigating proper-\nties. It would be important if for any problem the per-\nformance of a system could behave as a concave function\nof its structural complexity. Guided by this property the\nperformance global maximum for a problem could be ef-\nficiently found. It would be also beneficial if at the global\nmaximum the structural complexities of the system and\nthe problem could be related through an optimality con-\ndition.\nThe optimality condition might be interpreted as fol-\nlows: if through arithmetical interdependencies emerging\nin the hierarchical network between a computing system\nand a problem a new building block is formed at the high-\nest possible level, then the optimal performance takes\nplace. Or, in other terms, a computing system finds the\nsolution to a problem, once arithmetic interdependencies\nemerging between the system and the problem provide a\nchannel to obtain the desired information.\nIt is worth to note that since the correlation structures\nof a system are completely determined by prime integer\nrelations, which are equivalent to two-dimensional geo-\nmetrical patterns, the entropy of the system, measuring\nits information content, can be connected with the areas\nof the two-dimensional patterns. Thus, in our approach\nthere is a general connection between entropy and area.\nComputational experiments have been conducted [3]\nto test the navigating properties. In particular, an opti-\nmization algorithm A, as a complex system, of N com-\nputational agents minimizing the average distance in the\ntravelling salesman problem (TSP) is developed.\nThe\nagents work in parallel and start in the same city by\nchoosing the next city at random. Then an agent at each\nstep visits the next city by using one of the two strategies:\na random strategy or the greedy strategy.\nIn the solution of a problem with n cities the state of\nthe agents at step j, j = 1, ..., n−1 can be described by a\nbinary sequence sj = s1j...sNj, where sij = +1, if agent\ni, i = 1, ..., N uses the random strategy and sij = −1,\nif the agent i uses the greedy strategy. The dynamics\nof the complex system is realized as the agents step by\nstep choose their strategies and can be encoded by an\nN × (n −1) binary strategy matrix\nS = {sij, i = 1, ..., N, j = 1, ..., n −1}.\nWe try to change the structural complexity of the al-\ngorithm A monotonically by forcing the system to make\nthe transition from regular behaviour to chaos by period-\ndoubling. To control the system in this transition a pa-\nrameter v, 0 ≤v ≤1 is introduced. It specifies a thresh-\nold point dividing the interval of current distances passed\nby the agents into two parts, i.e., successful and unsuc-\ncessful. This information is provided for an optimal if-\nthen rule [1] that each agent uses to choose the next strat-"},{"page":4,"text":"4\negy. The rule relies on the Prouhet-Thue-Morse (PTM)\nsequence\n+1 −1 −1 + 1 −1 + 1 + 1 −1 ...\nand has the following description:\n1. if the last strategy is successful, continue with the\nsame strategy.\n2.\nif the last strategy is unsuccessful, consult PTM\ngenerator which strategy to use next.\nRemarkably, for each problem p tested from a class\nP it has been found that the performance of the algo-\nrithm A indeed behaves as a concave function of the\ncontrol parameter with the only global maximum at a\nvalue v∗(p). The global maximums {v∗(p), p ∈P} are of\ninterest to probe whether the structural complexities of\nthe algorithm A and the problem are related through an\noptimality condition. For this purpose strategy matrices\n{S(v∗(p)), p ∈P}\ncorresponding to the global maximums {v∗(p), p ∈P}\nare used and the structural complexities of the algorithm\nA and a problem p are approximated as follows. The\nstructural complexity C(A(p)) of the algorithm A is ap-\nproximated by the quadratic trace\nC(A(p)) =\n1\nN 2 tr(V 2(v∗(p))) =\n1\nN 2\nN\nX\ni=1\nλ2\ni\nof the variance-covariance matrix V (v∗(p)) obtained from\nthe strategy matrix S(v∗(p)), where λi, i = 1, ..., N are\nthe eigenvalues of V (v∗(p)).\nThe structural complexity C(p) of the problem p is\napproximated by the quadratic trace\nC(p) = 1\nn2 tr(M 2(p)) = 1\nn2\nn\nX\ni=1\n(λ′\ni)2\nof the normalized distance matrix\nM(p) = {dij/dmax, i, j = 1, ..., n},\nwhere λ′\ni, i = 1, ..., n are the eigenvalues of M(p), dij\nis the distance between cities i and j and dmax is the\nmaximum of the distances.\nTo reveal the optimality condition the points with the\ncoordinates\n{x = C(p), y = C(A(p)), p ∈P}\nare considered. The result has been indicative of a linear\nrelationship between the structural complexities and thus\nsuggests an optimality condition of the algorithm A [3]:\nIf the algorithm A demonstrates the optimal perfor-\nmance for a problem p, then the structural complexity\nC(A(p)) of the algorithm A is in the linear relationship\nC(A(p)) = 0.67C(p) + 0.33\nwith the structural complexity C(p) of the problem p.\nAccording to the optimality condition if the optimal\nperformance takes place, then in terms of the structural\ncomplexity the dynamics of the algorithm A is in a cer-\ntain relation with the structure of the problem p, i.e., the\ndistance network with the vertices as the cities and the\nedges specifying the pairwise distances.\nThe optimality condition is a practical tool.\nFor a\ngiven problem p by using the distance matrix we can\ncalculate the structural complexity C(p) of the problem\np and from the optimality condition find the structural\ncomplexity\nC(A(p)) = 0.67C(p) + 0.33.\nThen to obtain the optimal performance of the algorithm\nA for the problem p we need to tune the control param-\neter for the algorithm A to function with the structural\ncomplexity C(A(p)).\nIV.\nTHE ORDER OF THE PROCESSES AND\nEFFICIENT QUANTUM ALGORITHMS\nThe computational results point that by using self-\norganization processes of prime integer relations it may\nbe possible to design classical algorithms comparable to\nefficient quantum algorithms.\nQuantum algorithms rely on the practical use of entan-\nglement, whose sensitivity challenges the development of\nrelevant technologies. In a TSP quantum algorithm the\nwave function would be evolved to maximize through the\namplitudes the probability of the shortest routes to be\nmeasured. However, there is no general direction known\nfor the evolution to take in order to make the quantum\nalgorithm efficient. In this regard the majorization prin-\nciple seems to play an important role [4].\nIt provides\na local navigation, but without information about the\nwhole landscape.\nWhile the nature of quantum entanglement is yet to be\nunderstood [5] in our approach the nonlocal correlations\nare known from their origin in the self-organization pro-\ncesses of prime integer relations. The question is whether\nthis knowledge could be used to provide computational\nresources comparable to quantum computation.\nIn this paper we focus on whether such a resource could\nbe obtained from the nonlocal correlations although used\nin classical computation, but with the order of the pro-\ncesses preserved. If that could be possible, then the or-\nder of the self-organization processes would establish a\ngeneral direction for efficient computation. Remarkably,\nthe experiments raise the possibility that if the evolution\ngoes in this direction, then the performance landscape\nbecomes concave.\nIn the experiments we use the parameter to control\nthe correlation structures of the computing system with\ntheir consequences observed in the routes taken by the\nagents. To help in associations with the quantum case"},{"page":5,"text":"5\nthe average distance for a value v of the parameter can\nbe written as\n ̄D = 1\nN (γ1,...,n−1(v)d([1, ..., n −1 >) + ...\n+γn−1,...,1(v)d([n −1, ..., 1 >))\nwhere γi1,...,in−1(v) is the number of the agents followed\nthe route [i1, ..., in−1 >, d([i1, ..., in−1 >) is its distance\nand the n cities are labelled by 0, 1, ..., n −1 with the\ninitial city by 0. The interpretation of the coefficient\nγi1,...,in−1(v)\nN\nas the probability of the route [i1, ..., in−1 > suggests that\nthe minimization of the average distance considered in\nthe algorithm A can be connected with the maximization\nof the probability of the shortest route considered in TSP\nquantum algorithms.\nSignificantly, the experiments have shown that in the\ncase of the algorithm A the maximization of the prob-\nability turns out to be a one-dimensional concave opti-\nmization. In its course the computing system adapting\nto the problem got more complex or simpler until the\nglobal maximum is reached and the structural complex-\nities of the computing system and the problem become\nrelated through the optimality condition. More connec-\ntions might arise if the wave function could be involved\nin the description of the correlation structures.\nWe note that the algorithm A shares a common fea-\nture with Shor’s algorithm, which also relies on the PTM\nsequence [6].\nV.\nCONCLUSIONS\nWe have suggested that self-organization processes of\nprime integer relations could be used for information pro-\ncessing.\nIn particular, for a given problem self-organization pro-\ncesses of prime integer relations could be used to build\nthe correlation structures of a system in processing infor-\nmation demonstrating the optimal performance for the\nproblem. Remarkably, the processes can be equivalently\nrepresented by transformations of two-dimensional geo-\nmetrical patterns determining the dynamics of the sys-\ntem and revealing in the information processing its struc-\ntural complexity.\nThe information processing would be distinctive, be-\ncause self-organization processes of prime integer rela-\ntions can define the correlation structures of a system\nwithout reference to the distances, local times and sig-\nnals between the parts.\nComputational experiments testing competitive ad-\nvantages of the information processing have been pre-\nsented. They raise the possibility of an optimality condi-\ntion of complex systems: if the structural complexity of\na system is in a certain relationship with the structural\ncomplexity of a problem, then the system demonstrates\nthe optimal performance for the problem.\nThe optimality condition presents the structural com-\nplexity of a system as a key to its optimization. From\nits perspective the optimization of a system could be all\nabout the control of the structural complexity of the sys-\ntem to make it consistent with the structural complexity\nof the problem.\nImportantly, the experiments also indicate that the\nperformance of a complex system may behave as a con-\ncave function of the structural complexity.\nTherefore,\nonce the structural complexity could be controlled as a\nsingle entity, the optimization of a complex system would\nbe potentially reduced to a one-dimensional concave opti-\nmization irrespective of the number of variables involved\nits description. This might open a way to a new type of\ninformation processing for efficient management of com-\nplex systems.\n[1] V.\nKorotkikh,\nA Mathematical\nStructure\nfor Emer-\ngent Computation, Kluwer Academic Publishers, Dor-\ndrecht/Boston/London, 1999.\n[2] V. Korotkikh and G. Korotkikh, Description of Complex\nSystems in terms of Self-Organization Processes of Prime\nInteger Relations, in Complexus Mundi: Emergent Pat-\nterns in Nature, M. M. Novak (ed.), World Scientific, New\nJersey/London, 2006, pp. 63-72, arXiv:nlin.AO/0509008;\nV. Korotkikh and G. Korotkikh, On an Irreducible Theory\nof Complex Systems, InterJournal of Complex Systems,\n1751, 2006, arXiv:nlin.AO/0606023.\n[3] V. Korotkikh, G. Korotkikh and D. Bond, On Optimality\nCondition of Complex Systems: Computational Evidence,\narXiv:cs/0504092.\n[4] R. Orus, J. Latorre and M. A. Martin-Delgado, System-\natic Analysis of Majorization in Quantum Algorithms,\narXiv:quant-ph/0212094.\n[5] N. Gisin, Can Relativity be Considered Complete? From\nNewtonian Nonlocality to Quantum Nonlocality and Be-\nyond, arXiv:quant-ph/0512168.\n[6] K. Maity and A. Lakshminarayan, Quantum Chaos in\nthe Spectrum of Operators Used in Shor’s Algorithm,\narXiv:quant-ph/0604111."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"44 −34 −24 +14 ̸= 0. This hierarchical structure can specify a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"Pi, i = 1, ..., 16 whose changes ∆xi, i = 1, ..., 16 of the observ-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"ables xi, i = 1, ..., 16 relative to their reference frames can be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"given by the sequence ∆x1...∆x16 = +1 −1 −1 + 1 −1 + 1 +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"(t0) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"0, l = 0, 1, 2, 3 the geometrical patterns of the level l form","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"Wl of a geometrical pattern at level l, l = 1, ..., 4 equals the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"sum Wl = Wl−1,left + Wl−1,right = 2Wl−1 of the widths of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"patterns and Wl−1,left = Wl−1,right = Wl−1 as each geomet-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"rical pattern at level l −1 has the same width Wl−1 = 2l−1ε","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"and ti = εi, i = 0, 1, ..., 16. The height Hl of the geometrical","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"pattern equals Hl = Sl−1, where Sl−1 is the area of each of the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"height Hl can specify length scales of the level l, l = 0, 1, ..., 4.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"level l, l = 2, 3, 4, yet the boundary curve, as the graph of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"Sl = WlHl","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"1 (t)dt =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"the agents at step j, j = 1, ..., n−1 can be described by a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"binary sequence sj = s1j...sNj, where sij = +1, if agent","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"i, i = 1, ..., N uses the random strategy and sij = −1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"S = {sij, i = 1, ..., N, j = 1, ..., n −1}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"C(A(p)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"N 2 tr(V 2(v∗(p))) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"the strategy matrix S(v∗(p)), where λi, i = 1, ..., N are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"C(p) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"n2 tr(M 2(p)) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"M(p) = {dij/dmax, i, j = 1, ..., n},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"i, i = 1, ..., n are the eigenvalues of M(p), dij","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"{x = C(p), y = C(A(p)), p ∈P}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"C(A(p)) = 0.67C(p) + 0.33","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"C(A(p)) = 0.67C(p) + 0.33.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"̄D = 1","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":23753,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}