Effective information processing with pure braid group formalism in view of 2D holographic principle for information
aa r X i v : . [ m a t h . G M ] N ov Effective information processing withpure braid group formalism in view of 2Dholographic principle for information
Hubert Stokowski, Janusz E. Jacak and Lucjan Jacak
Abstract
The so-called holographic principle, originally addressed to the high energy physics, suggests more generally thatthe information inseparatly bounded with a physical carrier (measured by its entropy) scales as the event horizonsurface—a two-dimensional object. In this paper we present an idea of representing classical information in formalismof pure braid groups, characterized by an exceptionally rich structure for two-dimensional spaces. This leads to someinteresting properties, e.g. information geometrization, multi-character alphabets energetic efficiency for informationcoding characteristic or a group structure decoding scheme. We also proved that proposed pure braid group approachmeets all the conditions for storing and processing of information.
Keywords pure braid group, holographic principle, information processing, information geometrization
Introduction
The holographic principle, formulated on the basis ofthe Bekenstein entropy bound Bekenstein (1972); ’t Hooft(2009); Bousso (1999), strongly limits locality ’t Hooft(2009); Bousso (1999). This principle can be formulatedBousso (2002) in terms of a number of independent quantumstates describing the light-sheets L ( B ) (cf. Fig. 1)—thenumber of states, N , is bound by the exponential functionof the area A ( B ) of the surface B corresponding to the light-sheet L ( B ) , N [ L ( B )] ≤ e A ( B ) / . (1)It can be also expressed equivalently Bousso (2002), that thenumber of degrees of freedom (or number of bits multipliedby ln 2 ) involved in description of L ( B ) , cannot exceed A ( B ) / . Even though the complete holographic theory is notconstructed as of yet, this idea is considered as breakthroughon the way to quantum gravity—the holographic principlehighlights the lower two-dimensional character of theinformation (entropy) corresponding to any real 3D systemand in that sense refers to some mysterious 2D hologramcollecting all information needed to describe corresponding3D system.An interesting step toward an explicit formulationof the holographic theory was done by MaldacenaMaldacena and Strominger (1997); Witten (1998), whoproposed a holographic superstring theory formulation inthe anti de Sitter space. According to that model, the 3Dquantum gravity could be implemented by 2D quantumnonrelativistic hologram on the edge of the anti de Sitterspace (for the anti de Sitter space the boundary is a sphereat fixed time). The locally two dimensional spatial manifoldsfor the quantum hologram corresponding to 3D quantumrelativistic systems open, however, quite new possibilityfor interpretation of the nature of hypothetical holographiccounterparts of elementary particles. t L(B) BL(B) Figure 1.
Two light-sheets for 2D spherical surface B model ( B is the circle for 2D): t is the time, the cones created by light rayswith negative expansion define light-sheets suitable forcovariant entropy bound formulated by Bousso: the entropy oneach light-sheet S ( L ( B )) does not exceed the area of B , i.e., A ( B ) , S ( L ( B )) ≤ A ( B ) / . In the context of the holographic principle we proposeto grasp this two-dimensional information character witha mathematical formalism favoring two-dimensional spacesover higher-dimensional ones. Such a opportunity providesthe braid group formalism, which for two-dimensionalspaces is of rich infinite structure unlike for three- or
Department of Quantum Technologies, Faculty of FundamentalProblems of Technology, Wrocław University of Science and Technology,Wyb. Wyspia´nskiego 27, 50-370 Wrocław, Poland
Corresponding author:
Janusz E. JacakEmail: [email protected]
Prepared using sagej.cls [Version: 2016/06/24 v1.10]
Journal Title XX(X) higher-dimensional spaces. This will lead to formulation ofsome fundamental findings concerning the effectiveness ofinformation coding.From the informational application point of view wepropose to consider a more efficient system for codinginformation (based on pure braid group formalism) thancurrently widespread uniformed binary system, which hadearned its popular position because of the simplicity ofits realization with transistor-based processors. Nowadaysthis type of information coding is so popular that noother systems are considered to realize in the informationprocessing despite that no type of a binary-code exist innature.In this paper we present considerations of multi-characteralphabets in coding of classical information, which can befurther referred to the realization of neuron-like system ofdata storage and processing. It seems to be a promising wayto enhance the efficiency of the information transmissionrate.The main idea of the multi-character alphabets is based ona topological concept of pure braid groups, which has beenused in many applications Jacak et al. (2012). Topologicalproperties are highly useful to prove complex theorems ina simpler manner and solve complicated physical problemseasier. The original concept of information coding in thisfield seems to be promising. The idea of storing informationwith braid groups change the outlook on coding, becauseit refers not to the commonly known idea of making linksone-to-one, which represent a specific information. Braidgroups formalism proposed here lets to code information bychoosing how do the elements connect instead of which ofthem are connected. This formalism lets to gain even infiniteamount of data stored in a set of N elements, when thereare only N ! possible bits stored in this type of system in thebinary approach. Braid groups in information coding
For 2D plane R the braid group was originally described byArtin Artin (1947), and later vastly developed Birman (1969,1974); Birman et al. (1998); Birman and Brendle (2005);Kang et al. (1997); Dehornoy (1997); Guaschi (2002). TheArtin braid group, i.e., π ( Q N ( R )) is the infinite group,with generators σ i and its inverse σ − i (defining exchanges ofneighboring particles, i th particle with ( i + 1) th one, at someenumeration of particles, arbitrary, however, due to particleindistinguishability) cf. Fig. 2, σ i σ i − = e, (2)satisfying the conditions Artin (1947); Birman (1974), σ i σ i +1 σ i = σ i +1 σ i σ i +1 , for i N − , (3) σ i σ j = σ j σ i , for i, j N − , | i − j | > . (4)Among the different dimensional Euclidean spaces themost valuable is the two-dimensional space. For dim ( M ) > the braid group is equal to the permutation group S N . Thereason is all the closed loops in this type of space can becontracted into a point, no other 0-dimensional topologicaldefect(point) of M of the space can change it. In the 2-dimensional space the loops are uncontractible, thus the i +
1i i + i σ i-1 i +
1i i + i2 e Figure 2.
The geometric braid presentation of a full braid groupgenerator, its inverse element and a specific property in thetwo-dimensional space. braid group is a non-trivial infinite group. This impliesdifferent relationships, depending on the dimension of themanifold: ( σ i ) = e, for dim( M ) > , (5) ( σ i ) = e, for dim( M ) = 2 . (6)The pure braid group is the first group of homotopy π ( F N ( M )) = π ( M N \ ∆) , where F N ( M ) is a configura-tion space of N -braids defined as n -fold Cartesian product ofmanifold M reduced by ∆ -points, which represent diagonalpoints—topological defects of the manifold M . In a formalway generators of this group have to be introduced bygenerators of full braid group π ( Q N ( M )) = π (( M N \ ∆) /S N ) , where S N represents a permutation group, whichdetermines the equivalence class in the quotient structure. Itmeans, that the principal difference between full and purebraid groups is that in the full group trajectories can generatebraids in any way, with permutations between the beginningand the end, when the pure group is limited by the same setupof trajectories at the beginning and at the end of the braid.The pure braid group is generated by the l ij generators (cf.Fig. 3) formally expressed using elements of the full braidgroup as: l ij = σ j − σ j − . . . σ i +1 σ i σ − i − . . . σ − j − σ − j − . (7) i l ij j Figure 3.
The geometric braid presentation of a pure braidgroup generator l ij . Prepared using sagej.cls tokowski, Jacak and Jacak Through pure braid groups it seems possible to representthe multi-character alphabet coding system, where eachsymbol is represented by a different generator of a purebraid group. Representation of this type is useful only in thetwo-dimensional space, where the structure is non-trivial andthe braid is able to contain even infinite countable numberof code signs. To proof usability of this coding system itis necessary to show two principal conditions. First of allit is crucial to find at least two (in binary code) or moregenerators, which are able to represent alphabet characters ofthe particular alphabet. The second condition is the necessityto preserve the sequence of a code. It has to be shown thatthere is no possibility to change the order of used generatorsand it is impossible to untie the braid built by those.
Binary code information
The basic coding method used nowadays is the binarysystem. It is principal to show possibility of a building binarysystem in the formalism of braid groups to proceed with themulti-character alphabets. There is no possibility to showthis type of code with the 2-braid, because there is only onegenerator of the group in this case and its inverse generator: l = σ and l − = ( σ − ) . (8)Building the binary formalism with those two elements isimpossible, because the construction of a code with elementsinverse to each other would cause disentangle of the braid.The required procedure is adding trajectories to the braid togain more generators, which can possibly work as charactersin the coding alphabets. It can be proven that there exist apossibility to code in the binary system using a 3-braid with apure braid group. In this type of structure there are 3 possiblegenerators: l = σ , l = σ σ σ − , l = σ . (9)The inverse elements cannot be used to code information incombination with basic generators, thus they are not includedhere.Lets prove it is possible to code a binary informationthrough two operators of the 3-braid group. Its realizationwill be shown by taking two generators, which exchangebraids from the group with the specified one. It is trivial inthe 3-braid group, nonetheless this proof will be afterwardsexpanded into more complex structures. Lets take the 3-rd trajectory as the distinguished one and build a segmentof generators containing this trajectory. The l and l generators can be chosen as possible characters, it has to beproven there is no possibility to change the sequence of theaforementioned just through the braid group properties. Inthe geometrical braid representation it means that there is nopossibility to ’drag’ one entanglement through another in theorder.Showing that the braid generated by sequence of l l l preserves the original input order is enough for verificationthat any sequence built of those generators do preserve theinformation structure. The sequence realized by geometricalbraids has been shown in the Figure 4 A). The proof can bedone in two steps. The first one is a topological simplificationof the problem, by bringing each unperturbed trajectory into
123 l ≡ l ≡ l ≡ l ≡ l ≡ A) B)abc
Figure 4.
A) The binary code representation through a 3-braidpure group, B) the idea of topological simplification of theproblem. a point and representing each entanglement by a closed loop.In this representation the distinguished, 3-rd trajectory willbe transformed into a point and each element of the sequencewill be converted into a loop around this point, with thebeginning and the end in the point, which represents thesecond entangled trajectory. It is necessary to show all loopsgenerated by sequence elements are not homotopic and haveto cross each other in at least two points. The procedure isshown in the Figure 4 B).
Proof.
Suppose A and B are open neighborhoods ofthe point c , which boundaries represents distinguishedtrajectories and let a ∈ ∂A but a / ∈ clB and analogous b ∈ ∂B but b / ∈ clA , where ∂A and ∂B are boundaries of sets A and B respectively. It is therefore true that c ∈ A ∩ B = ∅ and neither A * B , nor B * A . Both A and B are simply-connected, not border and dense-in-itself sets.Since A and B are open sets, their intersection A ∩ B isopen as well, lets take a point q , which is a boundary pointof A ∩ B , let V be any open neighborhood of the q , then: ∀ q ∈ ∂ ( A ∩ B ) ∃ q ′ ,q ′′ ∈ V [ q ′ ∈ ( A ∩ B ) ∧ q ′′ ∈ ( X \ A ∩ B )] , (10)moreover, due to ∂ ( A ∩ B ) ∈ ∂A ∪ ∂B : ∀ q ∈ ∂ ( A ∩ B ) [ q ∈ ∂A ∨ q ∈ ∂B ] . (11)For all open neighborhoods V of the point q it can be written: ∀ q ∈ ∂ ( A ∩ B ) ∀ V ∃ q ′ ,q ′′ ∈ V { [ q ′ ∈ A ∧ q ′′ ∈ ( X \ A )] ∨ [ q ′ ∈ B ∧ q ′′ ∈ ( X \ B )] } . (12)Since A and B are dense-in-itself and simply-connected, A * B , B * A and A ∩ B = ∅ it has to be truth, thatthere exists a boundary point q i of A ∩ B and its any openneighborhood V i as: ∃ q i ∀ i ∈ (1;2 n ) ∀ V i ∃ q ′ i ,q ′′ i ∈ V i [ q ′ i ∈ ( A ∩ B ) ∧ q ′ i ∈ A ∧ q ′ i ∈ B ∧ q ′′ i ∈ X \ ( A ∩ B ) ∧ q ′′ i ∈ X \ A ∧ q ′′ i ∈ X \ B ] , (13)where n ∈ N thus: ∃ q i ∀ i ∈ (1;2 n ) [ q i ∈ ∂ ( A ∩ B ) ∧ q i ∈ ∂A ∧ q i ∈ ∂B ] . (14)Finally: ∂ ( A ∩ B ) ∩ ∂A ∩ ∂B = ∅ . (15)And there are even number of ∂A and ∂B intersections.It can be as well proven in language of topology andhomotopy, what is sketched below. Consider two loops A and Prepared using sagej.cls
Journal Title XX(X) B and three points of the manifold M : a , b and c . If the loop A contains the point a and surrounds the point c it cannot becontracted into a , due to a defect-like character of the point c in this context. Lets take a second loop B in the analogousmanner, but containing the point b instead of a . Neither theloop A surrounds the point b , nor the loop B surrounds thepoint a and none of them can be contracted because of thepoint c , which represents a topological defect of the manifold M . Moreover none of those loops can be expanded throughpoints a and b for loops B and A respectively. It is clear then,that those loops have to cross each other on the manifold M .It is fulfilled that the boundaries representing loops inthis approach intersect each other in each case, are nothomotopic and cannot be contracted into a point. Basingon the topological proof it is impossible to drag one braidgroup generator through another in the binary informationsequence. In more intuitive way one can undergo with pointsinto the ’bars’ in the 3-dimensional space as shown inthe Figure 5. It is clear that loops around the bars, whichrepresent different information symbols cannot be mixedwith each other. l ≡ l ≡ ≡ l ≡ l ≡ Figure 5.
The 3-dimensional representation of theindependence of each alphabet symbol of the binary code inthe braid groups formalism.
The other approach to the proof, basing on the braidgroups formalism itself can be necessary as well. It has tobe shown that there is no possibility to change the order offull braid group generators σ i , which are used in a realizationof pure braid group generators l ij assigned to alphabetcharacters. As it was said earlier it is enough to show there isno possibility to change the order in the sequence l l l .This sequence can be realized as follows: l l l = σ σ σ σ − σ . (16)Reducing the relation using the neutral element of thegroup from the equation (2) and the relation (3) turns outimpossible due to the relation (6) taking place in the two-dimensional space. Usage of defined relations leads onlyto complicating the braid, instead of reduction or changingthe order of the coded sequence. In result of this paragraphit has been formally shown that there exists a possibilityto code information through 3-braid group generators. Inthis formalism it is possible to use even three generators asindependent signs: l , l and l to define a ternary system,however the idea described here is easily expandable intomore complex braids. The proposed set of generators andassigned alphabet characters are listed in the Table 1. Table 1.
The binary system defined with 3-braid groupgenerators.
Braid group generator Binary symbol l l Ternary system
123 l ≡ l ≡ l ≡ Figure 6.
The 4-braid group concept of a ternary codingsystem. l ≡ l ≡ l ≡ Figure 7.
The scheme for a topological proof for theindependence of ternary system generators in a 4-braid system.
Expanding the proposed approach it is easy to define aternary system by adding one more trajectory to the braid.Lets define a 4-braid with 3 independent generators, whichcan be used as symbols. In the new 4-braid system bydistinguishing the 4-th trajectory one can choose at least 3independent generators, the concept is shown in the Figure6. The presented above proof can be easily expanded into the4-braid situation and show the independence of 3 mentionedgenerators. The only difference for systems with number oftrajectories N > is that one has to put attention to relation(4). Its consequence is impossibility to choose two generatorstangling up completely different trajectories. Choosing thistype of generators would cause a failure to preservation ofthe information sequence - dragging those braids would beunrestricted and will lead to wasting of the sequence order.However, this disadvantage does not affect the proposedcoding system. The generators of the chosen ternary systemhas been shown in the Table 2.For an intuitive topological proof it is comfortable tobase on the scheme shown in the Figure 7 and expandthe proof for 3 neighborhoods of a point-representation ofthe distinguished trajectory. For a braid group formalismit is enough to show the sequence of l l l l l l l Prepared using sagej.cls tokowski, Jacak and Jacak Table 2.
The ternary system defined with 4-braid groupgenerators.
Braid group generator Ternary symbol l l l l l , l l , l l , l l etc.) does preserve it. Multi-character alphabets
As it has been sketched above in introduced formalismit is possible to expand coding alphabets by adding moreand more trajectories to the braid and each addition has toexpand amount of generators eligible for coding informationat least by one. The consequence is very strong and leads tothe proposal of the multi-character alphabet definition usingthe 2-dimensional braid group formalism, where number ofalphabet signs increases at least linearly with the number N of trajectories in the N -braid. The discussion of addingmore than one generator with each new trajectory is open,but the main goal of this work is to show there is a possibilityto expand multi-character alphabets in the countable andinfinite manner. Adding more generators is possible, asit has been shown in the 3-braid case, but it is hard togeneralize into more complex braid structures, mainly dueto the relation (4). Possible representation of the N -characteralphabet coded using the (N+1) -braid group with the (N+1) -th trajectory distinguished has been shown in the Table 3. Table 3.
The N -character alphabet coded using the (N+1) -braidgroup formalism. Braid group generator N -alphabet symbol l NN +1 l N − N +1 l N − N +1 l N +1 N-1
Metrics and decoding distance
In the perspective of applications the decoding case can beconsidered. The most intuitive way in the pure braid groupsformalism, as it has a group structure, is to use the fullstring of inverse generators. Let S be a string of d binary-symbols in a pure braid sub-group representation and S − be its inverse string as shown in the Figure 8.
123 l ≡ l ≡ l ≡ l ≡ l ≡ l ≡ S S -1 Figure 8.
The pure braid group information decoding idea interms of geometric braids.
Through acting on the string S with its inverse S − it canbe found that the whole constructed braid is an equivalent to the neutral element of the group e —decoded string hasdisentangled completely with its inverse: S · S − ≡ l l l · l − l − l − ≡ e. (17)In general the string S of a coded information canbe represented in a formal way with pure braid groupgenerators: S = Y j ∈ a d l jn , (18)where a d is a sequence build of natural numberscorresponding to specific alphabet signs. The product Q j ∈ a d l jn is understood as a combination of followingpure braid group generators. The inverse string S − can bedefined as: S − = Y j ∈ a ′ d l − jn , (19)where the sequence a ′ d is given by: a ′ d : ∀ i ∈ (1 ,d ) a ′ i = a ( d +1 − i ) . (20)In terms of the information theory it is common to usespecific types of metrics, so called information distance.Using the classical binary system there is a well definedHamming distance Hamming (1950), which is constructedbetween two equal strings as a number of positions at whichthe corresponding symbols are different. Its generalizationon any string lengths has been shown by LevensteinLevenshtein (1966). In the case of braid-groups it is moreconvenient to define a different metrics, corresponding to theproposed decoding method.Lets consider a metrics, which correspond to the numberof equal characters at the end of two d -character stringsbuilt according to the specific braid sub-group with thedistinguished n -th trajectory. In case of two strings, whichlengths are not equal lets add its difference to the resultantdistance between information coded on those. The metricscorresponding to the inverse-braid decoding system for twostrings S and S can be proposed as: d ( S , S ) = f + d + | f − d | − x , (21)where d is the string’s S length, f is the string’s S lengthand x is a number of repeating characters at the end of those,formally: S = Y j ∈ a d l jn , S = Y k ∈ b f l kn , (22) a d and d f are d -element and f -element sequences consistingof natural numbers, corresponding with code characterscoded with l jn and l kn - pure braid group generators actingwith the n -th distinguished trajectory. Moreover ξ x is asubsequence of both a d and b f : x : ξ x : ξ x ∈ a d ∧ ξ x ∈ b f , thus : (23) ∀ i ∈ (1; x ) ξ i = a ( d − x + i ) ∧ ξ i = b ( f − x + i ) = ⇒∀ i ∈ (1; x ) a ( d − x + i ) = b ( f − x + i ) . (24) Proof.
It has to be proven that the proposed metrics meetsfour basic conditions:
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Journal Title XX(X)The non-negativity or separation axiom. If ξ x is as wella subsequence of a d as b f , so x d and x f , thus f + d > x and d ( S , S ) > is always fulfilled. The identity of indiscernibles. d ( S , S ) = 0 ⇔ f + d + | f − d | − x = 0 . (25)For f ∈ N and d ∈ N , which is determined by sequences a d and b f properties, expression f + d − | f − d | can beinterpreted as doubled maximum of a set ( d, f ) . f + d + | f − d | = 2 max ( d, f ) , so: max ( d, f ) = x, thus: x = f ⇔ f > d ∨ x = d ⇔ d > f, (26)moreover from (23): x f ∧ x d. (27)Finally one gets: x = f = d = ⇒∀ i ∈ (1; x ) a i = b i = ⇒ a d ≡ b f = ⇒ d ( S , S ) = 0 ⇔ S = S . (28) Symmetry.
According to the definition: d ( S , S ) = d + f + | d − f | − x , (29)is always equal to: d ( S , S ) = f + d + | f − d | − x . (30)Taking into account assumptions (23): S = Y k ∈ b f l kn , S = Y j ∈ a d l jn . (31) x : ξ x : ξ x ∈ b f ∧ ξ x ∈ a d , thus : ∀ i ∈ (1; x ) ξ i = b ( f − x + i ) ∧ ξ i = a ( d − x + i ) = ⇒∀ i ∈ (1; x ) b ( f − x + i ) = a ( d − x + i ) ⇔∀ i ∈ (1; x ) a ( d − x + i ) = b ( f − x + i ) , (32)finally: ∀ ( S ,S d ( S , S ) = d ( S , S ) . (33) Triangle inequality.
To proof the triangle inequality threedifferent strings have to be defined: S = Y j ∈ a d l jn , S = Y k ∈ b f l kn , S = Y m ∈ c g l mn , (34) a d , b f and c g are three different length sequences of naturalnumbers, corresponding with alphabet characters. In generaleach pair of sequences is able to have different number of thesame signs at the end, lets take x, y and z. For S and S : x : ξ x : ξ x ∈ a d ∧ ξ x ∈ c g , (35) ∀ i ∈ (1; x ) ξ i = a ( d − x + i ) ∧ ξ i = c ( g − x + i ) = ⇒∀ i ∈ (1; x ) a ( d − x + i ) = c ( g − x + i ) , (36) for S and S : y : ξ ′ y : ξ ′ y ∈ a d ∧ ξ ′ y ∈ b f , (37) ∀ i ∈ (1; y ) ξ ′ i = a ( d − y + i ) ∧ ξ ′ i = b ( f − y + i ) = ⇒∀ i ∈ (1; y ) a ( d − y + i ) = b ( f − y + i ) , (38)for S and S : z : ξ ′′ z : ξ ′′ z ∈ b f ∧ ξ ′′ z ∈ c g , (39) ∀ i ∈ (1; z ) ξ ′′ i = b ( f − z + i ) ∧ ξ ′′ i = c ( g − z + i ) = ⇒∀ i ∈ (1; z ) b ( f − z + i ) = c ( g − z + i ) . (40)It can be written that: d ( S , S ) + d ( S , S ) = f + d + | f − d | − y g + f + | g − f | − z g + d + | g − f | + | f − d | + 2 f − y − z . (41)From the absolute value properties: ∀ x i ∈ R | x + x | | x | + | x | , for x = g − f, x = − d + f | g − f | + | f − d | > | g − d | , thus: g + d + | g − f | + | f − d | + 2 f − y − z >> g + d + | g − d | f − y − z ) . (42)Two cases have to be considered. According to the definition,sequences a d and c g are the same on the x closing elements.If a d and b f are the same on the y last elements and:1) y x then z = y = ⇒ z x has to be fulfilled,because the same amount of closing elements have to beequal in pairs a d , b f and b f , c g . Moreover according to (37) y f , so ( f − y ) > . Then ( f − y − z ) > ( − x ) .2) y > x then z = x , to fulfill assumption that only xclosing elements of a d and c g are equal. ( f − y ) > isconserved, so ( f − y − z ) = ( − x ) .In both cases: g + d + | g − d | f − y − z ) >> g + d + | g − d | − x = d ( S , S ) , (43)thus: ∀ ( S ,S ,S ) d ( S , S ) d ( S , S ) + d ( S , S ) . (44)With the well defined metrics it is useful to consider adistance between the set of exemplary 3-bit sequence. Thecomparison has been shown in the Table 4.The Figure 9 shows a graphical intuition for understandingdistances in a simpler case - between all possible 2-bit binarysequences. To represent a 3-bit code it would be necessary touse a four-dimensional picture.The most valuable conclusion is for every 3-bit code thereis only one code in a distance of unity, two with a distanceof 2 and 4 with distance of 3. It is the significant difference Prepared using sagej.cls tokowski, Jacak and Jacak Table 4.
The comparison of information distances between all3-bit codes.
000 001 011 100 010 110 101 111001
00 1101 10
Figure 9.
The geometrical representation of distances betweenall possible 2-bit information using the proposed metrics. Solidline represent the distance of d=1, dashed line - d=2. between proposed metrics and e.g. the Hamming distance,where more codes are ’closer’ to each other. It is a valuableconclusion due to the error detection and the error correctionissues Wagner and Fischer (1974). In terms of the Hammingdistance for d=1 it is neither possible to detect nor correcterrors. For d=2 it is possible to detect them and for d=3also the correction is achievable. If it is possible to usethose relations in terms of the proposed decoding system andmetrics it can turn out much more effective.
Coding Efficiency
Figure 10.
The energy consumption versus number ofalphabet signs for f ( N ) proportional to N i for i = 1 , , , . Coding multi-character alphabets, which could be usedto store and process information in the 2-dimensional purebraid group formalism with possibly infinite number ofcharacters, engenders a new issue for the optimizationof the number of characters in the coding system. It isimportant to find an optimal number of alphabet signs forminimization of the energy efforts. This type of optimumshould exist, because the greater amount of characters leads
Figure 11.
The energy consumption versus number of alphabetsigns for f ( N ) proportional to N i for i = 1 / , / , / . to the reduction of the information length and, on the otherhand, adding more and more characters requires increasingof the energy consumption to ’remember’ all of them. Onecan try to model an analytic function suitable to predict theenergy effort necessary to take an advantage of N-characteralphabets versus the coding gain estimated for this amountof available signs, let g ( N ) be this function. To find anoptimum number N of alphabet signs it is important to takea function f ( N ) , which represent the energy efforts fora N -character alphabet and divide it by a function f ( N ) ,modeling the gain vs. the binary system efficiency. f ( N ) is an increasing function, because using higher number ofalphabet signs has to utilize the higher energy consumption. f ( N ) can be represented, with respect to the basics of theinformation theory as: f ( N ) = log N. (45)There is no formal function, matching the f ( N ) require-ments, but if it is indeed increasing the resultant g ( N ) function has to have one, trivial minimum for N > . g ( N ) = f ( N ) f ( N ) . (46)Some increasing functions can be proposed to examine thecourse of the function (46), a few basic ideas has been shownon the Figure 10, for f ( N ) = N i , where i = 1 , , , andon the Figure 11 for i = , , . For f ( N ) ∝ √ N theoptimum number of alphabet signs N is about 20, whichnaively corresponds to the number of syllables in the mostof the languages commonly used by humans. It can be aclue for the naturally preferable amount of characters in thealphabet and also for a possibly accurate model of the energyconsumption vs. number of alphabet signs as f ( N ) ∝ √ N in natural systems (cube root corresponds to a 3D volume). Conclusion
Presented approach emphasizes that every information mustbe connected with its physical carrier, and this relation(between information and its carrier) allows to discuss forexample energy balance of a information processing system(in case when one considers information without its carrier
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Journal Title XX(X) then in fact the binary system seems to be most optimaldue to its simplicity). On the other hand, the carrier mustbehave accordingly to its physical properties, which inview of a general character of the holographic principle,leads to quite interesting observation that every physicalinformation (information bounded to its carrier—in fact onlysuch information exists) scales as a two-dimensional objectand not a three-dimensional one, as it could be falselysuspected. Thus one can suspect that it would be desirableand effective to analyze information and its processingsystems with use of some mathematical description whichfavors two-dimensional spaces among all others.A multi-character alphabet coding using pure braidgroups has been introduced in context of the holographicprinciple elucidating two-dimensional information character(where complex properties of pure braid groups are onlypreserved in the dim( M ) = 2 ). If properly implementedthis could significantly improve the coding efficiency in theinformation technology. As it is hard to find in nature a trulybinary information processing system (other than artificiallyintroduced systems in computer science), a multi-characteralphabet based information processing system seems to bemore natural. Such system turns out more energy-efficientthan the binary system, which is commonly used nowadays.The proposed information decoding system forces an usageof a specific metrics, however it can be an advantage ofthe described information coding due to its potential in theerror-detection and the error-correction issue, since distancesbetween the sequences are usually high. The proposedformalism gives a possibility to connect the information withits physical carrier—the braid, which has not been proposedso far. Moreover, the described coding system seems to bea suitable way for a realization of neuron-like resonancecircuits for the data storage and is likely to explain workingprinciples of this type of connections in the human brain.The method described above introduces just one ofpossibilities for the coding of information with pure braidgroups generators. An equivalent method is using generatorsresponsible for tangling up the distinguished trajectory withother ones. The limit imposed on the pure braid groupgenerators can be as well a defect of this approach, becauseof the usage of a sub-group, as an advantage. It could beconsidered to preform an efficient error-correction systemusing a few equivalent representations of the alphabettransferring through different transmission channels. Otherapproaches can find more than N − generators useful forthe coding in the N -braid, as it has been shown for the3-braid, but the method proposed here cannot generalizethis type of generators due to the additional relation (4),appearing in systems with N > trajectories. Another wayto earn more possible characters for the coding system is arealization of more complicated form of generators, whichwould be able to preserve the necessary conditions. References
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Acknowledgements
Supported by the NCN project P.2011/02/A/ST3/00116