THIS paper analyzes the behavior of those complex

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 Itrisic Order Lexicographic Order Vector Order ad Hammig Weight Luis Gozález Abstract To compare biary -tuple probabilities with o eed to compute them we have defied a partial order relatio o the set { } of all biary -tuples: The so-called itrisic order relatio I this paper some properties of the itrisic orderig are derived These properties ivolve the lexicographic truth-table order i { } the vector order defied betwee the vectors of positios of -bits of the biary -tuples ad the umber of -bits i the biary -tuples ie the Hammig weights These results are illustrated through simple examples ad the itrisic order graph Idex Terms complex stochastic Boolea system Hammig weight itrisic order itrisic order graph lexicographic order vector order I INTRODUCTION THIS paper aalyzes the behavior of those complex systems which deped o a large umber of radom Boolea variables: The so-called complex stochastic Boolea systems hereafter CSBSs That is the basic Boolea variables of the system are stochastic o-determiistic ad they oly take two possible values or Usig the statistical termiology a stochastic Boolea variable ca be cosidered as a Beroulli variable Each oe of the 2 outcomes for a CSBS is give by a biary -tuple u u u { } of s ad s I the followig we assume that the Beroulli variables x x 2 x of the CSBS are statistically idepedet so that the occurrece probability of a give biary strig of legth u u u { } is give by Pr {u} p ui i p i ui i that is Pr { u} is the product of factors p i if u i p i if u i Example : Let 4 ad u { } 4 Let p p 2 2 p 3 3 p 4 4 The usig we have Pr { } p p 2 p 3 p 4 44 Oe of the mai questios i the aalysis of CSBSs cosists of determiig the orderig betwee the curret values of the 2 associated biary -tuple probabilities Pr {u} The simplest aswer to this questio amely computig all these 2 probabilities by usig ad orderig them i decreasig or icreasig order of their values is oly possible i practice for small values of However for large values of we eed alterative procedures for comparig the biary Mauscript received October 3 22 This work was supported i part by the Spaish Govermet Miisterio de Ecoomía y Competitividad ad FEDER through Grat cotract: CGL2-29396-C3- L Gozález is with the Research Istitute SIANI & Departmet of Mathematics Uiversity of Las Palmas de Gra Caaria 357 Las Palmas de Gra Caaria Spai e-mail: luisglez@dmaulpgces strig probabilities overcomig the expoetial ature of this problem For this purpose i [2] we have defied a partial order relatio o the set { } of all the 2 biary -tuples the so-called itrisic order betwee biary -tuples Usig the itrisic orderig we ca compare order two give biary -tuple probabilities Pr {u} Pr {v} with o eed to compute them simply lookig at the relative positios of the s ad s i the biary -tuples u v to be compared I this way for those pairs u v of biary -tuples comparable by itrisic order the orderig betwee their occurrece probabilities is always the same for all sets of basic probabilities {p i } i O the cotrary for those pairs u v of biary -tuples icomparable by itrisic order the orderig betwee their occurrece probabilities depeds o the curret values of the basic probabilities {p i } i The lexicographic order o the set { } is the usual truth-table order betwee biary strigs of legth The Hammig weight of a biary -tuple u { } is the sum of all its bits that is the umber of -bits i u The vector order is a total order relatio defied betwee the vectors of positios of -bits of the biary -tuples with the same Hammig weight The purpose of this paper is to preset the relatios betwee the itrisic orderig ad the three above cocepts lexicographic order Hammig weight ad vector order Some of these relatios especially those dealig with the Hammig weight ca be foud i [9] For this purpose this paper has bee orgaized as follows I Sectio II we preset all the backgroud about the itrisic order required to make this paper self-cotaied Sectio III is devoted to preset the relatios betwee the itrisic orderig ad the lexicographic order I Sectio IV the relatio betwee the itrisic orderig ad the Hammig weight is preseted The relatio betwee the itrisic orderig ad the vector order is aalyzed i Sectio V Fially coclusios are preseted i Sectio VI II THE INTRINSIC ORDERING A Itrisic Order Relatio o { } Throughout this paper the decimal umberig of a biary strig u is deoted by the symbol u ad we use the symbol to deote the equivalece betwee the biary ad decimal represetatios of a biary strig ie u u u u 2 i u i eg for 6 we have i 2 + 2 2 + 2 3 + 2 5 45 Accordig to the orderig betwee two give biary strig probabilities Pr u ad Pr v depeds i geeral o the parameters p i as the followig simple example shows Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 Example 2: Let 3 u ad v Usig we get the followig iequalities For p p 2 2 p 3 3 : Pr {u} 54 < Pr {v} 56 while for p 2 p 2 3 p 3 4 : Pr {u} 96 > Pr {v} 84 As metioed i Sectio I to overcome the expoetial complexity iheret to the task of computig ad sortig the 2 biary strig probabilities associated to a CSBS with Boolea variables we have itroduced the followig itrisic order criterio [2] deoted from ow o by the acroym IOC Theorem 2 The itrisic order theorem: Let Suppose that x x are mutually idepedet Beroulli variables whose parameters p i Pr {x i } satisfy < p p 2 p 5 2 The the probability of the biary -tuple v v v is itrisically less tha or equal to the probability of the biary -tuple u u u that is for all set {p i } i satisfyig 2 if ad oly if the matrix Mv u u u : v v either has o colums or for each colum i Mv u there exists at least oe correspodig precedig colum IOC Remark 2: I the followig we assume that the parameters p i always satisfy coditio 2 Note that this hypothesis is ot restrictive for practical applicatios because if for some i : p i > 5 the we oly eed to cosider the variable x i x i istead of x i Next we order the Beroulli variables by icreasig order of their probabilities Remark 22: The colum precedig to each colum is ot required to be ecessarily placed at the immediately previous positio but just at previous positio Remark 23: The term correspodig used i Theorem 2 has the followig meaig: For each two colums i matrix Mv u there must exist at least two differet colums precedig to each other I other words: For each colum i matrix M u v the umber of precedig colums must be strictly greater tha the umber of precedig colums Remark 24: IOC ca be equivaletly reformulated i the followig way ivolvig oly the -bits of u ad v with o eed to use their -bits Matrix Mv u satisfies IOC if ad oly if either u has o -bits ie u is the zero -tuple or for each -bit i u there exists at least oe correspodig -bit i v placed at the same or at a previous positio I other words either u has o -bits or for each -bit i u say u i the umber of -bits i v v i must be greater tha or equal to the umber of -bits i u u i The matrix coditio IOC stated by Theorem 2 or by Remark 24 is called the itrisic order criterio because it is idepedet of the basic probabilities p i ad it itrisically depeds o the relative positios of the s ad s i the biary -tuples u v Theorem 2 or Remark 24 aturally lead to the followig partial order relatio o the set { } [2] The so-called itrisic order will be deoted by ad we shall write v u or u v to idicate that v is itrisically less tha or equal to u or that u is itrisically greater tha or equal to v Defiitio 2: For all u v { } v u iff Pr {v} Pr {u} for all set {p i } i iff M u v satisfies IOC st 2 From ow o the partially ordered set poset for short { } will be deoted by I Example 22: For 3 we have ad because the matrices 3 4 4 3 ad do ot satisfy IOC Remark 23 Thus ad are icomparable by itrisic order ie the orderig betwee Pr { } ad Pr { } depeds o the parameters {p i } 3 i as Example 2 has show Example 23: For 6 we have because matrix 56 5 satisfies IOC Remark 22 Thus for all {p i } 6 i st 2 Pr { } Pr { } Example 24: For all the biary -tuples ad 2 are the maximum ad miimum elemets respectively i the poset I Ideed both matrices u u ad u u satisfy the itrisic order criterio sice obviously they have o colums! Thus for all u { } ad for all {p i } i st 2 { } { } Pr Pr {u u } Pr May differet properties of the itrisic order relatio ca be derived from its simple matrix descriptio IOC see eg [2] [3] [4] [5] Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 B A Graph for the Itrisic Order Now we preset the most commo graphical represetatio of our poset I { } The usual represetatio of a poset is its Hasse diagram see [2] for more details about these diagrams Specifically for our poset I its Hasse diagram is a directed graph digraph for short whose vertices are the 2 biary -tuples of s ad s ad whose edges go upward from v to u wheever u covers v deoted by u v This meas that u is itrisically greater tha v with o other elemets betwee them ie u v u v ad w { } st u w v A simple matrix characterizatio of the coverig relatio for the itrisic order is give i the ext theorem; see [4] for the proof Theorem 22 Coverig relatio i I : Let ad let u v { } The u v if ad oly if the oly colums of matrix Mv u differet from ad are either its last colum or just two colums amely oe colum immediately preceded by oe colum ie either Mv u u u 3 u u or there exists i 2 i st Mv u u u i 2 u i+ u 4 u u i 2 u i+ u Example 25: For 4 we have 6 7 sice M7 6 has the patter 3 2 sice M2 has the patter 4 The Hasse diagram of the poset I will be also called the itrisic order graph for variables deoted as well by I For small values of the itrisic order graph I ca be directly costructed by usig either Theorem 2 matrix descriptio of the itrisic order or Theorem 22 matrix descriptio of the coverig relatio for the itrisic order For istace for : I { } ad its Hasse diagram is show i Fig Ideed I cotais a dowward Fig The Itrisic Order Graph for edge from to because see Theorem 2 sice matrix has o colums! Alteratively usig Theorem 22 we have that sice matrix has the patter 3! Moreover this is i accordace with the obvious fact that Pr {} p p Pr {} sice p /2 due to 2! However for large values of a more efficiet method is eeded For this purpose i [4] the followig algorithm for iteratively buildig up I for all 2 from I depicted i Fig has bee developed Theorem 23 Buildig up I from I : Let 2 The graph of the poset I { 2 } o 2 odes ca be draw simply by addig to the graph of the poset I { 2 } o 2 odes its isomorphic copy 2 + I { 2 2 } o 2 odes This additio must be performed placig the powers of 2 at cosecutive levels of the Hasse diagram of I Fially the edges coectig oe vertex of I with the other vertex of 2 + I are give by the set of 2 2 vertex pairs { u 2 2 + u 2 2 u 2 } I Fig 2 we illustrate the above iterative process for the first few values of deotig all the biary -tuples by their decimal equivalets Basically we first add to I its isomorphic copy 2 + I This additio must be performed by placig the powers of two 2 2 ad 2 at cosecutive levels i the itrisic order graph The reaso is simply that 2 2 2 sice matrix M2 2 2 has the patter 4 The we coect oe-to-oe the odes of the secod half of the first half to the odes of the first half of the secod half : A ice fractal property of I! 2 3 2 3 4 5 6 7 2 3 4 5 8 6 9 7 2 3 4 5 Fig 2 The Itrisic Order Graphs for 2 3 4 Each pair u v of vertices coected i I either by oe edge or by a loger path descedig from u to v meas that u is itrisically greater tha v ie u v O the cotrary each pair u v of o-coected vertices i I either by oe edge or by a loger descedig path meas that u ad v are icomparable by itrisic order ie u v ad v u Lookig at ay of the four graphs i Fig 2 we ca cofirm the fact that ad 2 are the maximum ad miimum elemets respectively i the poset I see Example 24 Also Theorems 2 ad 22 are illustrated by Fig 2 The edgeless graph for a give graph is obtaied by removig all its edges keepig its odes or vertices at the same positios [] I Fig 3 the edgeless itrisic order graph of I 5 is depicted For further theoretical properties ad practical applicatios of the itrisic order ad the itrisic order graph we refer the reader to eg [2] [3] [4] [5] [6] [7] [8] [9] [] [] Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 2 3 4 5 8 6 9 6 7 7 2 8 3 9 2 4 2 24 5 22 25 23 26 27 28 29 3 3 Fig 3 The Edgeless Itrisic Order Graph for 5 III INTRINSIC ORDER AND LEXICOGRAPHIC ORDER The lexicographic order betwee biary -tuples is the usual truth-table order o { } deoted here by the symbol l begiig with the -tuple ad fiishig with the -tuple As is well-kow this orderig coicides with the atural orderig betwee the decimal equivalets of the rows biary -tuples of the truth-table That is u l v u v 5 Example 3: Let 4 u v The u < l v sice u precedes v i the truth-table or sice u < 3 v The lexicographic order is a ecessary coditio for the itrisic order More precisely we have the followig corollary of Theorem 2; see [3] for the proof Corollary 3: For all ad for all u v { } ie u v u l v 6 u v u v 7 However the ecessary coditio for itrisic order stated by Corollary 3 is ot sufficiet That is u l v u v as the followig simple couter-example ideed the simplest oe that oe ca fid! shows Example 32: For 3 u 3 v 4 we have see the digraph of I 3 the third oe from left to right i Fig 2 u l v However 3 4 sice matrix M4 3 does ot satisfy IOC However for some special biary -tuples u { } the ecessary coditio stated by Corollary 3 is also sufficiet Let us characterize i a very simple way such biary strigs First we must set the followig otatio Defiitio 3: For all ad for ay give biary - tuple u C u C u respectively is the set of biary -tuples v itrisically less greater respectively tha or equal to v ie C u {v { } u v } C u {v { } u v } Equivaletly accordig to Defiitio 2 C u ad C u ca be defied as C u {v { } Pr {u} Pr {v} {p i } i C u {v { } Pr {u} Pr {v} {p i } i st 2} st 2} Defiitio 32: For all ad for ay give biary -tuple u L u L u respectively is the set of biary -tuples v whose decimal equivalets are greater less respectively tha or equal to the decimal equivalet of u ie L u { v { } u v } L u { v { } u v } Equivaletly accordig to 5 L u ad L u ca be defied as L u {v { } u l v } L u {v { } u l v } With this otatio the implicatios 6 or 7 ca be simply rewritte as C u L u 8 ad the questio of characterizig the biary -tuples u for which u v u v is equivalet to characterize the biary -tuples u for which C u L u The followig theorem provides the aswer to this questio see [5] for the proof Theorem 3: Let ad u u u { } The C u L u if ad oly if u does ot cotai ay bit followed by two or more bits placed at cosecutive or o cosecutive positios ie u has the geeral patter u }{{}}{{}}{{} p q p + q + r + }{{} r 9 where ay but ot all! of the above four subsets of bits grouped together ca be omitted Example 33: For 4 ad u we have see the digraph of I 4 the right-most oe i Fig 2 C { 2 3 4 5} L sice u has the patter 9 To establish the dual result of Theorem 3 we eed the followig defiitio Defiitio 33: Let The Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 i The complemetary -tuple of a give biary -tuple u u u { } is obtaied by chagig its s by s ad its s by s u c u u c u u Obviously two biary -tuples are complemetary if ad oly if their decimal equivalets sum up to 2 ii The complemetary set of a give subset S { } of biary -tuples is the set of the complemetary -tuples of all the -tuples of S S c {u c u S } Now iterchagig the roles of u ad v i 6 we get v u v l u Usig the otatio itroduced i Defiitios 3 ad 32 the implicatio ca be rewritte as the followig set iclusio dual of the iclusio 8 C u L u Sometimes for some biary -tuples u the iclusio becomes the set idetity C u L u These biary strigs u satisfyig this ice property are characterized by the followig theorem which is the dual of Theorem 3 because the s are chaged by s ad the s are chaged by s i the correspodig positioal criteria The proof is straightforward usig Theorem 3 ad the fact that see [5] for more details C u c C uc L u c L u c Theorem 32: Let ad u u u { } The C u L u if ad oly if u does ot cotai ay bit followed by two or more bits placed at cosecutive or o cosecutive positios ie u has the geeral patter u }{{}}{{}}{{} p q p + q + r + }{{} r 2 where ay but ot all! of the above four subsets of bits grouped together ca be omitted Example 34: For 4 ad u 5 we have see the digraph of I 4 the right-most oe i Fig 2 C 5 { 2 3 4 5} L 5 sice u 5 has the patter 2 IV INTRINSIC ORDER AND HAMMING WEIGHT The Hammig weight or simply the weight of u is the sum of all its bits I other words the Hammig weight of a biary -tuple is the umber of its -bits ad it will be deoted by w H u u i i Example 4: For 7 we have w H 4 The itrisic order respects the Hammig weight More precisely we have the followig corollary of Theorem 2 see eg [3] [9] for the proof Corollary 4: For all ad for all u v { } u v w H u w H v Now we preset some relatios betwee the itrisic orderig ad the Hammig weight Our startig poit is Corollary 4 This corollary has stated that a ecessary coditio for u beig itrisically greater tha or equal to v is that the weight of u must be less tha or equal to the weight of v That is let u be a arbitrary but fixed biary -tuple The u v w H u w H v for all v { } 3 or equivaletly w H u > w H v u v For istace lookig at the digraph I 4 the right-most oe i Fig 2 we ca cofirm that ad that 4 3 w H 4 < 3 w H 3 3 2 w H 3 2 w H 2 However the ecessary coditio for itrisic order stated by Corollary 4 is ot sufficiet That is w H u w H v u v as the followig simple couter-example ideed the simplest oe that oe ca fid! shows Example 42: For 3 u 4 v 3 we have see the digraph of I 3 the third oe from left to right i Fig 2 w H 4 < 2 w H 3 However 4 3 sice matrix M3 4 does ot satisfy IOC or more easily sice 4 > 3; see Corollary 3 Moreover eve assumig that the two ecessary coditios stated by Corollaries 3 & 4 simultaeously hold this does ot imply itrisic order That is u < v ad w H u w H v u v as the followig simple couter-example ideed the simplest oe that oe ca fid! shows Example 43: For 4 u 6 v 9 Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 we have see the digraph of I 4 the right-most oe i Fig 2 u 6 < v 9 ad w H 6 2 w H 9 However 6 9 sice matrix M9 6 does ot satisfy IOC Moreover eve though assumig that the Hammig weight of u is strictly less tha the Hammig weight of v the two ecessary coditios stated by Corollaries 3 & 4 do ot imply itrisic order That is u < v ad w H u < w H v u v as the followig simple couter-example ideed the simplest oe that oe ca fid! shows Example 44: For 5 u 2 v 9 we have u 2 < v 9 ad w H 2 2 < 3 w H 9 However 2 9 sice matrix M 2 9 does ot satisfy IOC I this cotext two dual questios aturally arise They are posed i the two subsectios of this sectio First we eed to set the followig otatio Defiitio 4: For every biary -tuple u { } H u H u respectively is the set of all biary -tuples v whose Hammig weights are less greater respectively tha or equal to the Hammig weight of u ie H u {v { } w H u w H v} H u {v { } w H u w H v} A Greater Weight ad Less Probability Lookig at the implicatio 3 the followig questio immediately arises Ca we characterize the biary -tuples u for which the ecessary coditio 3 is also sufficiet? That is we try to idetify those bitstrigs u { } for which the set of biary -tuples v with weights greater tha or equal to the oe of u coicides with the set of biary -tuples v with occurrece probabilities less tha or equal to the oe of u ie u v w H u w H v ie C u H u The followig theorem provides the aswer to this questio i a very simple way Theorem 4: Let ad u u u { } with Hammig weight w H u m m The C u H u if ad oly if either u is the zero -tuple m or the m -bits of u m > are placed at the m right-most positios ie if ad oly if u has the geeral patter u m m 2 m m 4 where ay but ot both! of the above two subsets of bits grouped together ca be omitted Proof: Sufficiet coditio We distiguish two cases: i If u is the zero -tuple the u is the maximum elemet for the itrisic order as we have proved i Example 24 The C {v { } v } { } {v { } w H w H v} H ii If u is ot the zero -tuple the u has the patter 4 with m > Let v H u ie let v let a biary -tuple with Hammig weight greater tha or equal to m the Hammig weight of u We distiguish two subcases: ii-a Suppose that the weight of v is w H v m w H u The v has exactly m -bits ad m -bits Call r the umber of -bits of v placed amog the m right-most positios max { 2m } r m Obviously v has r -bits ad m r -bits placed amog the m right-most positios ad also it has m r -bits ad 2m + r -bits placed amog the m left-most positios These are the positios of the r +m r+m r+ 2m + r m+ m bits of the biary -tuple v Hece matrix Mv u has exactly m r colums all placed amog the m right-most positios ad exactly m r colums all placed amog the m left-most positios Thus Mv u satisfies IOC ad the u v ie v C u So for this case ii-a we have proved that {v { } w H v w H u m} C u 5 ii-b Suppose that the weight of v is w H v m + p > m w H u < p m The defie a ew biary -tuple s as follows First select ay p -bits i v say for istace v i v ip Secod s is costructed by chagig these p -bits of v by -bits assigig to the remaider p bits of s the same values as the oes of v Formally s s s is defied by { if i {i i s i p } v i if i / {i i p } O oe had u s sice w H s w H v p m w H u ad the we ca apply case ii-a to s O the other had s v sice matrix Mv s has p colums placed at positios i i p while its p remider colums are either or Hece M s v has o colums so that it satisfies IOC Fially from the trasitive property of the itrisic order we derive u s ad s v u v ie v C u So for this case ii-b we have proved that {v { } w H v > w H u m} C u 6 Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 From 5 & 6 we get {v { } w H v w H u m} C u ie H u C u ad this set iclusio together with the coverse iclusio C u H u which is always satisfied for every biary -tuple u; see Corollary 4 leads to the set equality C u H u This proves the sufficiet coditio Necessary coditio Coversely suppose that ot all the m -bits of u are placed at the m right-most positios I other words suppose that u m m Sice by assumptio w H u m the simply usig the ecessary coditio we derive that m m u ad the so that m m H u C u H u C u This proves the ecessary coditio Corollary 42: Let ad let u m m 2 m m where ay but ot both! of the above two subsets of bits grouped together ca be omitted The the umber of biary -tuples itrisically less tha or equal to u is C u m + m + + + Proof: Usig Theorem 4 we have C u H u C u H u {v { } w H u m w H v} {v { } w H v m m + } + + + m m + as was to be show B Less Weight ad Greater Probability Iterchagig the roles of u & v 3 ca be rewritte as follows Let u be a arbitrary but fixed biary -tuple The v u w H v w H u for all v { } 7 Lookig at the implicatio 7 the followig dual questio of the oe posed i Sectio IV-A immediately arises Ca we characterize the biary -tuples u for which the ecessary coditio 7 is also sufficiet? That is we try to idetify those bitstrigs u { } for which the set of biary -tuples v with weights less tha or equal to the oe of u coicides with the set of biary -tuples v with occurrece probabilities greater tha or equal to the oe of u ie v u w H v w H u ie C u H u The followig theorem provides the aswer to this questio i a very simple way For a very short proof of this theorem we use Defiitio 33 Theorem 42: Let ad u u u { } with Hammig weight w H u m m The C u H u if ad oly if either u is the zero -tuple m or the m -bits of u m > are placed at the m left-most positios ie if ad oly if u has the geeral patter u m m 2 2 m m 8 where ay but ot both! of the above two subsets of bits grouped together ca be omitted Proof: Usig Theorem 4 ad the facts that see eg [5] [7] we get C u c C uc H u c H u c C u H u C u c H u c C uc H u c u c has the patter 4 u has the patter 8 as was to be show Corollary 43: Let ad let u m m 2 2 m m where ay but ot both! of the above two subsets of bits grouped together ca be omitted The the umber of biary -tuples itrisically greater tha or equal to u is C u + + + m Proof: Usig Corollary 42 we get C u C u c C uc + + + m m + + + + m as was to be show Example 45: Let 5 Accordig to Theorem 4 the 6 biary 5-tuples u 2 m m 5 for which C u H u are: 3 7 5 3 Note that obviously {2 2 m } m {2 2 m } m The accordig to Theorem 42 the 6 biary 5-tuples u 2 5 2 m m 5 for which C u H u are the complemetary oes of the above 5-tuples : 3 3 28 24 6 Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 V INTRINSIC ORDER AND VECTOR ORDER Let ad let u be a ozero biary -tuple with Hammig weight w H u m > The the vector of positios of s of u is defied as the vector of positios of its m -bits with the covetio that these positios are arraged i icreasig order from the right-most possible positio to the left-most possible positio This vector will be deoted by [i i 2 i m ] i < i 2 < < i m so that { i {i i 2 i m } iff u + i i / {i i 2 i m } iff u + i 9 We use agai the symbol to deote the coversio betwee the ew vector otatio ad the biary ad decimal represetatios of the bitstrigs eg 5 4 3 2 [i i 2 ] 5 [ 4] 5 9 A ew order relatio betwee biary -tuples with the same weight m the so-called vector order is itroduced i the followig defiitio Defiitio 5: Let 2 ad let u v be two biary - tuples with the same Hammig weight w H u w H v m < m < ad with vectors of positios of s u [i i 2 i m ] v [j j 2 j m ] The we say that u precedes or is equal to v i the vector order deoted by u v v if ad oly if either or i p j p p m q mi {p { 2 m} i p j p } i q < j q Example 5: For 5 ad m 3 we have [ 2 3] 5 < v [ 3 5] 5 < v [3 4 5] 5 ie < v < v ie 7 < v 2 < v < v 28 The followig theorem provides us with a simple characterizatio of the itrisic order betwee two biary -tuples with the same weight usig their vectors of positios of s istead of their biary represetatios used i Theorem 2 IOC Theorem 5: Let 2 ad let u v be two biary - tuples with the same Hammig weight w H u w H v m < m < ad with vectors of positios of s The u [i i 2 i m ] v [j j 2 j m ] u v j p i p for all p 2 m Proof: Usig Defiitio 2 ad Remark 24 we have that u v iff matrix Mv u satisfies IOC iff either u has o -bits or for each -bit i u there exists at least oe correspodig -bit i v placed at the same or at a previous positio Now accordig to 9 sweepig the m -bits of u from left to right the last assertio is equivalet to sayig that j m i m j m i m j i ie j p i p for all p 2 m The followig corollary establishes the relatioship betwee the itrisic order ad the vector order Corollary 5: Let 2 ad let u v be two biary - tuples with the same Hammig weight The w H u w H v m < m < Proof: Let u [i i 2 i m ] u v u v v v [j j 2 j m ] be the vectors of positios of s of u ad v respectively The usig Theorem 5 we have u v j p i p for all p 2 m 2 We distiguish the followig two cases i If j p i p for all p 2 m the u v so that clearly u v v ii If j p i p for some p 2 m the the least idex q for which i q j q ecessarily satisfies i q < j q due to 2 Hece usig Defiitio 5 we get u v v The coverse of Corollary 5 does ot hold as the followig simple couter-example ideed the simplest oe that oe ca fid! shows Example 52: For 4 m 2 ad for u [i i 2 ] 4 [ 4] 4 9 v [j j 2 ] 4 [2 3] 4 6 usig Defiitio 5 we have u v v sice i < 2 j ad usig Corollary 3 we have see the digraph of I 4 the right-most oe i Fig 2 u v sice u 9 > 6 v VI CONCLUSION I this paper we have established three differet ecessary coditios for the itrisic order The first oe has ivolved the lexicographic truth-table order i the set { } of all biary -tuples The secod oe deals with the Hammig weight of the biary strigs The third oe has bee expressed i terms of the vector order defied betwee biary - tuples with the same weight We have provided differet simple couter-examples for provig that the above ecessary coditios for the itrisic orderig are ot sufficiet i geeral Moreover for the first two cases we have also established the geeral patters of the special biary - tuples u for which the correspodig ecessary coditios are also sufficiet ie they become equivaleces These patters have bee expressed by simple positioal criteria of the s & s i the correspodig biary -tuples Advace olie publicatio: 2 November 22

IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 REFERENCES [] R Diestel Graph Theory 3rd ed New York: Spriger 25 [2] L Gozález A New Method for Orderig Biary States Probabilities i Reliability ad Risk Aalysis Lecture Notes i Computer Sciece vol 2329 o pp 37-46 22 [3] L Gozález N-tuples of s ad s: Necessary ad Sufficiet Coditios for Itrisic Order Lecture Notes i Computer Sciece vol 2667 o pp 937-946 23 [4] L Gozález A Picture for Complex Stochastic Boolea Systems: The Itrisic Order Graph Lecture Notes i Computer Sciece vol 3993 o 3 pp 35-32 26 [5] L Gozález Algorithm Comparig Biary Strig Probabilities i Complex Stochastic Boolea Systems Usig Itrisic Order Graph Advaces i Complex Systems vol o Suppl pp -43 27 [6] L Gozález Rakig Itervals i Complex Stochastic Boolea Systems Usig Itrisic Orderig i Machie Learig ad Systems Egieerig Lecture Notes i Electrical Egieerig vol 68 B B Rieger M A Amouzegar ad S-I Ao Eds New York: Spriger 2 pp 3974 [7] L Gozález Duality i Complex Stochastic Boolea Systems i Electrical Egieerig ad Itelliget Systems Lecture Notes i Electrical Egieerig vol 3 S-I Ao ad L Gelma Eds New York: Spriger 22 pp 527 [8] L Gozález Edges Chais Shadows Neighbors ad Subgraphs i the Itrisic Order Graph IAENG Iteratioal Joural of Applied Mathematics vol 42 o pp 66-73 Feb 22 [9] L Gozález Itrisic Order ad Hammig Weight Lecture Notes i Egieerig ad Computer Sciece: Proceedigs of the World Cogress o Egieerig 22 WCE 22 4-6 July 22 Lodo UK pp 783-788 [] L Gozález Itrisic Orderig Combiatorial Numbers ad Reliability Egieerig Applied Mathematical Modellig to be published [] L Gozález D García ad B Galvá A Itrisic Order Criterio to Evaluate Large Complex Fault Trees IEEE Trasactios o Reliability vol 53 o 3 pp 297-35 24 [2] R P Staley Eumerative Combiatorics vol Cambridge MA: Cambridge Uiversity Press 997 Advace olie publicatio: 2 November 22