Encapsulation theory: the transformation equations of absolute information hiding.

Transcription

1 1 Encapsulation theoy: the tansfomation equations of absolute infomation hiding. Edmund Kiwan * Abstact This pape descibes how the potential coupling of a set vaies as the set is tansfomed, that is, as elements ae added and emoved. The equations govening the changes caused by these tansfomations ae deived and biefly analysed. Keywods Encapsulation theoy, encapsulation, potential coupling, tansfomation equation. 1. Intoduction The potential coupling of a set was intoduced in [1], which deived the equations fo the potential coupling of any given set of absolute infomation hiding. These equations, howeve, wee static, offeing no insight into the evolution of a set ove time. This pape addesses this evolutionay aspect by deiving the equations which descibe not the oveall potential coupling of a set but the changes in potential coupling as a set undegoes an abitay seies of tansfomations. This pape consides sets of absolute infomation hiding only. 2. Standad deviation Befoe examining the tansfomation equations themselves, let us pefom some expeiments whose esults we shall compae with those we might intuitively expect. Poposition 1.11 in [1] showed that, given two othewise equivalent sets, both of which unifomly distibuted in violational elements, the set whose infomation hidden elements ae non unifomly distibuted ove disjoint pimay sets can neve have a potential coupling of less than that of the set with unifomly distibuted infomation hidden elements. This may be undestood qualitatively by consideing the intenal potential coupling of a disjoint pimay set which, as also shown in [1], was shown to be popotional to the squae of the numbe of elements in that disjoint pimay set. Thus conside a set of unifomly distibuted elements whee each disjoint pimay set has 10 elements; each disjoint pimay set will have an intenal potential coupling of 90 (= ). An element moved fom one disjoint pimay set to anothe will (in a sense we shall late define pecisely) incease the, "Non evenness," of the distibution: now one disjoint pimay set will have 11 * Edmund Kiwan Revision 1.10 Novembe 14 th (Oiginal evision 1.0 witten Januay 12 th 2009.) axiv.og is ganted a non exclusive and ievocable licence to distibute this aticle; all othe entities may epublish, but not fo pofit, all o pat of this mateial povided efeence is made to the autho and the title of this pape. The latest vesion of this pape is available at [2].

2 elements and an intenal potential coupling of 110 (= ), wheeas the dono disjoint pimay set will have an intenal potential coupling of 72 (=9 2 9): moving this element has caused an oveall intenal potential coupling incease of 2. Loosely speaking, being popotional to the squae of the numbe of elements in a disjoint pimay set, the intenal potential coupling tends to amplify deviations fom unifom distibution, so the moe nonunifomly distibuted a set is, the geate its potential coupling. Can we investigate this elationship moe fomally? Can we igoously measue this, "Unevenness?" Indeed we can, by using a tool of the statistician: the standad deviation. The standad deviation measues how widely spead the values in a dataset ae. We shall use it fist to measue how widely spead the numbe of infomation hidden elements pe disjoint pimay set is, that is, to measue the hidden element distibution. If we take a set of disjoint pimay sets whee x i is the numbe of hidden elements in the i th disjoint pimay set and whee x is the aveage numbe of hidden elements pe disjoint pimay set, then the standad deviation is defined by the equation: = 1 2 x i x i=1 The standad deviation of the hidden element distibution fo a unifomly distibuted set is 0; this figue then ises as the set becomes inceasingly non unifomly distibuted. Instead of examining how the potential coupling behaves as the standad deviation of the hidden element distibution inceases, howeve, it is useful to instead examine how the isoledensal configuation efficiency (also defined in [1]) behaves, as the configuation efficiency, being defined between 0 and 1, helps to nomalise the tend fo sets of diffeent cadinalities. Thus, wheeas we expect that the potential coupling of a set will ise as the standad deviation of its hidden element distibution inceases, we expect the configuation efficiency of that set to fall as its standad deviation inceases. Finally, we need only state the actual means of inceasing the non unifomity of a set's distibution. We shall begin not with a pefectly unifomly distibuted set but with a set of, say, 100 disjoint pimay sets, each disjoint pimay set having one violational element, and a andom numbe between 0 and 30 of infomation hidden elements. Being non unifomly distibuted the set will have a non zeo standad deviation of hidden element distibution. We shall then take one hidden element fom a disjoint pimay set and move it to an abitaily designated taget disjoint pimay set. We shall ecod the change in the hidden element distibution standad deviation and its esulting change in configuation efficiency. We shall then move a second hidden element fom a disjoint pimay set into the taget disjoint pimay set and pefom the measuement again. This shall be epeated until all the hidden elements of the set ae in the taget disjoint pimay set, thus maximising the hidden element distibution standad deviation. Figue 1 shows the esulting configuation efficiency plotted as a function of the changing hidden element distibution standad deviation. 2

3 3 Figue 1: Isoledensal configuation efficiency as a function of inceasing standad deviation of the hidden element distibution. This figue shows the expected esult: the isoledensal configuation efficiency falls as the hidden element distibution standad deviation inceases, i.e., as the set becomes inceasingly non unifomly distibuted in hidden elements. To show a slightly boade example of this tend, Figue 2 shows a futhe ten andomly geneated sets subjected to the same expeiment. Figue 2: Isoledensal configuation efficiency as a function of inceasing standad deviation, multiple sets. We shall now examine the five tansfomation equations and attempt to confim the above esults in tems of the appopiate equation. 3. The tansfomation equations Fom the point of view of investigating changes in potential coupling thee ae only two fundamental

4 tansfomations we can make to any set: we can add a given numbe of violational elements to a disjoint pimay set and we can add a given numbe of infomation hidden elements to a disjoint pimay set. Let us denote this, "Given numbe," m. It may seem that we could also tansfom a set by adding a disjoint pimay set itself, but in encapsulation theoy dependencies can only be fomed between elements, not disjoint pimay sets, and so adding any numbe of empty disjoint pimay sets cannot change the potential coupling of that set. Of couse, by definition, no element can exist outside a disjoint pimay set, so these tansfomations theefoe pesume, whee necessay, the existence of an empty disjoint pimay set into which new elements may be intoduced. Although only two tansfomations ae fundamental, it is possible to deive a futhe thee tansfomations fom these two fundamental tansfomations. These thee deived tansfomations both cove common changes to sets and yield insight into the natue of the changing potential coupling The thee deived tansfomations ae: moving m violational elements fom one disjoint pimay set to anothe, moving m infomation hidden elements fom one disjoint pimay set to anothe, and conveting m infomationhidden elements in a disjoint pimay set into m violational elements. We note that m may be negative and we establish the convention that adding a negative numbe of elements to a disjoint pimay set may be intepeted as emoving m elements fom that disjoint pimay set. Whee m is negative, it may not exceed the numbe of elements that actually eside within a disjoint pimay set: no disjoint pimay set may contain a negative numbe of elements at any time. Befoe poceeding, we ecall the definitions of the tems fom [1]: G : a set as defined in [D1.1] [D1.5] of [1]. V : The numbe of violational elements in set G. K x, K s and K t : disjoint pimay sets in set G. K x is used when only one disjoint pimay set is involved. Tanslation tansfomations involve two disjoint pimay sets: K s is the souce disjoint pimay set fom which elements ae moved, K t is the taget disjoint pimay set to which elements ae moved. K x : The total numbe of elements in disjoint pimay set K x. v K x : The numbe of violational elements in disjoint pimay set K x. n : the total numbe of elements in a set. Finally, we intoduce two functions. The function s G will epesent the change of potential coupling of set G due to the application of some tansfomation. The function h(k x ) will epesent the numbe of infomation hidden elements in disjoint pimay set K x. Box 1 lists the five tansfomation equations. 4

5 5 1. s G =m nk x V v K x m 1 The infomation hiding violation tansfomation equation (see poposition 3.11), which gives the change of potential coupling of a set G when m violational elements ae added to disjoint pimay set K x. 2. s G =mv v K x 2 K x m 1 The infomation hidden tansfomation equation (see poposition 3.17), which gives the change of potential coupling of a set G when m infomationhidden elements ae added to disjoint pimay set K x. 3. s cumulative G=mh K t h K s The infomation hiding violation tanslation tansfomation equation (see poposition 3.18), which gives the change of potential coupling of a set G when m violational elements ae moved fom souce disjoint pimay set K s to a diffeent taget disjoint pimay set K t. 4. s cumulative G=m 2 K t 2 K s v K s v K t 2m The infomation hidden tanslation tansfomation equation (see poposition 3.19), which gives the change of potential coupling of a set G when m infomation hidden elements ae moved fom souce disjoint pimay set K s to a diffeent taget disjoint pimay set K t. 5. s cumulative G=m n K x The convesion tansfomation equation (see poposition 3.20), which gives the change of potential coupling when m infomation hidden elements ae conveted into violational elements. Box 1: The five tansfomation equations. 4. Reflections on the equations 4.1 The non consevative tansfomation equations Conside the fist two tansfomation equations. These ae the fundamental equations and they ae non consevative in that they change the total numbe of elements in the set; the othe thee equations ae consevative in that they do not change the total numbe of elements in the set. Pehaps the most inteesting aspect of the two non consevative tansfomation equations is that it is tivial to show (by subtacting the second fom the fist) that adding a violational element to a set causes a lage incease in potential coupling than adding a hidden element, as we intuitively expect. 4.2 The tanslation tansfomation equations The thid and fouth equations ae deived fom the fist two. These equations ae tanslation equations in that they show how potential coupling changes as elements ae tanslated o moved fom one disjoint pimay set to anothe. We shall examine them in evese ode The fouth equation Conside the fouth tansfomation equation, descibing the change of potential coupling as infomation hidden elements ae moved between disjoint pimay sets. This is the equation govening the changes that we found in section 2, whee all the hidden elements of a set wee incementally tanslated fom thei oiginal disjoint pimay sets to a specific taget disjoint pimay set, theeby maximising the standad deviation of the hidden element distibution. If we look at the tems of the fouth equation, we see that thee ae thee components of the potential

6 coupling change (ignoing the common scaling m facto): (i) (ii) (iii) 2 K t 2 K s v K s v K t 2m The 2m component is clealy independent of the disjoint pimay sets affected by the tansfomation. Component (i) is the diffeence in the total numbe of elements (multiplied by two) between the souce and taget disjoint pimay sets. Component (ii) is the diffeence in the numbe of violational elements between the souce and taget disjoint pimay sets, though in the opposite sense of component (i) in that component (i) is taget minus souce but component (ii) is souce minus taget. The inteaction between these two components is complicated, but in ou expeiment in section 2, we moved moe and moe hidden elements into a single, taget disjoint pimay set, causing the taget disjoint pimay set to become inceasingly lage while its violational elements emained unchanged: thus component (i) gew to dominate component (ii) and epeated tanslations inceasingly added to the potential coupling of the set. Inceasing the potential coupling of a fixed numbe of elements by definition deceases the set's configuation efficiency and this is pecisely figues 1 and 2 show. The evese is also tue: moving infomation hidden elements fom a lage to a smalle disjoint pimay set must necessaily decease the potential coupling of a set and thus incease the configuation efficiency. This explains why a set of unifomly distibuted hidden elements cannot have a potential coupling geate than one of non unifomly distibuted hidden elements (all else being equal) The thid equation Wheeas the fouth tansfomation moved infomation hidden elements between disjoint pimay sets, the thid tansfomation moves violational elements between disjoint pimay sets. We might expect this equation to yield esults quite simila to the fouth equation given that they ae both tanslation tansfomations, but this is not the case. To investigate this cuious diffeence we shall pefom anothe expeiment. In section 2 we plotted the falling configuation efficiency of a set as its hidden elements wee inceasingly piled into just one disjoint pimay set. Let us pefom a simila expeiment but this time we shall incementally move only the violational elements into one disjoint pimay set. (A mino diffeence in pocedue must be obseved: evey disjoint pimay set must contain at least one violational element as othewise it is uncontactable by elements in othe disjoint pimay sets, so instead of moving all violational elements fom the souce disjoint pimay sets, we shall move all but one violational element. This diffeence in itself should not significantly alte the outcome.) Let us again take a set of 100 disjoint pimay sets. In section 2 we put one violational element in each disjoint pimay set and put a andom numbe between 0 and 30 of infomation hidden elements in each disjoint pimay set. Fo ou new expeiment we shall do the opposite, putting one infomation hidden element in each disjoint pimay set and putting a andom numbe between 1 and 30 of violational elements in each disjoint pimay set. In section 2, it was the standad deviation of the hidden element distibution that we measued; this time we shall measue the standad deviation of the violational element distibution: the numbe of violational elements pe disjoint pimay set. Incementally moving one violational element into one taget disjoint pimay set will epeatedly incease the standad deviation of the violational element distibution of the entie set. The question is: how will the configuation efficiency change as the standad deviation of the violational element distibution inceases? The esult is shown in figue 3.

7 7 Figue 3: Isoledensal configuation efficiency as a function of inceasing standad deviation of the violational element distibution. Figue 3 eveals two aspects that call fo explanation but only one is elevant hee. Fistly, the low the configuation efficiency of this set may be stiking but this is simply due to the way we ceated the set. The set was ceated with just one hidden element in each disjoint pimay set and up to 30 violational elements in each disjoint pimay set; the set is theefoe composed ovewhelmingly of violational elements and as such exploits little encapsulation: hence the low configuation efficiency. Moe elevant to ou expeiment and pehaps moe supising is the esult that in ou sample set the configuation efficiency is independent of violational distibution standad deviation; in othe wods, the potential coupling of the set is unchanged by moving all the possible violational elements into one disjoint pimay set. Why is this esult so diffeent fom the fist expeiment with infomation hidden elements? The answe lies in the thid equation. Let us again beak the equation into its component pats; we see it is just the simple diffeence between two tems (ignoing the common scaling m facto) : (i) h K t h K s This is much easie to intepet than equation fou. Component (i) is just the diffeence between the numbe of hidden elements of the taget and souce disjoint pimay sets. In ou expeiment all the disjoint pimay sets have the same numbe of hidden elements, thus the diffeence between the numbe of hidden elements in any two disjoint pimay sets is zeo. So it doesn't matte how many violational elements ae moved between disjoint pimay sets: these tanslations cannot change the potential coupling and cannot change the configuation efficiency of the set. This explains the unchanging configuation efficiency of figue Non unifomity In both expeiments pefomed so fa, one set of elements was unifomly distibuted: in the fist expeiment, each disjoint pimay set had just one violational element wheeas the hidden elements wee non unifomly distibuted; in the second expeiment, each disjoint pimay set had one hidden element wheeas the violational elements wee non unifomly distibuted. To model moe, "Real wold," poblems, we must examine sets whose hidden elements and violational elements ae both non unifomly distibuted.

8 Let us e visit the fist expeiment and look at the tanslation of hidden elements in a set again of 100 disjoint pimay sets with each disjoint pimay set having a andom numbe between 0 and 30 of hidden elements and a andom numbe between 1 and 30 of violational elements. Befoe we do so, howeve, we shall attempt to pedict the esults by examining the tanslation tansfomation equation fo hidden elements, the fouth equation in box 1. As we noted befoe, the dominant component of tanslation tansfomation equation fo hidden elements is simply the total numbe of elements in the taget minus the total numbe of elements in the souce. As we ae choosing at andom the taget disjoint pimay set into which all the hidden elements will be moved, then this disjoint pimay set will initially have between 0 and 60 elements in total (thee will be at most 30 hidden and at most 30 violational elements). It is theefoe conceivable that the fist souce disjoint pimay set chosen fo a tanslation will have moe elements than ou taget disjoint pimay set, and so the total numbe of elements in the taget disjoint pimay set minus the total numbe of elements in the souce disjoint pimay set will be negative; this negative change in potential coupling implies that the configuation efficiency of the set would initially ise. As moe tanslations ae pefomed, howeve, we should quickly each a situation, in a andomly distibuted set, whee the numbe of hidden elements in the taget disjoint pimay set becomes geate than the numbe of elements in any othe single disjoint pimay set; this will cetainly be the case when the taget disjoint pimay set contains 61 elements and will pobably be the case much soone. Afte this point, all the hidden element tanslations will incease the potential coupling and incease the standad deviation of the hidden element distibution. This will yield a pictue vey simila to that aleady shown in figue 1: the configuation efficiency will fall as the hidden element distibution standad deviation inceases. The esults ae shown in figue 4. 8

9 9 Figue 4: Isoledensal configuation efficiency as a function of inceasing standad deviation of the hidden element distibution fo a set with non unifomly distibuted hidden and violational elements. Thee ae thee aspects of figue 4 that equie explanation but only two ae elevant to ou expeiment. Compaed with figue 1, the initial configuation efficiency of the set in figue 4 is athe low, but this is due to the set's now containing multiple hidden and multiple violational elements andomly distibuted: any such set with a non tivial numbe of disjoint pimay sets will usually have a configuation efficiency of aound 0.5. (Recall that the set in figue 1 had only one violational element pe disjoint pimay set: it was extemely well encapsulated and hence its configuation efficiency was much highe than ou latest set.) The second and most inteesting aspect of figue 4 is that, as suspected, the configuation efficiency of a set of non unifomly distibuted hidden and violational elements falls with inceasing hidden element distibution standad deviation, just as was the case with the non unifomly distibuted hidden elements and unifomly distibuted violational elements of the fist expeiment. Thee only emains to be explained why the teminal configuation efficiency of the set in figue 4 is not as low as that in figue 1. The explanation comes again fom the fouth equation. The lagest change of potential coupling occus when the diffeence in total numbe of elements between the souce and taget disjoint pimay sets is maximised. In ou fist expeiment, all the disjoint pimay sets contained only one violational element but in this latest expeiment thee wee always a andom numbe of violational elements left behind when the hidden elements wee extacted, thus the diffeences in total numbe of elements between taget and souce disjoint pimay sets wee not as geat as those in the fist expeiment which in tun causes the potential coupling to ise by a lesse amount than in the fist expeiment. This diectly tanslates to the configuation efficiency's not falling as fa in ou latest expeiment. As befoe, meely to demonstate the tend, figue 5 shows ten andomly geneated sets, each constained as was the set in figue 4, subjected to epeated hidden tanslation tansfomations.

10 10 Figue 5: Isoledensal configuation efficiency as a function of inceasing standad deviation of the hidden element distibution fo a set with non unifomly distibuted hidden and violational elements, multiple sets. Let us also e visit the second expeiment and look at the tanslation of violational elements in a set again of 100 disjoint pimay sets with each disjoint pimay set having a andom numbe between 0 and 30 of hidden elements and a andom numbe between 1 and 30 of violational elements. Befoe we do so, howeve, we shall attempt to pedict the esults by examining the tanslation tansfomation equation fo violational elements, the thid equation in box 1. As we saw, the equation showed us that the change in potential coupling geneated by the tanslation depends only on the diffeence between the numbes of hidden elements in the taget and souce disjoint pimay sets. In ou pevious violation tanslation expeiment, that diffeence was zeo, and so moving all the violational elements to one taget disjoint pimay set had no effect on the set's potential coupling o configuation efficiency. In ou andomly geneated set, howeve, the diffeence will usually be non zeo and so thee will usually be a change of potential coupling It will not, howeve, esemble the potential coupling change caused by hidden element tanslations. In hidden element tanslations, the change in potential coupling was popotional to the diffeence of the numbe of elements in the taget and souce elements, and this change gew inceasingly lage as the taget element gew inceasingly lage. The vey act of tanslating a hidden element to the taget disjoint pimay set inceased the change in potential coupling caused by tanslating all subsequent hidden elements to the taget disjoint pimay set. The thid equation exhibits vey diffeent behaviou. The change in potential coupling is popotional to the diffeence in the numbe of hidden elements only and moving violational elements does not change the numbe of hidden elements in souce o taget disjoint pimay set. Thus epeated violational element tanslations will not geneate potential coupling changes popotional to the inceasing size of the taget disjoint pimay set: the change in potential coupling is fixed by the choice of souce and taget disjoint pimay sets. Also, suppose the taget disjoint pimay set has 15 hidden elements; then tanslating violational elements fom a souce disjoint pimay set with fewe than 15 hidden elements will cause an incease of potential coupling and tanslating violational elements fom a souce disjoint pimay set with moe than 15 elements will cause a decease of potential coupling So unlike the hidden element tanslations, which afte an initial time wee guaanteed to only incease potential coupling, violational element tanslations can lead to small inceases of potential coupling which can then be offset by subsequent small deceases of potential

11 coupling 1 In othe wods, although the inceasing violational distibution standad deviation will usually change the configuation efficiency, we do not expect configuation efficiency to change by anything like as much as was caused by the hidden element tanslations: the configuation efficiency should be quite insensitive to inceasing violational distibution standad deviation. The esults of epeated violational element tanslations of a set non unifomly distibuted in both hidden and violational elements is shown in figue Figue 6: Isoledensal configuation efficiency as a function of inceasing standad deviation of the violational element distibution fo a set with non unifomly distibuted hidden and violational elements. As figue 6 indeed shows, configuation efficiency does change with inceasing violational distibution standad deviation, but only negligibly. Figue 7 shows the esults fo multiple andomlygeneated sets. Figue 7: Isoledensal configuation efficiency as a function of inceasing standad deviation of the 1 This justifies the notion of the inteface epositoy in compute pogamming, a subsystem holding only public intefaces that act as facades to vaious othe subsystems of hidden implementations.

12 violational element distibution fo a set with non unifomly distibuted hidden and violational elements, multiple sets The fifth equation The fifth equation in box 1 is the convesion tansfomation equation, which gives the change of potential coupling when m infomation hidden elements ae conveted into violational elements. The equation simply states that when a hidden element in a disjoint pimay set is conveted into a violational element in that same disjoint pimay set, then the potential coupling must ise. This just confims ou expectation that inceasing the access to an element within a disjoint pimay set must aise the potential coupling of a set. The evese is also tue: the change of potential coupling in conveting a violational element to a hidden element can be calculated by changing the sign of the m, which then shows that such a convesion must educe the potential coupling of a set. 5. Conclusions Systems evolve. To contol this evolution means be able to deteministically pedict the affects of changes befoe those changes occu. Fo systems that can be modelled by sets, this pimaily means pedicting the potential coupling of the set befoe the changes occu. This pape poposed that the evolution of a set may be modelled as an abitaily complex seies of tansfomations that may be applied to that set. The two fundamental tansfomations wee then established that descibe all changes to a set and the two potential coupling equations coesponding to those tansfomations wee poposed. Thee futhe equations wee deived fom these fundamental equations to descibe the moe common changes that sets undego. 6. Related wok 7. Appendix A 7.1. Definitions Note that in these definitions the symbol means, "Maps to." [D3.1] Given pimay set Q with disjoint pimay set K in set G as defined in [D1.1] [D1.5] of [1] and the tansfomation T, the change of intenal potential coupling Q effected by applying T to Q is given by equation: Q= T Q Q The change of intenal potential coupling G effected by applying T to G is given by equation: G = T G G [D3.2] Given pimay set Q with disjoint pimay set K in set G as defined in [D1.1] [D1.5] of [1] and the tansfomation T, the change of extenal potential coupling Q effected by applying T to Q is given by equation: Q= T Q Q The change of extenal potential coupling G effected by applying T to G is given by

13 equation: G= T G G [D3.3] Given pimay set Q with disjoint pimay set K in set G as defined in [D1.1] [D1.5] of [1] and the tansfomation T, the change of potential coupling s Q effected by applying T to Q is given by equation: s Q= s T Q s Q The change of potential coupling s G effected by applying T to G is given by equation: s G = s T G s G [D3.4] Given the set G as defined in [D1.1] [D1.5] of [1] with an i th disjoint pimay set K i of K i elements and extenal infomation hiding violation of v(k i ), let T p be the violational tansfomation * * which maps K i onto K i whee K i diffes fom K i by m violational elements whee m K i o: T p K i,m ={K i G : v K i v K i m} Whee T p is applied to just the x th disjoint pimay set K x of G, the tansfomation becomes: And: T p x,g, m ={G T p G : v K i v K i m i=x ; v K i v K i i x} T p x,g, m ={G T p G :V T p V } Note that as m may be positive o negative, K i * may have moe o fewe violational elements than K i. [D3.5] Given the set G as defined in [D1.1] [D1.5] of [1] with an i th disjoint pimay set K i of K i elements and extenal infomation hiding violation of v(k i ), let T z be the hidden tansfomation * * which maps K i onto K i whee K i diffes fom K i by m infomation hidden elements whee m K i and whee the violational elements emain unchanged, that is: T z K i, m ={K i G : K i K i m ; v K i v K i } Whee T z is applied to just the x th disjoint pimay set K x of G, the tansfomation becomes: T z x, G, m ={G T z G : K i K i m i=x ; K i K i i x ; v K i v K i i} Note that as m may be positive o negative, K i * may have moe o fewe infomation hidden elements than K i. [D3.6] The infomation hiding function of set G as defined in [D1.1] [D1.5] of [1], witten h(g), is the function that maps the infomation hidden elements of that set to thei own set. The infomation hiding of set G is the cadinality of its infomation hiding function, h G. The infomation hiding of disjoint pimay set K is the cadinality of its infomation hiding function, h K. The infomation hiding of G divided by the numbe of disjoint pimay sets is called the specific infomation hiding of G. Given that a set consists entiely of infomation hidden and violational elements, then by definition: (i) (ii) G = h Gp G K = h K p K 7.2. Popositions The popositions ae oganised as follows. Popositions establish some geneal esults concening the sum of changes of potential coupling and the application of tansfomations to sets. 13

14 Popositions establish the fundamental tansfomation equation fo the application of the violational tansfomation to a set. Popositions establish the fundamental tansfomation equation fo the application of the hidden tansfomation to a set. Popositions establish the thee deived tansfomation equations fo the tanslation and convesion tansfomations. All popositions elate to sets of absolute infomation hiding only. 14 Poposition 3.1. Given disjoint pimay set G and the tansfomation T, the change of potential coupling effected by applying T to G is given by: s G = s in G s ex G By poposition in [1]: s G= s i n Gs ex G (i) Let K * = T(K). Theefoe: s G s G * = s i n G * s ex G * (ii) Also be definition: s G = s T G s G (iii) Substituting (i) and (ii) into (iii) gives: s G = s G * s G = G * s ex G * G G = G * Gs ex G * G = G s ex G Poposition 3.2. Given the disjoint pimay set K and the violational tansfomation T p defined in [D3.4], then the numbe of elements in T p (K) diffes fom the numbe of elements in K by m, o: T h K, m= K m Let K contain a infomation hidden elements and definition: K =av K (i) v K violational elements. Thus, by Let K * =T p (K,m). By definition T p leaves the numbe of infomation hidden elements unchanged, theefoe: K * =av K * (ii)

15 15 Also by definition of T h : Substituting (iii) into (ii) gives: Substituting (i) into (iv) gives: v K * = v K m (iii) K * =av K m (iv) T h K, m= K * = K m Poposition 3.3. Given the set G as defined in [D1.1] [D1.5] of [1] with an i th disjoint pimay set K i and given that a paticula x th disjoint pimay set K x only is subject to the violational tansfomation T p defined in [D3.4], the numbe of violational elements in T p (G) is given by: By poposition in [1]: v T h x,g, m = V m = V = i=1 v K i v K i v K x (i) Let G * =T p (x,g,m) and let K i* =T p (K i,m). By definition: By the definition of T h : Substituting (iii) into (ii) gives: Also by the definition of T h : Substituting (v) into (iv) gives: Substituting (i) into (vi) gives: = V * = i=1 v K i * v K i * v K x * (ii) i x: v K i v K i (iii) V * = v K i v K x * (iv) i=x: v K i v K i m (v) V * = v K i v K x m (vi)

16 16 v T h x,g, m = V * = V m Poposition 3.4. Given the disjoint pimay set K and the hidden element tansfomation T z defined in [D3.5], then the numbe of elements in T z (K) diffes fom the numbe of elements in K by m, o: T z K, m= K m Let K contain a infomation hidden elements and definition: K =av K (i) v K violational elements. Thus, by Let K * =T z (K,m). By definition T z leaves the numbe of violational elements unchanged, theefoe: Also by definition of T z : Substituting (iii) into (ii) gives: Substituting (i) into (iv) gives: Poposition 3.5. K * =a * v K (ii) a * =a m (iii) K * =a mv K (iv) T z K, m= K * = K m Given the set G as defined in [D1.1] [D1.5] of [1] with an i th disjoint pimay set K i and given that a paticula x th disjoint pimay set K x only is subject to the hidden element tansfomation T z defined in [D3.5], the numbe of violational elements in T z (G) is given by: v T z x,g, m = V Let G * =T z (x,g,m) and let K i* =T z (K i,m). By definition: By the definition of T z : Substituting (ii) into (i) gives: V * = i=1 v K i * (i) i : v K i * = v K i (ii) v T z x,g, m = v K i = V i=1

17 Poposition 3.6. Given the set G as defined in [D1.1] [D1.5] of [1] with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements and intenal potential coupling s in (Q i ), the change of intenal potential coupling when the numbe of violational elements in K i changes by m whee m K i is given by: =2m K i m 2 m Let T p be the violational tansfomation defined in [D3.4] and let K i* =T p (K i,m). By definition [D3.1], the change of intenal potential coupling effected by applying T p to Q i is: 17 By poposition 1.2: = * (i) * = K i* K i* 1 (ii) By poposition 3.2: K i* = K i m (iii) Substituting (iii) into (ii) gives: Q * i =K i m K i m 1 2 = K i m K i K i m K i m 2 m = K i K i 1 2m K i m 2 m (iv) But: K i = K i K i 1 (v) Substituting (v) into (iv) gives: K * i = K i 2m K i m 2 m K * i K i =2m K i m 2 m And theefoe by (i): K i =2m K i m 2 m Poposition 3.7. Given the set G as defined in [D1.1] [D1.5] of [1] with an i th pimay set Q i and an i th disjoint pimay set K i of intenal potential coupling s in (Q i ), the change of intenal potential coupling of the entie set G when the numbe of violational elements in a paticula x th disjoint pimay set K x changes by m whee m K x is equal to the change of intenal potential coupling of K x, o: G= s in By poposition of [1], the intenal potential coupling of G is the sum of the intenal potential

18 18 coupling of all its disjoint pimay sets: = s i n G= s i n i=1 s in (i) Let T p be the violational tansfomation defined in [D3.4] and let K i* =T p (K i,m). Futhemoe, let T p apply to the x th disjoint pimay set only such that K x* =T p (K x,m) and G * =T p (x,g,m). By definition: s i n G * = = i =1 * * s in * (ii) By definition [D3.3], the change of intenal potential coupling of G effected by applying tansfomation T p to G is given by: Substituting (i) and (ii) into (iii) gives: G= i=1 x G= G * G (iii) * s in * i=1 x (iv) But as T p is only applied to K x then all disjoint pimay sets except K x ae unchanged, o: And theefoe: Substituting (v) into (iv) gives: G= i=1 x K i * =K i i x K i * = K i i x (v) s in * i=1 x = K x * K x (vi) By definition [D3.1], the change of intenal potential coupling effected by applying T p to K x is then: Substituting (vii) into (vi) gives: K x = * (vii) G= s in Poposition 3.8. Given the set G as defined in [D1.1] [D1.5] of [1] with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements and extenal potential coupling, the change of extenal potential coupling when the numbe of violational elements in K i changes by m whee m K i is given by:

19 19 =m V m v K i Let T p be the violational tansfomation defined in [D3.4]; let K i* =T p (K i,m) and let G * =T p (G,m). By definition [D3.2], the change of extenal potential coupling effected by applying T p to K i is then: By poposition 1.4 in [1]: = * (i) * = K i* V * v K i * (ii) By poposition 3.3: Substituting (iii) into (ii) gives: V * = V m (iii) * = K i* V m v * (iv) By poposition 3.2: K i* = K i m (v) Substituting (v) into (iv) gives: Q * i = K i m V m v K i m = K i m V m v K i m = K i m V v K i = K i V K i v K i m V m v K i = K i V v K i m V m v K i (vi) But: = K i V v K i (vii) Substituting (vii) into (vi) gives: * = m V m v K i Q * i =m V m v K i And theefoe by (i): =m V m v K i Poposition 3.9. Given the set G as defined in [D1.1] [D1.5] of [1] of n elements with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements and extenal potential coupling, the change of extenal potential coupling of the entie set G when the numbe of violational elements in a paticula x th disjoint pimay set K x changes by m whee m K x is given by: G=mn m K x s ex

20 By poposition of [1], the extenal potential coupling of G is the sum of the extenal potential coupling of all its disjoint pimay sets: = G= i =1 s ex (i) Let T p be the violational tansfomation defined in [D3.4] and let K i* =T p (K i,m). Futhemoe, let T p apply to the x th disjoint pimay set only such that K x* =T p (K x,m) and G * =T p (x,g,m). By definition: G * = i=1 * 20 By definition [D3.2]: = * s ex * (ii) G= G * G (iii) Substituting (i) and (ii) into (iii) gives: G= * s ex * i =1 x By definition [D3.2]: = Q * x (v) Substituting (v) into (iv) gives: G= * i =1 x (iv) s ex (vi) By poposition 1.4 in [1]: = K i V v K i (vii) If we conside K i whee i x then as T p is applied only to K x then: K i* = K i i x (viii) Substituting (viii) into (vii) gives: Q * i = K i V * v K i (ix) By poposition 3.3: V * = V m (x) Substituting (x) into (ix) gives: Q * i = K i V m v K i = K i V K i v K i m K i = K i V v K i m K i (xi)

21 21 But: Substituting (xi) into (xii) gives: = K i V v K i (xii) * = m K i As this holds i x we can take the sum ove all pimay sets except x: By definition, So: * = = Substituting (xiv) into (viii) gives: m i=1 m K i i=1 x K i (xiii) n= K i = K i K x n K x = * = i=1 x K i (xiv) m n K x Substituting (xv) into (vi) gives: G= * = mn m K x (xv) mn m K x = mn m K x s ex Poposition Given the set G as defined in [D1.1] [D1.5] of [1] of n elements with an i th disjoint pimay set K i of K i elements and extenal potential coupling s ex (K i ), the change of extenal potential coupling of the entie set G when the numbe of violational elements in a paticula x th disjoint pimay set K x changes by m whee m K x is given by: G=mn m K x m V m v K x Let T p be the violational tansfomation defined in [D3.4] and let it apply to the x th disjoint pimay set only. By poposition 3.8, the change of extenal potential coupling of K x by the application of T p to K x is given by:

22 =m V m v K x (i) By poposition 3.9, the change of extenal potential coupling of the entie set G by the application of T p to K x is given by: Substituting (ii) into (i) gives: Poposition G=mn m K x s ex (ii) s ex G =mn m K x m V m v K x Given the set G as defined in [D1.1] [D1.5] of [1] of n elements with an i th disjoint pimay set K i of K i elements, the change of potential coupling of the entie set s G when the numbe of violational elements in a paticula x th disjoint pimay set K x changes by m whee m K x is given by: s G =mn m K x m V m v K x m 2 m Let T p be the violational tansfomation defined in [D3.4] and let K i* =T p (K i,m). Futhemoe, let T p apply to the x th disjoint pimay set only such that K x* =T p (K x,m) and G * =T p (x,g,m). Fom poposition 3.7, when the numbe of violational elements in K x changes by m, the change of intenal potential coupling of the entie set is given by: G= s in (i) By poposition 3.6, when the numbe of violational elements in K x changes by m, the change of intenal potential coupling of K x is given by Substituting (ii) into (i) gives: =2m K x m 2 m (ii) G=2m K x m 2 m (iii) Fom poposition 3.10, when the numbe of violational elements in K x changes by m, the change of extenal potential coupling is given by: G=mn m K x m V m v K x (iv) By poposition 3.1, the change of potential coupling of G is given by: Substituting (iii) and (iv) into (v) gives: s G = s in G s ex G (v) s G =mn m K x m V m v K x 2m K x m 2 m = mn m K x m V m v K x m 2 m 22 Poposition Given the set G as defined in [D1.1] [D1.5] of [1] with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements and intenal potential coupling s in (K i ), the change of intenal potential

23 23 coupling when the numbe of infomation hidden elements in K i changes by m whee m K i is given by: =2m K i m 2 m Let T z be the hidden tansfomation defined in [D3.5] and let K i* =T z (K i,m). By definition [D3.1], the change of intenal potential coupling effected by applying T z to K i is: By poposition 1.2 in [1]: = * (i) s i n * = K i* K i* 1 (ii) By poposition 3.4: K i* = K i m (iii) Substituting (iii) into (ii) gives: Q * i =K i m K i m 1 2 = K i m K i K i m K i m 2 m = K i K i 1 2m K i m 2 m (iv) But: = K i K i 1 (v) Substituting (v) into (iv) gives: Q * i = 2m K i m 2 m Q * i =2m K i m 2 m And theefoe by (i): =2m K i m 2 m Poposition Given the set G as defined in [D1.1] [D1.5] of [1] with an i th pimay set Q i and an i th disjoint pimay set K i of intenal potential coupling, the change of intenal potential coupling of the entie set G when the numbe of infomation hidden elements in a paticula x th disjoint pimay set K x changes by m whee m K x is equal to the change of intenal potential coupling of K x, o: G= s in By poposition in [1], the intenal potential coupling of G is the sum of the intenal potential coupling of all its disjoint pimay sets: s i n G= s i n i=1

24 = s in (i) Let T z be the hidden tansfomation defined in [D3.5] and let K i* =T p (K i,m). Futhemoe, let T z apply to the x th disjoint pimay set only such that K x* =T p (K x,m) and G * =T z (x,g,m). By definition: = s i n G * = i =1 s i n * * s in * (ii) By definition [D3.1], the change of intenal potential coupling of G effected by applying tansfomation T z to G is: Substituting (i) and (ii) into (iii) gives: G= i=1 x G= G * G (iii) * s in * i=1 x (iv) But as T z is only applied to K x then all disjoint pimay sets except K x ae unchanged, o: And theefoe: Substituting (v) into (iv) gives: G= i=1 x K i * =K i i x * = i x (v) s in * i=1 x = * (vi) By definition [D3.1], the change of intenal potential coupling effected by applying T z to K x is then: 24 Substituting (vii) into (vi) gives: = * (vii) G= s in Poposition Given the set G as defined in [D1.1] [D1.5] of [1] with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements and extenal potential coupling, the change of extenal potential coupling when the numbe of infomation hidden elements in K i changes by m whee m K i is given by: =m V m v K i Let T z be the hidden tansfomation defined in [D3.5]; let K i* =T z (K i,m) and let G * =T z (G,m).

25 By definition [D3.2], the change of extenal potential coupling effected by applying T z to K i is then: 25 By poposition 1.4 in [1]: = * (i) * = K i* V * v K i * (ii) By poposition 3.5: Substituting (iii) into (ii) gives: V * = V (iii) * = K i* V v K i * (iv) By definition of T z : v K i * = v K i (v) Substituting (v) into (iv) gives: Q * i = K i* V v K i (vi) By poposition 3.4: K i* = K i m (vii) Substituting (vii) into (vi) gives: Q * i = K i m V v K i = K i V K i v K i m V m v K i = K i V v K i m V m v K i (viii) But: = K i V v K i (ix) Substituting (ix) into (viii) gives: * = m V m v K i Q * i =m V m v K i And theefoe by (i): K i =m V m v K i Poposition Given the set G as defined in [D1.1] [D1.5] of [1] of n elements with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements and extenal potential coupling, the change of extenal potential coupling of the entie set G when the numbe of infomation hidden elements in a paticula x th disjoint pimay set K x changes by m whee m K x is given by: G= s ex

26 By poposition in [1], the extenal potential coupling of G is the sum of the extenal potential coupling of all its disjoint pimay sets: = G= i =1 s ex (i) Let T z be the hidden tansfomation defined in [D3.5] and let K i* =T z (K i,m). Futhemoe, let T z apply to the x th disjoint pimay set only such that K x* =T z (K x,m) and G * =T z (x,g,m). By definition: G * = i=1 * 26 By definition [D3.2]: = * s ex * (ii) G= G * G (iii) Substituting (i) and (ii) into (iii) gives: G= * s ex * i =1 x By definition [D3.2]: = Q * x (v) Substituting (v) into (iv) gives: G= * i =1 x (iv) s ex (vi) By poposition 1.4: Q * i = K i* V * v K * i (vii) If we conside K i whee i x then as T z is applied only to K x then: K i* = K i i x (viii) Substituting (viii) into (vii) gives: By poposition 3.5: * = K i V * v K i (ix) Substituting (x) into (ix) gives: V * = V (x) * = K i V v K i (xi) But by poposition 1.4 in [1]: Substituting (xii) into (xi) gives: = K i V v K i (xii)

27 27 * = As this holds i x we can take the sum ove all pimay sets except x: * = Substituting (xiii) into (vi) gives: G= (xiii) Poposition Given the set G as defined in [D1.1] [D1.5] of [1] of n elements with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements and extenal potential coupling, the change of extenal potential coupling of the entie set G when the numbe of infomation hidden elements in a paticula x th disjoint pimay set K x changes by m whee m K x is given by: G=m V m v K x Let T z be the hidden tansfomation defined in [D3.5] and let K i* =T z (K i,m). Futhemoe, let T z apply to the x th disjoint pimay set only such that K x* =T z (K x,m) and G * =T z (x,g,m). By poposition 3.14, the change of extenal potential coupling of K x by the application of T z to K x is given by: =m V m v K x (i) By poposition 3.15, the change of extenal potential coupling of the entie set G by the application of T z to K x is the same as the change of extenal potential coupling of K x o: Substituting (i) into (ii) gives: Poposition G= (ii) G=m V m v K x Given the set G as defined in [D1.1] [D1.5] of [1] of n elements with an i th pimay set Q i and an i th disjoint pimay set K i of K i elements, the change of potential coupling of the entie set s G when the numbe of infomation hidden elements in a paticula x th disjoint pimay set K x changes by m whee m K x is given by: s G =m V m v K x 2m K x m 2 m Let T z be the hidden tansfomation defined in [D3.5] and let K i* =T z (K i,m). Futhemoe, let T z apply to the x th disjoint pimay set only such that K x* =T z (K x,m) and G * =T z (x,g,m). Fom poposition 3.13, when the numbe of infomation hidden elements in K x changes by m, the change of intenal potential coupling of the entie set is given by: G= s in (i)

28 28 By poposition 3.12, when the numbe of infomation hidden elements in K x changes by m, the change of intenal potential coupling of K x is given by Substituting (ii) into (i) gives: =2m K x m 2 m (ii) G=2m K x m 2 m (iii) Fom poposition 3.16, when the numbe of infomation hidden elements in K x changes by m, the change of extenal potential coupling is given by: G=m V m v K x (iv) By poposition 3.1, the change of potential coupling of G is given by: Substituting (iii) and (iv) into (v) gives: Poposition s G = s in G s ex G (v) s G =m V m v K x 2m K x m 2 m Given the set G as defined in [D1.1] [D1.5] of [1] of n elements, the cumulative change of potential coupling of the entie set s cumulative G when m violational elements ae moved fom a paticula souce disjoint pimay set K s to a diffeent taget disjoint pimay set K t is given by: s cumulative G=mh K t h K s Let T p1 be the violational tansfomation defined in [D3.4] and let T p1 apply to the s th disjoint pimay set K s only such that K s* =T p1 (K s,m) and G * =T p1 (s,g,m). T p1 will emove m violational elements fom K s. Let T p2 be the violational tansfomation defined in [D3.4] and let T p2 apply to the t th disjoint pimay set K t whee t s only such that K t* =T p2 (K t,m) and G ** =T p2 (t,g *,m). T p2 will add m violational elements fom K t. Let us define the violational tanslation tansfomation T Tp as the combination of the two tanslations T p1 and T p2 such that: T Tp (G)= T p2 (T p1 (G)) As these tansfomations ae linea, the change of the potential coupling of T Th is equal to the sum of the changes of the potential coupling of the tansfomations T p2 and T p1, o: s cumulative G= s G s G * (i) Consideing elements emoved fom a disjoint pimay set as negative, then T p1 will add m elements to G. By poposition 3.11, the change of potential coupling in G caused by the emoval of these m violational elements fom K s is given by: Substituting m fo m in (ii) gives: s G =mn m K s m V m v K s m 2 m (ii) s G = mn m K s m V m v K s m 2 m (iii) By poposition 3.11 again, the change of potential coupling in G * caused by the addition of m

29 29 violational elements to K t is given by: s G =mn * m K t m V * m v K t m 2 m (iv) G * has m fewe elements than G and they ae all infomation hiding violational, and thus both v(g * ) and n * ae changed in compaison with G, such that: Substituting (v) and (vi) into (iv) gives: V * = V m (v) n * =n m (vi) s G * =m n m m K t mv m m v K t m 2 m = mn m 2 m K t m V m 2 m v K t m 2 m Substituting (iii) and (vii) into (i) gives: = mn m K t m V m v K t m 2 m (vii) s cumulative G= mn m K s m V m v K s m 2 m mn m K t m V m v K t m 2 m = m K t v K t K s v K s (viii) By definition [D3.6]: K = h K v K K v K = h K (ix) Substituting (ix) into (viii) gives: s cumulative G=mh K t h K s Poposition Given the set G as defined in [D1.1] [D1.5] of [1], the cumulative change of potential coupling of the entie set s cumulative G when m infomation hidden elements ae moved fom a paticula souce disjoint pimay set K s to a diffeent taget disjoint pimay set K t is given by: s cumulative G=m 2 K t 2 K s v K s v K t 2m Let T z1 be the hidden tansfomation defined in [D3.5] and let T z1 apply to the s th disjoint pimay set K s only such that K s* =T z1 (K s,m) and G * =T z1 (s,g,m). T z1 will emove m infomation hidden elements fom K s. Let T z2 be the hidden tansfomation defined in [D3.5] and let T z2 apply to the t th t s disjoint pimay set K t only such that K t* =T z2 (K t,m) and G ** =T z2 (t,g *,m). T z2 will add m infomation hidden elements fom K t. Let us define the hidden tanslation tansfomation T T as the combination of the two tanslations T z1 and T z2 such that: T T (G)= T z2 (T z1 (G)) As these tansfomations ae linea, the change of the potential coupling of T T is equal to the sum of the changes of the potential coupling of the tansfomations T z2 and T z1, o: