Encapsulation theory: radial encapsulation. Edmund Kirwan *

Transcription

1 Encapsulation theoy: adial encapsulation. Edmund Kiwan * Abstact This pape intoduces the concept of adial encapsulation, wheeby dependencies ae constained to act fom subsets towads supesets. The pape shows how hieachically encapsulated sets can be viewed as tees and investigates how potential coupling changes as the banches of these tees ae manipulated. Keywods Encapsulation theoy, adial encapsulation, potential coupling. 1. Intoduction In [1] an encapsulation context was defined as a set of constaints on the fomation of potential odeed pais within an encapsulated set. The absolute infomation hiding context was intoduced wheeby a violational element is accessible to all elements in all othe disjoint pimay sets. Aguably the simplest and most widely employed, this context is not, howeve, the only context imaginable. Otheist. The key point of these othe contexts is that they enable that which is fobidden by absolute infomationhiding: they allow violational elements which ae accessible to some but not all of the elements in othe disjoint pimay sets. Violational access becomes a elative athe than an absolute popety: hence these ae the elative infomation hiding contexts, o simply, "Relative contexts." Relative contexts diffe fom one anothe only in the means they employ to manifest this elative element access, and thee is no limit to numbe of such manifestations. To take the computing envionment, fo example, we can imagine that the pogam units of a system ae named (as they usually ae) and theefoe we could contemplate the constaint that dependencies between pogam units must flow alphabetically. Pogam unit Anold may depend on pogam unit Melissa; pogam unit Zeba may not depend on pogam unit Benda. Despite the cassness of thiample, such an unusual dependency constaint nevetheless establishes a elative context: violational elements accessible to some ae not accessible to othes (those alphabetically, "Up steam"). When faced with multiple elative contexts, one must usually choose between suitos; seldom do the * Edmund Kiwan 009. Revision 1.0, Decembe 0 th 009. (Revision 1.0: Decembe 0 th, 009.) All entities may epublish, but not fo pofit, all o pat of this mateial povided efeence is made to the autho and title of this pape. The latest evision of this pape is available at [].

2 constaints of one elative context ovelap with those of anothe to the degee that both may eign consistently ove the same system. Occam's azo is invoked: all else being equal, the elative context defined by the simplest, most economic constaints might be consideed the best choice. This pape intoduces such a elative context, the adial encapsulation context, defined by a single constaint: dependencies may only fom fom subsets to supesets.. Radial encapsulation Fist, let us look at an absolute infomation hiding context. Figue 1 shows two pimay sets, a and b, both containing two elements, one violational (the black element) and one hidden (the white element) note that the seconday sets V and H ae not shown but undestood. Also shown ae the potential dependencies fom the elements in set a on those elements in set b, thus we see two dependencies on the violational element in b. Figue 1: An absolute infomation hiding context with two pimay sets. Recall that the absolute context is embodied in the definition of the extenal potential coupling set (see definition [D1.9] in [1]): =K i V K i (i) This tells us that, in the absolute context, dependencies may fom fom the elements of any disjoint pimay set to the violational elements of evey othe disjoint pimay set. In tems of softwae, this says that all pogam units of any given subsystem may fom dependencies on all the public pogam units of all othe subsystems in the system. Hence the two dependencies shown in figue 1 ae pefectly legal. How would this simple configuation look in a adial encapsulation context? The defining constaint of adial encapsulation is also embedded in the definition of the extenal potential coupling set (see definition [D8.1]): =K i V Q i (ii) Despite the appaent similaity between equations (i) and (ii), these sets ae vey diffeent. In (i), as mentioned, dependencies may fom towads evey othe disjoint pimay set; in (ii), howeve, dependencies may fom on towads those disjoint pimay sets that ae not subsets of the pimay set in question, Q i. To put it anothe way, dependencies may only flow fom subsets into supesets, o moe fomally that an odeed

3 pai x a, y b epesents a legal dependency if and only if a b. It thus appeas that the two dependencies potayed in figue 1, though legal in the absolute context, would be illegal in the elative context. The potential dependencies fom b to a, also, would be illegal: b is not a subset of a no is a a subset of b; this would leave us with two sets which could not inteact and a tivially degeneate system. So how can we allow inteaction between these two sets whilst especting adial encapsulation? The answe is to hieachically constuct the two sets, as shown in figue. Figue : Radial encapsulation with two pimay sets. In figue, set a is defined as a subset of set b. Now, the elements in a ae fee to fom potential dependencies on the violational elements in b without tansgessing the constaint that dependencies can only fom fom subset to supeset. Of couse, the elements in b may not, still, fom potential dependencies towads the elements in a, as those dependencies would fom fom supeset into subset, but a degee of inteaction is now possible. Encapsulation contexts whose dependency constaints ae ealised by the patial o hieachical odeing of sets ae called hieachical encapsulation contexts, a subcategoy of the elative encapsulation contexts; the adial encapsulation context is a yet futhe subcategoy of the hieachical encapsulation contexts. (The, "Radial," tem, incidentally, comes fom the patten made by the potential dependency aows in figues such as the one above: the aows always adiate outwads, neve inwads.) Befoe we examine the popeties of adially encapsulated sets, and how they diffe fom those of, "Nomal," absolute infomation hiding contexts, let us look at an altenative epesentation of the set in figue. Puely fo convenience, instead of dawing set a inside set b, we can depict a as being on top of b, as shown in figue 3.

4 Figue 3: Radial encapsulation with a defined within b. In figue 3, it is undestood that dawing a on top of b implies that a is a subset of b, and b is a supeset of a. Viewed so, the adial encapsulation context allows only those dependencies that flow, "Downwads." 3. Radial banches Figue 4a shows thee sets adially encapsulated. Figue 4a: Thee adially encapsulated sets. The aows of figue 4a now show the geneal diection in which potential dependencies fom between violational elements contained with each of the sets (not shown), though note that elements in a may fom dependencies not just towads elements in the supeset b, but towads elements in c too, thus the aows epesent a simplified view of the potential dependencies. The oganisation of sets shown in figue 4a is

5 called a banch and the length of the banch is the numbe of its compised sets. Let us now populate the banch of figue 4a with elements and attempt to calculate its potential coupling, see figue 4a. Figue 4b: Thee adially encapsulated sets populated with elements. The intenal potential coupling in the elative context is the same as that in the absolute context. The potential couplings of the thee sets ae: Set c: The intenal potential coupling is 5x4=0. This set has no extenal potential coupling, as thee is not set "below" it to which it can fom dependencies. Thus the potential coupling of c is 0. Set b: The intenal potential coupling of this set is 3x=6. The extenal potential coupling is the numbe of elements in this set multiplied by the numbe of all the violational elements (the black elements) it can see in all othe sets. As the only set on which this set can fom dependencies on is set c and thee ae only two violational elements in c, then the extenal potential coupling is 3x=6. Thus the potential coupling of b is 6+6=1. Set a: The intenal potential coupling of a is 3x=6. The extenal potential coupling is the numbe of elements in a multiplied by the numbe of violational elements in b and c, which is +=4; so the extenal potential coupling is 3x4=1. Summing the potential couplings of all thee sets, we find that the potential coupling of this banch is 44. Noting that thee ae 11 elements in this banch, we might also ask whethe we could educe the potential coupling of this banch by changing the numbe of sets and e distibuting the elements. Does thee exists a specific numbe of sets pe banch that minimises the potential coupling of that banch? This question is answeed in the next section. 4. Minimised banches Conside a banch of 100 elements. Let us plot the potential coupling of this banch as the elements ae unifomly distibuted ove inceasing numbes of disjoint pimay sets (i.e., ove an inceasingly long

6 banch) subject to the constaint of a specific violation of one (i.e., thee must be one and only one violational element pe disjoint pimay set, d=1). Thus, the fist banch will consist of a single set with a potential coupling of Next, the 100 elements will be spead ove a banch of two sets, giving a potential coupling of 4950, and so on. Figue 5: Potential coupling of an inceasingly long, banch (d=1). Fom figue 5, we can see that that potential coupling of this banch of 100 elements is minimised when the length of the banch is slightly less than twenty. It can be poved that the potential coupling of a unifomly distibuted banch occus when the numbe of disjoint pimay sets in the banch is given by (see poposition 8.3): = n ild d Fo n=100 and d=1, the numbe of disjoint pimay sets is 14. A significant diffeence aises hee between the absolute and the adial encapsulation contexts. Given that all disjoint pimay sets must contain at least one violational element, the minimum potential coupling of any numbe of elements occus when thee is just one violational element pe disjoint pimay set; having one violational element pe disjoint pimay set futhe implies that the set is unifomly distibuted in violational elements. Within the absolute encapsulation context, it was poved that a set unifomly distibuted in violational elements has a minimised potential coupling when it is also unifomly distibuted in hidden elements (see poposition 1.11 in [1]). This, howeve, is not the case in the adial encapsulation context consisting of a single, banch: given a banch unifomly distibuted in violational elements, it is possible to educe that banch's potential coupling by distibuting its hidden elements non unifomly. Deiving the equation fo the configuation which offes this minimum potential coupling is descibed in [3]; this pape will concen itself with unifomly distibuted configuations.

7 5. Radial tees We have thus fa consideed only single banches. A set, of couse, may have many immediate subsets, thus foming multiple banches, o a tee, see figue 6. Figue 6: Multiple banches. Figue 6 shows a tee of two banches shaing the same oot set. The two banches ae {a, b, e} and {c, d, e}. The aows in figue 6 again show the geneal diection of dependencies fom elements within the sets to elements in othes sets, thus elements in a can depend on elements in both b and e. We wish to calculate the potential coupling of such multi banch sets. The inclusion exclusion fomula of set theoy descibes how to calculate the total numbe of elements given a numbe of intesecting sets; a simila equation applies to banches, too (banches being sets themselves), and given that the potential coupling of a set is defined in tems of the numbe of its elements, then it can be shown that the potential coupling of a tee of a banches is given by: a a a 1 s B i i = s B i s B i B i 1 =1 This gives us the means to investigate how multiple banches affect oveall potential coupling and aises two, geneal questions concening a set of multiple banches. The fist question is: how does the potential coupling of a single banch change as it is split into two banches whee the split occus at eve lowe sets? Let us take 500 elements evenly distibuted ove a banch of 10 disjoint pimay sets, whee we shall numbe the oot set as 1 and the leaf set as 10; thoughout these expeiments, we shall keep the specific violation of 1, d=1. We shall then ecod its potential coupling. We shall then conside two banches, the fist being 10 sets long, the second being a banch of only 1 set, and shall join this banch to the fist banch by adding it as a subset of the ninth set; thus we will have two banches 10 sets long, both having 9 sets in common and diffeing only in thei leaf set. Again, we shall ecod the potential coupling. We shall then conside the fist banch again of 10 sets and a second banch of sets, adding this second banch to the fist

8 as a subset of its eighth set, and so on. The pocedue is illustated in figue 7. Note that, with each iteation, we evenly edistibute all the elements ove all sets. Figue 7: Redistibuting elements ove two banches. Plotting the potential coupling at each iteation then shows how this splitting of a banch in two affects the system, see figue 8. Figue 8: Potential coupling as the banch split moves down the fist banch. The x axis of figue 8 shows not the set at which the split between the two banches occus, but the distance fom the leaf set; thus as x=0, thee is only one banch; at x=1, the second banch (one set long) is split fom the fist banch at ninth set; at x=, the second banch (two sets long) is split fom the fist banch at the eighth set, etc. We can see fom figue 8 that the potential coupling falls monotonically as the two banches ae split futhe down the sets of the fist banch. In othe wods, to minimise the potential coupling of the two banches, banches should be joined at the oot set. The second question we face concening a set of multiple banches is: how does the potential coupling of

9 a set change as we incease the numbe of banches into which the elements ae distibuted? Let us take 500 elements evenly distibuted ove a banch of 5 disjoint pimay sets, again with d=1. This banch has a potential coupling of Let us then take two banches of 5 sets each, again with 500 elements unifomly distibuted ove both banch and disjoint pimay set, and with both banches joined at the oot set. This has a potential coupling of Let us continue like this, inceasing the numbe of banches but maintaining both the numbe of elements and the numbe of disjoint pimay sets pe banch; figue 9 shows the potential coupling as the numbe of banches inceases. Figue 9: Potential coupling as a function of inceasing numbe of banches. Figue 9 shows that the moe banches we have, the lowe the potential coupling. Unlike the numbe of sets pe banch, as shown in figue 5, thee is no specific numbe of banches that minimises the potential coupling of ou system. Cetainly, at least concening the application of encapsulation theoy, an abitaily lage numbe of banches may not be pactical. As applied to softwae development, fo example, the above gaph suggests splitting a system into as many banches as possible, yet each banch can only communicate with othes via the single disjoint pimay set of elements at the oot of all banches, and in the expeiment above, the numbe of elements in the oot set fell as moe and moe banches wee added, thus endeing the expeiment slightly unealistic. Nevetheless, the above shows that, in geneal, banching is a good stategy fo educing potential coupling, as can be shown by a simple compaison: poposition 1.14 of [1] shows that the minimum, isoledensal potential coupling achievable within the absolute encapsulation context is given by: s ild G =n nd 1 d If we substitute into this equation the figues used in the expeiment above (n=500, d=1) we find that the minimum potential coupling achieved is As can be seen fom figue 9, adial encapsulation can achieve a potential coupling fa below anything possibly achievable by absolute encapsulation.

10 6. Conclusion. This pape intoduced adial encapsulation and showed how the potential coupling of an encapsulated set, adially encapsulated, changed by splitting the set into banches. 7. Appendix A Many of the popositions pesented hee necessaily contast the potential coupling of absolute infomation hiding with that of adial infomation hiding. To avoid confusion we must adapt notation slightly. Wheeve absolute infomation hiding is concened, we shall append the wod, "Absolute;" wheeve adial infomation hiding is concened, no exta wods shall be appended. Thus, the potential coupling within the adial context is simply called, "Potential coupling," wheeas the potential coupling within absolute infomation hiding is called, "Absolute potential coupling." Similaly, wheeve s is used to denote potential coupling we shall intoduce the supescipt, "ab," to indicate potential coupling of an absolute infomation hiding context; the absence of such a supescipt will indicate potential coupling of adial infomation hiding. Fo example, the adial infomation hiding extenal potential coupling of encapsulated set G will be denoted G ; the absolute infomation hiding extenal potential coupling will be denoted s ab ex G. 7.1 Definitions [D8.1] Let (Q i ) be the set of odeed pais fomed by the Catesian poduct of all elements in the disjoint pimay set of Q i with all elements of V that ae not also membes of the pimay set Q i. This is called the extenal potential coupling set of Q i ; the extenal potential coupling of Q i is cadinality of this set. Thus: =K i V Q i [D8.] Given pimay set Q i, tee T of Q i, witten T (Q i ), is set of all disjoint pimay sets that ae subsets of Q i. Thus: T ={ K j :K j Q i } Fo example, if Q j Q i and Q k Q j, then T ={K i, K j,k k }. T (Q i ) is epesented in figue 10.

11 Figue 10: The tee T (Q i ). The pimay set Q i of T (Q i ) is also called the oot pimay set. A pimay set of a banch with no subsets is called a leaf set. A banch Q i, witten B(Q i ), is a tee whose evey pimay set is the oot set, a leaf set o has only one "paent" supeset and one "child" subset, o: B ={K j : K j Q i K j, Q k,q m,q k Q m Q k Q i Q m Q i Q k Q m Q m Q k } Thus figue 10 is also a banch. 7. Popositions Poposition 8.1 Given an encapsulated set G and given that the i th pimay set is Q i, the extenal potential coupling of Q i is a subset of the absolute potential coupling of Q i, o: s ab ex By definition [D1.9] in [1], the absolute potential coupling set of Q i is the set of odeed pais fomed by the Catesian poduct of all elements in the disjoint pimay set of Q i with all elements of V that ae not also membes of the disjoint pimay set of Q i : s ab ex =K i V K i (i) By definition [D8.1], the potential coupling set of Q i is the set of odeed pais fomed by the Catesian poduct of all elements in the disjoint pimay set of Q i with all elements of V that ae not also membes of the pimay set Q i : =K i V Q i (ii) By definition [D1.5] the disjoint pimay set K i is the set of all elements in Q i but not in any pimay

12 subset of Q i, thus K i is a subset of Q i : K i Q i (iii) Fom (iii) it follows that: V Q i V K i (iv) Take the Catesian poduct of K i with both sides (iv) gives: Substituting (i) and (ii) into (v) gives: K i V Q i K i V K i (v) s ab ex Poposition 8. Given an encapsulated set G and given that the i th pimay set is Q i, the extenal potential coupling set of Q i is disjoint the extenal potential coupling set of all othe pimay sets, o:,q i Q j : = By poposition in [1], the extenal potential coupling set of Q i is disjoint the extenal potential coupling set of all othe pimay sets, o: Q i, o:,q i Q j :s ab ex s ab ex = (i) By poposition 8.1, the extenal potential coupling of Q i is a subset of the absolute potential coupling of s ab ex (ii) Substituting (ii) into (i) gives;,q i Q j : = Poposition 8.3 Given an encapsulated set G and given that the i th pimay set Q i, the extenal potential coupling of G, G, is given by: G= By definition [D1.1] of [1], the extenal potential coupling of G is: G = Taking the cadinality of both sides of (i) gives: i =1 (i)

13 = G= 1 i =1 1 (ii) By poposition 8., the extenal potential coupling set of any pimay set is disjoint fom evey othe extenal potential coupling set, thus: Substituting (iii) into (ii) gives: 1 1 =0 (iii) G= i =1 Poposition 8.4 Given an encapsulated set G and given that the i th pimay set Q i, the extenal potential coupling set of Q i is disjoint fom the intenal potential coupling set of Q i, o: : s i n = By poposition of [1], the extenal potential coupling set of Q i is disjoint fom the intenal potential coupling set of Q i, o: :s ab ex s ab i n = (i) As thee is no diffeence between the intenal potential coupling and the absolute intenal potential coupling, then: s i n =s ab in (ii) Fom poposition 8.1, the extenal potential coupling of Q i is a subset of the absolute potential coupling of Q i, o: Substituting (ii) and (iii) into (i) gives: s ab ex (iii) : s i n = Poposition 8.5 Given an encapsulated set G and given that the i th pimay set is Q i, the extenal potential coupling set of

14 Q i is disjoint fom the intenal potential coupling set of all othe pimay sets, o:,q i Q j : s i n = By poposition of [1], the extenal potential coupling set of Q i is disjoint fom the intenal potential coupling set of all othe pimay sets, o:,q i Q j :s ab ex s ab i n = (i) As thee is no diffeence between the intenal potential coupling and the absolute intenal potential coupling, then: s i n =s ab i n (ii) Fom poposition 8.1, the extenal potential coupling of Q i is a subset of the absolute potential coupling of Q i, o: Substituting (ii) and (iii) into (i) gives: s ab ex (iii),q i Q j : s i n = Poposition 8.6 Given an encapsulated set G and given that the i th pimay set is Q i, all extenal potential coupling sets ae disjoint fom all intenal potential coupling sets, o: : s i n = Fom poposition 8.4, the extenal potential coupling set of Q i is disjoint fom the intenal potential coupling set of Q i, o: : s i n = (i) Fom poposition 8.5, the extenal potential coupling set of Q i is disjoint fom the intenal potential coupling set of all othe pimay sets, o:,q i Q j : s i n = (ii) Fom (i) and (ii) it follows that: : s i n =

15 Poposition 8.7 Given an encapsulated set G of pimay sets, and given that the i th pimay set is Q i, the potential coupling of Q i is the sum of the extenal potential coupling sets of Q i and the intenal potential coupling of Q i, o: s = s i n By poposition [D1.10] of [1], the potential coupling set of Q i is given by: s =s i n (i) Taking the cadinality of both sides of (i) gives: s = s in = s i n s i n (ii) By poposition 8.6, all extenal potential coupling sets ae disjoint fom all intenal potential coupling sets, o: : s i n = (iii) Substituting (iii) into (ii) gives: s = s i n Poposition 8.8 Given an encapsulated set G of pimay sets, the potential coupling s G of G is the sum of the intenal potential coupling of G and the extenal potential coupling of G, o: s G= s i n G G By definition [D1.13] of [1], the potential coupling set of G is given by: s G =s i n G G (i) Taking the cadinality of both sides gives of (i) gives: s G= s i n G G = s i n G G s i n G G (iii)

16 By definition [D1.11] of [1], the intenal potential coupling set of G is given by: s i n s i n (iv) G = i =1 By definition [D1.1] of [1], the extenal potential coupling set of G is given by: G = (v) Thus fom (iv) and (v): s i n G G = s i n i =1 (vi) Fom poposition 8.6, all extenal potential coupling sets ae disjoint fom all intenal potential coupling sets, o: : s i n = (vii) Substituting (vii) into (vi) gives: s i n G G = (viii) Substituting (viii) into (iii) gives: s G= s i n G G Poposition 8.9 Given an encapsulated set G of pimay sets and given that the i th pimay set is Q i, the potential coupling s G of G is the sum of the intenal potential coupling of all Q i and the extenal potential coupling of all Q i, o: s G= { s i n } By poposition 8.8, the potential coupling s G of G is the sum of the intenal potential coupling of G and the extenal potential coupling of G, o: s G= s i n G G (i) By poposition of [1], the intenal potential coupling of G, s i n G, is given by: s i n G= s i n (ii)

17 By poposition 8.3, the extenal potential coupling of G, G, is given by: G= (iii) i =1 Substituting (ii) and (iii) into (i) gives: s G= s i n = { s i n } Poposition 8.10 Given an encapsulated set G of pimay sets and given that the i th pimay set is Q i, the potential coupling s G of G is the sum of the potential couplings of all Q i, o: s G= s By poposition 8.9, the potential coupling s G of G is the sum of the intenal potential coupling of all Q i and the extenal potential coupling of all Q i, o: s G= { s i n } (i) By poposition 8.7, the potential coupling of Q i is the sum of the extenal potential coupling sets of Q i and the intenal potential coupling of Q i, o: s = s i n (ii) Substituting (ii) into (i) gives: s G= s Poposition Given a unifomly distibuted banch B in encapsulated set G of disjoint pimay sets and given that the i th disjoint pimay set K i contains K i elements and has an infomation hiding violation of d, the extenal

18 potential coupling of B, B, is given by: B= n 1 d By definition [D8.1], the extenal potential coupling of Q i is the numbe of odeed pais that may be fomed fom this disjoint pimay set towads the violational elements of all othe disjoint pimay sets that ae not a subset of Q i, o: Conside the oot pimay set Q 1. = K i V Q i (i) As Q 1 is the oot pimay set, howeve, all othe disjoint pimay sets in this banch ae subsets of Q 1, thus the extenal potential coupling of the oot disjoint pimay set is 0. K 1. The extenal potential coupling of the Q is then the numbe of odeed pais that may be fomed towads The extenal potential coupling of the Q 3 is the numbe of odeed pais that may be fomed towads K 1 and K, etc. Thus: Q 1 =0 Q = K v K 1 Q 3 = K 3 v K v K 1 Q 4 = K 4 v K 3 v K v K 1... Q = K v K 1 v K...v K v K 1 As B is a unifomly distibuted set then by definition [D1.14] in [1]: v K 1 = v K =...= v K =d (ii) (i) Substituting (ii) into (i) gives: Q 1 =0 Q = K d Q 3 = K 3 d Q 4 = K 4 3d... Q = K 1 d By poposition in [1], the extenal potential coupling of set B, witten B, is the sum of the extenal potential coupling of Q i, o: (iii)

19 B= (iv) Substituting (iii) into (iv) gives: B= K dk 3 d K 4 3d...K 1 d (v) As B is a unifomly distibuted set then by definition [D1.14] in [1]: K 1 = K =...= K = n (iv) Substituting (iv) into (v) gives: B= n d n d n 3d... n 1 d = n d = n 1 d i = n d 1 = n 1 d Poposition 8.1. Given a unifomly distibuted banch B in encapsulated set G of n elements and of disjoint pimay sets, with each disjoint pimay set having an infomation hiding violation of d, the potential coupling of B, s u B, is given by: s u B=n n 1 1 d By poposition in [1], the potential coupling of set B, witten s B, is the sum of the intenal and extenal potential coupling of B, o: s B= s i n B B (i) By poposition 1.6 in [1], the intenal potential coupling of B, s i n B, is given by: s i n B=n n 1 (ii) By poposition 8.11, the extenal potential coupling of B, B, is given by: Substituting (ii) and (iii) into (i) gives: B= n 1 d (iii)

20 s u B=n n n 1 d 1 = n n 1 1 d Poposition Given a unifomly distibuted banch B in encapsulated set G of n elements and of disjoint pimay sets, with each disjoint pimay set having an infomation hiding violation of d, the numbe of disjoint pimay sets ild that isoledensally minimises the banch's potential coupling is given by: = n ild d By poposition 8.1, the potential coupling of B is given by: s u B=n n 1 1 d (i) To find the numbe of disjoint pimay sets that isoledensally minimise the potential coupling, diffeentiate (i) with espect to and set to zeo: s B= u [ n n 1 d 1 ]=0 n 1 1 d =0 n d =0 n d =0 = n d = n ild d Poposition Given a unifomly distibuted banch B in encapsulated set G of n elements and of disjoint pimay sets, with each disjoint pimay set having an infomation hiding violation of d, the banch's minimum isoledensal potential coupling is given by:

21 s ild B= n 8 dn d By poposition 8.1, the potential coupling of a unifomly distibuted banch is given by: s u B=n n 1 1 d (i) By poposition 8.13, the numbe of disjoint pimay sets ild that isoledensally minimises this banch's potential coupling is given by: Substituting (ii) into (i) gives: = n ild d n s ild B=n n d (ii) n d 1 d 1 = n n d n 1 d n d d = n dn 1 dn d = n dn d = n dn 1 d 8. Refeences [1] "Encapsulation theoy fundamentals," Ed Kiwan, [] "Encapsulation theoy: adial encapsulation," Ed Kiwan, [3] "Encapsulation theoy: the minimised, unifomly violational adial banch,"