A new way to obtain homology groups in Binary 2D images using membrane computing

A NEW WAY TO OBTAIN HOMOLOGY GROUPS IN BINARY 2D IMAGES USING MEMBRANE COMPUTING DANIEL DÍAZ-PERNIL 1 , MIGUEL A. GUTIÉRREZ-NARANJO 2 , PEDRO REAL 1 , AND VANESA SÁNCHEZ-CANALES 1 Abstrat. Membrane Computing is a omputational mo del inspired in the struture and funtion of living ells and tissues. In this paper we use Membrane Computing tehniques to solve the Homology Groups of Binary 2D Image (HGB2I) Problem . This is a lassial problem in Homology Theory whih tries to alulate the numb er of onneted omp onents and the representative urves of the holes of these omp onents of a given binary 2D image. To this aim, we use a family of P systems whih solves all the instanes of the problem in the framework of Tissue-like P systems with atalysts . Introdution Natural Computing studies new omputational paradigms inspired from Nature. It abstrats the way in whih Nature ats , oneiving new omputing mo dels from natural phenomena in physis, hemistry and biology 1 . The eld is growing rapidly and urrently there are many op en researh lines based on Nature, oering solutions to many lassial omputational problems from a new p ersp etive. From this new paradigm, we an ite areas suh as Cel lular Automata [29℄, Geneti algorithms [15℄, Neural Networks [21℄, DNA-based moleular omputing [1℄, Swarm Intel ligene [11℄ or Membrane Computing [24, 25℄ among others. All these omputational paradigms have in ommon the use of an alternative way of eno ding the information, adapted to the bio-inspired substrate and the use of intrinsi parallelism of natural pro esses. In this pap er we use tehniques from Membrane Computing 2 . This new researh area was b orn from the assumption that the pro esses taking plae within the ompartmental struture of a living ell an b e interpreted as omputations [25℄. In partiular, it fo uses on membranes, whih are involved in many reations taking plae inside various ompartments of a ell. Biologial membranes are muh more than mere barriers that dene ompartments. They at as seletive hannels of ommuniation b etween dierent ompartments as well as b etween the ell and its environment [2℄. The omputational devies in Membrane Computing are alled P systems . Roughly sp eaking, a P system onsists of a membrane struture, in the ompartments of whih one plaes multisets of ob jets whih evolve aording to given rules. In the most extended mo del, the rules are applied in a synhronous non-deterministi maximally parallel manner, but some other semantis are b eing explored. Membranes divide the Eulidian spae into regions, whih ontains ob jets. Ob jets evolve aording to lo al reation rules. Suh ob jets an b e desrib ed by symb ols or by 1 An intro dution on Natural Computing an b e found in [16℄. 2 We refer to [25℄ for basi information in this area, to [27 ℄ for a omprehensive presentation and the web site [30℄ for the up-to-date information. 1 2DANIEL DÍAZ-PERNIL 1 , MIGUEL A. GUTIÉRREZ-NARANJO 2 , PEDRO REAL 1 , AND VANESA SÁNCHEZ-CANALES 1 strings of symb ols from a given alphab et. These ob jets an also pass through membranes, under the ontrol of sp ei rules. We will fo us here on a P system mo del alled (b eause of their membrane struture) Tissue-like P systems . This P system mo del has two biologial inspirations (see [20℄): interellular ommuniation and o op eration b etween neurons. The ommon mathematial mo del of these two mehanisms is a net of pro essors dealing with symb ols and ommuni- ating these symb ols along hannels sp eied in advane. The ommuniation among ells is based on symp ort/antip ort rules 3 . Symp ort rules move ob jets aross a membrane together in one diretion, whereas antip ort rules move ob jets aross a membrane in opp osite diretions. Homology theory is a branh of algebrai top ology that attempts to distinguish b etween spaes by onstruting algebrai invariants that reet the onnetivity prop erties of the spae. The eld has it origins in the work of Poinaré. Homology groups (related to the dierent n -dimensional holes, onneted omp onents, tunnels, avities, et.) are invariants from Algebrai Top ology whih are frequently used in Digital Image Analysis and Strutural Pattern Reognition. In some sense, they reets the top ologial nature of the ob jet in terms of the numb er and harateristis of its holes. This is not the rst time in whih life-based metho ds are applied to Algebrai Top ology. In 1996, J. Chao and J. Nakayama onneted Natural Computing and Algebrai Top ology using Neural Networks [7℄ (extended Kohonen mapping). Some years after, K.G. Subramanian et al . presented in [6 , 5℄ two works where Digital Image and Natural Computing were linked. In 2009, Christinal et al . presented in [9℄ a new way to obtain the Betti numb ers of a binary 2D and 3D digital image. In this pap er, we present a new tehnique to get not only the Betti numb ers, we obtain the representative urves of the homology group H1 of a binary 2D digital image to o. The neessary time to alulate the homology groups of 2D digital images with basi P systems dened in Setion 2 is logarithmi with resp et to the input data ( O(n) ). This involves an improvement with resp et to the algorithms development by S. Peltier et al . in [28℄, where they use irregular graphs pyramids with a time omplexity of O(n5/3) . The pap er is organized as follows: rstly, we present our bio-inspired formal framework. Next, we briey reall the Homology Groups of Binary 2D Image (HGB2I) Problem . In the next Setion, we present our solution in the framework of tissue-like P systems with atalysts. Finally, some onlusions are presented. 1. Formal Framework In tissue-like P systems the membrane struture is a general undireted graph. The edges of suh graph are not given expliitly, but they are dedued from the set of rules. From the seminal denition of tissue P systems, several researh lines have b een develop ed and other variants have arisen (see, for example, [3, 4, 14 , 18, 26℄). In this pap er, we endow tissue-like P systems with atalysts. Catalyti P systems were intro dued in [24 ℄. The main feature of these P systems is the presene of ob jets in membranes suh that they are not onsumed by the appliation of the rule, but their presene 3 This way of ommuniation for P systems was introdued in [22℄. A NEW WAY TO OBTAIN HOMOLOGY GROUPS IN BINARY 2D IMAGES USING MEMBRANE COMPUTING3 in the membrane is neessary for the triggering (see, e.g., [12, 13, 19, 17℄. Next we provide the denition of tissue-like P systems with atalysts: Denition 1.1. A tissue-like P system with atalysts of degree q≥1 is a tuple of the form Π = (Γ,E, w1,...,wq,R, i0), where: (1) Γ is a nite alphab et, whose symb ols will b e alled ob jets. (2) E ⊆ Γ is a nite alphab et representing the set of the ob jets in the environment available in an arbitrary large amount of opies. (3) w1,...,wq are strings over Γ representing the multisets of ob jets asso iated with the ells in the initial onguration. (4) R is a nite set of atalyti rules of the following form: (cat |i, u/v, j) for i, j ∈ {0,1,2,...,q}, i 6=j and cat, u, v ∈Γ∗ . The length of a atalyti rule is dened as |u|+|v| . The atalyst cat is mo died by the appliation of the rules and cat and v an b e empty. (5) i0∈ {0,1,2,...,q} denotes the output region, whih an b e the environment ( i0= 0 ) or the region inside a ell ( 1≤i0≤q ). Informally, a tissue-like P system with atalysts of degree q≥1 an b e seen as a set of q ells (eah one onsisting of a single membrane) lab eled by 1,2,...,q . The ells are the no des of a virtual graph, where the edges onneting the ells are determined by the ommuniation rules of the system, i.e., as usual in tissue-like P systems, the edges linking alls are not provided expliitly: If a rule (cat |i, u/v, j) is given, then ells i and j are onsidered linked. The appliation of a atalyti rule (cat |i, u/v, j) onsists on the trade of the multiset u (initially in the ell i ) against the multiset v (initially in j ). The trade an also b e b etween one ell and the environment, lab elled by 0. The rule is applied if in the ell with lab el i the ob jets of the set cat are present (atalyst). If the atalyst is empty, then the rule is alled a ommuniation rule . In our denition, all ob jets in the alphab et an at as atalysts, dep ending on the applied rule. Rules are used as usual in the framework of membrane omputing, that is, in a maximally parallel way (a universal lo k is onsidered). In one step, eah ob jet in a membrane an only b e used for one rule (non-deterministially hosen when there are several p ossibilities), but any ob jet whih an partiipate in a rule of any form must do it, i.e., in eah step we apply a maximal multiset of rules. A onguration is an instantaneous desription of the P system and it is represented as a tuple (w1,...,wq) . Given a onguration, we an p erform a omputation step and obtain a new onguration by applying the rules in a parallel manner as it is shown ab ove. A sequene of omputation steps is alled a omputation . A onguration is halting when no rules an b e applied to it. The output of a omputation is olleted from its halting onguration by reading the ob jets ontained in the output ell. Example 1.2. Let us onsider the following tissue-like P system with atalyst of degree 3, Π = (Γ,E, w1, w2, w3,R, i0) where Γ = {a, b, c, d, e} , E={d} , w1=a , w2=bd and w3=ce . 4DANIEL DÍAZ-PERNIL 1 , MIGUEL A. GUTIÉRREZ-NARANJO 2 , PEDRO REAL 1 , AND VANESA SÁNCHEZ-CANALES 1 The set R has four rules: R1≡(c|2, b/a, 1) R2≡(e|3, c/d, 2) R3≡(3, d/a, 0) R4≡(3, d/b2,0) The output region is i0= 3 . Aording to the set of rules, ell 2 is linked to ells 1 and 3 and ell 3 is linked to the environment. The omputation starts from the initial onguration C0= (w0, w1, w2) . Rule R1 annot b e applied in this initial onguration, sine the atalyst c do es not app ear in ell 2. Rules R3 and R4 annot b e applied, sine ob jet d is not plaed on ell 3. In this initial onguration we only an apply rule 2, sine ob jets e and c are plaed in membrane 3 and d is plaed in membrane 2. After applying this rule we obtain a new onguration. C1= (w′ 0, w′ 1, w′ 2) with w′ 1=a , w′ 2=cb and w′ 3=de . Now, rule 2 an b e applied by interhanging ob jets a and b , b eause the atalyst c is plaed on ell 2. The ommuniation rules 3 and 4 an b e applied, sine d app ears in ell 3 and the environment ontains ob jets a and b , but only one of them an b e applied, and it is non-deterministially hosen. In we ho ose rule 3, then the new onguration is C2= (w′′ 0, w′′ 1, w′′ 2) with w′′ 1=b , w′′ 2=ac and w′′ 3=ae . No more rules an b e applied and the omputation nishes. The output of the omputation is the multiset in ell 3 in the halting onguration w′′ 3=ae . If we ho ose rule 4, the obtained onguration is the same, but w′′ 2=b2e . 2. Calulating Homology Groups In a binary 2D image, the omputation of homology groups an b e redued to a pro ess of blak and white onneted omp onents lab eling. The dierent blak onneted omp onents are the generators of the 0 -dimensional homology group of the blak part of the image whereas the losed blak urves surrounding the dierent white onneted omp onents of the image are the generators of its 1 -dimensional homology group. In order to formalize this problem we will onsider a ordered 4 set of n2 pixels P={(i, j) : 1≤i, j ≤n} . An image on P with olors in the nite set C is a mapping I:P→ C . As usual, suh image an b e written as a set of pairs ((i, j), x) where i, j ∈ {1,...,n} and a=I(i, j) . To simplify we denote aij . As we onsider in this pap er to work with binary images, we take C={b, w} where b o dies blak and w white. The new tehnique for P systems presented in this pap er onsist to assign to eah ob jet o difying a pixel a lab el. So, we our ob jets pass to b e of the form (aij,(i, j)) . We will see b elow how to use these lab els to solve our problem with P systems. From a formal p oint of view we an formulate our problem as follows, Homology Groups of Binary 2D Image (HGB2I) Problem: Given a binary 2D digital image, alulate the numb er of blak onneted omp onents and the representative urves of the holes of these omp onents. We have deide to onsider in this pap er the 4-adjaeny for blak pixels and the 8adjaeny for white pixels. The only one reason to do this hoie is the omputational omplexity from a membrane mo dels p oint of view. It is easier for this typ e of mo dels works with 8-adjaeny for blaks and 4 for whites. Moreover, in this last ase the systems are very similar to the systems show b elow. Then, we have deided to present the membrane solution of HGB2I problem for the 4-adjaeny for blak pixels and the 8-adjaeny for white pixels. 4 We onsider the lexiographi order for N2 in P . A NEW WAY TO OBTAIN HOMOLOGY GROUPS IN BINARY 2D IMAGES USING MEMBRANE COMPUTING5 2.1. A membrane solution of HGB2I Problem. In order to provide a logarithmi- time uniform solution to the our problem, we design a family of tissue-like P systems with atalyst, Π . Given an image I of size n2 , we take the system of the family Π(n) to work with I . The input data (image I ) is o died by a set of ob jets Aij with A=B∨W and 1≤i, j ≤n . The family of P systems is dened as follows: Π(n) = (Γ,Σ,E, ω1, ω2, ω3, ω4,R1,...,R19,{R15,R17}>R10, iin, i0) where: •Γ = {zi: 1 ≤i≤n+ 6} ∪ {Bij, bij, b′ ij,¯ bij, Wij, wij, w′ ij,(bij,(k, l)),(wij ,(k, l)) : 1 ≤i, j, k, l ≤n} ∪ {(pij,(0,0)),(pji,(0,0)) : i= 0, n + 1,0≤j≤n+ 1} ∪ {Aijkl, Zijkl : (1,1) ≤(i, j)<(k, l)≤(n, n)} . •Σ = {bij, wij : 1 ≤i, j ≤n} . • E = Γ −Σ . •ω1=∅ . •ω2={z1} . •ω3={z1,(pij,(0,0)),(pji,(0,0)) : i= 0, n + 1,0≤j≤n+ 1} . •ω4=∅ . • The sets of rules are:  R1≡(1, Aij/aija′ ij,0) for 1≤i, j ≤n . For eah pixel of our input image we generate two opies with these rules.  R2≡(1, aij/λ, 2) for 1≤i, j ≤n .  R3≡(1, a′ ij/λ, 3) for 1≤i, j ≤n . With the last two typ es of rules, the rst opies of our pixels are sent to ell 2 and the seond opies of our pixels are sent to ell 3. From this p oint, we have two parallel pro esses. A pro ess asso iated to ell 2 (to alulate H0 ) and the seond one asso iated to ell 3 (to alulate H1 ):  Rules asso iated to the H0 pro ess: ◦R4≡(2, zi/zi+1,0) for 1≤i≤n+ 1 . These rules generate a ounter that will b e used in the output of the system. ◦R5≡(2, bij/(bij,(i, j)),0) for 1≤i, j ≤n . These rules add lab els to blak pixels in order to work with them. ◦R6≡(2,(bij,(k, l))(bi′j′,(k′, l′))/(bij,(k, l))(bi′j′,(k, l))Aklk′l′,0) for (1,1)≤ (k, l)<(k′, l′)≤(n, n) , 1≤i, j, i′, j′≤n and (i, j) , (i′, j′) adjaent pixels. ◦R7≡(2,(bij,(k, l))(bi′j′,(k′, l′))/(bij,(k′, l′))(bi′j′,(k′, l′))Ak′l′kl,0) for (1,1) ≤(k′, l′)<(k, l)≤(n, n) , 1≤i, j, i′, j′≤n and (i, j) , (i′, j′) adjaent pixels. The two last typ es of rules hange the lab els of adjaent pixels, we need all the adjaent blak pixels to have the same lab el, so we will know that they are all in the same onneted omp onent. ◦R8≡(Aijkl|2,(bi′j′,(k, l))/(bi′j′,(i, j)),0) for 1≤i, j, k, l, i′, j′≤n . In these rules we intro due atalysts, and pro ess b eomes faster. The 6DANIEL DÍAZ-PERNIL 1 , MIGUEL A. GUTIÉRREZ-NARANJO 2 , PEDRO REAL 1 , AND VANESA SÁNCHEZ-CANALES 1 atalyst has b een reated when the pixel lab eled by (k, l) traded its lab el for (i, j) , so (i, j) and (k, l) are adjaent pixels and other pixels with these lab els an b e hanged. ◦R9≡(zn+2|2,(bij,(i, j))/λ, 4) . With these rules we send one pixel for eah onneted omp onent to the ell 2.  Rules asso iated to H1 pro ess: ◦R10 ≡(3, zi/zi+1,0) for 1≤i≤n+ 5 . This rule ounts the numb er of steps of the pro ess. We will use this to start the Deleting Stage after n+ 2 steps, and the Segmenting Stage after n+ 4 steps. ◦R11 ≡(3, wij/(wij,(i, j)),0) for 1≤i, j ≤n . These are the only rules used in the Label Al loation Stage . These rules add lab els to white pixels in order to work with them. ◦R12 ≡(3,(wij ,(k, l))(wi′j′,(k′, l′))/(wij,(k, l))(wi′j′,(k, l))Zklk′l′,0) for (1,1) ≤ (k, l)<(k′, l′)≤(n, n) , 1≤i, j, i′, j′≤n and (i, j),(i,′j′) adjaent pixels. ◦R13 ≡(3,(wij,(k, l))(wi′j′,(k′, l′))/(wij,(k′, l′))(wi′j′,(k′, l′))Zk′l′kl,0) for (1,1) ≤(k′, l′)<(k, l)≤(n, n) , 1≤i, j, i′, j′≤n and (i, j),(i′, j′) adjaent pixels. These two set of rules are used in Label Conversion Stage to ompare two adjaent white pixels, and hange the lab el of one of them. We need all the adjaent white pixels to have the same lab el. ◦R14 ≡(Zijkl|3,(wi′j′,(k, l))/(wi′j′,(i, j)),0) for 1≤i, j, k, l, i′, j′≤n . The atalyst Zijkl ats to b eome the pro ess faster. It has b een reated when the pixel lab eled by (k, l) traded its lab el for (i, j) , so (i, j) and (k, l) are adjaent pixels and other pixels with these lab els an b e hanged. ◦R15 ≡(zn+3|3,(pij,(0,0))(wkl,(k′, l′))/(pij,(0,0))(pkl,(0,0))Z00kl,0) for (i, j),(k, l) 8-adjaent pixels, 0≤i, j ≤n+ 1 , 1≤k, l, k′, l′≤n . These rules are used in Deleting Stage to delete white pixels whih are out of the onneted blak omp onent. By using 8-adjaeny, we b eome outer white pixels into pink pixels, in order to dierentiate them from the interior white pixels (holes). We will refer to the ob jets pij as pink pixels. ◦R16 ≡((Z00ij|3,(wi′j′,(i, j))/(pi′j′,(0,0)),0) for , 1≤i, j, i′, j′≤n . A new atalyst ats in the same way, trading white exterior pixel for pink pixels. In this way, the Deleting Stage takes only 2 step. ◦R17 ≡(zn+5|3,(wij,(i′, j′))bkl/(wij,(i′, j′))¯ bkl,0) for 1≤i′, j′, i, j, k, l ≤n and (i, j),(k, l) 8-adjaent pixels. In the Segmenting Stage a blak pixel is marked if it and a white pixel are 8-adjaent pixels. It starts after n+ 2 steps. ◦R18 ≡(3,¯ bij/λ, 4) for 1≤i, j ≤n . At the end, in the Answer Stage , blak marked pixels are sent to membrane numb er 4, so we obtain whih blak pixels are ontaining the holes. A NEW WAY TO OBTAIN HOMOLOGY GROUPS IN BINARY 2D IMAGES USING MEMBRANE COMPUTING7 ◦R19 ≡(zn+6|3,(wij,(i, j))/λ, 4) for 1≤i, j ≤n . We want to obtain the numb er of holes to o, so these rules send one white pixel for eah hole to membrane numb er 4. •iin = 1 is the input ell. •i0= 4 is the output ell. We will also use priorities among rules. Rules from sets R15 and R17 are applied b efore rules from the set R10 . Figure 1. A simple example of the pro ess to obtain H0 Eah system of the family implements the following stages: (1) Input Stage : When the ob jets Aij (with A=B∨W ) arrive to ell 1 the rst typ e of rules are applied. We hange these ob jets by ob jets aij and a′ ij (with a=b∨w ). In the next step, we send ob jets aij to ell 2 and ob jets a′ ij to ell 3. From this p oint we have two parallel pro esses: The rst pro ess o ur in ell 2, and it is dediated to obtain the numb er of blak onneted omp onents ( H0 ). The seond one is lo ated in ell 3 and is dediated to generate the representative urves of the holes of the blak onneted omp onents. Moreover, system will give us the numb er of these holes ( H1 ). (2) H0 Proess : It b egins when ob jets aij arrive to ell 2. (a) Label Al loation Stage : Cell 3 trades ob jets bij against others with the form (bij,(i, j)) with the environment. The white ob jets are not transformed. (b) Label Conversion Stage : We an ompare the blak adjaent pixels by using atalyst, and we trade the lab el of the greatest pixel against the lab el of the other pixel; i.e. (i, (bij,(i′, j′))(bkl,(k′, l′))/(bij,(i′, j′))(bkl,(i′, j′))Ai′j′k′l′, j) , where (i, j) and (k, l) are adjaent pixels. Moreover, we an see a new ob jet arriving to ell i . It is a atalyst and it is used to o dify if two lab els must b e ompared. Later, they are onneted, and one of them an b e hanged by the other one, as we an see in the Figure 1. () Answer Stage : In the step n+ 2 , the ob jet zn+2 arrives to the ell 1 due to the ounter. It is used by the system as a atalyst, and the ob jets with the form (bij,(i, j)) are sent to the output ell representing eah one to a blak onneted 8DANIEL DÍAZ-PERNIL 1 , MIGUEL A. GUTIÉRREZ-NARANJO 2 , PEDRO REAL 1 , AND VANESA SÁNCHEZ-CANALES 1 omp onent. The P system have used n+ 2 steps to obtain the numb er of blak onneted omp onents of an n2 image. Figure 1 shows a sequene of ongurations of the pro ess to obtain H1 for input data given by the onguration C0 of the piture. Figure 2. Representative ongurations of a simple example to obtain H1 (3) H1 proess : It b egins when ob jets a′ ij arrive to ell 3. (a) Label Al loation Stage : Cell 3 trades ob jets w′ ij against others with the form (wij,(i, j)) with the environment. (b) Label Conversion Stage : We ompare the lab el of two white adjaent pixels, and we trade the lab el of the greatest pixel against the lab el of the other pixel; i.e., we use rules with the form (i, (wij,(i′, j′))(wkl,(k′, l′))/(wij,(i′, j′))(wkl,(i′, j′))Zi′j′k′l′, j) , where (i, j) and (k, l) are adjaent pixels. Moreover, we an see a new ob jet arriving to ell i , Zi′j′k′l′ . It is a atalyst and is used to o dify when two lab els must b e ompared. Then, the lab els are onneted, and one of them an b e hanged by the other one, as we an see in C7 in the Figure 2. () Deleting Stage : Initially, system keeps in ell 1 a set of ob jets o difying the frame of the input image ( p0i, pn+1i, pi0, pin+1 for i= 0,...,n+ 1 ) with the lab el (0,0) asso iated. When the input data is intro dued in the system, the white pixels not ontained inside of blak onneted omp onents are sent to the environment to trade against of ob jets with the form of the frame. We need a linear numb er of steps with resp et to n to eliminate all the p ossible white pixels. We an see the result in C11 in the Figure 2. A NEW WAY TO OBTAIN HOMOLOGY GROUPS IN BINARY 2D IMAGES USING MEMBRANE COMPUTING9 (d) Segmenting Stage : This part b egins when deleting stage nishes due to the ounter zi (rules R1 ). If there are white pixels in ell 1 in this step are in a hole. The P system takes pairs of adjaent pixels, one blak and the other white, adding a mark to the blak pixels of these pairs. Then, we have marked the blak pixels adjaent to a hole. We need a onstant numb er of steps to segment an image with P systems. Figure 2 shows in C12 how the holes of the image are o died. (e) Answer Stage : We send the marked blak pixels to output ell in the following step to b e marked. So, we obtain, the representative urves of the holes in the image I . We also send white pixels whih keep their lab els, there is only one pixel for eah onneted white omp onent, ie, for eah hole in the image. We only need one step more with resp et to the segmenting stage. Figure 3. Initial and fourth onguration 2.2. Example. In this setion we will show with a simple example our tehnique to obtain the homology groups H0 , and H1 of a 2D image by using membrane omputing. Let us onsider the rst image in Figure 3 whih have 9×9 pixels. The blak pixels represent two onneted omp onents, the white pixels represent the exterior of the omp onents and the holes, one in eah onneted omp onent. In the initial onguration we have 4 ells. There is one ob jet Aij (with A=B∨W ) in ell 1 o difying eah pixel. At the b eginning of the pro ess the system dupliates these elements and sends one opy to the membrane 2 on order to alulate H0 , and the other opy to the membrane 3 in order to alulate H1 . These two pro esses are working at the same time, so we obtain a parallel system. The pro ess to alulate H0 is the following one: The ell 1 has the ob jet z1 working as a ounter, it allow us start a dierent stage after the neessary numb er of the steps. In the