ACKERMANN S FUNCTION AND NEW ARITHMETICAL OPERATIONS

Web PDF version -2004-09-15 ACKERMANN S FUNCTION AND NEW ARITHMETICAL OPERATIONS CONSTANTIN A. RUBTSOV GIOVANNI F. ROMERIO Abstrct. Ackermnn s function implies the existence of n infinite spectrum of new rithmeticl opertions (hyperopertions) belonging to the Grzegorczyk Hierrchy. Two of these hyperopertions, tetrtion nd zertion, together with their inverses re given specil ttention. Tetrtion (power tower, or superpower) hs recently been described in the scientific literture nd hs ttrcted the ttention of both creer nd mteur mthemticins; zertion is less well known. However, the new rithmeticl opertion, clled zertion, follows nturlly from generlizing formuls used in itertive clcultions of both known nd new inverse opertions. Zertion expnds our pproch to mthemticl concepts such s infinity nd continuity nd its ppliction for describing discontinuities is shown in prcticl exmples. The possibility of using tetrtion to represent very lrge numbers is outlined. Questions requiring further reserch re lso rised. Mthemtics Subject Clssifiction (2000): 20k30 homomorphisms; 03D99 computbility nd recursion theory; 11A25 rithmetic functions; 11H99 computtionl number theory; 68Q15 complexity clsses. 1. INTRODUCTION 1.1. Algorithmic Complexity Clsses. In the frmework of rtificil intelligence (AI) nd problem solving sciences, lgorithm efficiency is nlyzed by mesuring how much the durtion of process, generted by n lgorithm, grows s function of the mount of dt on which the lgorithm itself opertes. Let the mount of dt (mesured, for instnce, in bits) be clled x nd the length of the process (i.e. the lgorithm execution time, or the mount of process steps, or the totl number of bits describing the process) be clled y. Then, vrious levels of lgorithm complexity cn be ssessed ccording to the types of expressions y = f(x), s seen in problems complexity theory. Normlly, if n lgorithm opertes on lrger number of dt, the corresponding process requires lrger number of steps to be executed, i.e. the process length y lso increses. In prctice, we hve situtions such s those shown in the following plots of the y = f(x) functions. Constntin A. Rubtsov. pr. Vlutin, d. 12 - Belgorod Russin Federtion Giovnni F. Romerio. vi Torino, 48-12037 Sluzzo (CN) - Itly

y = f(x) 2 x In prticulr, the digrm shows vritions of liner type (y = x), s well s of the qudrtic (y = x 2 ) or cubic (y = x 3 ) types. The digrm lso shows series of exponentil (or iterted exponentil) vritions, growing much fster thn the power plots. In principle, we cn hve the following mjor clsses of vritions: Clss P The process/dt plots re of type y = k x, y = k x 2, y = k x 3,., or of generlized polynomil type y = x n + bx n-1 + cx n-2 + k. Algorithms of this type re sid to belong to the clss of polynomil type, corresponding to reltively esily solvble problems. Clss EXP The process/dt plots re not of the polynomil type but they cn be described by exponentil functions, such s y = exp(x). In prctice, process length y cn be represented by using the powers of 2 of the lgorithm length x, such s y = 2 x, or by using the powers of e. This clss represents exponentil or intrinsiclly complex lgorithms, corresponding to the clss of intrinsiclly difficult problems, which normlly strongly resist computer solutions. Clss H-EXP - AI specilists hve found lgorithms identified by the y = 2^(2^x) plot (or biexponentil), or even by tri-exponentil or tetr-exponentil or, generlly, k-exponentil expressions. Some plots of this kind re indicted in the digrm. We my observe tht these plots increse much more rpidly thn ny plot of polynomil or of simple exponentil type. A detiled clssifiction of H-EXP (hyper-exponentil) lgorithms could help in defining the rnks of intrinsicl problem complexity. Clss 0 - A clss of lgorithms in which the process length (corresponding to the execution time) is lwys infinite. In this cse the computer will never stop nd, therefore, it will never deliver the solution. This is the clss of unsolvble problems, i.e. of problems for which the solution, in the frmework of the existing nd dmitted hypotheses, is considered to be nonexistent. 1.2. Iterted exponentitions. It is interesting to note tht, in order to define the subclsses of the H- EXP clss, some expressions tke the form: (1) y = e ^(e ^(e ^(e ^..(e ^ x)))) [e ^ iterted k times, on x] nd these re outside the rnge of normlly ccepted elementry rithmeticl opertions, s defined in clssicl Algebr. These kinds of expressions (i.e. iterted exponentitions) hve been identified, in the literture, s power towers, superpowers or, simply, towers. L. Stockmeyer nd A. Chndr, of the Thoms Wtson IBM Reserch Centre, hve nlyzed expressions of this type. 2

3 They lso reported, in pper published in 1989 [1], tht Albert Meyer [2] proved, in 1972, tht the S1S lgorithm hs complexity greter thn exponentil. In prticulr, he hs shown tht the lgorithm deciding bout the truth of sttement with length x of the S1S lnguge hs complexity tht increses more thn the k-exponentil function of x (for ny k). 1.3. Algorithmic Informtion. As clerly stted by Stephen Wolfrm [3], it is generlly ccepted tht the description of piece of dt cn lwys be done with progrm for reproducing it. Andrei Kolmogorov nlyzed the problem of defining the informtion crried by sequence of bits, representing process, with different pproch from tht of Shnnon s informtion theory, where the informtion contents of signl is given by the logrithm, to the bse 2, of the reciprocl of the configurtion probbility. Shnnon s informtion quntity, thus defined, supposes the existence of communiction chnnel, linking the Source of informtion with the Receiver nd tkes into ccount the probbility tht the Receiver cn detect the signls sent by the Source. Kolmogorov s objective ws, however, to try nd define nother informtion quntity tht could possibly mesure the bsolute internl order of process, independently of ny trnsmission chnnel. For this, he hd the ide of connecting process with the shortest possible lgorithm which could reproduce it. If no such lgorithm shorter thn the process exists, then we must resort to rndom process. The shortest possible lgorithm, ssocited with process, is clled its Algorithmic Informtion, which, in wy, highlights the internl redundncy of the process itself. Chrles Bennet clled it the process Logicl Depth. It is interesting to note tht the definition of the Algorithmic Informtion of process coincides with the definition of the Algorithmic Complexity of the progrm tht hs generted it (see 1.1.). 1.4. Ackermnn s Function (AF). Ackermnn s Function [4], recursive function tht, while Turing computble, grows fster thn ny primitive recursive functions, cn be used to estblish link between clssicl elementry rithmeticl opertions (ddition, multipliction, exponentition) nd to nlyze opertions such s: (2) y = ^( ^( ^( ^..( ^ )))) [ ^ iterted k times, on ] These re rther similr to expressions (1), except for the fct tht iterted exponentition, in this cse, lwys opertes on the sme mgnitude which is the bse of the exponentil. AF itself demonstrtes the existence of n infinite series of hyperopertions, including the clssicl rithmeticl opertions, s well s tower power nd other iterted exponentitions, ech one indexed by positive integer vlue of prmeter s, clled the hyperopertions rnk. This series is known s the Grzegorczyk Hierrchy [3]. 1.5. Ω-Mpping. The first uthor (C. A. Rubtsov), while nlyzing AF, hd the ide of defining the properties of new hyperopertion of rnk zero (s = 0), i.e. with rnk lower thn the rnk of ddition, becuse it ppered to be utomticlly implied by some AF vlues. This new opertion ws nmed the zero-rnk opertion or zertion. The properties of zertion hve been presented in pper [5], concerning New Mthemticl Objects, following some ides presented since 1989 [6]. Pper [5] nlyses the properties of zertion nd of the new set of numbers tht it genertes. This, together with the outline of the bsis concepts of the homomorphisms theory, expnds our representtion of the spectrum of existing mthemticl objects (http://numbers.newmil.ru). Ω- mpping formlism cn be used to discover the properties of hyperopertions such s zertion nd tetrtion, together with those of their inverse opertions. 1.6. Extension of Tower to the Rels. The second uthor (G. F. Romerio) hs, since 1986, been ttrcted to the problem of nlyzing the properties of the tower opertion, defined within the set N of nturl numbers, in order to extend it to the set R of rel numbers. This problem is similr to tht of extending the clssicl fctoril opertion: n! with n N to ny number x, belonging to the set of rel numbers. This question ws solved by the definition of the Gmm Function, such s: x! = Γ(x + 1) with x R. Furthermore, the solution to our problem of extending the tower opertion (lso clled tetrtion) to the rels is lso similr to the clssicl question of extending the vlidity of expression n (n N) to x, 3

4 where we hve x R. This problem hs lso been solved long time go, strting by identifying n expression with rtionl exponent, such s ½ (with 0 < 1/2 < 1 ; i.e.: 1 < 1/2 < ), with the squre root of ( ). Unfortuntely, the homologue opertion, t the tower/tetrtion opertion rnk, ws never given mening until now. Therefore, there is still need, for instnce, to clrify the exct mening of tetrted-½ which, incidentlly, must not be confused with wht we might cll the super-squreroot of (see prgrphs 6.3 nd 6.4). 1.7. Objectives. The immedite objective of this study is to drw the ttention of mthemticins to the possibility of defining new elementry functions bsed on the properties of hyperopertions nd their inverses, s well s to suggest possible prcticl pplictions nd strtegies for future reserch. 2. ITERATIVE EXTENSIONS OF TRADITIONAL ALGEBRAIC OPERATIONS 2.1. Itertions of clssicl opertions. We wish to exmine the possibility of introducing n extensible series of lgebric opertions, by using the constructive itertive ppliction of pproprite opertors, ech one corresponding to one of the trditionl rithmeticl opertions (ddition nd multipliction), together with exponentition. For this purpose, let us define the following opertors: +, such tht: + x = + x Opertor + (Addition), such tht: x = x =. x Opertor (Multipliction) ^, such tht: ^ x = ^ x = x Opertor ^ (Exponentition) These three opertors ct on the opernd situted to the right nd cn be iterted, s follows: + + + x = + + ( + x) = + ( + ( + x)) = + ( + ( + x)) = +++x x = (. x) = (. (. x)) =. (. (. x)) =...x ^ ^ ^ x = ^ ^ ( ^ x) = ^ ( ^ ( ^ x)) = ^ ( ^ ( ^ x)) = As we know, in the first two cses (ddition nd multipliction), the prentheses cn be removed in the finl result, becuse of the ssocitivity of the two opertions. In the cse of exponentition, this cnnot be done nd the order defined by the prentheses remins unltered. Suppose we now pply the + opertor, by repeted itertions, to the sme numericl vrible. The result of ech itertion of ddition, which is n opertion identified s the first rnk in the hierrchy (s = 1), will be represented by multipliction, which is n opertion of the second rnk (s = 2): Rnk s = 1, iterted dditions, represented by multiplictions (s = 2): + =. 2 (3) + + =. 3...... + + + +...+ (n times) =. n (-times-n) We now go on to describe the second iterted opertion, in which we use opertor. This consists of the itertion of multipliction (s = 2), represented by exponentition (s = 3): Rnk s = 2, iterted multiplictions, represented by exponentitions (s = 3):. = ^ 2 (4).. = ^ 3.............. (n times) = ^ n (-power-n) The itertions referred to bove cover nd represent the three clssicl lgebric opertions (ddition, multipliction nd exponentition). 2.2. Tetrtion. However, the itertion process cn be continued, by defining n dditionl hierrchicl level (s = 4), which represents iterted exponentition. In this cse we get: Rnk s = 3, iterted exponentitions, represented by new opertion clled tetrtion (s = 4) or power tower, or tower, for which we choose opertor symbol # : x 4

5 ^ = # 2 (5) ^ ^ = # 3......... ^ ^ ^ ^...^ (n times) = # n (-tower-n or -tetrted-n) In both exponentition nd tetrtion, the ssocitive property is not fulfilled. These opertions re, therefore, intended to be executed with priority to the right. Tetrtion derives its nme from the fct tht it ppers t the 4 th rnk in the hyperopertions hierrchy, in which ddition, multipliction nd exponentition re identified by rnks 1, 2 nd 3, respectively. The first super-exponentil opertion (-tetrted-n or -tower-n) cn be represented in vrious wys, such s: # n = n or lso s n. The ltter is derived from the representtion of stndrd exponentition by one rrow i.e.: ^ n = n = n, s proposed by Knut (1976), nd it hs the dvntge of being pplicble to ll the hyperopertions with rnks > 3. The rrows symbolism for s < 3 fils. Nottion n ws proposed by Rucker in 1995. We shll see tht the itertion process cn continue, for rnks s > 4, nd tht we cn define series of opertions (hyperopertions) with no upper limit to the vlue of the rnk. 2.3. Zertion. The problem of the existence of opertions with hierrchicl rnk s < 1, nd prticulrly for s = 0 (the zero-rnk opertion, or zertion) ws studied by one of us [5], in 1986. Leving the justifiction for it until lter stge, we shll t this point simply present here the scheme of this new opertion, the itertion of which cn be represented by the ddition opertion. The process requires the definition of new opertor, with rnk zero :, such tht: x = x Opertor (zertion) The scheme so obtined is s follows (with priority to the right): = + 2 (6) = + 3......... (n times) = + n (-plus-n) This opertor defines new lgebric opertion, with hierrchicl rnk s = 0, for which the ssocitive property is lso not fulfilled. An pproprite nme for this opertion is zertion. 2.4. The Grzegorczyk Hierrchy. The fmily of opertions mentioned in the previous prgrphs of this section (zertion, ddition, multipliction, exponentition nd tetrtion) belong to n infinite series of hyperopertions, clled the Grzegorczyk Hierrchy, nmed fter prof. Grzegorczyk, distinguished Polish philosopher of Logic (See lso [7]). The hierrchy cn be globlly described by introducing generlized hyperopertor of rnk s s tht will operte on its right in the following scheme: s, such s: s x = s x hyper-opertor of rnk s: s Here, s is n infixed form of the hyper-opertor of rnk s, tht genertes the following scheme: for s=0 s b = 0 b = b zertion for s=1 s b = 1 b = + b ddition (sum) (7) for s=2 s b = 2 b =. b multipliction (product) for s=3 s b = 3 b = ^ b exponentition (power) for s=4 s b = 4 b = # b tetrtion (tower, super-power) We shll see tht rnk s cn ssume the vlues of ll nturl numbers, thus completing n infinite series of hyperopertions. Indeed hyperopertion s b could probbly lso be defined for negtive or non-integer vlues of rnk s. So, we cn foresee the mthemticl existence of opertions, not only with negtive s, but lso with n exotic rnk s = 1,5, between ddition (s = 1) nd multipliction (s = 2), or with rnks s = π, or s = e, or even s = - i (!!). But this my well require lot of dditionl reserch work. 5

6 3. ITERATION AND RECURSION 3.1. Itertions. In the mthemticl procedures pplied to computer science nd rtificil intelligence (A.I.) there re formlisms using concepts such s itertion nd recursion [11]. We intend to tke this into ccount in our nlysis of the problems described in the previous section. We define the m-fold ppliction of function y = f(x), nd we denote it s y [m] = f [m] (x), the itertive ppliction, m times, of function f(x) to the vrible x (or m - 1 times to function f(x)). For instnce, if: y = f(x) we hve: y [2] = f [2] (x) = f [f(x)] nd: y [m] = f [m] (x) = f [f(..f(x))] pplied m times to x. It is very interesting to see, then, wht hppens if, by definition, we choose for f(n) series of functions Φ s (n), such s: Φ 1 (n) = 2 + n Φ 2 (n) = 2. n Φ 3 (n) = 2 ^ n Φ 4 (n) = 2 # n (iterted exponentition, or tetrtion). 3.2. An exmple of nested ddition. Concerning Φ 1 (n), we cn esily see tht, if: Φ [1] 1 (n) = 2 + n = 2. 1 + n we hve: Φ [2] 1 (n) = 2 + (2 + n) = 2. 2 + n nd: Φ [3] 1 (n) = 2 + (2 + (2 + n)) = 2. 3 + n nd, in generl: Φ 1 [k] (n) = = 2. k + n or: Φ [m-1] 1 (n) = 2 (m 1) + n nd, finlly, for n = 2: Φ [m-1] 1 (2) = 2 (m 1) + 2 = 2. m Therefore: (8) Φ [m-1] 1 (2) = Φ 2 (m) = 2. m We cn therefore see tht the nested itertion of function Φ 1 (n) = 2 + n on itself, m-1 times, produces (for n=2) new function Φ 2 (m) = 2. m. 3.3. An exmple of nested multipliction. In similr wy, we cn proceed with function Φ 2 (n) = 2. n. In fct, if: Φ [1] 2 (n) = 2. n = 2 1. n we hve: Φ [2] 2 (n) = 2. (2. n) = 2 2. n nd: Φ [3] 2 (n) = 2. (2. (2. n)) = 2 3. n nd, in generl: Φ [k] 2 (n) = = 2 k. n or: nd, finlly, for n = 2: Φ [m-1] 2 (n) = 2 m-1. n Φ [m-1] 2 (2) = 2 m-1. 2 = 2 m In conclusion: (9) Φ [m-1] 2 (2) = Φ 3 (m) = 2 m = 2 ^ m Agin, the itertive ppliction of function Φ 2 (n) = 2. n on itself, m-1 times, produces (for n=2) new function Φ 3 (n) = 2 ^ n. 3.4 Recursive extensions. We cn esily verify tht it must lso be: (10) Φ 3 [m-1] (2) = Φ 4 (m) = m 2 = 2 # m And, s foreseen, the process cn continue, s we hve lso previously foreseen. It is possible to put together these vrious results by writing, in generl: (11) Φ s (n)= Φ s-1 [n-1] (2) This expression cn be considered both s the result of the itertive ppliction of n opertor nd s recursive function, since it is function tht refers to (nd opertes on) itself. A celebrted exmple of two vrible recursive function is Ackermnn s Function (Willim Ackermnn, 1886-1992, see [4]), which hs gret importnce in theoreticl computer sciences nd is very relevnt to the subjects we re exmining in this pper. 6

7 4. ACKERMANN S FUNCTION 4.1. Ackermnn s function. Ackermnn s Function A(s,n) is described by the following recursive tble, the elements of which (written in mtrix boxes ) cn be built up s follows: the first box of the tble, with co-ordintes s=0 nd n=0, contins vlue A(0,0) = 1; ll the boxes in row s=0 contin numbers obtined by stting: A(0,n) = n + 1, in other words, they contin vlues: A(0,0) = 1, A(0,1) = 2, A(0,2) = 3, etc.; in row s=1, nd for n 1, ech box A(1,n) of the row contins number found in column n of the top row (s=0), where n is equl to the contents of box A(1,n-1), i.e. t the immedite left of A(1,n); e.g. the contents of box A(1,2) is equl to the content of box A(0,3) = 4, which is equivlent to sying tht: A(1,n) = A(0,n+1); for ll the boxes in column n=0, we hve A(s,0) = A(s-1,1), s highlighted in the tble, for exmple, concerning elements A(1,0) = A(0,1) = 2; in ll the other boxes, s indicted for box A(2,2), we hve: A(s,n) = A(s-1,(A(s,n-1)). A(s,n) n=0 n=1 n=2 n=3 n=4 n=5 n=6 s=0 A(0,0) A(0,1) A(0,2) A(0,3) A(0,4) A(0,5) A(0,6) 1=2 3-3 2=2 4-3 3=2 5-3 4=2 6-3 5=2 7-3 6=2 8-3 7=2 9-3 s=1 A(1,0)=A(0,1) A(1,1)=A(0,2) A(1,2)=A(0,3) A(1,3)=A(0,4) A(1,4)=A(0,5) A(1,5)=A(0,6) A(1,6)=A(0,7) 2=2+3-3 3=2+4-3 4=2+5-3 5=2+6-3 6=2+7-3 7=2+8-3 8=2+9-3 s=2 A(2,0)=A(1,1) A(2,1)=A(1,3) A(2,2)=A(1,5) A(2,3)=A(1,7) A(2,4)=A(1,9) A(2,5)=A(1,11) A(2,6)=A(1,13) 3=2.3-3 5=2.4-3 7=2.5-3 9=2.6-3 11=2.7-3 13=2.8-3 15=2.9-3 s=3 A(3,0)=A(2,1) A(3,1)=A(2,5) A(3,2)=A(2,13) A(3,3)=A(2,29) A(3,4)=A(2,61) A(3,5)=A(2,125) A(3,6)=A(2,253) 5=2^3-3 13=2^4-3 29=2^5-3 61=2^6-3 125=2^7-3 253=2^8-3 509=2^9-3 s=4 A(4,0)=A(3,1) A(4,1)=A(3,A(4,0)) A(4,2)=A(3,A(4,1)) A(4,3)=A(3,A(4,2)) A(4,4)=A(3,A(4,3)) A(4,5)=A(3,A(4,4)) A(4,6)=A(3,A(4,5)) 13=2#3-3 65.533=2#4-3 2#5-3 2#6-3 2#7-3 2#8-3 2#9-3 s=5 A(5,0)=A(4,1) A(5,1)=A(4,A(5,0)) A(5,2)=A(4,A(5,1)) A(5,3)=A(4,A(5,2)) A(5,4)=A(4,A(5,3)) A(5,5)=A(4,A(5,4)) A(5,6)=A(4,A(5,5)) 65.533=2$3-3 2$4-3 2$5-3 2$6-3 2$7-3 2$8-3 2$9-3 The definition of Ackermnn s Function cn be summrised s follows: A(0,n) = n + 1 (12) A(s,0) = A(s-1,1) A(s,n) = A(s-1,A(s,n-1)) 4.2. Hyperopertions progression. When we nlyze the tble, we find extremely interesting structures in the elements contined in the vrious boxes. For instnce, it is surprising to find the following pttern: (13) o in row s=0, by definition: A(0,n) = n + 1 o in row s=1: A(1,n) = 2 + (n+3) 3 = n + 2; o in row s=2: A(2,n) = 2. (n+3) 3 = 2n + 3; o in row s=3: A(3,n) = 2 ^ (n+3) 3 = 2 n+3 3; o in row s=4: A(4,n) = 2 # (n+3) 3 = n+3 2 3; o in row s=5: A(5,n) = 2 $ (n+3) 3, etc.. With the provisionl exception of row s=0, we could re-define Ackermnn s Function s follows: (14) A(s,n) = 2 s (n+3) 3 In other words, by chnge of vrible, we hve: A(s,n-3) = 2 s n 3 or: 2 s n = A(s,n-3) + 3 nd, remembering (10), we cn lso write: (15) Φ s (n) = 2 s n = A(s,n-3) + 3 = Φ [n-1] s-1 (2) 7

This formul, verified for s>0, puts ech hyperopertion 2 s n into binry reltion with element A(s,n-3). of Ackermnn s Function. 8 4.3. Zertion. The cse of the vlues in row s=0 deserves seprte considertion. In fct, to keep the sme pttern, we should lso hve similr formul in the (13) system. By putting s=0 in expression (14) nd remembering (13) we gin obtin wht we hve clled zertion (see [5] nd [6]): A(0,n) = 2 0 (n+3) 3 = n + 1 (zertion) which gives: (16) 2 (n+3) = n + 4 nd, finlly: (17) 2 n = n + 1 (for: n 3) But, since we lredy know tht: (18) 2 2 = 2 + 2 nd tht: n n = n + 2 we cn strt using these expressions in order to find out the first properties of the zertion opertion. 4.4. Recursive properties. We my conclude, in fct, tht zertion, ddition, multipliction, exponentition, tetrtion (etc.) re generlized nturl hyperopertions tht belong to the clss of recursive functions, for which we know the following reltions (the first line being presented s n homomorphic extension of the others): s = 1 Addition: x + 0 = x ; x + (y + 1) = x (x + y) s = 2 Multipliction: x. 0 = 0 ; x. (y + 1) = x + (x. y) s = 3 Exponentition: x ^ 0 = 1 ; x ^ (y + 1) = x. (x ^ y) s = 4 Tetrtion: x # 0 = 1 ; x # (y + 1) = x ^ (x # y) or, generlly: (19) x s (y + 1) = x s-1 (x s y) 4.5. The Grzegorczyk Hierrchy. Therefore, from the definition of Ackermnn s function, new rithmeticl opertions utomticlly follow. In prticulr, we should like to stress the following two: A(0,n) = 2 (n+3) 3 = n + 1 (zertion) A(4,n) = 2 # (n+3) 3 = n+3 2 3 (tetrtion) The terms tetrtion nd zertion nd their opertor symbols ( # nd ) re proposed by the uthors. W. Ackermnn hs ctully estblished the existence of n infinite spectrum of rithmeticl opertions, sometimes referred to s the Grzegorczyk Hierrchy. Conventionl mthemtics nottion breks down t this point nd something new needs to be specificlly devised, s we shll see, in order to represent very big numbers. We shll describe in detil the tetrtion nd zertion opertions, in the following sections. 5. TETRATION 5.1. Construction of Tetrtion opertion. As we hve seen in n elementry wy in section 2, the construction principles governing opertions such s multipliction nd exponentition re well known: (20). n = + + +.. + ----- n times ------- (21) ^ n =....... ----- n times ------ where: n N (Set of nturl numbers). By nlogy with these formuls nd reclling (5), we cn write similr formul for tetrtion: (22) # n = ^( ^ (..( ^ )) -------- n times --------- with: n N. 8

9 We must however observe tht both exponentition nd tetrtion re not commuttive, i.e. tht: (23) ^ n = n n ^ = n # n = n n # = n We shll see tht they don t stisfy the ssocitive property either nd we cn nticipte tht ll opertions with rnks different from those of ddition nd multipliction hve properties which re different from those vlid for the two first bsic elementry opertions. 5.2. Right nd Left Towers. The question to be exmined here is how we cn represent wht we cn cll the compction of n opertion such s: c = ^. It is power tower built up by elevting number to itself. As we hve seen, we cn define new opertion tht we hve clled tower or tetrtion, tht we shll write s follows: (24) c = # 2 = ^ = with : rel number nd tht we could lso show s: c = # 2 = 2 = 2, to be red: -tower-2. As we hve lredy sid, symbol # 2 is used in this pper. It is the first thing tht comes to mind, if we need cler nd free symbol. As we hve lso seen, symbol 2 ws proposed by Rucker (1995) nd ws creted by Knut (1976). The word tower mens precisely tht, but it is lso inspired by n ssonnce with power, since tower is hierrchiclly following the power opertion. The mening of the word tetrtion hs lredy been clrified. The result of tetrtion opertion such s c = # 2 is evidently rel number to the power of itself, which gives n exponentil tower t two levels. If the floor levels (the extension, or the height of the tower) re only two, there re no problems. When this is not the cse, we hve to devise wy of performing this opertion. Indeed, given the non commuttivity of exponentition, the two following expressions re different: ( ) ( ). (Plese note! - For = 2, we exceptionlly hve 2 ^ (2 ^ 2) = (2 ^ 2) ^ 2 = 16, but this is only coincidence). By using the sequentil nottion of stndrd pocket clcultors, we my lso write: (25) ^ ( ^ ) ( ^ ) ^ But the non commuttivity lso implies the non ssocitivity of the opertion. In order to give mening to n expression such s: z = ^ ^ ^ = we hve to choose procedurl rule, otherwise the opertion is not defined. We my in fct hve: priority to the right: z = ^ ( ^ ( ^ )) or, priority to the left: z = (( ^ ) ^ ) ^ (but other configurtions re lso possible). 5.3. Definitions. As we see in the following exmple, strting from power tower with ny number of levels nd with priority to the left, we finlly obtin n inhomogeneous tower, with only three levels, but with priority to the right. It is inhomogeneous, or incomplete, becuse the lst exponent of the tower is different from. This, however, is generlly the cse: since, if (n levels): z = (((( ^ ) ^ ) ^ ) ^.) ^ = (((( ) ) ). ) [n times] we lwys hve:..... [ n 1. times] ( 1 ) (26) z = = n with n: positive integer number Which mens tht ll the left homogeneous towers (with ny number of levels) re collpsible to become right (inhomogeneous) towers, with only three levels. This suggest tht only the right towers re ctully new fundmentl elementry opertions, not utomticlly reducible to others, wheres ll the left towers re to be considered s bnl cses of reducible expressions. We re therefore justified, s shown in section 2, in going on to use n exponentil opertor tht cts on its right, s follows: z = ^ x = ^ x = x with : positive rel number. (The of > 0 condition is chosen for ske of simplicity nd to sty in touch with relity!) 9

10 By continuing the itertion of ^, equivlent to the use of opertor exp ( ), exponentil to the bse of opernd ( ), we re ble to crete complete homogeneous towers (with identicl elements) with priority to the right, for instnce, s in the following exmple (with 5 levels): i,e.: nd we cn gree tht: z = ^ ^ ^ ^ = ^ ( ^ ( ^ ( ^ ))) = z = exp (exp (exp (exp ()))) (27) z = exp (exp (exp (exp ()))) = (without prenthesis nd with priority to the right) In generl, for the right towers, the following expression: ( ( ( ) ) ) (28). z = exp ( (exp ())) = [n times] n times is not reducible. We cn therefore estblish tht this is the true tower, superpower, or tetrtion opertion (with priority to the right), which is shown s follows (in the cse of n itertions): (29) z = # n [-tower-n] (with : positive rel nd n: positive integer). And we hve: : bse of the tower n: height of the tower or super-exponent. 5.4. The super-exponentil function. From expression z = x # y, we hve two min situtions, depending on to which of the x, y vlues is the independent vrible (the other one being prmeter). We cn hve: - z = # y (with > 0 nd rel number, the super-exponentil function), or - z = x # n (with integer n 0, the super-power or tower function). From expression (27), with y = m (nturl), we hve: we my write: z(m+1) = z(m) = exp ( (exp ())) = m times........ [m times] = # m [m+1 times] = # m = # (m+1).... nd lso: z(m -1) = [m - 1 times] = log ( # m) = # (m-1). In other words, we cn sy tht opertor exp ( ) elevtes the tower s height by one unit nd tht opertor log ( ) lowers the tower s height by one unit; i.e. we hve the following importnt recursive properties: (30) # (m+1) = # m = ^ ( # m) # (m -1) = log ( # m) As fr s the tetrtion opertion is concerned, with bse = constnt, we cn esily clculte the vlues of z = # y, for smll vlues of y, strting from the fct tht we must lwys hve: z(2) = # 2 =. Therefore: (31) # 1 = # 2 = # 3 = etc. but lso: # 0 = log ( # 1) = log = 1 nd: # -1 = log ( # 0) = log 1 = 0 nd, finlly: # -2 = log ( # -1) -. The fct remins tht function z(y) = # y (the super-exponentil function) cn be clculted only for integer vlues of y (nd for y -2). The dependent vrible z(y) is therefore quntified discontinuous entity. Any effort to obtin something like continuous plot fces serious problem when it comes to extending tetrtion to rel numbers. Another importnt mtter is tht ll the vlues of z, for y > 5, re finite (for y finite). This is lmost unbelievble, becuse the vlue tht we cn clculte for 2 # 5 = 2 ^ 65.536 is lredy virtully unimginble. 10

11 Let us think of the immensity of number such s 10.000 # 10.000! This fct suggests tht tetrtion could be n excellent tool for representing very lrge numbers (See 7.4). 5.5. The Tower function. Nevertheless, the plot of function z(x) = x # n (with n positive integer), i.e. the tower function with super-degree n, cn esily be obtined by using very simple pocket clcultor, t lest for x 0. For the vlues of x < 0, the dependent vrible z(x) ssumes complex vlues. In the next figure, we plot the vlues of functions z = x # n, tht cn be clled super-power functions, for integer vlues of the super-degree prmeter n = 0, 1, 2, 3, 4, 5, 6... nd for x > 0. 6. THE SUPER-ROOT 6.1. Inversions. As lredy stted in expressions (23), exponentition nd tetrtion re not commuttive opertions, i.e.: x ^ y = x y y ^ x = y x (exponentition) x # y = y x y # x = x y (tetrtion). Let us now compre the two bove-mentioned z functions (of vribles x, y) such tht, by nlogy with: z = x ^ y = x y [power or exponentil] we hve: z = x # y = y x [tower or super-exponentil]. Let us then try to extrct the vlues of x nd y, i.e. to find the inverse functions of z. We hve: - in the cse of the power-exponentil: (32) x = y z = y-rt(z) [y-th root of z] y = log x (z) [logrithm bse x of z] - in the cse of the tower-super-exponentil: (33) x = y z = y-srt(z) [y-th super-root of z] y = slog x (z) [superlog bse x of z] Therefore, we hve the following inverse functions : - for y = n (constnt): (34) z = x ^ n = x n n-th power of x => x = n z n-th root of z z = x # n = n x n-th tower of x => x = n z n-th super-root of z - for x = (constnt): (35) z = ^ y = y exp, bse, of y => y = log z logrithm, bse, of z z = # y = y superexp, bse, of y => y = slog z superlog, bse, of z 11

12 Consequently, the non-commuttivity of the opertions implies tht there must be two different inverse tower opertions, designted s super-root nd super-log. 6.2. The Squre-root. In this section we propose to investigte the possibility of finding some lgorithms for clculting the super-root (order n) of number z, defined s number x tht stisfies the following expression: x # n = z => x = n z = nth-srt z (nth-super-root) In prticulr, we shll try to solve the problem where n = 2, i.e. in the cse where: x # 2 = z => x = 2 z = s-sq-rt z (super-squre-root) The nlysis of this problem strted in 1986 [5] with the review of n itertive formul used since ncient time for clculting the clssicl squre root of number. In fct, if z is the squre of x, i.e.: z = x. x = x ^ 2 = x 2 then, x cn be obtined with: (36) x =sqrt z ( p + z/p) / 2 where number p is first pproximted vlue for the squre-root of z, with z > 0. The result is obtined by n iterted ppliction of the formul, by systemticlly stting x p. It cn be verified with pocket clcultor tht the itertions converge very rpidly to the vlue x = sqrt z. This formul cn be proved s follows. Supposing p to be n pproximte solution, we shll certinly hve: z = x 2 = x. x = z/p. p In fct, more pproximte vlue for solution x could be obtined using the rithmetic men between the two mgnitudes z/p nd p. After putting the rithmetic men s the new pproximte vlue p nd fter certin number of itertions, we get z/p = p = x, the required vlue for the squre-root of z. This itertive formul hs been known for more thn two thousnd yers nd ws pprently used in ncient Greece. 6.3. The Super Squre-root (First lgorithm for clculting the ssqrt ). If we consider the opertions of immeditely higher rnk, i.e. the rnk of exponentition insted of multipliction, we cn suppose tht the following expressions would be vlid: If: z = x ^ x = x # 2 = 2 x then: (37) x = ssqrt z (p. log p z) or: x = ssqrt z (p. p z ) where p, gin, is first pproximte vlue for x, super-squre-root of z. The procedure is indeed similr to tht used for finding the squre-root of z, by incresing ll the opertions rnks by one unit nd by using the geometric, insted of the rithmetic men. In this cse, we observe tht log p z nd p z re the inverse opertions of z = p ^ x (with x = log p z) nd of z = x ^ p (with x = p z ), both corresponding to expression z/p, used for finding the squre-root of z. Here, gin, it cn esily be verified tht the ppliction of one of the lst two formuls (fter repetedly exchnging x p) for the clcultion of the geometric men (of either log p z or p z with p), gives result rpidly converging to the super-squre-root of z, i.e. to x = ssqrt z. Actully, formul (37) ws tenttively dmitted s working hypothesis, by using homo-morphicl mpping between expressions (36) nd (37). It cn esily be verified tht formul (37) rpidly converges, for vlues of the rgument z > 1,7 ( p > 1,6). 6.4. A second lgorithm for clculting the ssqrt. Another lgorithm for clculting the super-squre root of number is bsed on different principle. In fct, very importnt reltion cn be found in n extreme cse, in which n is unlimited, i.e. where the vlue of n. Let us consider the following expression: (38) z = lim x # n =. x x x ( times!) = h(x) 12

13 We propose to exmine the problem of the vlues of function z = h(x) for ll possible vlues of vrible x (rel > 0). Let us tke the logrithm (bse e) of z in (38):. x. x x x ln z = ln ( lim x # n) = ln x = x. ln x = h(x). ln x = ln h(x) ( times) i.e.: (39) ln z = h( x). ln x = ln h(x) which gives: ln x = (ln h(x))/ h(x) = (ln z )/ z = ln (z ^ (1/ z )) nd, then: (40) x = z z ) But, s we remember, from expression (38): x # n = x # = h(x) z = lim n we hve: x = g(z ) = lim z = z Therefore: (41) z z In other words, the -th super-root (s = 4) of quntity z is equl to the z-th (clssicl, s = 3) root of z. This is n importnt result, tht deserves further creful nlysis, since it ppers tht similr formul is lso vlid for the other hyperopertion levels. From formul (41) it follows tht, in generl, we cn write: x = x x or, tht: n lim x = x x Formul (41) llows us to clculte the vlue of the super-squre root of number, for z < 1.7. In fct, let us put: 2 x = y => 2 y = x Then, by pplying formul (41) to rgument 1/y, we hve: 1/ y = 1/y 1/y = (1/y) y = 1 / y y = 1 / 2 y which implies: (1 / 2 y) = 1 / y or, to be more precise: lim [ n (1 / 2 y)] = 1 / y i.e.: lim [ n (1 / x)] = 1 / y or : y = ssqrt (x) = 1 / lim [ n (1 / x)] nd, in order to be consistent with the nottion of (37), we cn write: (42) x = ssqrt (z) = 1 / lim [ n (1 / z)] This is the second formul for clculting the super-squre root of z (for z < 1,7). 6.5. A more generl formul. Let us now reconsider expression (42), tht we cn write s: (: z) # = : ( 2 z ) (with : z = 1/z, reciprocl of z) which cn lso be written s: 2 z = 1 / [(1/z) # ]. In other words, the super-squre-root of z is the reciprocl of the infinite tower of 1/z. This lst formul cn be further developed by expressing the infinite tower of 1/z by using function h(1/z), s defined by expression (38), e.g. by: h(1/z) = (1/z) # nd cn be expressed s: h(1/z) = - w[-ln(1/z)] / ln(1/z) where w(u) is the solution of: u = w e w nd it is clled is the Lmbert function or Product Log (see Appendix A-06). Therefore we hve tht: 2 z = 1 / h(1/z) = - ln(1/z) / w[-ln(1/z)] 13

14 And, finlly, by chnging the vribles, we obtin: 2 (43) x = ln(x) / w[ln(x)] (with x > e 1/e, x 1) This is the third (more generl) formul cn be dopted for the clcultion of the super-squre-root of number x. Condition x e 1/e is lso geometriclly justified by the grphicl inversion of function: x = y # 2 (y = 2 x ). The grph of y = 2 x is shown in the following figure. The super-squre-root of x for x < e 1/e (0,692 201 ) involves complex solutions. The unique rel vlue of ssqrt(e 1/e ) is 1/e. For e 1/e < x 1, the super-squre root hs two rel vlues. For x > 1, it is continuous incresing function nd we hve: lim ssqrt( x) =. In prticulr, vlue w(1) = w 1 = x 1,567 143 is the omeg constnt, obtined for w. e w = 1. For x = e, we hve: 2 e = 1/w 1 = 1,763 220. 7. THE SUPER-LOG 7.1. The Super-logrithm. The tetrtion opertion is only well defined, in the field of rel numbers, for integer vlues of the super-exponent, greter thn -2. In other words, in expression: (44) z = x # y = y x = x y (x-tetrted-y) tetrtion is only vlid for super-exponent y = n, with n > -2 (for n = -2, the vlue of z is unlimited, i.e. z - ). For the domin of y < -2, the estimtion of z requires us to use the logrithms of negtive numbers. We shll exmine here the ppliction of expression (44) to the cse in which the bse x is rel number, lrger thn 1, i.e., for: x =, with > 1. The bove-mentioned expression will be trnsformed, in this cse, s follows: (45) z = # y = y = y From expression (45), nd tking lso into ccount definitions (33) nd supposing we know the vlue of z, we cn extrct y, s follows: (46) y = slog z (the super-log, bse, of z) 14

15 The super-log, bse, of z is the vlue, or the height, of the super-exponent y tht should be ssigned to number, for obtining number z. From expression (44) nd bering in mind its restrictions, we must ccept tht quntity y hs been defined only for vlues of z for which y is n integer number > -2. But, since z is supposed to be ny rel number, the problem posed here is how to extend the vlidity of the super-log definition so s to eliminte this restriction, if possible. We shll, in prticulr, see tht the extension of the superlog to the rels is closely connected to the solution of the problem of extending tetrtion itself to the rels ( 7.3). 7.2. Incomplete Towers. Furthermore, in the recent scientific literture, we cn find expressions such s: p (47) z = = ^ ( ^ p). This (generlized) tetrtion ppers s n incomplete tower opertion, incomplete becuse the lst exponent of the iterted exponentition is different from bse. However, this kind of expressions is commonly found in scientific texts concerning, for instnce, problem-complexity theory, gme theory or in some AI developments. How cn we incorporte this incomplete tower into kind of generlized tetrtion? In order to investigte the possibility of doing this, let us gin use the following exponentition opertor: (48) ^ p = ^ p n nd: ^ p = ^ ( ^ ( ^ ( ^.. p))) with n itertions of ^ Let us now suppose tht quntity p is rel number lrger thn 1 nd smller thn. This is often the cse in expressions similr to wht we hve clled incomplete towers. If this is not so, we cn trnsform these expressions into cnonicl form, with 1 < p <. Let us now define nother new tool, the logrithm opertor, complementry to ^, i.e.: (49) log p = log p nd: log n z = log (log (log (log.. z))) with n itertions Now, if we hve: n z = ^ p we could lso put: z = # (n + q). with: 0< q < 1 In fct, the incomplete tower hs n extension (its super-exponent) tht must be higher thn n, otherwise we would hve hd p = 1 (q = 0), nd must be less thn n + 1, otherwise we would hve hd p = (q = 1). In other words, we cn write: (50) n z = ^ p = # (n + q). But, by itertively pplying the log opertor, we lso hve: log n z = p = log n [ # (n + q)] = # (n + q n) i.e.: (51) p = # q with 0 < q < 1 nd 1 < p < q = slog p for ny n. 7.3. Strtegy for extension to the rels. Formuls (50) nd (51) provide strtegy for defining generlized tetrtion opertion z = # y (s well s continuous super-log function), extending its vlidity to ll the rel vlues of its super-exponent rgument y nd, so, including the results of the incomplete tower opertion. Once vlue for p is chosen, depending upon the cnonicl form of the incomplete tower, vrible q is clculted s the super-log, bse, of p. Knowing p nd q, if they cn be estimted, would therefore be essentil for the extension of tetrtion to the rels. Reclling the tetrtion expression z = # y, vrible p will be used to define the vlue of z nd vrible q will fix the y coordinte. The problem is therefore to clculte q s the super-log, bse, of quntity p (with 1 < p < ), bering in mind tht the result of the opertion for q should be such tht 0 < q < 1. 15

16 In some cses these difficult clcultions (ssuming tht they re ctully possible) re not necessry, s in the following exmples of super-logs, bse 2, concerning certin known integer quntities: slog 2 16 = 3 becuse: 2 # 3 = 16 slog 2 65.536 = 4 becuse: 2 # 4 = 65.536 However, there is the further possibility of simply indicting the super-exponent extension, without ctully performing the clcultion of the non integer vlue of the super-exponent n + q of tower, or of super-log, i.e. by exctly estimting the vlue of q (with n integer nd 0 < q < 1). This cn be done, very precisely, by using n opertor tht would merely highlight the vlue of p (with 1 < p < ). This opertor is the so-clled conctention opertor, from now on shown s n sterisk (*), with reference to the second formul of the (49) set nd to (50), s follows: n (52) ^ p = ( # n) * p = # (n + q). Formul (52) shows n incomplete tower built up from iterted exponentitions of number, on itself, nd terminted by lst exponent (the super-exponent extension) equl to p. In other words: ( # n) * p = ^ ( ^ ( ^ ( ^.. p))) or: ( # n) * p = ^ ^ ^ ^ p (with n times nd priority to the right!) 7.4. Tetrdic Representtion of numbers. The few exmples shown in Appendix A-03, re sufficient to give n ide of how tetrtion cn be used to represent very big numbers in wht we could cll Tetrdic Representtion. D. W. Lozier nd P. R. Turner hve published ppers ([8], [9], [10])describing number formt clled Symmetric Level-Index (SLI), in which numbers re stored in the form:. P e N = e ^ (e ^ (e ^ p))= e, where p is frction from 0,000 to 0,999... nd there re s mny e's s necessry. For exmple: 10 = e ^ (e ^ 0,834 032...) = 2 e * 0,834 032 143 = e ^ (e ^ (e ^ 0,471 239 ) = 3 e * 0,471 239 The dvntge of this proposed system is tht there will not be ny computing overflow or underflow if we perform finite number of opertions such s + - nd /. In one of their rticles, Lozier nd Turner proposed formt tht uses 3-bit level field with 2 sign bits nd the remining bits (59, if it is 64-bit word) for the frction p. This lets us represent numbers s high s the number represented by tower of seven e s. This is the highest known number tht cn be hndled by computer numberrepresenttion system. However, there is lso computing softwre clled Hyperclc tht goes even higher. 8. ZERATION 8.1. Homomorphisms of inverse opertions. In section 2 we introduced zertion, the new opertion with rnk lower thn tht ddition, in the hyperopertion hierrchy. The bsic chrcteristics of such opertion re described in formuls (16), (17) nd (18) nd should be consistent with tble (6). This mens tht we must hve: - from (6): = + 2 (n times) = + n - from (17): 2 = + 1 (for > 2) In order to discover the properties tht zertion must hve, for it to be comptible with known opertions, let us consider expression (42): (53) n lim z = z z [= g(z)] This formul cn esily be compred with similr expressions used in other mthemticl fields, such s: (54) lim n z = z / z [= 1] nd: lim z / n = z z [= 0] This comprison is mde becuse ll the (53) nd (54) expressions involve the inverse opertions (of the root type) of the homologue hyperopertions of three pirs of contiguous rnks (s=4, s=3), (s=3, s=2), (s=2, s=1). Therefore, s working hypothesis, it cn be ccepted tht n equivlent formul might be vlid involving inverse opertions of the (s=1, s=0) rnk pir. 16

17 Indeed, the inverse opertions, of the root type (left-inverse opertions) of the hyperopertions of rnks 4, 3, 2, 1 nd 0 cn be shown s follows: (55) s = 4 z = x # n => x = n z s = 3 z = x ^ n => x = n z s = 2 z = x. n => x = z / n s = 1 z = x + n => x = z - n s = 0 z = x n => x = z n In the lst line, we implicitly defined the inverse opertion of zertion, indicted with the Delt opertor. First of ll, we observe tht expressions (53) nd (54) imply the clcultions of lim x nd, then, tht in expression (54) the sme clcultion, for the rnk s=0, is not there. In other words (nd respecting the homomorphic mppings) we should hve: (56) lim z - n = z z [= - ] Expression (56) mens, if it is cceptble, tht - should operte s the unit element for the zertion opertion, i.e. we should hve:. 1 = + 0 = (57) ( - ) = (unit element) We must observe tht, if we hve x = z z = -, then we must lso hve z (- ) = z, i.e. quntity - indeed cts s the unit element for the zertion opertion. 8.2. Homomorphisms of lgorithms. At this point, let us recll the formuls used to clculte, by itertive ttempts, the super-squre root nd the squre root of number (36 nd 37). From: z = x ^ x = 2 x we hve (from 37): x = ssqrt z (p. log p z) [with x p], or: x = ssqrt z (p. p z ) [with x p]. Also, from: z = x. x = x 2 we hve (from 36): x = sqrt z ( p + z/p) / 2 [with x p]. As we hve lredy noted, formuls (37) nd (36), the vlidity of which we cn verify, re linked together through homomorphic mpping tht reltes opertions of contiguous rnks, by decresing their rnk by one unit. In fct, tetrtion corresponds to exponentition, multipliction to ddition nd, s fr s the inverse opertions re concerned, logrithm nd root correspond to the division opertion. At this point, we cn tenttively pply the sme mpping model, by decresing the opertions rnk by one unit (with sqrt corresponding to ½, ddition to zertion, division to subtrction), nd so obtining: from: z = x + x = x. 2 by nlogy with (37, 36), if z is n even integer number, the following formul is vlid: (58) x = z / 2 (p (z p)) 2 [with x p]. with p Z (Z: set of integers) By chnge of vribles (x-p p), we cn ssume tht the following reltion is lso verified: x = z / 2 ((z p) p) 2 nd conclude tht, within the vlidity of (58) zertion must be commuttive. However, it is not ssocitive. 8.3. Properties of Zertion. We cn therefore put together the opertionl properties tht, in our working hypothesis, should be stisfied by zertion, grouped into new set of formuls: (59) - from (6): = + 2... (n times) = + n - from (17): 2 = + 1 (for > 2) - from (58): b = b (commuttivity) - therefore: 2 = + 1 (for > 2) - from (57): (- ) = (- : unit element) 17

18 We must lso remember expression (19), for s=1: x + (y + 1) = x (x + y) (y > 0) nd, by putting x = nd remembering (59): ( + y) = ( + y) = + y + 1 = ( + y) + 1 for ny y, by putting + y = d >, we must hve: d = d = d + 1 (d > ) From these formuls we conclude tht: (60) b = mx (,b) +1 (for b) nd: b = + 2 = b + 2 (for = b) nd: (- ) = (- ) = In conclusion, the properties of zertion ( b) should be those shown in the following tble ( b = b ): (61) b = + 1 for: > b b = b + 1 for: < b b = + 2 = b + 2 for: = b b = for: b = - b = b for: = - 8.4. Grph of Zertion function. As n exmple, the grph of zertion function, defined s: y(x) = x = x cn be shown, for = 2, in the digrm of the next figure (y = 2 x). The digrm shows the following chrcteristics of the zertion plot: the zertion grph shows constnt brnch, y = 3, for x < 2. In the generl cse, i.e. for y = x, this brnch is y = + 1, for x < ; the zertion grph is liner (with 45 slope), y = x + 1, for x > 2. In the generl cse, if we put y = x, there is second brnch, y = x + 1, for x >, with discontinuity in the tngent in x = ; there is discontinuity of y, for the vlue x = 2, where we hve y = 2 + 2 = 4. In generl, for y = x, the vlue of y for x = is y = + 2 nd the limit of y, for x, (i.e.: y = + 1), is different from y() = + 2; the y = x + 1 stright line is chrcteristic of ll the zertion opertions of type y = x = x, disregrding the vlue tht constnt cn ssume; it could be considered s the support stright line for ll the zertion functions, defining the second brnch of ll the grphs (the first brnch is defined by the vlue of the constnt ); the vlue of y = x, for x = -, i.e. y = (- ), is nother discontinuity of the zertion function, not shown by the grph, since we hve: y = (- ) = (insted of + 1), nd (- ) x = x (insted of x + 1). This fct, s derived from Rubtsov s demonstrtions (see [5],[6]), must be crefully emphsized. In conclusion, the zertion function (y = x, with : rel constnt), prt from the discontinuity of tngent, is chrcterised by two other mjor vlue discontinuities: - for x = where y jumps up by one unit; - for x = - where y jumps down by one unit. 18

The zertion grph (y = x = x ) tkes the form of broken stright line, with strong discontinuity t point x =. 19 9. PRACTICAL APPLICATIONS OF ZERATION 9.1. Discontinuities. In the physicl nd technicl fields, the mjority of nlyzed events re connected with modifictions to the internl structure of subject of reserch (submitted to the ttention of n observer) nd, s corollry, with modifictions to functionl reltions of the subject with its environment. In most cses these modifictions involve sudden chnges in the mesured physicl mgnitudes. Modern mthemticl methods re not perfectly suitble for describing digitl processes. As rule, discontinuous functions, impulse nd step functions re pproximted by Stieltjes integrls, Fourier series, generlized by functions nd differentils, s defined in the frmework of the Lurentz theory. Use is lso mde of vrious lineriztion models, spline pproximtions, etc., to describe these processes. However, ll procedures for dpting such mthemticl models still require rigorous demonstrtion [21]. But, in some cses, we need to use opertions tht llow us to express ll the segments of broken liner, or discontinuous, process with single function. Among the existing tools, prticulrly used in digitl signl nlysis, there re the well-known Dirc s impulse function nd Heviside s unitry step function, which, however, re not defined by mens of elementry opertions. In the following prgrphs, we shll see how it is possible to introduce, in n elementry nd logicl wy nd without hving to rely on merely d-hoc definition, set of such functions, once zertion is ccepted s one of the elementry hyperopertions. 9.2. The Step function. Heviside s unitry step function z = H(x) is defined s follows: z = H(x) = 0 for: x < 0 z = H(x) = 1 for: x 0 Function H(x) is well known tool used in digitl circuit nlysis nd it hs very simple, continuous nd regulr Lplce trnsform, function 1 / p: px e H( x) dx = 1/ p (p: complex vrible) 0 In order to see how zertion cn llow us to void n d-hoc definition of H(x), let us consider expressions (60) nd (61), re-interpreted s follows, where n independent vrible x is supposed to be zerted to constnt, rel number, : x = mx (x, ) + 1 if. x x = mx (x, ) + 2 if : x =. 19