Informatica 2 1 (1997) 79-114 7 9 Informational Graph s Anton P. Železnikar An Active Member of the New York Academy of Sciences Volaričeva ulica 8 SI—1111 Ljubljana, Slovenia Email: anton.p.zeleznikar@ijs.si Keywords: graph definition; informational gestalt, graph, operand, operator, star gestalt, transition; parenthesis pairs; primitive parallel formula system; serial-circular formula system Edited by: Jifi Šlechta Received: August 6, 1996 Revised: December 3, 1996 Accepted: December 17, 1996 Informational graph seems to be one of the most basic underlying structures (circuits [1, 4], frontal lobes functions [10], schemata [15], impressions, etc.) for the concept and possibilities ofinforming and its understanding. The graph behaves as a regular parallel informational system of formulas (entities) with its own possibilities of informational spontaneity and circularity. By means of informational graph, it is possible to explain the origin ofthe so-called informational gestalt and, besides, the arising of informational formulas especially concerning the so-called causality in regard to the position of the formula parenthesis pairs. Another view of the graph lies in the moving along the arrows in the graph, that is, a formula construction, when choosing a path and setting parenthesis pairs in the emerging well-formed formula, in a spontaneous and circular way. This approach, together with the arising of the graph itself, can represent one of the keystones of the informational arising of formulas, the vanishing of their parts, and the changing of the structure during the informational moving through the graph. The paper shows how the informational graph can be understood by the phenomenalism of informational gestalts exerting the causal possibilities of formulas with the same length but differently displaced parenthesis pairs. Several examples are formalized. Introdliction original (initial) formulas. As such, it appears as a schematic pattern of operands and opera-Informational graph1 is a graphical imitation (in-tors, in which the user can set parenthesis pairs formational presentation, and circuits [4], frontal Spontaneously, getting an arbitrary causal depen­lobes functions [10], schemata [15] in the neuro- dence of operands (informational entities) within logical sense) of a serial informational formula sys- j^ e graph's pattern2 . tem (a system of parallel serial formulas) or also Informational graphs can be comprehended as •a parallel system of basic (atomic) informational generalizations of informational formulas (as par-transitions (without any parentheses pairs). In- alle l system s 0f cer tai n serial formulas) from formational graph performs like an imprint along whic h variou s formula reconstructions are possi­which different formula interpretations are pos- bl6 j th e num be r of which depends on the involved sible. In this sense, an informational graph pre- informational operators, that is, on the formula serves the sequence (direction) of the occurring in-lengthm Th e numb e r of possible graph interpre­formational operands and operators, but does not consider (that is, ignores) the parenthesis pairs of 2Informational phenomenalism joins the terms repre­ senting phenomenology, ontology"and causation in regard 1This paper is a private author's work and no part of to an informational entity. An informational operator \= it may be used, reproduced or translated in any manner exerts an existential (Being-like) as well as causal property whatsoever without written permission except in the čase ofthe operand(s), to which the operator belongs (connects of brief quotations embodied in critical articles. them). 8 0 Informatica 21 (1997) 79-114 tations by formulas grows rapidly by the number of the occurring binary operators in the formula as we shall show in the study which follows. We shall learn also in which way the most rational (unique) formal description of an informational graph is possible and how interpretations of the circular graphs by circular formulas can carry a substantial degree of expressional redundancy. On the other side we have to determine a set of new concepts concerning graphs and their inter­pretations especially by informational gestalts. It has to be answered rigorously what does an infor­mational graph represent and how can it be used for the generation (emerging) of different formula interpretations. For the sake of the understanding clarity we can introduce special primitive graph­ical symbols by which graphs of any complexity can be presented in the form of graphical sketches (schemes, circuits). For example, complex circu­lar informational graphs can be studied from the different points of view. On one side, such a graph appears as a relatively clear picture to the user; on the second side it can be described formalh/ in the most rational form by a parallel system of the primitive informational transitions; on the third side, it can be expressed by a parallel svstem of arbitrary serial and circularly serial formulas, con­sidering only those of them which in a given čase represent the reality (rationality) of the problem. An informational graph represents ali possible interpretations which number depends solely on the involved binary operators. Additionally, as interpretations of the graphs, the so-called star gestalts of graphs can be introduced which illus­trate the moving through the graph from an initial operand, constructing serial and circularly serial formulas of different lengths, for example from the length i. = 1 (basic or atomic transition) on to an arbitrary length L = n. Thus, systems of formulas belonging to a circular star gestalt from a circular graph can be constructed (by a parenthesizing) in an arbitrarily lasting way. Informational graphs are visual (texts, images), acoustic (voices, music, noise), tactile (Braille script), taste (food evaluation), smelling (perfume competition), etc. They are static and dynamic. Some practical examples of static informational graphs are literature texts, artistic pictures, draw­ings, music notes, etc. Examples of dynamic in­formational graphs are theatre performance, TV A.P. Zeleznikar and radio transmissions (mixed visual and acous­tic), everyday happenings in characteristic situa­tions (common patterns of behavior and uncom­mon reactions), etc. For aH these graphs it is typical that they are not 'parenthesized'. Under­standing of the mentioned phenomena (together with Parenthesizing, causation, interpretation) is left to the observer. It means that the causal structuring of a graph belongs to the domain of the observer and that different observers can causally-differently structure one and the same static or dynamic informational graph. 2 Elements of Informational Graph s Elements of informational graphs are circles (or ovals, if their informational markers are longer or complex) and arrows (vectors). Parallel and al­ternative input and output buses contribute only to the compactness of graphs. Circles (operand atoms) are marked by operands which represent informational entities in informational formulas. Arrows (operator atoms) are marked and un­marked. Unmarked arrows represent the most general operator |= and are used also in situations where ali of the unmarked arrows belong to one and the same operator, e.g., to the implication operator =$• in the global implication axioms of Hilbert [8]. The symbolic atoms of informational graphs are presented in Fig. 1. operand a operator |= operator |=p Figure 1: Graphical sijmbols for informational operands and operators (atomic elements). The symbols in Fig. 1 can be connected in the form of a graph representing a formula or formula system, but without parenthesis pairs. In this way the order of operands and operators remains preserved (exactly in cases of a single formula, and to some extent in cases of formula systems). In Fig. 2 some elementary graphs for the most basic phenomenal occurrences of an informational entity a are presented. These occurrences are (1) operand (entity) a as such, INFORMATIONAL GRAPHS Informatica 21 (1997) 79-114 8 1 a (a j a| = & (1) (2) •\(XJ \=ot a \= a (3) aN« •©— (tf) . N a (5) Figure 2: Informational graphs for the basic (phe­nomenalistically structured) informational formu­las. (2) cVs externalism a \= (informing for others), (3) a's internalism |= a (informing for itself), (4) cv's metaphysicalism a (= a (informing in or within itself), (5) a's phenomenalism (a \=; \= a) (informing as such), and (6) for the sake of clarity, the entire modus agendi3 of an informing entity a, that is, its phenomenalistic and simultaneously metaphysicalistic informing as a system of (a\=;\=a;a\=a). (1) Figure 3: Informational graphs representing (phe­nomenalistically) the structure of data 5. Data, marked by S, is a characteristic informa­tional entity which can be graphically presented by a particularized graph being essentially differ­ent from the modus agendi belonging to a gen­eral informational entity, marked by a. The data 3Modus agendi of an informational entity, exerting an entity characteristic internalism, externalism, and meta­physicalism, has the potentiality to be compared with the Self behaving self-sensitive and self-active in regard to its environment (exterior) and interior (the subject regarded as the object of its own activity). graph is sketched in Fig. 3, where the feedback loop is marked by the equality operator '=' , real­izing the metaphysical situation 5 = 5, belonging to data. In čase (2) one can introduce also the explicit operator of non-informedness of data 5, that is \L 5. In this čase, the non-informedness of 5 means the absence or impossibility of any exte­rior or interior impact on data 5. In the čase of input and/or output parallelism concerning an informational entity a, we intro­duce, for the sake of graphical transparency, a kind of input and output lines (parallel buses), respectively. Some specific cases are presented in Fig. 4. (1) «1= •©• h« " 1 (2) /»i N «2 ^2 /32 N —{Ly­ (3) •0 ­ M 5, Figure 4: Informational graphs representing characteristic cases of informational parallelism, where vertical double-bars (||) are parallel input and output collection lines of operand (entity) a. The graph in Fig. 4, čase (1), shows how the parallel decomposition of operand a might be senseful, that is, in the form of implication a =L-. ; 82 Informatica 21 (1997) 79-114 A.P. Zeleznikar which expresses the a's unlimited input (internal­istic) and output (externalistic) potentiality. The unique description of the graph is presented by a system of the incomplete basic transitions, that is, a ^ (N a; \= a; • • •; |= a; a \=; a \=; • • •; a |=). Otl)«1 Ka 2 «2 On) (1) (2) Čase (2) in Fig. 4 can be described formally as /7N«i;\ \ / A N \ |="2; A N N« N \\ N «m/ / VAN/ But, as we see looking at the graph, there exists stili another formula interpretation of the graph, in the form / / A H\\ t=«2; A N N a |= \H"m/ v vA i=yy The unique description of the graph by elemen­tary transitions is in the form of the parallel for­mula system, which is f\=ai\ N a2\ 1= «m; \ a\ \= a; «2 N «; "m |= a a NA; a NA; ot NA ; V A N; A h; AN y Čase (3) in Fig. 4 includes a loop of parallel operands and a possible formal description of this graph is /NTI;\ // /ČIN\\ \ N 72; , N a\= N« Vi=w vv \8q\=JJ J The reader can guess in which way stili four other interpretations of the graph (3) in Fig. 4 are pos­sible. On the other side, the very unique graph description, including ali possible graph interpre­tations, is the parallel formula system /N71 ; N 72; •; N 7P; \ 71 N« ; 72 N «; • ••; 7P N «; a N <*i; a\= 52; •• • ; a N 5q'i \* i N« ; h N «; • • ; sq N « / Let us show two examples which explain the graphical presentation of parallelism. Figure 5: Graphically, the parallel connection of operands a\, OL ,P\=(-r\=P). ©-d H (ž>-©-i (3) Figure 6: The joining oftivo serial graphs given by formula systems (1) and (2) into a unigue serial-parallel graph determined by the adeguate formula system (3). Fig. 6 shows in which way two circular serial graphs (1) and (2) can be joined into the circular serial-parallel graph (3). In this way the graphi­cal transparency of the resulting formula system is essentially improved. The serially circular for­mula system, being parallelized by joining graphs (1) and (2), means a (graphically) equivalent form INFORMATIONAL GRAPHS ^Nfr^^ 7)^(^( 7 TNf t V/3N7/ čH^llM; a: 0 N d;')1= 0 V / This formula describes the joined systems (1) and (2) in Fig. 6 (compare to the eight formulas). There exists a sort of meaning equivalence (op­erator ==) between the transition formula system describing to the graph (3) in Fig. 6 and the serial-parallel formula system on the right side of oper­ator ==. Similar role as the parallel informational op­erator ||=, that is semicolon ';'> na s the alterna­tive informational operator |=aiternativeiy-to> that is comma ','• In an informational graph we use a specially marked alternative bus (instead of par­allel bus) to distinguish the particular čase of al­ternativism. For instance, if in cases of Fig. 6, ali semicolons are replaced by commas, the alterna­tive situation is presented in Fig. 7. '7l=(0l=7),> a h (/? N «0, (7 1= /3) h 7, (1) (2) (/3 h «) (= /3, (/3 h 7) N /3, ./»1= (7 M) , ®-i ®-®n v (3) v y^A •© A A Figure 7: T/ie joining of two serial graphs given by formula systems (1) and (2) into a unique serial-alternative graph determined by the adequate for­mula system (3). After the discussion by examples we have to answer the question, what does an informational Informatica 21 (1997) 79-114 8 3 graph mean at ali, what is its informational rep­resenting potentiality? 3 Informational Graph as an Informational Entity After the introductory interpretation concerning the informational graph on one side and the infor­mational formula (also formula system) phenom­enalism on the other side we have to construct a precise definition of the informational graph and its meaning, and deduce some consequences of this definition. Further, how can an informational graph arise informationally? To answer this ques­tion we must determine which kind of informa­tional entity or even entities (operands, formulas) does the informational graph represent. DEFINITIO N 1 (A General Graph Determination) An informational graph is a connection of in­formational operands (entities) and informational operators (entities' informational properties) not containing the parenthesis pairs of a formula. Operands are marked circles or ovals represent­ing arbitrary informational entities. Operators are unmarked or marked arrows (vectors) represent­ing adequate informational properties of entities which they connect. An informational graph as a structure of connected and unconnected (iso­lated) graphical elements can represent any pos­sible well-formed informational formulas—serial, parallel, and circular—which can be constructed by a consequent use of the parentheses pairs. • DEFINITIO N 2 (A General Primitive Parallel For­mula Svstem) A general primitive parallel formula system (GPPFS , for short), marked by ip', is a par­allel sequence (system) of the most simple inter­nalistic, externalistic, transitional (serial, serially circular) formulas of length t = 1 (having a single operator) and unconnected (marker) formulas of length L = 0, concerning the informationally in­volved operands. Such primitive formulas are, for example, H ; [internalistic (input) formula] [externalistic (output) formula] [transition formula as a serial or serially circular formula] [unconnected marker formula] v: 84 Informatica 21 (1997) 79-114 t, o, 1 TliT2i / € V [primitive operands domain] v J where set V represents ali possible primitive (the most elementary) operands or operand markers. Thus, for example, ^(l',0,T1,T2,v) ^ I i \= Ti; Ti |= T2; j \T 2 |= o; v J marks a general form of GPPFS. • According to the last definition we can represent an informational graph graphically, as already done in the previous section in Fig. 2, or also sym­bolically, that is by formulas or formula systems, using an appropriate and unique (well-formed) informational symbolism. However, there exists only one form for unique graph representation by informational formulas without parenthesis pairs. THEORE M 1 (A Graph Interpretation by the Gen­eral Primitive Parallel Formula System) An arbi­trary informational graph (drauiing), being for­mally marked by ®(vj (7i>72, • • • ,7n7; vi, v2, •••,«„„)) concerning the connected operands 71,72, • " , 7n7 and the unconnected (informationally isolated) operands vi, v2, • • •, vUv, can uniquely be described by the primitive parallel formula system ip' of the form 2, •••,!>„„ ) — ( 1= 7i; Tj 1=; 7P N= 7«; \ 7*1 lj,7pi7? G {7l, 72, • • •,7n7 } ; Preliminarily unconnected operands v\\ v2; •••; vHv function as entities which could become con­nected in the course of further decomposition of the system ip'. • Proo f 1 (Graph & and Formula System ip') Each graph © can completely be described by the ad­equate formula system ip'. The proof of the the­orem is the following. For any two by the arrow A.P. Zeleznikar connected marked circles (or ovals) in the direc­tion from j p to j q , there is, evidently, 7P |= 7g, where the arrow represents the operator |=, be­ing marked or unmarked. For each arrow in the graph, there is exactly one basic transitional for­mula. In this way, ali operator connections of the graph can be formalized. In principle, a graph can include also unconnected (isolated) operands which are not connected elsewhere. The input arrows leading to some operand markers are for­malized in the form |= 7,, while output arrows leading from some operands have the form 7, |=. This completes the proof of the theorem. • The next question we have to solve is what an informational entity, that is, formula or formula system, does an informational graph, formally de­termined in Theorem 1, imply. We have to prove the informational transformation possibilities of the parallel system in the previous theorem. Ev­idently, a parallel informational system hides the power of another parallelism of serial and circular structure which has a causal origin. PRESUMPTIO N 1 (Interpretation Possibilities of the Graph) An informational graph represents a parallel system of serial and serial-circular for­mulas of different lengths in which operands can be variously interwoven and the parenthesis pairs in the occurring formulas can be arbitrarily dis­placed. This means that the parallel system of the most basic transition formulas, representing the graph (as in Theorem 1), can be informationally transformed in a causally more transparent and for the human observer more understandable par­allel system of serial and/or circular-serial formu­las of different lengths. In this way transformed formula system brings to the surface the explicit possibilities of a causal structure of the occurring formulas which, then, can be chosen as the best fitting ones for the description of an informational situation and attitude. • Serial and circularly serial formulas will be su­perscribed by the formula length n and n + 1, respectivelv, and subscribed by the formula index i in an interval 1,2, • • • ,N_> and 1, 2, • • • , N^ , re­spectivelv, where N_> and N^ , respectivelv, the so-called numeri causae (causal numbers), will depend on the length of a serial and circularly serial formula. Usually, subscript 1 will be given to a serial formula INFORMATIONAL GRAPHS "^(CT , CTi,o-2,---,crn_i,CTn) ;=± ((• • • ((CT 1= °l) \= °i) 1= • • • *n-l ) 1= °n) where er, oi , 02) • • • J cn _i , crn are serial operands, and to a circular formula (((• • • ((w |= wi) |= w2) |= • • • wn_i) |= wn) |= u) where (w,wi,W2,i,, !Wn -i,wn are circular serial operands. The subscribing proceeds consequently (systematically) from 1 to N_^ and 1 to N , re­spectively. DEFINITIO N 3 (A Serial Formula and Its Gestalt) By a serial formula ,ip let us mark any well-formed arrangement of the operands, operators and parenthesis pairs, for which "(^(^CTi,--- ,0-„_i,CT„), > ; Yl+(cr>0'l! • " , ^n-b^n ) ie{l,2,...,JNU ; ^ } = ^ T (2 „") ; 2 (o"> °"i > • • • > tfn-i, o-n)y where n is the length of serial formula being equal to the number of binary operators |= in the formula, i is a systematic (causal) number­ing of serial formulas of length n, N+ is the number of ali possible serial formulas obtained from a serial formula by the displacements of the parenthesis pairs, and T(.ip (a, a\, • • • , an _i , an)) is the so-called gestalt of the serial formula rep­resenting the parallel system of ali formulas of length n obtained from the original formula by ali possible displacements of the parenthesis pairs in the original formula. As evident, operands and operators remain on the initialh/ fixed po­sitions, according to the original (initial) formula "(/^(O-JČTI,---,ern_i,crn). D Informatica 21 (1997) 79-114 85 DEFINITIO N 4 (A Circularly Serial Formula and Its Gestalt) By a circularhj serial informational for­mula . tp let us mark any serially and circularly well-formed arrangement of the operands, opera­tors and parenthesis pairs, for which n+l O , N .ip^iu,^,---,Un-l,LL>n) G t n+1 ° ( \ \ n+l O , s n + l O , \

i, u2, ••• , ojn-i, uin) is in their lengths l_^ and l , that is, t — L_ + 1. Thus, -gestal t T\J cludes 2L. + 1 formulas of length L_^; gestalt T^jip^iu, ux, w2, • • • , un-i, w„)J of a circular formula includes, analogously to the serial structure of its circularity, f2l_ o o formulas of length l formulas of length L = l^ +1 ; and 8 6 Informatica 21 (1997) 79-114 the so-called circular gestalt of the form r { jV^ K wi> ^2, ••• , un-i, un)J m­cludes, considering ali operands in the cir­cular structure (loop, cycle), formulas of length L_^ — L_^ + 1. In this respect, for the gestalts of a serial and cir­cularly serial formula, and for the circular gestalt of a circularly serial formula, the corresponding numeri causae are the following: l (2L, _ . ftt-> \ _L_ (2n\1++1 \Li I ~ n+l U/ ' TO I1 fll?s _i _ /2n+2\. (U^ _J&+1\t° / n+2 ^ n+l) ' o „„ o C2f-> ~\ _ n+ l (2n+2\ °N° i,a>2,---,uin, that is, n+ l O / . .. . (p^(u},Ui,U}2,- • • ,Un-l,Vn) => /n+l O/ , \ ^^(^l,^,--- ,w„,w); > n+ l O / x Vjn^^n,^,^!,---,Wn_2,Wn_i)y In a seriallv circular formula, the circularitv con­cerns ali operands, that is, u,UJI,UJ2, • • • ,u)n-i,u)n. D A.P. Železnikar Wn-1 j ^n ) & Figure 8: The serially circular graph representing n+ž ( n+i) sena ^2 / circular formulas of length i = n + l. Proo f 2 (Circular Informing of Operands in a Cir­cular Formula) For a serially circular formula n+ l O / x iV^^^l'""" )Wn-l,Wn) ^ ((• • • ((J (= Wi ) | = • • • Vn-l) |= Wn ) we can construct the circular graph in Fig. 8. In the loop of the graph, for the remain­ing operands OJ\,---,uin-i,ujn, the existence of the serially circular formulas (viewed as by the parenthesis-pairs-displaced specific functions of n + l operands) n+ l O / N J1 ^(wi,w2 ,-'- ,w„,w); "}2V" (^2, w3, • • • ,un,uj, wi); n+ l O / v is evident. This proves the validity of the theo­rem. D THEORE M 3 (A Graph Interpretation by the Cir­cular^ Serial Formula System) An informational graph (draming in Fig. 9), being formallv marked by ( / io , ^l , L2, ' • • ^nii \ \ oo,oi,o2 ,--> °n0'i 0~pO,0~pl,0~p2, i °~pmv j e , LO qnq i V \V0,Vl,V2,--,^n u / / concerns the systematically grouped and also over­lapping — input (internalistic) operands, marked byl = {l'0,l'l,l'2,-m' i^nj ; — the output (externalistic) operands, symbol­ized byO = {o0, ox, o2, • • • , on w gn ? \ J<70 "^ 5 w gn<, >w k = 0 , 1, •• • ,71 , / / i/ie elements of the connected operands are rep­resented by the union set C = lUOUS U Cl, then Cn U = 0. Evidently, the different sorts of graphs corresponding to the input, output, serial, and serially circular gestalts can mutually (spon­taneously, arbitrarily) overlap. D Serial graphs with operands ap0, • •• ,crpmp Input Outpu t O graphs graphs Circular serial graphs with t = (-o operands uq0,---,u)qrig oo h h n »i N =2] 1= 'n, °n„ \= Unconnected graphs with operands v0,---,vUv O O o Figure 9: A complex (circularly parallel-serially structured) graph concerning input (internalis­tic operands), output (externalistic operands), se­rial (transitional), serially circular, and isolated operands, corresponding to the gestalt in Theo­rem 3. Proo f 3 (Covering the Graph by Serial and Circu­lar Formulas) It is to stress that any operand of the system formula Y^_>. appears in the graph one time only. This situation leads to the over­lapping of input, output, serial, and serially cir­cular graphs. Overlapping means the occurrence of one and the same operand in several formulas concerning the input, output, serial, and serially circular arrangement of operands and operators. The proof of the theorem can be realized by the inspection of the graph. In this situation, serial and circular formulas can be constructed, concerning different paths and loops, respectively, with the aim, to cover the entire graph by this formula description technique. In this process, any form of the formula, irrespective of the set­ting of the parenthesis pairs within it, can be cho­sen. Certainly, for each chosen path or loop of the graph, a formula and to it corresponding gestalt can be determined. Evidently, by such a tech­nique, the graph with aH its circles, ovals, and arrows can be covered, that is described by for­mulas in whole. It is evident that for, the iso­lated operands and for internalistic and external­istic formulas, there is {vi;v2;---\vnv) ^T{vi;v2;---;vnv) and (H H) ^ T(\= Li); (0j |=) ^ T{0j h ) 8 8 Informatica 21 (1997) 79-114 respectively. Thus, the number of causal varia­tions of systern ..$ is 2rnr B 2nq + 2 L ' +E ^0mp + l\mpJ ^ ng + 2 V ng + 1 This completes the proof of the theorem. • DEFINITIO N 5 (Graphical Equivalence of Formulas and Formula Systems) Two formula systems (or formulas), marked by i and $2, are said to be graphically equivalent, if they describe (cover, have, possess) exactly (completely) one and the same informational graph <3. In this čase, we in­troduce $1 =« $2 where = 0 marks the operator of the informational graphical equivalence. The meaning of this graph­ical equivalence can be expressed by the formula ($ i = e $ 2 ) - (<5($i ) = 0 ($2)) in which the general informational equivalence (operator = ) can be used. For a formula system (or formula) $1 and its graph, and a formula sys­tem (or formula) $2 and its graph, the following is alternatively (the comma) informationalb/ im­plied: (<3 ( o' p = 0,l,---,A\ t WqQ,iOgi,U>g2,-'• ,&qnq', 9 = 0,1,--- ,B ; \v0,vi,v2,---,vnv A.P. Zeleznikar (Theorem 1), consisting of input operands 1^, out­put operands 0{0, serial (elementary transitional) operands a^, circular (elementary transitional) operands u ^ and isolated operands viv, which can (with ezception of the isolated operands) arbitrar­ily overlap one another, on the one side, and the graphically equivalent parallel circular serial for­mula system ..^ , on the other side. In this sys­tem, there ezist circular serial formulas (of length 2 and greater), tnhich can be derived from the graph. There is 01 Moreover, tp means nothing else than the prim- o itive parallelization of ,,$_,., by which serial and o circular serial formulas of ,.$_^ are decomposed into primitive (basic) transitions. In this way, H*°) to' inhere / t-o,i'i,i<2, • • • ,tfit ; 00,01,02,-•• ,o n o O"p0,O-pi,crp2,--• j °~pmv j o' t P = 0,1,---,A\ qnq i 9 = 0,1 , ••• ,B; / f\=tr; r = 0, l,---,nt ; os (=; s = 0, l,---,n0; °~pi |= °p,i+l> p = 0,l,---,A\ i - 0, l,---,mp - 1; Wqj f= Ugj+V, Wqnq |= U}qo] 9 = 0,1,-• • ,B; 3 = 0,1,---,nq-l; \vu; « = 1,2, ,n. The obtained serial and circular serial basic tran­sitions are seen in the lower formula array. D Proof 4 The proof of the last theorem is evident and proceeds from the previous definitions and theorems. For a graph 0 , there is evidently, INFORMATIONA L GRAPH S OQ,Oi,02,-•• ,Ono; 0~pQ, O-pl, (°"0) 01, • • • i Cn) which length is n. In a pure INFORMATIONAL GRAPHS Informatica 21 (1997) 79-114 9 1 serial formula of length n there are exactly n bi­nary operators and n + 1 serial operands. As de­scribed, parenthesis pairs can be displaced in a spontaneous manner, giving to the formulas dif­ferent causal meanings (the corresponding causal subscript i of formula "95^. Thus, the formu­las can be systematically numbered (subscribed), where, for example, iV-+ (°'o,0'i.--- ,CTn) ^ ((• • • (cr0 |= cri) )= • • • crn_x) (= an)\ ((• • • (o-o h °"i) 1= • • • ovi-i \= o-n)); f 2n \ P_.{&0,V2,---,crn) ^ 1 ' (O-O N (CT1 1= • • • (°~n-l 1=an) • • O) Thus, the gestalt of any of these formulas is a parallel system of N_> — ^j-j- (^ ) serial formulas, that is, in a shortened symbolic form, _ 2 V H r(>J &-+; l-(2n) HTU for 1 < i < iV,. Each of these formulas deter­mines one and the same graph 0 . On the other hand, a graph (5, corresponding to a formula nip_^ can be described, according to Theorem 1, by formula tp'. This proves the the­orem. • 4.4 A Circular Serial Formula and Its Graph DEFINITIO N 8 (A Pure Serially Circular Formula) A pure serially circular formula, ". ip , is deter­ mined in the following manner: n+l O / s J I ip (Lj0,wi,---,L*;„) ^ (= wt; u>o \=; (wt,u;0 G {w0,wi, • • • ,un}); ((• • • (w0 1= wi) (= • • • o;n_i) 1= u)n) \= w0 where the serially circular formula of the form ((•••(wo |= wi) |= ••• w„_i) |= w„) |= w0 can be substituted by any other serial formula with displaced parenthesis pairs, that is by a formula which is graphically equivalent to the original for­mula (Definition 5). Formula subscript j system­atically varies in the interval 1 < j \ < ^ ^ (2JT+I2) • D L L L w nU n T Figure 11: Two graphical interpretations cor­responding to formula system in Definition 8. The equivalent bottom graph explicates the parallel character of the serially circular formula. 4.5 Primitive Circular Serial Parallelism, and Gestalt of Circular Serial Formula For the circular serial parallelism, one of the ba­sic transition formulas must informationally con­nect the 'last' operand with the 'first' one, that IS, "las t N "first- THEORE M 6 (Graphical Equivalence of a Circular Serial Formula Gestalt and a Pure Circular Serial, Primitive Parallel Formula Svstem) The graphical eauivalence of the form n + l O1 , N determines one and the same graph for the gestalt of a circular serial formula of length n + l and the corresponding primitive circular parallel for­mula system. This eauivalence shows the power of the basic transition formulas (of the length 1, and informing in parallel) compared to long circu­lar serial formulas of the length n + l, belonging to the gestalt T(" . ip (UQ,UI, • • • ,wn)J. • Proof 6 We have to prove the graphical equivalence of the both circularly structured parallel formula systems in the theorem. Gestalt T(n .(p (UQ,UI, • • • ,un)\ is a paral­lel system of the serial formulas of the form 92 Informatica 21 (1997) 79-114 .ip (OJQ,U>I,-• • ,tjjn) which length is n + 1. In a pure circular serial formula of the length n + 1 there are exactly n + l binary operators and the n + 1 serial operands (one of them appears twice, that is, the title operand of the circular formula, at the beginning and at the end of the formula. As described, parenthesis pairs can be displaced in a spontaneous manner, giving to the formulas different causal meanings (the corresponding causal subscript i of formula . (p (UJO, CJI, • • • ,ujn). Thus, the formulas can be systematically numbered (subscribed), where, for example, n+ l O , >. (((• • • (wo \= wi) |= • •' w n-i ) |= w„) |= oj0); n+l O , -. 2(p_t[ujo,ui,- •• ,w„) ^ ((• • • (w0 (= wi) |= • • • tt»n_i) |= (ujn f= w0)); n+l O 1 r2n+2^->0,W2,--' ,W„) «nU+i J (w0 |= (wi |= • • • (w„_i |= (w„ |= w0)) • • •)) Thus, the gestalt of any of these formulas is a parallel system of N_^ = ^ ^ ( n+i ) c i rcu l a r serial formulas, that is, in a shortened notation, t>:)­ n+l O n+l O n+l O iY> ; 2^ ; ' _1 C*n+2\ V-> n^Z\ n+l ) for 1 < j < N . Each of these circular formulas determines one and the same graph (5. On the other hand, graph (3, corresponding to a formula " . ip can be described, according to Theorem 1, by formula " ip . This proves the theorem. D 4.6 Circular Gestalt The circular gestalt of a circular formula con­cerns the parity of the formula operands. Within the cycle of the formula graph, each operand can be rotated to the title (initial, leftmost) posi­tion, and in this situation, it can function also as the leftmost and simultaneously as the rightmost operand in the circular formula. A.P. Zeleznikar THEORE M 7 (Graphical Equivalence of Primitive Circular Parallel Formula Svstems) If the expres­sion

n) represents a primitive circular parallel formula sijstem (of basic n + l transitions), then n+l O' / s

i,U>i+l, • • • ,(Jn,U(),U)l, • • • ,Ui-l) corresponds to a gestalt T of the circular serial formula n+l O , . (p [U>i,Uli+1,- • • ,OJn,U0,OJl,- • • ,Ui-l) The parallel sijstem of such gestalts is called the the circular gestalt T of a circular serial function n+1 ° i \ mu . (p (UJO, u>i, • • • ,u)n). lhere is •pO/n+l O , A \ ^^(^»^i+l)" -,Wn,Wo,Wl,--- ,Wj_l)J =* (r (T^ ^°'Wl'''' 'w"7 ' ^ r \7iVt (wi >w2, • • • , w„,u0)j ; ,/n+l O, w i+i: • • • ,wn,wo,a;i,- • • , ,)); \r(T^(u;n'a;o'a'1''" 'w«-i)J 7 D Proof 7 In a parallel formula system, formulas can occur in an arbitrary sequence. The formula ordering does not influence the informing of the system. Therefore, instead of ( a f= /3) \\= (7 f= d), simply the semicolon system (a \= /3; 7 (= 6) can be used (taking a semicolon instead of |f=), where semicolon performs as an associative operator of parallel occurrences of formulas in an informa­tional system. The proof of the first part of the theorem con­sists of the graphical equivalence of informational systems of the form INFORMATIONAL GRAPHS Informatica 21 (1997) 79-114 93 /o>0|=wi; \ fui \=u>i+i; \ toi \= w2; k>i+l | = <*>i+2', Ui+l p U>i+2] w0 p wi; un-i p un; U>;_2 (= Wj_i ; for i = 0,1,2, • • • , n. The right formula system is nothing else than the parallel reordered left for­mula system. So, n+l O'/ v

0 j ^1 , • • • , ix>i-.)) n+l O', x V3 , (wo,wi,---,wn) There also is, evidently, ^ ( n J• i V : ^( w ^^ w ^+l,•• • i Wn , W0, ^1 , • • • ,Wj-l)J n+l O'/ s = ,5 i,---,Ui-i), the cy­cle can be repeated several times, and can be­gin at an arbitrary operand and end by an an­other (arbitrary) operand. Such a čase is not particularly interesting in the sense of a star Informatica 21 (1997) 79-114 A.P. Zeleznikar c, Hausbilden c, h ausbilden' T ' " M* M* •( e etwa s j *\ a ander e J 1 N.c entvvorfen —,c ^H ^ bft ^ N*al s ' c-o^-® ' verstehen 6etwas F* * ^anderej -K R \p 'Verstandenei MR^S ; "A-R verstehen ^ ' M Centworfen V;M p t <->Seinkonnen> '-'SeinkSnnen C-D\ l ^Verstandene \ ( ^Verstandene '-'Seinkonnen R fiir " i I R sondern J F * verstehen V \ -^Ausarbeitung/ F ' entwerfen ' R als '-'Seinkonneni V C V; V dE; V R ausbilden ^'i * R ausbilden -"i ausbilden ' V^±A; v * R -^-Ausbildung! v R eignen *verstandene i V CA; v R eetwasj R verstehen > V; R entwerfen / ,0' Understanding ..V_^ of something a requires the additional transition a\=V to the system .V , causing {.V_^ (a), and thus aH operands of system .V^ become functions of a, that is, take the general functional form (p(a) [20]. 5 Complexly Interweaved Circular Informational Graphs Real informational systems are complexly circu­larly interweaved. This is a condition sine qua non, for only circular systems have the poten­tiality of emerging from that what already is, to that what unforeseeably could arise. In the most normal situation, each occurring operand (repre­senting an informational entity) is circularly con­nected in one or more loops (circular graphs), and Informatica 21 (1997) 79-114 each loop is in some way connected to ali other loops in the system. For such a system we say that it is completely circularly connected or informa­tionally closed. In some way, each operand of such a system informs (indirectly via other operands) ali other operands and vice versa: each operand is indirectly (or, in particular cases, directly) in­formed by ali other operands. This means that a singular operand impacts the system entirely and is entirely impacted by the system in which it oc­curs. This type of informational closeness does not mean that system operands cannot be impacted from the exterior and that they cannot impact the exterior operands (entities). According to in­formational axioms [21], this property of infor­mational openness or independence of operands is given, in general, to any informational entity. Only in clearly explicated cases, it can be deter­mined in which čase an entity does not inform an another entity. Besides, there can exist operands which do not appear in a loop, for instance, merely in linearly serially structured formulas, where the last (end) operands of such formulas function like final desti­nations, informing for the sake of its own purpose, as a kind of final receptors. In some cases, such final informational destinations can be foreseen for a later looping into the system, when their in­forming will begin to impact the other entities of the system. In the described complexly interweaved circu­lar informational system, presented by the cor­responding graph, the only senseful and signifi­cant function of each operand is to be circularh/ connected to the system, that is, to play a devel­opmental role of the system by its arising and di­minishing6 . Otherwise, the existence of an uncon­nected or partly connected informational item is not within an informationally senseful and signif­icant function of system development, emergence, and function. In the same sense, parallel informational graphs can exist, but, in a senseful situation, they must be in some way connected. Isolated graphs are presentations of possible informational informing and as such, that is mutually isolated, they per­form as a kind of informational reductionism. 6For instance, in the sense of an estimate of the precision or certainty or definiteness according to [11]. A.P. Zeleznikar This especially holds in the cases when informing of system entities is studied, grasped, and finally presented under essentially limited circumstances. Sooner or later, the need for informational com­plexity in the form of a serial, parallel, and circu­lar phenomenalism comes to consciousness. Both in the living and artificial systems, as well, the facts of this informational complexity can be con­sidered. 6 A Classification of Informational Graph s 6.1 Isolate d Graph s Isolated graphs perform as informationaily iso­lated entities. It simply means that they are not connected to and from the other graphs or en­tities. They usually emerge as a consequence of the so-called reductionistic reasoning, where each graph, as such, represents a reductionistic situa­tion of a particularly isolated view or interest. Isolated graphs as informational entities are in no way senseless since they can become, through the emergence of conceptual and informing sys­tems, parts of systems, also with essential in­formational modification, further decomposition, connectivity, and the like. As such, they can be­come suitable for the so-called bottom-up com­position. In this sense, isolated informational graphs hide the potentiality for their future in­volvement into a linearb/ (serially), parallel (si­multaneously), and circularly (cyclically, with re­gard to a loop structure and organization) con­nected informational realm. By definition, isolated informational graphs do not loose the axiomatically given property of the input (internalistic) and output (externalistic) possibility of operands, that is, their connection to and from the potential environment. Consider­ing this situation for isolated graphs, the following definition becomes meaningful. DEFINITIO N 9 (Isolated Graph) An isolated infor­mational graph is an arbitrary organized graph of operand circles and/or ovals and them connecting operator arrows in a serial (linear), parallel, and circular way, but not connected to or from other INFORMATIONAL GRAPHS informational graphs. Operands of the graph re­tain their potentiality for undetermined internal­istic and externalistic informing, from and to the virtual graph environment. Isolated graphs and operands are graphically presented as circles and ovals without arrows. • As a consequence, each finite informational graph is isolated, that is obtained on the basis of a re­ductionistic approach. However, it hides the pos­sibility for its further decomposition of operands and operators [26]. Graphs can contain isolated operands and isolated subgraphs. 6.2 Primitive Transition Graph A primitive transition of the form a \= /3, and its decomposition, [26] is the key form of any in­formational system, particularly of the primitive serial and serially circular parallel systems of ba­sic transitions ( ntp' and ip J. Its operand and operator decomposition possibilities were exhaus­tively treated and discussed in [26]. DEFINITIO N 10 (Transition Graph) A transition informational graph consists of two operand cir­cles or ovals, connected by a single operand arrow. This arrangement of both operands and an arrow is treated as an informational unity, that is, as a transition from the first operand to the second one. • 6.3 Pure Serial Graph Pure serial graph is a graphical presentation of the formula . (p with n + 1 operands and n binary operators. Pure serial graph represents a system of different formulas .(p of length n, where 1 < 1 -n+l\nl­ 6.4 Pure Circular Serial Graph A pure circular serial graph is a graphical pre­sentation of formula . (p with n + 1 operands and n + 1 binary operators. A pure circular serial graph represents a system of different formulas B+V° of length n + 1, where 1 < j < ^ CnL) • 6.5 Parallelism of Graphs The reader can see that informational graphs are by arrows connected circles and/or ovals. Both Informatica 21 (1997) 79-114 97 circles/ovals and arrows are marked: circles/ovals by operands and arrows by operators. An un­marked arrow represents the operator |= (an oper­ational joker). At the first glance, such connected circles/ovals in the graph give the impression that the system of operands is informing in a serial (also circularly serial) manner. To exceed this surface impression, the reader should stay aware that any basic informational transition, a (= /?, with its left part a and its right part /?, simulta­neously means the detachment [23], in the form "M a; P This detachment says, that a and (3 inform inde­pendently and in parallel to the transition a \= (3. This detachment must be understood recursivelv, irrespective of the complexity of both a an (3, which can represent arbitrary formulas or systems of formulas. Each graph represents parallel informing of ali operands and aH possible subformulas and formu­las which proceed from the graph in the sense of their informational well-formedness, that is as se­rial formulas n(p and . ip , primitive formula systems ip' and (p and ali the possible for­mulas and formula systems of these formulas and formula systems. Using the last rule of the parallel detachment of subformulas recursively, one can determine how many parallel processors are needed for a simula­tion of formulas "tp and .

1. Let this hold for any two loops and let ali the occurring loops cover aH the operands of graph 0 . Evidently, under these circumstances, each operand is transitionally connected with the remaining operands of the graph. This means not only that an operand circularly transitionally in­forms the remaining operands, but also that it is circularly transitionally informed by the remain­ing operands. • Evidentlv, according to the proof, a loop can be arbitrarily structured, from the shortest one, con­sisting of two operands to the longer ones. The next request is, that ali loops of the graph must be mutually connected, and no loop group must be isolated. For loops of the graph the same holds as for the graph operands. This leads to the conclusion that every operand is in some way circularly connected within the graph. A completely (circularly) structured graph guarantees an equal informational treatment of any occurring operand and its possibility to become, in some particular context, the title operand of the system, when it observes and is ob­served as a significant entity, magnifying its phe­nomenalism within the system presented by the INFORMATIONAL GRAPHS graph. 7 Wha t Would Mean to Understan d an Informational Grap h and t o Progra m It s Possibilities? An informational graph is the most powerful pre­sentation of an instantaneous informational sys­tem since it contains ali the possibilities of the instantaneous informational system in respect to its (integral) gestalt (gestalts of the possible loops with already occurring graph operands and oper­ators). To program ali such possibilities of an instan­taneous graph means simply to process in parallel not only ali the operands in parallel, but also to have separate processors for any subformula of the system and, at last, for the system as an entirety. We have seen how a primitive parallel (transition) formula system does not represent (e.g., simulate) only the concrete system of formulas, but also ali the other possibilities which the formula system with its operands and operators could represent. This situation is not always an adequate one, al­though its simulates the aH possible situations of the instantaneous system. The next problem of an informational graph, representing a system, is its emerging, being con­ditioned by the informational arising of the sys­tem it represents. Informational arising is a con­sequence of different decompositional processes concerning operands (informational entities) to­gether with their operands. It means that a graph changes its graphical structure during the informing of the system. An informational graph is nothing other than a particular (representa­tional) informational entity, which underlies the principles of informational externalism, internal-ism, and metaphysicalism. One can imagine a decomposition of a graph in a similar manner as the decomposition of an operand or operator [24, 26]. Between two operands of the graph, connected by an operator arrow, the decomposition means, that instead of a single arrow, a new subgraph arises, being ade­quately inserted into the graph. Let a \= /3 be a transition and A (a |= /3) its decomposition in the form Informatica 21 (1997) 79-114 99 ((•••((« 1= 7i) 1= TU) |="-7n-i)|=7n)|=/ 3 This decomposition of a |= /3 can be grasped in the following way: 1. operand a-decomposition is framed as ((...((a|=7l)(=72)l=...7n_1)|=7n) M 2. operand /3-decomposition is framed as ((•••(( «1= 7l)l=72)|=---7n-l)|=7n) M and, finally, 3 . operands a-/3-decomposition or the original operator (=-decomposition is framed as ((•••(( OL |= 7l )|=72)|=---7n-l)N7n) N /3 8 Informational Graph s for Informational Being-in and Informational Being-of What are informational graphs for informational Being-in and informational Being-of and how could they be reasonably interpreted? In concern to these relatively simple informational cases we can discuss the meaning of the occurring gestalts, circular gestalts, and star gestalts. 8.1 Informational Being-in In [19], the informational Being-in or informa­tional inclusion (operator c ) was defined in the following way. DEFINITION 13 (Informational Includedness) Let the entity a inform within the entity /3, that is, a C /3. This expression reads: a informs ivithin (is an informational component or constituent of) /3. Let the following parallel system of included­ness (Being-in) be defined recursively: //3 h «; (ac/3) ^±Def UM ; \3(aC/?) . where for the extensional part E(a C /3) of the includedness a C /3, there is, 100 Informatica 21 (1997) 79-114 /f(/?f=«)c/n\ («N/3)cA 2( a C /3) € V ] {/3\=a)Ca,( VI (a H /3) C a J/ The most complex element of this power set is denoted by Cases, where E{a C /3) ^ 0 and 0 denotes an empty entity (informational nothing), are excep­tional (reductionistic). • If one looks into this definition, irrespective of the complexity and recursiveness of the defini­tion, (s)he can observe the informational inter­play solely between two entities, that is, a and /3. Informational includedness means that both a and /3 are under mutual informing and observing. Within this interplay two informational operators appear: |= and C. Let us perform the primitive parallelization II' of the definition, that is, '/3\=a; j3\=a; a\=(3; n' | a \= /3; s^K« ; a c/3 ,2g>C/J). and //3N«; « c/3' n'(SJ^( a C /3) ^ a C a; a H # This parallelization delivers according to the def­ CTj8.a 3( a C /3) = sL>c/3) Figure 13: Informational graphs for the simplest and the most complez čase of informational in­cludedness. inition of the informational includedness 16 differ­ent graphs, where the minimal and the maximal configuration is presented in Fig. 13. A.P. Zeleznikar For the so-called zero or minimal option (H (a C /3) = 0), there is, (a C /3) ^ (P \= a; a \= /3) The remaining S-parts are: S L7_ 77_ "& "ot ^ a 'zja —a i —/3) "/3,a> — > —a; <—0> "/S.a' s/3 s/9 s-3 s/3 ra8,a s/?>" s^. a The minimal graph configuration must be in­cluded in ali the H-parts. 8.2 Informational Being-of In [20], the informational Being-of or informa­tional function (expressed as (p(a)) was defined in the following way. DEFINITIO N 14 [Informational Function] Let entity ip be an informational function of the entity a, that is, (p(a). This expression reads: K f a)Ctp)*± (v 1= (v Kf«)) c v V ((v Kf a) h v) c y> / and, for the second informational includedness, according to [19], /tp \=of (a \=^-€)(B) Figure 14: Informational graph for the čase of informational function: (A) for (p(a), (B) for the functional transition (p \= a, and (C) for the def­inition of informational function ip(a). inition (p{a>), the standardized informational tran­sition of the form (p (= a is introduced, with the meaning ) Hof «; (v K f ot)\=ip;\ a \= (ip \=o{ a);

(

) , and the main implication, called the informational detachment (operator in the form m an of a fraction line, denoted by 5 informa­tional graph). Formally, the rule of informational modus ponens, pMP, is a formula of the form a: a /? Pup(a,P) — /3 with the graph in Fig. 16, corresponding to the primitive rule parallelization, that is, / ot |(= a; \ , ( a; a ==> /3 n' { S ) In this primitive parallelization scheme, the semi­colon in the premise was replaced by operator of parallelism |f=, to make the primitive formula sys­tem more transparent (semicolons are used as sep­arators between elementary transition formulas). By the rule of modus ponens, operand /3 is de­tached from the rule premise, that is, follows as a conclusion. What is characteristic for this rule of inference is its linear serial structure in concern to a, but circular structure in concern to /3, being evident from the graph in Fig. 16. It will be presented how other rules of inference are much more in­formationally circularly structured and, that in­formational circularity pervades the entire living and artificial (also mathematical) informational realm. Thus, modus ponens, as one of the firmest inference rules in mathematics, is pseudolinear (linear consequential) and, in fact, violates it­self (in a hidden form) the mathematical princi­ple of straightforwardness. Namely, the detached operand (3 is, according to the modus agendi of an informational entity, circularly decomposed in the most consequent form % (or /3 = > /5). What could bring a real surprise into the phi­losophy of modus ponens is a mathematically and informationally legal rearrangement of the rule INFORMATIONAL GRAPHS premise. The semicolon represents a conjuctive connective in mathematics and a parallel connec­tive in informational theory: both underlie the so-called associative law. This means, that operands on the left and the right of semicolon can be ex­changed. This sort of legal conclusion causes a rule of modus ponens in the form a P; a PLr(">fl^ P If this rule with exchanged premise parts is math­ematicallv unacceptable, it just means that be­hind the philosophv of modus ponens something remains unexplained (hidden, unrevealed), and that the semicolon, also in mathematics, occu­pies an looselv determined logical connective (sep­arator), being formallv excluded from the valid (transparent) field of legal intelligibilitv. It should mean that parallelism does not enter into the regular mathematical discourse, although it func­tions as the most normal phenomenon in logical concluding. Figure 17: Informational graph for another for­mal interpretation of informational modus po­ nens. This disputable rule of modus ponens delivers the graph in Fig. 17 or the primitive paralleliza­tion scheme a P;g\ _ n' P a V 0 J As the reader can see, the local (inner) loops for operands a and P disappeared, and instead of them two other loops including both operands a and P appear. Preciselv the graphs for the first and the second formal presentation of modus po­nens explicate the essential difference of them and bring up the dispute of the ultimate and everlast­ing validitv of this type of concluding. A radicallv different concept of modus ponens concerns the decomposition of both operands a Informatica 21 (1997) 79-114 103 and p. As well a as /3 hide a decompositional na­ture A a and Ap, which decide about the neces­sary informational (modus ponens) adequateness between a and p. Thus, the new inference rule p^Ap(a,P, Aa, Ap) must be understood as A/9(/9) P The graph for this rule of modus ponens is pre­sented in Fig. 18. IN 3 A h* N* -TA Figure 18: Informational graph for decompo­sitional formal interpretation of informational modus ponens. 9.4 Grap h Investigatio n for Informationa l Modu s Tollens The informational modus tollens already arises from a more informationally slippery ground, with interweaved informational loops which some­times might perform explicitly tautologically. Two possible graphs for modus tollens are pre­sented in Fig. 19 Figure 19: Informational graphs for informational modus tollens. 9.5 Graph for Informational Modus Rectus The aim of informational modus rectus pMR is detaching intention ^intention of an informational entity a within a transitional intentional inform­ing of a to an entity /3. E.g., an exterior ob­server, say P, of a who observes the a's inten­tional informing of P, comes to the conclusion that there exists an a>'s intention in the informing 104 Informatica 21 (1997) 79-114 by which it informationally impacts (3. The re­sult of this intentional informing is observed (see­able, comprehensible, intelligible) in j3 as an in­formational Being-in [19]. Intention of entity a appears as a general circular decomposition, e.g. as A_^ (ijntention («)) or as a metaphysicalistic de­composition M" (tintention(a))­ The rule of informational modus rectus has the form PMR \ a i Pi '•intention j ^_ + ) ^ " S (( « |=intend /3) '•intention («)) A_,(iintention(a) ) C /3 The graph for this rule is presented in Fig. 20. Primitive parallelization ' . o" p-inten d {'intention J h* Figure 20: Informational graph for informational modus rectus. n ' [PUR (<*) & intention , A ° ) j fa\\= a; a Hntend /?; N P '* ''intention! '•intention ^» p Ol', OL A O A: K *intention > A^ a ! a \ intentio n t= * C P J is an exact description of the graph in Fig. 20. One can see how modus rectus emerges through a particular moving along the graph paths (ar­rows). This formula can be understood as the one of the possibilities of belonging to the infinite star gestalt formula systern, which arises through ali the possible moving along the graph paths. Many other rules for detaching (extracting) the intentional informing of an entity could be con­structed considering various views and beliefs of intentionality as an informational phenomenal­ism. 9.6 Grap h for Informationa l Modu s Obliquu s An informational modus obliquus [18, 23] is a broad informational concept for concluding and A.P. Železnikar inference which has its roots in the Latin conver­sation practice (e.g., slanting, sideways, oblique, indirect, covert, envious discourse). The modus obliquus (MO) concerns indirect adjustment of an absurdly experienced, individually motivated, felt, emotional, interested, etc. consciousness in­forming. MO as informational detachment of an informational item out of an absurd, disapproved, distrust informational situation ušes the inference and concluding forms and processes, even in its conscious realm of informing, in the realm of un­awareness, illiteracy, doubt, and falsity. This is the absurd attitude of the MO itself with the aim to surpass the absurdness of a complex informa­tional situation. The reader will agree that setting an informa­tional graph for various possibilities of MO would require a separate and exhaustive study. But some simplified concepts of MO can be presented by informational graphs and the corresponding formula systems. As a form of the rule using indirect informa­tional content and meaning, MO obviously devi­ates from a direct or intentional line (e.g., the line with modus rectus) of discourse, performing roundabout or not going straight to the point. In this respect it performs within a speech game in which behind-the-scenes intentions, views, and purposes remain hidden and must not be dis­closed (e.g., in political, ideological, clan-like, de­ceptional public discourse). As an indirect form of inference, it involves concluding with "com­monly noninforming" (secretly, unconsciously in­forming) entities. But on the other side, right in this manner, MO can reveal information being not openly shown to the participants of a com­plex, yet not essentially disclosed discourse. Let a mark a complex controversial and unex­amined informational entitv. The controversy of a means that a clear absurd informational item babsurd in a can be identified. In this situation, babsurd has to be informationally decomposed (an­alyzed, synthesized, interpreted, transformed) in the circular form A_^ (babSUrd(a)) with the aim to deliver the conclusion cconc]usion in such a way that absurd is informationally included in the conclusion. In this way, the possible scheme for Cconclusion detachment from the controversial ori­gin entity a is INFORMATIONAL GRAPHS Informatica 21 (1997) 79-114 105 PM O [ a ' ^absurd i cconclusionj ^_ j j l»_j j ^ Va r^absurdl / "absurdI3 ) ) C. 3 ; ^ ^ (.t>absurdlaJJ ^ cconclusion '^_j (^absurdi 3 / / t- Cconclusion In this rule, N marks the so-called negated circular serial decomposition of the absurd informational entity babSUrd(c). This circular decomposition is something like (•••((bab surd(a)^"i(a))^«2(a))--^ «n(a) ) Y" babsurd(a ) where some derivatives ai(a) informationally counterinform (informationally oppose) in respect to babsurd (a)- It means that this decomposition is a circular serial function of the form denoted by n+l O

«j+i(a),--' 5«n(a), babsurd(a), Ji —* «i(a) , o2(a),--- ,«i-i(a)) The primitive circular parallel system (Cconclusion J Figure 21: An informational graph for possible informational modus obliguus. ^ \PMO [ a , babsurd , C conc i us i on , A_^ , IM^ J J ^ a j ' a |^absurdl y "absurd i ^absurd F * a C a; a|HA^; a ^_ > F1 $ "absurd i babsurd l= *> ^ Cconclusion a * Cconclusion! l\T •J. Dabsurd) ^absurd K a; (") \ 3 C Cconciusion / is the exact description of the informational graph in Fig. 21. Formulas marked by (*> and C*' in the primitive system are superfluous and are used solely for seeing the circular continuity of the sys­tem. 9.7 Graph for Informational Modus Procedendi A goal or aim of informing of an entity, as some­thing essentially different in regard to informa­tional intention, can become the subject of the detachment, for instance, in a strategic environ­ment. The question is what could a system of goals within a strategy be at ali? The Latin pro­cedo has the meaning of to go forth or before, ad­vance, make progress; to continue, remain; and to go on. Let an informational strategy entity s include a hidden goal system ggoai- Strategy s causes a system of consequences c (s), elsewhere, such that c (s) can be transparently observed. The goal sys­tem ggoai performs as a cause of c (s), that is, as a system of the form C (s ) C S ; C ( s ) |= cauS e ggoai The |= cause operator has to be particularized to operator => caus e with the meaning implicatively causes. Detachment of ggoai must remain within a care­ful (goal-consequent) decomposition G_^ ofconse­quences c, (s,), that is, as G^ (c (s)). Modus procedendi is a rule (pMPr) for the de­tachment of goal-directed organization ggoai of the strategy informing entity s , through the observ­ing of c (s). Thus, 106 Informatica 21 (1997) 79-114 A.P. Zeleznikar PMPr(s>ggoal,c,G"j ^ ggoal C S ; ggoal G" (c(s)) ggoal The graph for this inference rule is presented in Fig. 22. Primitive parallelization of the rule gives K Figure 22: Informational graph for informational modus procedendi. n'(pMpr(s,ggoai,c,G° J ^ /ggoal C S; S JN ggoalA ggoal »'cause ^ ^ j ^ ^ F > c j Vc K s; J A more firm and direct inference rule would be that of modus ponens, e.g., s; s ggoal PMP(S> ggoal) — ggoal from which c and G are excluded. But with —> modus rectus the preceding informational com­ponents can be considered in a slighth/ different way (see Fig. 23, for instance, in the form Figure 23: Informational graph for the čase when modus procedendi is replaced by slightly modified modus rectus. PMR (s >c > ggoal , A_ J ?=i S! ((S Ngoaic(s)) =^g goal (s) ) A ^ (ggoal(s)) C C (s) Here, decomposition A_^ is a circular serial de­composition concerning the goal as a function of the strategy, that is, as ggoai(s). Parallelization of the presented rule for modus rectus is n '(P M R( S , C > ggoal, A" | s c /s IN s; f^goal i i c K s; s ^ ggoal i S ggoal N *s i A" A_> N * ggoal j ggoal N * s > \ s Cc ; c N* s / 9.8 Graph for Informational Modus Operandi An informational modus operandi detaches the (inner, own) operational capabilities of an entity a in the form of an informational function of the entity in the general form ip(a) (e.g., informa­tional Being-of in [20]). This function is usu­ally called the entity informing and marked by 3a. But informing 3a, as an operational property of the entity a, performs by itself as an infor­mational entity within a, as an a' s includedness, 3a C ct (e.g., informational Being-in in [19]). A modus operandi concerns the decomposition of an entity in respect of its interior informing. Informing 3a is only the initial step in the de­composition process when circular informing of the form (a \= 3a) \= a and/or a \= (3a N a) comes to the surface. The deeper operational detachment of further inner components of a can deliver not only the other inner components but also the informational structure (formula) of them. One of the possible forms of the opera­tional detachment concerning an entity is the so-called rule p^ 0p (a , 2fa, iQ, LQ, ca, La, ea) of meta­physicalistic modus operandi (metaphysicalistic decomposition M^ or /Zj-decomposition) in the form a' la a'Ca> PMOP^'' a>*"a> ((((((« N ?*) N ia) N Ca) N Ca) N La) N »a) N a ^OLI *cn *~ac) CQ:, v^ct? ^ot where the number ji of possible causal interpre­tations (because of the definite particularization of operators ';'> := ^ !an d E and the rule firmness) INFORMATIONAL GRAPHS Informatica 21 (1997) 79-114 107 varies within the interval 1 < ji < 132 (= }(g2)). The possible interpretations concern the meta­physicalistic formula ^ (a, 3a,\a, €a, ca, (((Ca |= cQ) p La) p e«) p La, P a P ia) 1= •Joti I \S-a P CQ ) P *^ai V (Ca P ea) h La / *^OLI *ai ^a : Ca , Ca j CQ This rule of the multiloop detachment could be systematically marked, according to the six loops, as P MO p l ai Joti l a i {'On CQ, W*! e a I where the second subscript q of fj,jpq corresponds to a subloop in the main loop. There is to stress that operands below the de­tachment line are separated by commas. This means that each of operands is detached sepa­ratelv, and that there is not meant the parallel system of the form 3a; ia; Ca; cQ; Ča; eQ. This situation causes a complex and circularly inter-weaved graph structure (not so easily drawn on a piece of paper). Thus, r Metaphysicalisti c formula 1fj, ( a,3 a ,\ a ,L a , ca, &a, c a ) alread y considers th e semanticall y determine d compo ­ nent s (operands ) of informing, counterinforming , an d em ­ bedding . a _^ /a a\ Ki^Kf i) where detachments % and - are understood as the basic transitions at the parallelization of the rule. The system of the basic transitions for the discussed metaphysicalistic modus operandi is OL | p O , JQ. , C a , CQ.J Oi )• Ct, J a , CQ. , CQ , ~ II p p || 0. . ®-i •Jg.-i *-a; ^ a J a | P ^-ai ^a> c a I P ^a ! ~ . ^ ­ O! p Ja j J a p ^ai la p *-a! *-a p Ca! Ca P *~aj *-a P ^a\ C Q p O ; ca |= j a \ za p LQ ; i a p JQ ; Ca P *-aj Ca p ^ a The system in the first frame includes 35 basic transitions and represents the part of the rule out­ 7 O side the formula 1/i_^ (o:,3a, ia, L a , ca, La,Ca) , wit h exception of parallel (semicolon) operators of the formula. In the second frame the "feedback" tran­ 7 O sitions of the formula xn occur. Finally, on the basis of the described parallel system, the graph of the rule can be drawn. 9.9 Graph for Informational Modus Vivendi An informational modus vivendi [18] concerns the information of life (e.g., autopoietic informational entity a) in environmental, individual, techno­logical, and social circumstances. Informational problems of social transition [26] are typical forms of modi vivendi when the governing totalitar­ian ideological pattern of understanding has to be replaced with a more flexible and life-suited paradigm of social and environmental survival. The basic living information—conscious as subconscious—existing everywhere the life arises, may be recognized as autopoietic or self-organizing information. It does not organize only the organism for the suited behavior in life cir­cumstances but organizes, through the pressure of the environment, also the information itself for a proper organism behavior, for instance, building up the so-called metaphysicalism \i of autopoietic entity a, marked \ia or together with sensory in­formational entity aa. 108 Informatica 21 (1997) 79-114 It is to understand that, in the beginning, a and aa impact the arising of fia, and then fia essentially impacts the emerging of both a and aa, thus a basic circular system of the form ((a \= aa) (= Ha(o)) \= « is reasonable. Within this cycle, metaphysical-ism (J,a(cr) is a constitutive part of a, e.g. na{(j), which has to be. extracted from a by a modus vivendi, being essential for survival and adapta­tion of a. in the environment. This rule must be­long to a itself as a particular informational en­tity, being able to identify instantaneous /iQ( fJ,a(cr)) Ha{a) representing a kind of modus ponens in the frame­work of modus vivendi. 9.10 Graph for Informational Modus Necessitatis By a necessitv, the 'must' is compelled. In this respect, the meaning of the verb must becomes necessity by itself in the realm of the theory of informational. Necessity as informational phe­nomenon is a pressure of circumstances, which can be grasped also as an essential impossibility of a contrary informational position. It appears as an urgent informational experience, emotion, A.P. Zeleznikar memory, need and desire (the consciousness of the must), in such a way, that it as a particular infor­mational entity cannot inform outside itself, that is, in an another direction. Within the causalism, necessity can be comprehended as an inevitable informational consequence. Modus necessitatis is that principle of inference which in several situations can coincide with dis­cussed principles hidden in other modi informatio­nis. Intentionality, which comes to the surface in modus ponens and modus rectus, can be grasped as a particular čase of necessity, being consistentb/ bound to the so-called logical mind of human, in the sense of the "best" rationalistic tradition. A more detailed discussion (also formalized) is given in [18]. The attentive reader is already able to proceed into the philosophy of informational, its formalization, and construction of informational graph by himself/herself. 9.11 Graph for Informational Modus Possibilitatis Possibility is a modal informational phenomenon which opposes the instantaneous reality (essen­tialness, existentiality) and necessity. A modality (potentiality) by itself is a mood of revealing of experience, emotion, consciousness, and in more general form, of Being, thinking, occurrence; it is a mood of game with conditionalities. In logic, modality of propositions means the degree of trustability of propositions in regard to possibility, existence, and to necessity. Such a proposition of the possibility is, for instance, a |=Can_be P, read as a can be p, or information-ally consequently as a informs that there could be (3. An asserting (existential) proposition is, for example, a = /3 with the meaning a is /3 or, consequently formally, a |=3 /3. An apodictic (ne­cessity) proposition is a (=must_be P- Modus possibilitatis is that modus which opens the realm of informational potentiality, including views of exaggeration, inauthenticity, unreliabil­ity, controversialism, but also unreasonableness, insolence, and mania. AH this concerns the infor­mationally active (intense) and quality creative possible consciousness searching. In this respect, modus possibilitatis can become bound to modus obliquus and modus vivendi, but also to the non-informing (negative) possibilities of modus neces­sitatis. A basic rule for such a kind of inference INFORMATIONAL GRAPHS is described in [18] in the form {a, P); (a, /3 \= a) \=n 7 7 K 71,72,-•• ,7» where a is the subject entity, /3 its exterior im­pact, 7 an entity induced as possibility, compo­nents 71, 72, • • • , 7n its informational derivatives, while \=„ represents a possible informing (possi­bility operator). 10 An Informational Čase for Strategy Decision Making Maruyama (1993) has invented a practical com­puter supported scheme for simulation of strategy decision making for business executives and gov­ernmental planners. Let us show this concept of a typical computer supported simulation program in the realm of an informational formula system and the corresponding graph, where substantial conceptual changes in understanding the informa­tional nature of the decision making problem take plače. The informing scheme of the problem can be presented in an informationally condensed form in Fig. 25 and parallelized according to the par­ticular entities e, f, g, h, j , t, m, n, r, D, r., 38 . Accord­ing to the graph in Fig. 25, the adequate formal description of the graph (9 becomes, considering the gestalt F of the formula system, 8Maruyama (1993) has given the following meaning to the operand markers (informational entities in our čase): e—exchange rate (value of county's own cur­rency); f—infiation (priče); g—government subsidy of inefficient firms; f) — import restriction; j — amount of import; t—efficiency of business; m—money supply; n—nationalism; r—interest rate; 0 — international balance of payment; r.—restriction of investment from foreign coun­tries; 3 — foreigners' deposits in banks and purchase of gov­ernment securities, stocks and other investment. As under­stood, the model was reduced to the corresponding numer­ical values and by three influence parameters (particular­ized operators between operands) by which the character of the impact onto the informed operand has been qualified. Informatica 21 (1997) 79-114 109 /|=e,f,fl,f),j,6 > m,n,t,t>,y,3; W e,f,0,f),j,e,m,n,t,t),r, 3 |= ; HA 0; 0 (t>Hn)h(? ;);(j?; )M ; V V(^i=f)i=t / / The scheme in Fig. 25 presents a complex and completely circularly structured informational graph (Subsection 6.7, with additional input and output paths for each involved entity). 11 An Example of Association Grap h Informational connection means entities being op­erationally linked and joined together through causal or logical relations or sequences. By infor­mational association such a connection between informational entities is meant which has a re­lation in similarity, contrariety, contiguity, cau­sation, etc. For example, a thought (process) is linked in the mind or memory with the other thoughts in the process of thinking (e.g. associa­tive components, an associative system, encoding consciously apprehended information, in [10], pp. 137-138). Further, the sensory and the motor ar­eas of the cortex are supposed to be connected with ideation, etc. For these associative phenom­ena a general type of the informational circular graph can be constructed in which the so-called parallel association arrays occur as presented in Fig. 26. Let us describe the graph in Fig. 26 in more de­tail, concerning a process of thinking association. Let be given a sentence in which operands (lan­guage entities like nouns, adjectives, pronouns, etc.) 110 Informatica 21 (1997) 79-114 A.P. Zeleznikar Figure 25: A parallelized scheme of the graph concerning an example of strategic decision making, according to Maruyama (1993), and with ezplicit input and output informing of the 12 involved infor­mational entities, e, f, g, h, j , t, m, n, t, o, f, 3. —ki rv 1—• -*\oqt-­ -*\aij-— — * J /•v" ' 1—ta. \ 2 J 1-*{oJ\* —*{a V-­ \~niJ Figure 26: The graph of parallel associative arrays in an associative serial loop. a i > a2>''' > ani J 22 2 al> a2> ' ' ' ) ari2' al> a2' >"nfc perform as certain sentence words, but between them the operators (language entities like verbs, adverbs, prepositions, etc.) are set. In the graph, operators are presented by vectors and the nature (meaning) of them depends on operands which they connect. A strict causal scheme of an associative medi­ation of the sentence, presented by the graph in Fig. 26 is / / //«1;\ /«i _1;\\ o fc-i. a. a a h H H VI Vl a.2 1 H \ Nrn! J i aj2e{a\,al,---,a2 n2}­ Ni?2^ \Nl ) F2> ' ' ' ' Nm2J i aj* e{af,a§,--- ,a*J ; |_Pfcc /U-fc l_fc |_fc \ Fgfe t\Fi)F2i'' iFmfcj are on disposal. In a natural language, such choices are nothing other than the adequate syn­onym and antonym word entities, by which the association process in the next associative cycles can come to the surface. It is evident that through such an informational processing the very ini­tial sentence can not only meaningly change in a substantial way, but can, through the use of antonyms and again synonyms, pass through var­ious mutually oppositional meanings. This pro­cess reminds on or approaches to a real associa­tive mediating in the living brain when linguistic thinking is performed on the conscious level (and, for example, by the use of dictionaries). It is important to stress that a cyclic graph hides the informing of an undetermined length. It only insists to make at least one informational cy­cle. Afterwards, the informing can stili be cyclic, but it can also stop at any operand or operator entity, not closing the ongoing cyclic informing. In this čase the part of the last cycle is serial. On the other hand, to some extent, cyclic informing is causal, depending on the concrete form of the cyclic formula, which can not be directly recog­nized from the graph. 12 Conclusio n Problems of informational graphs reveal the com­plexity of informational phenomenalism and make the appearahce of circularity and possible spon­taneity of emerging and arising informational en­tities more transparent as a pure informational formula and formula system approach could do in such an evident way. On the other hand, graphs as graphical informational entities can have their own informational presentation and can perform as regular informational entities (systems). 112 Informatica 21 (1997) 79-114 The history of the informational theory (since 1987) has gone through substantial principled (axiomatic) and formalistic innovations, so today it can fit the most pretentious requirements for the formalization in the area of consciousness phe­nomena (e.g., formalistic and graphical treatment of psychological, psychiatric, understanding, eco­nomical informational models, presented in [25]). The exposed formalism together with informa­tional graphs (a kind of conscious imprints, ex­pressed as the extremely possible parallel form) seems to be appropriate for defining, handling and observing the problems of consciousness with var­iouš consciousness components and systems con­cerning, for instance, experience, emotion, mem­ory, association, qualia, sensitivity, awareness, attention, intention, significance, meaning, dis­course, understanding, self, subconsciousness, un­consciousness, etc , being connected into a com­plex system of consciousness (conscious thinking). In the discussed sense, the informational graphs could be incorporated into the 'hot' (T < 0_) the­ory of the brain and society ([12, 13]) in a fuzzy disperse pattern. Wheeler [16] has argued persuasively that physics stands to learn a great deal about the world by looking it in terms of information. Infor­mation occupies a wonderfully ambiguous plače somewhere between the concrete and the subjec­tive [6]. He suggested [17] that information is fun­damental to the physics of the universe so that in a double-aspect theory, proposed by Slechta [14] and Chalmers [2, 3], information has both physi­cal and experiential aspects. Hameroff and Pen-rose [5] stress how experiential phenomena and the physical universe are inseparable (e.g., the duality of energy and information in [14]), and this may imply a necessary non-computability in conscious thought processes; and they argue that this non-computability must also be inher­ent in the phenomenon of quantum state self­reduction—the 'objective reduction'. Besides others, the theory of the informational fulfills these requirements and the concept of in­formational graph not only widens the instrumen­tality of the theory but makes the formalistic ap­proach more evident (technical). A.P. Železnikar References [1] Structure and Functions of the Human Prefrontal Cortex. 1995. J. Grafman, K.J. Holyoak, & F. Boller, Eds. Annals of the New York Academy of Sciences 769: i-ix + 1-411. The New York Academy of Sciences. New York. [2] CHALMERS , D. 1996 Facing up the Problem of Consciousness. Journal of Consciousness Studies 2: 200-219. [3] CHALMERS, D. 1996. The Conscious Mind. Oxford University Press, New York. [4] CUMMINGS, J.L . 1995. Anatomic and Be­havioral Aspects of Frontal-subcortical Cir­cuits. In [1]: 1-13. [5] HAMEROFF , S. & R. PENROSE . 1996. Con­scious Events as Orchestrated Space-Time Selections. Journal of Consciousness Studies 3: 36-53. [6] HAUSLADEN, P. , B. SCHUMACHER, M. VVESTMORELAND, & W.K . WOOTERS . 1995. Sending Classical Bits via Quantum Its. In Fundamental Problems in Gjuantum Theory: A Conference Held in Honor of Professor John A. Wheeler: 698-705. D.M. Greenberger & A. Zeilinger, Eds. Annals of the New York Academy of Sciences 755 i ­xiv + 1-908. The New York Academy of Sci­ences. New York. [7] HEIDEGGER , M. 1986. Sein und Zeit. Sechzehnte Aufiage. Max Niemeyer Verlag. Tubingen. [8] HILBERT , D. und P . BERNAYS. 1934. Grundlagen der Mathematik. Erster Band. Die Grundlagen der mathematischen Wis­senschaften in Einzeldarstellungen. Band XL. Verlag von Julius Springer. Berlin. [9] MARUYAMA, M. 1993. A Quickly Under­standable Notation System of Causal Loops for Strategic Decision Makers. Cybernetica 36: 37-41. [10] MOSCOVITCH , M. & G. WINOCUR . 1995. Frontal Lobes, Memory, and Aging. In [1]: 119-150. INFORMATIONAL GRAPHS [11] SLECHTA, J. 1996. On the Generalized Un­certainty Relations in the Quantum Statis­tical Theory of the 'Hot' (T < 0_) (Living) Systems (Brain), Human Society and Living Metabolism (Physics). In Knowledge Trans­fers 96. A. Behrooz, Ed.: 16-23. pAce. Lon­don. [12] SLECHTA, J. 1989. Brain as a 'hot' Cellu­lar Automaton. In Proceedings of the 12th International Congress on Cybernetics: 862­ 869. International Association for Cybernet­ics. Namur, Belgium. [13] SLECHTA, J. 1992. Society as a 'hot' Cellu­lar Automaton. In Proceedings of the 13th International Congress on Cybernetics: 405­ 409. International Association for Cybernet­ics. Namur, Belgium. [14] SLECHTA, J . 1993. On a Quantum-statistical Theory of Pair Interaction between Memory Traces in the Brain. Informatica 17: 109­ 115. [15] STUSS , D.T. , T . SHALLICE, M.P . ALEXAN­DER, & T.W . PICTON . 1995. A Multidis­ciplinary Approach to Anterior Attentional Functions. In [1]: 191-211. [16] WHEELER , J.A. 1990. It from Bit. In Com­plexity, Entropy, and the Physics of Informa­tion. W.H. Zurek, Ed. Addison-Wesley. [17] WHEELER, J.A . 1990. Information, Physics, Quantum: The Search for Links. In Com­plexity, Entropy, and the Physics of Infor­mation. W.H. Zurek, Ed. Addison-Wesley. [18] ZELEZNIKAR, A.P . 1989. Informational Logic IV. Informatica 13: 6-23. [19] ZELEZNIKAR, A.P . 1994. Informational Being-in. Informatica 18: 149-173. [20] ZELEZNIKAR, A. P . 1994. Informational Being-of. Informatica 18: 277-298. [21] ZELEZNIKAR, A.P . 1994. Principles of a For­mal Axiomatic Structure of the Informa­tional. Informatica 18: 133-158. [22] ZELEZNIKAR, A.P . 1995. A Concept of In­formational Machine. Cybernetica 38 : 7-36. Informatica 21 (1997) 79-114 113 [23] ZELEZNIKAR, A.P . 1995. Elements of Meta­mathematical and Informational Calculus. Informatica 19: 345-370. [24] ZELEZNIKAR, A.P . 1996. Informational Frames and Gestalts. Informatica 20: 65-94. [25] ZELEZNIKAR, A.P . 1966. Organization of Informational Metaphysicalism. Cybernetica 37: 135-162. [26] ZELEZNIKAR, A.P . 1996. Informational Transition of the Form a \= (3 and Its De­composition. Informatica 20: 331-358. Appendix A A Genera l Overview Concernin g For­mul a Markers , Graphs , Causa l Coeffi­cients, etc . In this papers several characteristic formula mark­ers, other symbols, and informational graphs have been introduced. To keep them in mind together with their meaning a special table (on the next page) could be helpful. It can serve the reader as a dictionary of the main notions and concepts presented in the paper. Figure 28 is a comprehensive table reminder of the most often used graphs and their informa­tional symbols belonging to formulas. The reader will easily find the entities concerning informa­tional axiomatism, serialism, circularism, causal­ism, parallelism, and gestaltism, together with the corresponding graphs. Markers of specific formu­las and formula systems used in the paper are sys­tematically structured and can be unambiguously distinguished from one another. These markers are shortcuts for standardized formulas and for­mula systems and will be used, from now on, al­ways consequently in the same form. Complex symbols in Fig. 28 can be typograph­ically standardized for later use. For instance: T n}_{J}\!\varphL{_{_{\!\to}}} ^ T{\:\:\:n}}-{-{N_{_{_{\!\!\to\!}}}}}% \!\varphi.{.{.{\!\to}}} ^ "{"{\,n+l}}.{-{N44\!\!\to}}"{"{\circlearrowleft% }}}}\!\varphi({\circlearrowleft}}-{.{.{\!\to}}}n+V" etc. 114 Informatica 21 (1997) 79-114 A.P. Zeleznikar Informational Axiomatism Serial Formula Systems /n+1 Q\ n+i o ^/ n \ M ip 1; Extemalism Internalism Metaphysicalism Phenomenalism a\= \=a a\=a (a\=? Parallel Formulas n , n+1 O' -H a -H o; -K a 0-v> ; v.. rf-v:) Informational Serialism ?_> —((••• ((« N «l) N «2) N • • • an-l) N* "n); 2V_» —((••• ((« N «l) r=a2) N • • •an-2) N* (an-1 N On))! 1 f2n\ l — (a N* («11= («2 H • • • («n-i 1= «n) • • •))) (Gl) •K*n-2j "/an-l j "/«; ©—< rCf^ («, ai, a2, • • • 1 «n | ^ L ^ 5 2V_; ' 1 (2n\ ,^-> ' 1 /2n\^ ­S+Tl,n J -1 nTU n) n+1 O Circular Serialism 1 _» — (((• ••((<* h « 0 N «2) t= • • • «n-i) N «n) N* «); n+1 O / x n+1 O . (p (a, «i, • • • ,an); 'V_ — (((• "•((«!= «1) 1= «2) |= • •• a n-2 ) |= «n-l ) N* («n 1= «)); -L-(2n+2\ l — n+2 ^n+1^ (G2) 02)—- • • • —«/«»1-2ij «/o!n-iJ «(an n+1 O n+1 O n+1 O n+1 O 1^-»' 2^-+ ' ' _l(2n+2\_.^-tV. ' 1 /'2n+2\ S+lZVn+lJ 1 H+1H n+1 ,/ Circulating the Main Operand a. j (ao = a) n + 1 O ;,V_>(Q;j.Q!i+i) -,ocn-i,otn,a,ai,---,aj-i); !<«_,•< AT°; j = 0,1,••• ,n (G3) •("j+ U Otn-l) >{an) *Ca) *(ai) ' *(aj-l) <\rO _ n+1 (2n+2\ Iy-* ~~ n+2 V n+1 / 1 ( iY'_i(Q;j>Q;J^-l!••• )«n-i,«n,a,ai,---,"i-ij) ^ Circular Causalism (r( " ; ^ ("j, ay+i, • • • , an-i,an, a, ai, • • • , ay-i)J ; j = 0,1,2, • • • , nj Primitive Informational Parallelism n'(" rCV-^ajai)"" ,an_i,an)) C VO^aii'' ' i"n-i,an); n'rV°(a,ai,' " ^^^(a^aii« ! |=a2 ;an -i [=aB ;an (=a); (G2 G3) n+1 O' , ' \_ i TT//' n +1 O/ \ \ ' V? {a,ai,---,an-.i,an) ^1 1 I . (^ (a,ai, • • • ,an-i,an)j; ! ^ V (a,ai,---,an-i,an)J C