Informational Erames and Gestalts Anton P . Zeleznikar. An Active Member of the New York Academy of Sciences Volaričeva ulica 8 SI 61111 Ljubljana, Slovenia (anton.p.zeleznikar@ijs.si) Keywords: informational frame—arrangement, concatenation, demarked, disharmonious, edge syn­tax, harmonious, operator composition, parenthesized, possibility, syntax; informational framing; in­ formational gestalt—axiomatic, causal, circular, inference, informational machine, intelligent, me­taphysicalistic, parallel, phenomenalistic, reverse, serial, star; informational gestaltism; number of formulas in a gestalt Edited by: Jifi Šlechta Received: September 28, 1995 Revised: November 6, 1995 Accepted: November 20, 1995 This article deals with two characteristic and widely useful notions, called the informa­tional frame (of representation) and the informational gestalt. A preliminary discussion ofboth notions was already presented in [17]. Informational framing is nothing else than a part of informational gestaltism by which various causal possibilities of formulas come into existence. Although a frame is everything which can be put in a frame within a well-formed informational formula, the concatenation of frames must preserve the so-called possibility of a frame to be a part of a well-formed formula. On the other hand, the gestalt structure represents a parallel array of informational formulas, that is, an informational system of causally different formulas proceeding from a given formula. To this, circular gestalting can induce the reversely circular properties, so different forms of gestalts become possible. The most complex and free gestalt called star gestalt is a consequence of an initial circular formula and' its graph, where operator transitions from one to the next operand are possible in an arbitrary manner of repeated looping. 1 Introduction \= to an arbitrarily complex part of the formula, ) (= a . In a similar way, the principle of fra­ say Both the notion of informational frame and infor­ming can be applied to elements of a demarked mational gestalt1 have been introduced in a super-formula [12], for example, [7] or to any complex ficial form, in connection with the informational part of the demarked formula, e.g. .|=o ABeing-of [17] as a phenomenon of informational gestalt of any informationalformula is everything functionalism. In this article, both informational formally, especially causally, hidden behind the frame and informational gestalt will be tackled structure of the formula which represents an ar­in a more fundamental manner using some parti­bitrarily complex informational entity. We could cular means of informational formalism to make say that the gestaltistic nature of a serial formula them informationally accessible and formally ef­comes in the foreground when this formula is rou­fective. ghly sketched by the corresponding informational An informational frame is everything which can graph (a graphical scheme of operand circles and be framed within an informational formula, from operator arrows) in which no parenthesis pairs or a single parenthesis ) , operand [a ] or operator demarcation points of a formula are considered. Therefore, the gestalt of an informational entity 'This paper is a private author's work and no part of remains the hidden other, formallv, the adequa­ it may be used, reproduced or translated in any manner tely transformed formula system concerning the vvhatsoever without written permission except in the čase of brief quotations embodied in critical articles. original operands and operators in an unchanged order within a formula, describing the entity in question. In general, an informational entity can appear (be understood, interpreted, grasped) as a gestalt of different possible entities (various for­mulas), strictly corresponding to the initial order of the operand-operator structure of the entity in question leaving parenthesizing or demarcation completely open. As the reader will see, the informational frame (of representation [4]) can be any part of an in­formational formula (system) and, in this respect, the frame does not follow a structurally limited and traditionally organized syntactic formalism. The concept of the informational frame shows also how traditional syntactic schemes can be ma­naged in a natural frame-appearing form leaving open the concatenation of frames in, certainly, a regular form. Any part of the formula means that the formula can be broken off. at any plače and that a part of it does not necessarily represent an operand or operator structure. On the other hand, an informational gestalt will represent the possibility of the variety or, the whole variety or the whole variety of an operand, formula or formula system in a structural sense, where different structures (possible formulas to a given formula) can be forecasted automatically to some determined extent. 2 Two Cases of Informational Enframing The usefulness of framing was demonstrated for a verbal čase concerning the predicate of existing of something in [18]. Framing of the informational entities can become an effective and transparent aid in transforming linguistic forms (especially, sentences) into informational ones (e.g., cogniti­vely relevant internal states, representations) and vice versa. 2.1 Enframing of a Sentence and Its Translation Let us take an example of framing a sentence ([5] p. 161) in German, its formal transcription, and retranslation into English. The enframed original German sentence is A.P. Zeleznikar Die Rede ist mit ^, Befindlichkeit und Verstehen existenzial gleichursprunglich The dot at the end of the sentence will be re­placed by the semicolon at the end of a formula, when the formula appears in a parallel informa­tional system. In the last frame the main ope­rator composition (verbally, ist existenzial gleic­hursprunglich mit) is split within the two frames marked consecutively by i and, to this, stili ver­bally transposed. In this sense, the substitutional enframed German sentence, also suitable for for­mal translation, becomes Die Rede ist existenzial gleichursprunglich mit Befindlichkeit und Verstehen Let us introduce the following marks for the ope­rands and operators in the given two sentences: t for 'Rede' (discourse <$), b for 'Befindlichkeit' (state-of-mind a) and o for 'Verstehen' (under­standing v) as operands, and (=ex for 'ist exi­stenzial' ('is existential' or 'informs existentially', operatdr (=exist), (=gi-ur for 'ist gleichursprunglich' ('is equiprimordial' or 'informs equiprimordially', operator (=eq-prim) and |=mit for 'ist mit' ('is with' or 'informs with', operator f=with) as operators. An interpretation of the first enframed form is where (t(=mit M ) when the German 'und' ('and') was interpreted by comma (a short form) and by semicolon (a long form) within a parallel system of two formulas. Both kinds of expression are equivalent (informa­tional operator =) . The second form of the sentence enframing (frame-informational interpretation) delivers, for­mally, t ( Nex ° Ngl-ur) ° Nmit &. t> being a parallel system (as shown in the preceding čase) in regard to Befindlichkeit b and Verstehen t). The first formula is externalistically open via the operator composition [=ex o (=gi-Ur while in the second formula this composition is brought into a formula interior operator composition with Nmit) tha t iS, (Nex°Ngl-ur)°Nmit­ Let us take the second form of the enframing of the German sentence (the last formula) with the English-adequate operand and operator markers, tha t is, P (Nexist ° Neq-prim)° Nwith &, V Considering this formula, the English translation of the German sentence becomes (in an adequa­tely enframed form) Discourse is existentially equiprimordial with state-of-mind and understanding This sentence appears in the English translation of [5], tha t is in [6] (p. 203). A 'weaker' (so-called literal) translation would follow the first form of the German sentence enframing, tha t is, Discourse is with fa state-of-mind and\ understanding existentially equiprirnordial Another scheme of framing for the discussed sen­tence, depending upon an intuitive understan­ding, could take the enframing form Discourse is existentially equiprimordial with state-of-mind and understanding In the parallel approach, as formally expressed, the and disappears and so two parallel sentences are coming to the surface, tha t is, Discourse is existentially equiprimordial with state-of-mind Discourse is existentially equiprimordial uiith understanding Now, a dot appears at the end of each of the two enframed sentences. In the formal čase, we have got the possibility with the comma between a and v, or the semicolon between the two formulas. It is to stress tha t substantial differences can exist between the operators subscripted by \ and fo because the first one concerns internally the state­of-mind while the second one pertains internally to understanding. Here, the informational cha­racter of an equally marked operator in different contexts comes to the worth. Namely, in an in­formational transition of the form a (= /?, where a is the informer and /3 is the observer, operator \= is to be understood as operator composition (= a o^ , where \=Q has to be decomposed accor­ding to the a' s informingness and \=p according to the /3's informedness. The demonstrated translation of the German sentence into informational formula was-only tha t which could be called the first informational approximation. Introducing the concept of the so-called informational graph it is possible to give several kinds of interpretation of an informatio­nal formula. In Fig. 1 an informational graph of the widening of this approximation is presented Figure 1: A graphical interpretation ofthe bidirec­tionally and circularly (5-loop) structured parallel decomposed system v concerning the input entity a. where the semantic scope of the word 'gleichur­spriinglich' (equiprimordial) is taken into consi­deration. Let us mark the operator composition ((=exo[=g^ur)o|=mit corresponding to 'ist existen­zial gleichurspriinglich mit' by operator (=c. By this operator the arrows between entities t, b, t) in Fig. 1 are marked. The vertical input and output arrows are unmarked and represent the general operator N- Which is the parallel informational system de­scribing the graph in Fig. 1? If discourse (die Rede) r concerns an entity a, that is t(«) , the so-called t-solution of the system graphically pre­sented is (a (=r; |= b, ti; \ t, b,tt (=; v (a) t N e b; b h=c t>; o (=e r; \t|= c o ; t)(=cb; b\=ctj In the first row of the parallel array the input pa­ths are located and in the second row the output paths. The third and the fourth row represent the main two loops (the left and the right one) of the graph. These two loops, joined, hide three more elementary loops existing between operands r and b, b and t>, and t) and t. Thus, one of the so-called serial type splutions (a specific one) for x(a) can be expressed as (a (= r; |= b, ti; \ t, b,t> [=; t Ne (b K (o K t)); t1 (a) t K (o Ne (b Ne t)); t Ne (b Ne t); b Ne (»Neb); VtiNe(tNeO) / This solution formula system considers the input and output strueture of the graph in Fig. 1 in the first and the second row, two long-size (bidireeti­onal) loops in the third and fourth rov/, and three short loops in the last two rows, respectively. The discussed examples of the natural text translation into informational formulas, and vice versa, demonstrate the hidden (intuitive) impor­tance of the informational framing. As a con­sequence, framing concerns the so-called gestal­ting, where as the gestalt of a serially struetured formula the parallel system of ali possible formu­las is meant, emerged from an initial formula by ali possible displacements of the parenthesis pairs. 2.2 Sentential Paradigm and Possible Frames and a Gestalt The ideas for the subsequent example of using informational frames and a gestalt go back to Fo­dor [2, 3] being sketehed in a condensed form in Churchland ([1], pp. 385-388). We will study this particular čase on the basis of the graph presented in Fig. 2. This graph is an exact representation of the parallel formula system (as a free-standing sy­stem without a specific marker) consisting of pri­mitive transitions (from an operand to andther), which is, i N X ; \ N i*; i* N Lt; d N 7*; li N * N «. / This array includes three types of transitions: the input, forvvard, and backward ones. In the first row, the only input transition informs the internal state about the sentence. In the second and third row seven forward transitions are listed. In the last two rows, seven backward (feedback) transi­tions appear, enabling a complex cycling of the systern when the input sentence is identified in a processing way by the arising representation pv and content ct. On the basis of this graph we will present a solution for content ct and some possible, for the discussion relevant frames. The gestalt and possi­ble interpretations of content ct will be presented (and discussed later on). The concept of sentential attitudes aa belongs to the autonomy of human being; on the other side, the logical inference defends co-evolution and interdependence, that is, informing of enti­ties, in informational terms. Beliefs, desires, tho­ughts, intentions, etc. inform between a being's internalism and externalism and play a crucial role in causation of living behavior. Sentential at­titudes approach the nature of the so-called repre­sentations and of the rules that govern the transi­tions between representations. Ali this sounds in­formationally familiar where sentential attitudes and representations may funetion as distinguished informational entities. There exists an interplay of the sentential at­titudes and representations, a parallel and serial informing among entities. The theory postulates that sentential attitudes can also play a role in nonconscious processes and, in this way, can be involved in cognitive processes, e.g. in the form internal identification identification representation-content state informing counterinforming embedding Figure 2: A sententialbj paradigmatic, metaphysicalistically (7-tuple-loop) structured internal state system i giving the input sentence(s) a the sentence(s) representation pL{o~) and sentence(s) internal content (understanding with meaning) ct(a). of inductive and deductive logic and decision ma­king (the theory of internal states). Let operand a denote a sentence or a set of sentences, and operand p a representation or a set of representations. Transitions of sentences <7j |= (pi |= p j) Other entities are the sentential attitude, (o (ai,...,an); t ^ (ti,.. .,L„); aa(ou ...,an) i*(c(0 <—• cr); Pi v± (pu..-,Pn); \Ct # C t (ii),...,C 4 (A n ) / / and the internal state t. Cognitively relevant internal states can be comprehended as t # ti,. . . ,4m . An internal state i has, in general, a content c (4) (or in a short form, ct), thus, cL ^ ct(ti) , ... , ct(in) , where content components can be identified via sentences, that is, ct(ii) v^ o~im , • • • ; ct(im) # o-m In this čase, operator '^ ' can be replaced by ope­rator |=identically which reads 'informs identically' (that is, circularly between the #-involved ope­rands). Such an operator is bidirectional (consti­tuting a loop, graphically), that is, = ^i.\l/i) r^identically fj j 0"t r identically Ci\H ) The identification, it(ct(ij) <—>CT,-), is possible by virtue of the isomorphism (operator U—>•') be­tween n and o~{. Let us discuss the solution for content cc(a) on the basis of the graph in Fig. 2, where ct(<7) ^ (a |=i ; 3*'i=[(]*i=[(]c*i= 3. h(i.l=(*.l= (%l=*))); 0(€t h TO h csj)K(h h (c h t= (L N a \= la) \= la) \= CQ) La\=0! )) (((( a \=la)\=*(Sa\=( ( ((( C* h Z« ) N O t= Ca) (= la) h* {L<* N (Sa h «)) between operands a at the beginning and a at the end, a at the beginning and sa, and Ta and ia, respectively. In the first two cases, the operator frame is an entity of three parts, namely, the left, middle and the right one, the third čase has a two-part operator frame. • The last example shows how parenthesized fra­mes can become badly transparent and mutually dependent. Definition 3 [Operator Frame in a Demarked For­mula] Operator frame in a demarked formula is the part betvoeen arbitrary two operands. • Exampl e 2 [Some Operator Frames of a Demar­ked Formula] Let the formula in Example 1 be expressed in the demarked form, that is, « \=la-\= t-a-\=Ca-\= la-\=-La \= •L» \= a The enframed examples, discussed in Example 1, become evidently, a a; la " F" " ^"a r~ • ^a F L<* |=.e f=a; a 7ot-|=­ a |= Xa .\=ia.\=Ca.\= ja.\=.La\=. Each čase, irrespective of the operator frame structure (and position), possesses only one de­marked frame. D 4.4 Harmonious Informational Frames We have to determine harmonious and disharmo­nious parenthesized and demarked informational frames. Which are the various possible frame forms and in which manner do they appear as parenthesized and demarked frames? Which are the advantages of frames in one or the other form? At the first glance, it seems that harmonious frames always appear in one piece, tha t is, they are not split within a formula. We shall see how this principle can have different consequences comparing the parenthesized and the demarked frames. 4.4.1 Harmonious Parenthesized Frames Parenthesized frames in formulas of Example 1 are ali harmonious. A frame, although split in two or three parts, is harmonious if it is functionally closed. Definition 4 [Harmonious Parenthesized Frame] A split, or unsplit, parenthesized frame is harmo­nious if it is functionally closed. The functional closeness of a parenthesized frame means that af­ter concatenation of its parts the resulting frame presents a uiell-formed formula, that is, an ope­rand frame. In the procedure of frame concate­nation the empty parenthesized parts of the form () are replaced by an empty part (informational nothing), X. At such places, the rule () <— A is applied. Also, the outmost parenthesis pair can be omitted. D The concatenation of frame parts in Example 1 gives N ?a) \= la) \= Ca) (((( \= la) \= ta) (= Ca) (((( )) \= la) |=* V* 1= ( ((( )h*a)NCo)| = which results in three well-formed formulas, tha t is, ((((h la) \= t j \= Ca) \= la) h* (Sa \= (e }=)); ' (((([= I^|= ta) |=C^|=7a)|=*(^l=); One can see how the concatenated frame-harmonious structures are reduced in respect of the 'goal' entities, so tha t they preserve the ne­cessary formula well-formedness. Example 3 [Split Parenthesized Frames] Split pa­renthesized frames can be harmonious and dishar­monious. They are divided in at least two parts and each part of a split harmonious frame is dis­harmonious. But the parts of a harmonious frame can be concatenated into a unique (well-formed) frame formula. For instance, the split frames in parenthesized formulas a h č)l= 7) M ((a |= (3 ) \= 7) \= S; ((°M),H7[)ll=* are ali harmonious. On the other hand, the frame parts are ali disharmonious; tha t is, (( ; M)h7 ) ; •((«!=; )l=7) ((«M)f= ;. ) The reader can recognize the disharmonious structures of the listed frames by himself/herself. D 4.4.2 Harmonious Demarked Frames What is the difference between a harmonious pa­renthesized and demarked frame in concern to a frame splUting? Definition 5 [Harmonious Demarked Frame] An unsplit demarked frame is harmonious if it is functionally closed. The functional closeness of a demarked frame means that the frame itself pre­sents a demarked uiell-formed formula. D Because the demarcation point replaces the pa­renthesis pair, the split harmonious frames in čase of parenthesized formulas appear as unique fra­mes in cases of demarked formulas. This is quite true for Example 2, where the demarked harmo­nious frames are p: Xa . \= ia .f=LQ . |= p= XQ. . |= ia . p= La . p= ya . p= • ; • f— '•ar " P1 ^a • F The rightmost operator combination . C • in the second frame means . C (), so, it can be redu­ced into . C for the sake of simplicity. Similarly, the leftmost operator combination . C in the third frame can be replaced by operator C to keep the frame in a common form. 4.4.3 A Syntax Comparison between Harmonious Parenthesized and Harmonious Demarked Frames How can harmonious parenthesized frames be re­cognized at once? The answer is: by stating that they represent well-formed formulas. The correct form of a formula can be proved by the usual syntax analysis, taking into account the general, context-free grammar for well-formed informatio­nal formulas. It is instructive to determine such grammars for parenthesized and demarked infor­mational formulas. Designing a syntax, one can consider that a formula or formula system is nothing other than an operand. It means that the initial (starting) grammatical variable in a formula development (generation) is the operand, symbolized gram­matically (and as a terminal) by o. A gene­ral context-free grammar for the parenthesized formula systems can be constructed by the fol­lowing items (syntax categories and terminals): o as operand; (= as operator; o as a separa­tor in an operator composition; semicolon ';' as the operator of formula parallelism; comma ',' as the operator of alternativeness; and '( ' and ')' as parenthesis pair. A preliminary context-free grammar is a construct G = (N,T,R,o), where N = {o, \=} is the alphabet of nontermi­nals, T - {'(', ')' , o, ';', ',', o, \=} denotes the ter­minal alphabet, R is ase t of context-free rules (see below) and o marks the initial symbol. The set of rules is determined by two syntax rules, which are o f- o |= | (= o |o (= o | (o) | o; o | o,o \= <- h°h 1(1=) In čase of formula systems using the demarca­tion points instead of the parenthesis pairs, the demarked grammar is G' = (N, T' ,R*, o), the al­phabet of terminals is T' = {'.'o, ';', ',', o, \=} and the rules of R' are o <— o(=|(=o|o(=o|.o|o.|o;o|o , o N <- N°NI-NI N- Rules with demarcation point in the second line (operator composition čase) can be used only in such a way that the point is inside of an operator composition. For example, in an operator gene­ration process, there is (= -» (=o^ -» N -0 ^ "^ l=c'l=,0h:: -* (=o[= .o. (= -» (=o (= .o. [=o (= where the end result would correspond to the ope­rator composition (\= o (=) o ([= o |=). Symbol -» represents the derivation (generation) step. 4.5 Disharmoniou s Informationa l Frame s The concept of the disharmonious informational frame (DIF) enables the treatment of arbitrary for­mula parts which do not fit harmonious informa­tional frames. DIFS , in this way, contribute to the possibility to treat arbitrary parts of formulas as entities which may, in special cases, be of essential interest. Definition 6 [Disharmonious Frame] An infor­mational frame is disharmonious if it is a part of a tvell-formed formula or formula system but it does not represent a well-formed formula by itself. a Disharmonious frames are parenthesized incom­pletely within a parenthesized formula and de­marked within a demarked formula. 4.5.1 Disharmonious Parenthesized Frames In a parenthesized formula, disharmonious paren­thesized frames are the most arbitrary (enframed) entities. They can be parts of harmonious fra­mes on one side, on the other side they can em­brace any imaginable sequence of adequately ser­ried operands, operators and parentheses, which do not constitute a harmonious frame. So, both types of frames exclude each other. Definition 7 [Disharmonious Parenthesized Fra­me] An informational frame is a disharmonious parenthesized frame if it is a part of a well-form­ed parenthesized formula or formula system but it does not represent a well-formed parenthesized formula by itself. • By this definition, a disharmonious parenthesized frame is not an arbitrary sequence of operands, operators and parentheses, but is an arbitrary sequence of the mentioned entities which consti­tute a part of a well-formed parenthesized formula or formula system. For example, frames ( , ) , etc. are disharmonious pa­ (h H h (h renthesized frames because they can be completed to the harmonious parenthesized frames. 4.5.2 Disharmonious Demarked Frames In a demarked formula, disharmonious demarked frames are enframed entities being arbitrary parts of the formula. They can represent parts of de­marked harmonious frames and, in this way, can include any imaginable sequence of adequately composed operands, operators and demarcation points, which in this sequence appear in a harmo­nious frame. Demarked harmonious, and demar­ked disharmonious frames, exclude each other. Definition 8 [Disharmonious Demarked Frame] An informational frame is a disharmonious de­marked frame if it is a part of a voell-formed de­marked formula or formula system but it does not represent a voell-formed demarked formula by it­self. • By this definition, a disharmonious demarked frame is not an arbitrary sequence of operands, operators and demarcation points, but is an ar­bitrary sequence of the mentioned entities which Informatica 20 (1996) 65-94 75 constitute a part of a well-formed demarked for­mula or formula system. E.g., frames [7|, I. (= L etc. are disharmonious de­ [B h-h - -h marked frames because they can be completed to the harmonious demarked frames. 4.5.3 A Comparison between Disharmonious Parenthesized and Disharmonious Demarked Frames The comparison betvveen disharmonious paren­thesized and 'disharmonious demarked frames concerns the so-called of the frame's left and right edge development (see Subsubsection 5.6.1 and Table 1). In designing a frame (harmonious as well as disharmonious one), the designer (desi­gning entity) proceeds from that part of the frame which already exists, developing the frame at its edges in such a way that the emerging disharmo­nious frame will become a part of a possible har­monious frame or of well-formed formula. Syntax rules for generation of disharmonious parenthesi­zed and disharmonious demarked frames differ, of course, essentially betvveen the parenthesized and the demarked čase. 4.6 Functional Frame s Functional frames are characteristic in such a way that they can be clearly recognized in comparison with other formula frames. Informational func­tion as an informational operand has the form («) vihere a is an argument formula and (p is a func­tion representative informing upon the argument [17]. D The following comment could be useful: frames \~Č7\ and (a) are harmonious and can appear as such also outside a functional context. On the other side, the possibility of framing both the function entity (p, and the function argument a, makes them visible for further informational in­vestigation. 4.6.2 Disharmonious Functional Frames A disharmonious functional frame offers an espe­cially characteristically visible čase which expli­citly concerns the syntactic structure pertaining solely to the concept of informational function. Definition 10 [Disharmonious Functional Frame] Characteristic disharmonious functional frames take the general forms as )( , )( , N)( , )( H , N(N , v)(N , etc. Theu must satisfv the operator syntax descri­bed in Subsection JtJh3. • In a similar way, frames for the demarked opera­tor compositions can be determined. Definition 12 [Harmonious Demarked Frame of Operator Composition] Harmonious demarked fra­mes of an operator composition can take the ge­neral forms t=°N » ho-f=°h , h°t=-° t= h°h-°-h° h etc. They must satisfy the operator syntax descri­bed in Subsection 4-4-3'• D In a syntactically regular way, harmonious opera­tor composition frames can be arbitrarily adequa­tely nested. 4.7.2 Disharmonious Frames for Operator Compositions Which are the main forms of disharmonious pa­renthesized and demarked operator composition frames? At least, a harmonious operator compo­sition frame must begin and end by an operator |= (which, obviously, follows-from the previous defi­nition). A general answer is given in the form of the following two definitions for a parenthesized and demarked čase, respectively. Definitio n 13 [Disharmonious Parenthesized Fra­me of Operator Composition] Disharmonious pa­renthesized frames of an operator composition can take the general forms 0. )° . °( . )°( > N° . °(N h)°(h , °h)°(h°, °N°(N°h etc. • On the other hand, frames of disharmonious de­marked operator compositions are determined in the following manner. Definitio n 14 [Disharmonious Demarked Frame of Operator Composition] Disharmonious demar­ked frames of an operator composition can take the general forms [o],^,^,^, o^ h |= .o . [=o , (=o f= .o.\= o[=.o . \=o\=. etc. • A Frame-analytical Com­prehension of Informational Transition and Possible Fram e Concatenatio n 5.1 Operato r Fram e Informational operator is tha t entity-which appe­ars between two informational operands, forming the so-called basic informational transition, irre­spective of the complexity of operands. On the other hand, the operator can be understood as an arbitrarily complex entity, composed of many other entities, tha t is, operators, operands, and parenthesis pairs, which do not constitute a well-formed formula. In such a sense, operator can be Informatica 20 (1996) 65-94 77 Left Right Oper­ (a) oper--oper­ ator and and Left Left Right Right Sepa (b) oper-oper­oper­oper­ rator and ator ator and Figure 3: Frame sequences for transition of type (a) a \= /? and (b) a |=o(= /3. comprehended as a the most essential informatio­nal frame—the operator frame—which has to be studied carefully, and exhaustively. Informational transition, of the form (a) a \= /3, or, operator-compositionally, in the form (b) a \= o [= (3 can be comprehended by the general schemes in Fig. 3, respectively. How can infor­mational frames be consistently (well-formedly) joined together? Let us start with the following frame structu­res concerning the transition cv (= /3. There are, evidently, some possible frame configurations: a \= /?; « K° N /3; a K 1 ° 1 H P a \=a o N/3/9 ; a \=a o | N0 01 (=« 0 H /? Let us study particular framings of informational transition. 5.2 Fram e Concatenatio n an d Fram e Parallelis m Under certain circumstances, informational fra­mes can be concatenated into new frames and can be stacked in a parallel manner within frames and, then, the stacked frames concatenated, etc. Definition 15 [Frame Concatenation] Two fra­mes (pa, and 4>@, can be concatenated buiiding up a frame (j), that is, (p # a4>[)!tfboth a,\, a,2, '•••, 4>a,k and ptl, (f>pft, •••, pikl tively, can be concatenated into frame array (j) voith frame components 4>\,2,'' • i^k *'" such a way, that (j> v^ <}>a!3, where 4>i 4>a,l P,l 4>2 4>a,2 4>P,2 4>a,2 4>(3,2 -that is, k <}>a,k 4>(i,k 4>a,k P,k (and 4>i # 4>a,ip,i for i-1,2,••-,k). D 5.3 Framing the Transition a \= /3 In the framed transition a \= /3, the question what could an operator (= represent, and which is the degree of its complexity in a serial and parallel sense, comes to the surface. Initially, |= is an arbitrarily structured operator and its structure has now to be clarified. Evidently, the serial parenthesized decomposi­tion of a transition a (= j3 can have the form • a, r)\=* (•••(( a /?)•••) ) (fr (=(&!=•• (Pn\= The complex operator frame is split into three parts and one can understand how a symbol (= between operands can become as complex as pos­sible. Operator |=* is the main operator and si­gnals the transition decomposition process is ne­ver ended. Simultaneously, it marks, how the a­part of length La — m of the operator belongs to a and how the /3-part of length L@ = n belongs to (3. Thus, the length of decomposed transition is La^p = m + n + 1. The transition gestalt includes (see Subsection 6.3) 1 /2 m + 2n + 2> N,a\=p m+n+2\ m + n + 1 , possible decompositions of transition a (= (3 of length La + Lp + l. The demarked form of the discussed operator decomposition of transition has a compact shape, with only one (unsplit) frame, that is, h ai . h « 2 • h ••« m h­ a P Pi h •P2 h h•Pn h A.P. Zeleznikar where the čare of parenthesis pairs is left over the the mechanism of the demarcation point. The next enframing shows a clear separation between the left, and the right, part of transition a (= (3 in regard of the main operator (=*. There is respec­a (•••( ( K \= •••otm) (Pi h (P2 h /> )-». . •••09» 1= where (f>a is a harmonious left frame and 4>p a har­monious right frame. Within these frames, frame pairs and (4>a,\, <^a,r) and (cf>/3t\, <5^/?,i) are disharmo­nious. Additionally, the main operator does not belong explicitly either to operand a or to ope­rand /3. It stays between both of them and can be decomposed in a further way in the left, and the right, direction. The demarked čase of the discussed example brings a clear evidence how the informer, and the observer, part in a transition can be separated. There is a h P • \=---otm Certainly, there can exist several different, that is parallel, decompositions of transition a (= /3. In this čase, instead of asingle operator enframed formula, there are, say, k parallel formulas of the form (= «u ) N «12) 1= •••«lm ) •(( a h* P )•••)) (fti h (Pu N • • (An 1= (• • |= a2i) \= a22 ) (= •••«2m ) (•• •(( a p )•••)) (P21 \= (P22 \= • • •(ftn|= (= atki) \= «12) |= •••Oik m) (• •(( a K P )•••)) (Pki \=(Pk2 h" (Pkn \= which results into a compact interpretation of the transition decomposition by <*im) h* Wu \= (#12 N • • • (#ln h (•• •(( )• ••)) (= «2i) h «22) h (• •(( «2m) (=* (#21 )• ••)) a h(/322N---(^2n h # (• )• •(( ••)) h (&2 h •••(&« h The reader can comprehend in which sense the complexity of a transition operator (= can deve­lop. In the last enframing example the split parts of a parallel decomposed transition operator are separately enframed, so that operands a and /3 appear only once, like in an informational graph. In the last complex parallel čase, the transition gestalt includes (see Subsection 6.3) 1 2mJ + 2n t+2> N.oN? L f^ mi + n{ + 2\ ro,- +ra,- + 1 possible decompositions of transition a (= # of parallel lengths la ^ + Ipj + 1 for i = 1,2,..., k. As already ascertained, the demarked form does not need an explanation concerning the pa­rallel frames within a frame. The demarked for­mula system for multiple operator (e.g. interpre­tative) decomposed transition is (without serially split frames) (= «H . |= a i 2 • f= • • • «lm . N -/3ll h .#12 h ••••Pm h 4>i (= «2 1 • \= «2 2 • (= • • • «2m-|= -#21 a (=-#22r=--"#2n| = rf. : |= <*fcl • N ak2 • N • • • Oikm- t = -#A:1 h-#* 2 !=••••#* » 1= In a compact, however virtually artificial way, the separation into the left and the right part fra­mes can be expressed in the form Informatica 20 (1996) 65-94 79 1= an ) 1= «12) (• • « . .' Q, l ( = •••«im ) L r Ni ( • •«^ , a, l a | =f= a 2 i ) | = «22 ) •••Q!2m ) 4>l r K • ( • •K**, Q, l 3,1 : |= ttfci) N afc2) H | = •••Oikm) 0^ . (#11 N (#12 h •••(ftn h i>\. *P,1 ) • (#21 H (#22 1= %,r ) • •••(#2n| = '^, r 0i , # ' vp,\ )• "P^.r (#fcl N (#Jfe2 N •••(/3kn\= <&, ^/3,1 The reader can find the demarked form of the last formula system in Section 11 and Subsections 5.4 and 5.5 where the philosophy around transition a f= o (= /3 is debated. A kind of the system simplification will become evident. 5.4 Framing the Transition a (=o(=# At the first look, there is a minimal difference between the presentation of transition decompo­sition a (= /3 and a (= o (= /3. However, as it is pointed out in Subsection 5.5, the difference is es­sential, because in the second čase the informer part (a) and the observer part (/3) can be sepa­rated up to the operator composition separator 'o'. In general, for the parenthesized čase, the split, composed operator (|=o|=) frame example is (=ai)(=a2 ) |= • ••Oim) ( = (•••(( a #)•••) ) h (#1 N (#2 N " •(#n h The demarked form of the same formula becomes \= ax. (= a2 . f= ••otm.\= a # l=-#il=-#2k-•••!=•# » h The next two examples of possible enframing come near to the goal of separation between the informer and the observer. In čase of parenthesi­zed formula, one obtains the enframing \= «i) |= a2) a (•• •(( (= •••«m) N r ' 1 '9(3,t 0/3,1 and in čase of demarked formula the enframing \=ai.\=a2 P a • ( = • • • « m • ( = •••t=-/?n h comes into the separation foreground. Further examples, discussed in Subsection 5.3, can easily be constructed for the a\=o\= /3 čase. 5.5 Interpreting the Transition Framing a \=a o \=p f3 Framing of the form a \=a o \=p (5 is essen­tial for a proper understanding and informational regularity of the separator 'o' , functioning as a regular operator. Evidently, ( a K o hj/?)s=((«l=«)°(h9)3)) In this formula, subformulas (a \=a) and ([=g (3) are well-formed formulas, and between them an informational operator, tha t is, also, operator 'o' , can appear. In this way, separator 'o ' is a regular informational operator. On the other hand, in a t=a0 N 3 /?! entity 'o ' is a member of the opera­tor composition \=Q o \=p, tha t is, the separator between operators \=a and [=g. In both cases, the meaning of 'o ' is a sort of informational concate­nation between the informing operand a and by it informed operand (3. 5.6 A Consistent Concatenation of Frames in Complex Transition Formulas How can arbitrary frames be linked together to keep the possibility tha t a final frame concate­nation will represent a well-formed formula? A A.P. Zeleznikar particular question concerns the frame concate­nation which would lead to an operator frame be­tween arbitrary two operands in a serial (or se­rially circular) formula. What are the characte­ristics of such an disharmonious (irregular, non­well-formed) operator frame? 5.6.1 A Conditional Frame Edge Syntax How can frames be composed beginning from an initial frame H? Let the initial frame tt be repla­ced by a single basic alternative frame which is nothing else than a general symbol appearing in a well-formed formula. Thus, we introduce the initial replacement rule in the form H <-a | |= | o | ( | ) | , | ; where a represents ali possible operands, (= ali possible operators, o composition operator for operators. We can also introduce operand in such a way, tha t the final result of different frame concatenation delivers, at the end, a vvell-formed informational formula, after an arbitrary initial rule K <— (j) was chosen. A complete collection of frame edge syn­tax rules is listed in Table 1. In this table, the aesthetical (obligatory) space symbol, u , is introduced, which explicitly marks the usual space between formula components. As one can see, there are seven syntactic types of symbols as given by the H-rule. The use of rules from Table 1 is conditional. The condition which must be satisfied at the generation of frame is a syntactically correct part of the arising informational formula. Thus, additional conditi­ons in the form of context dependent rules can be constructed in the following way: We must not forget tha t the rules can be applied only at the edge of the current frame . 5.6.2 Explanations Concerning the Use of Rules in Table 1 Explanations concerning the use of concrete rules in Table 1 is necessary. As said at the very be­ginning of this section, the application of concrete 0 1 2 3 4 5 6 7 8 9 10 11 12 # 0 K 4-a h o ( ) ) J 1 a 4-«h -Na-( a a) a , jU « a; ;u« «(.... «u(, ) a ) u a 2 f- h «h h« (h h) h, ,uh h; ;uh h( ")N" fo oh 3 o «-oh h° oh) (ho o( )° a h° o[= a au(h° oh)u« )°h 4 ( must be a part of a well-formed formula. We will refer to particular rules by (x, z) markers, for example, (2,12) |= <— o(=. The most interesting cases are those in which for a given edge symbol several ru­les for this symbol could be applied, but only one (or several) can generate a well-formed formula. The use of a certain rule depends on the context on the right or the left side of the edge symbol. Additional explanations to the use of some no­nevident rules (replacements) for the edge sym­bols are given in Table 2. The application of ru­les is conditional in the sense tha t the result of a symbol replacement must stay within the well-formed formula. The preceding (already existing) context and the intention of a formula develop­ment determine the choice of a concrete rule. 5.6.3 Examples : th e Applicatio n of Conditiona l Fram e Edg e Synta x Let us show several examples -of the discussed frame syntax. This syntax enables a straightfor­ward generation of frames from the left to the ri­ght and vice versa, but also from wherever in the middle of an arising formula and then proceeding on its left and its right side. Within such a frame generation the way to the well-formed formula as the final result must be considered, tha t is, the de­sign of an adequate (syntactically correct) frame We can distinguish two characteristic cases of the design by the conditional frame syntax. In the first čase, we are confronted with the forma­tion of a complex operator composition, for which certain conditions of generation have to be satis­fied. In the second čase we discuss a general čase and point out the conditions where syntax rules could violate the emerging of a well-formed for­mula. Exampl e 4 [Generation of a Complex Operator Composition] An operator composition can be be­gun by several rules of Table 1. The beginning an d symbol is N and rules ft f-h ^ <— o are both adequate. The used rules can be marked by (x : y <— z) where x is the line number, and y and z are the column numbers in Table 1. The shor­tened marker is simply (x,z) and marks uniquely the rule of the table. Thus, we will use the mar­ked deduction arrow of the form \—'-}. In this way, the final form of a frame can be generated in the following way: K^NlJ4|=o2^|=o|=,H (h°h ^ ((h°h ^ ((h°h) ^ ((hoh)o ^4 ((hoh)oh^ ((h°h)oh) & ((hoh)oh)° ^ ((hoh)oh)oh ^ ((h°h)oh)oh* ^ a((h°h)oh)oh^ with the current frame at the end of the deduc­tion chain. • Exampl e 5 [General Formula Generation Viola­ting the Formula Syntax] How, by using the ru­les, the well-formedness of the emerging formula can be violated? Somebody being acquainted or having the feeling of formula well-formedness can immediately sense the mentioned violation. According to the Table 1, the follovving illegal (syntactically incorrect) frames (derivation re­sults or their parts) can be generated: (x, z) Rule Explanation (1,3) a <— (a a begins a function argument formula or a regular serial subformula (1,4) a a i—> a) i—> \aa\ J; 0,1 2,1 . . 6.1 Introduction N i—> a i—> aa v> 0,2 , 2,5 , Gestalts are a kind of interpretative possibilities K>—•> K M (=[77] to a given formula. As one will learn, gestalts can 0,1 1,1 , 1,. , , !>2 . be classified in various directions, the formal and N i—>• a i—> a |=i—>• \= a |= the applied ones. In some respect, the concept of v> 0,1 1,2 , 1,2 informational gestalt approaches the concept of K«N an informational graph, but in a formal, especially .. 0,1 1,3 , 1,3 ., 4,3 ,. . 4,9 is i—> a \—>• ( a i—> ((a i—> (|= ( a causal and circular sense. i-4 . o([=(a^ o((=( a) 6.2 Informationally Phenomenalistic Gestalts etc. Through these examples one can understand Informational entity on the formalistic level is no-the conditionality of the edge syntax. In this thing else then an informational formula. The way, the intermediate results of formula gene­simplest form of a formula is a marking operand ration must be proved on syntactic correctness, which marks an entity. Thus, in the very begin­ However, this mode of correct frame generation ning of our discourse we have to put the question enables a spontaneous approach in emerging of concerning the gestalt of a formula (or formula sy­formulas, connected with semantic (interpretive) stem) on an intuitive level. Later, we will answer concepts of the spontaneously arising formulas. • the question in an informationally formalistic way, tha t is, by an adequate formula expression for in­formational gestalts. Gestalt of a formula describes the entire pos­sible structure of causality in the framework of the informational logical consistency, that is, the well-formedness of formulas which follow (can be derived) from the original formula by ali possible displacements of the parenthesis pairs. For example, a sentence in a natural language is a grammatically correct sequence of words and each word in the sentence performs (more exac­tly, informs) as an autonomous and with other words informationally connected entity, tha t is, formally, as an informational operand or informa­tional operator (the property, quality of operands which it concerns). Such a sentence hides ali pos­sible causal choices (cases, example) of the sen­tence and this sentence presentation potentiality is called the gestalt of the sentence. In this way, a sentence can also be understood as an informational graph, which is an ordered structure (sequence, loop) of operands and opera­tors without any parenthesis pairs (grouped infor­mational connections) between words (operands and operators). In a practical čase of a sentence understanding, only few of the possible cases of the setting of parentheses pairs are realized (con­structed) by the observer of the sentence, tha t is, only those which fit in the best possible manner the given discourse of involved observers. 6.3 Definition of the Gestalt of a Formula and of a Formula System In this subsection, we have to define gestalts con­cerning a serial and circular formula, and gestalts of parallel formula systems. Definitio n 1 7 [Length of a Serial Formula] The length L of a serial formula is an integer being equal to the number of binart/ informational ope­rators (of type \=) in the formula. • For the length L, unary operators in a formula do not count. They represent the so-called interna­lism (input) or externalism (output) of operands occurring in the formula. For instance, the length of formula «1= ((«i h;f=«i) f=«2) is, evidently, L = 2, where operand ot\ disposes of its own input and output. Evidently, the last Informatica 20 (1996) 65-94 83 formula can be expressed by a parallel formula system in the form a )= [ax \=a2); « i N t= « i where from the original serial formula the unary parts are removed. Definitio n 1 8 [Gestalt of a Serial Formula] Let a \= (al N («2 |= • • • («n-l t= «n) • • •)) be a serial formula

^ of length L — n. Then, the parallel formula svstem r(^) # (a \= (ax |= (a2 |= '• • • (an_i \= an) • • -));\ [a (= ai ) (= (a2 (= • • • (an -i (=«»)• • •) \ (•••((« |= ai ) (= a2) f= •••a„_i ) \= an J consisting of exactly "In"N, r(v-0 n + 1 l n formulas obtained from formula tp^ by ali possible replacements of the parenthesis pairs, including ip^., is called the gestalt of serial formula ip^. • A circular serial formula ° ) ^ _>(a, o>i,..., an) of the length n could be possibly parallelized (when looking into its details on the operand level, and ignoring the set parenthesis pairs)? Such a view, ignoring the parenthesis pairs, searches for ali pos­sible causal cases of a serial formula presentation when operands and operator keep their places and meanings of the original formula, but the informa­tional relations concerning the parenthesis pairs are changed in ali possible manners. Definition 23 [Parallelization of a Basic Serial Formula] Let Pi mark parallelization and let (a, ai,... , a„ \«n-l 1= OinJ is called the parallelization of the serial formula. D On the other hand, the discussed serial formula (a, «i,... , an) has the standardized gestalt T(->.(a, <*!,..., an)) =© T((p^(a,al,...,an)) O This definition shows that a gestalt of a serial for­mula represents, according to the causal possibi­lities (causal structure), nothing more than the parallelization of the formula. As a consequence, the last definition and the previous ones deliver, evidently, for basic serial formulas the following graphical equivalences: r(v>->) = « v«; r(vM) =0V||; Informatica 20 (1996) 65-94 For circular serial formulas, there is, analogously, r(o. r(v?) =0 vP; ,o u(vo) =0i>a); r(^) =&. are paral­lel and serial formulas according to Definition 20 and Definition 18, respectively. Then, VM =^s^ ; v-> esys r() Operator >$ reads Hmplies graphically' and t • i r ti operator Lsys means 'is a component of the sy­stem'. • Similar definition can be set in concern to circular formulas. Definition 26 [Graphical Implication betvveen Se­ rialization of a Parallel formula and Circular Serial Formula belonging to the Gestalt] If (p1-* and e °) Operators => g and Gsys means have the same meaning as in the preceding. definition. • This completes the discussion on the.possibilities of drawing the serial consequences (possible inter­pretations) from parallel systems. In this respect, a serialization on the basis of parallelism has the role to investigate the details, possible particula­rities and the like. 6.5 Axiomatic Gestalts The axiomatic gestalts include ali the formulas obtained by the possible replacements of paren­thesis pairs in given axioms. In mathematics, axi­oms function as the given initial and hypothesized theorems and as given rules of inference (method, deduction, induction). It would be extremely in­teresting to make a look into the background of the axiomatic gestalts and to observe ali the possi­ble 'axiomatic' formulas, that is those, proceeding from the given axioms by their 'gestalting'. Mathematical axioms [7, 8] can be rewritten in an informational form. Let us take the implica­tion axiom set ([7], p. 66) in the form 1) a=^ {P=> a); 2) (a =>(«=•/3) ) = » (a =>/3); 3) (a = > /3) ==* ((/3 =^j)^(a 7)) For seeing the possibilities in the sense of a ge­stalt, informational graphs in Fig. 4 can be used. These graphs are uniquely described by the cor­ 1) Figure 4: A graphical interpretation of the impli­cation axioms where graphs give an insight to the corresponding gestalts r(y)1), r( /3; /3= > a); 1) 2) (a =>• a; « = > /3; /3 =$• a); 3) (a =j> /3; /3 = » /3; /3 = > 7; 7 a; a = > 7) The last system of formulas represents the so-called parallelization of axiom formulas by the most elementary informational-implication tran­sitions of the form L ==> n. In the axiomatic gestalts corresponding to the axiom formulas ip\, if2 an d (/53 in 1), 2), and 3), the A.P. Zeleznikar number of the possible formulas is N = -n^(() where L is the number of : -operators in a for­mula. Thus, n**) 4 = 2; JVX = L(<) = 2; r(y2 ) /a = ( 0 = (/3 =*(«=>/3)) ) ; \ (a-. > (/3 =>(«=*/3)) ; V (((«=* a) = » /3) a) = P) 4 = 4;JVa = I ® 14; (a • (P (0 = (7 = (a = 7))))l\ (a >(/ 3 >(7 : V((((« /3) = » 7) = > a) = » 7 / h'-5; iV3 Ž(1.°)=42 Novikov ([8], p. 75) replaces axioms 2) and 3) by a single axiom of the form 4) (a=>(/3=» 7 ))= » {(<*=> P) ==> (or=>7)); L4 = 6; iV4 = i(1 6 2) = 132 The graph of this axiom is presented in Fig. 5 and, as one can see, differs from graphs 2) and 3) in Fig. 4, substantially. The basic parallel formula Figure 5: A graphical interpretation of the impli­cation axiom 4) where the graph gives an insight to the corresponding gestalt T (^4). system for this graph is 4) (a P; P a; p 75 7 a: a: *7 ) The gestalt T(«)); a (7 a) describes the situation in a particular čase. 6. 6 Seria l an d Reversel y Seria l Gestalt s A serial decomposition of an informational entity roots in the causal nature of entities and its infor­mational components. For instance, the analysis of an entity progresses into the direction of disco­vering its informational details, stepping deeper and deeper into the structure. But, commonsen­sically, when reaching a deep informational detail, the process can be reversed, so that the analysis proceeds in the opposite direction, for example, verifying the obtained analytical (decompositio­nal) results and accomplishing them informatio­nally. On the level of conscious thought, such a forward and backward informational processing is thoroughly possible. Let us discuss in short the čase of a serial and, simultaneously, reversely serial decomposition of an entity, marked by a\, as shown in Fig. 6. At the first look, this structure is a multiloop @nt^ti: a «i+i :::=! @ Figure 6: A graphical interpretation of the seri­ally and reversely serially [\n(n — l)-tuple-loop] structured decomposed system for entity ai. one. There is, for example, the longest cycle [Anax = 2(n— 1)], in which operands appear in the sequence aii, a2, «i, ai+i, an, a.n—1 i <*;+i, cti,... , a2 , <*i, and there are ali the possible other shorter cycles [L = 1, 2,... , 2n - 3]. A short analysis shows that the decomposition structure in Fig. 6 has a number of loops (L) L = -n(nv — 1); 2 that is, numerically, n 1 2 3 4 5 6 7 8 9 10 L 0 1 3 6 10 15 21 28 36 45 A parallel formula system for the graph in Fig. 6 is, certainly, «i (= a2) ai (= ai+i; Otn-l \= «n ! a2 (= «i ; ai+1 \= oti\ an \= an _i Let us define, precisely, the kinds of different "loops" and their numbers according to the graph, formula, and causal situation. Definitio n 2 7 [A Concept of Graphical, Formula, and Causal Loop for Simultaneously Serial and Re­verselv Serial Čase] Let (5 be a graph in Fig. 6. Let us distinguish three kinds of loops: 1. A graphical loop is a loop visible through the circumspection of the graph which, regar­dless of its circular structure, is considered as graphically different from aH the other possi­ble loops in the graph. 2. A formula loop is a loop vohich, in any appro­priate form (arbitrary displacements of pa­renthesis pairs and arbitrary choice of the leftmost operand of the loop) corresponds to the graph loop. 3. A causal loop follows from a formula loop by an arbitrary displacement of the parenthesis pairs. The are three different numbers of loops for the graph in Fig. 6: 1. The number L# of ali graphical loops is 1 L<& = 2n(n~ 1 ) 2. The number Lv of ali formula loops amounts to Lv = 2n2 3. The number L^n, of ali causal loops attains 2nl ^—v 1 / 2ti M) where ti marks the length of the correspon­ding formula loop. Evidently, Lvn corresponds to the number of ali the formulas included in ali the gestalts correspon­ding to the graph in Fig. 6. • 6.7 Circular and Reversely Circular Gestalts Circularly serial decomposition of an informatio­nal entity roots in the causal and metaphysica­listic nature of informational entities and its in­formational components. The reversal circularity brings something new into the discourse of the possible circular structures and its practical im­plications. Let us see in short the čase of a circularly serial and, simultaneously, circularly reversely serial de­composition of an entity, marked by aj , as shown in Fig. 7. A parallel formula system for the graph Figure 7: A graphical interpretation of the cir­cularly serially and circularly reversely serially [\n{n + l)-tuple-loop] structured decomposed sy­stem for entity ct\. in Fig. 7 is (oti \= a2; a2 \= «i; \ a2 \= a3; a3 (= a2\ «71-1 f= 1= «n-i ; \an \= cti; QH\= an J The n longest serial loops and their counterloops are determined as ai \= (aH-l N -' • («n-l 1= (an 1= K h («21= •••(«.•-! 1= «.•)•• •))))•••); ai (= (a2-_i (= • • • (a2 |= («1 h (an \= (an_i j= • • • (ai+ i (=«»)•• •)))) • • •); i = 1, 2,. . ., n respectively. The length of each of 2ra circular formulas is n and, thus, the gestalts for these for­mulas only include, altogether, 2ra f2n\ n + l\n J formulas (with ali the possible displacements of the parenthesis pairs). Evidently, according to Definition 27, for the circular čase, there is A.P. Zeleznikar Lg = |n(n + 1); L " = 2(n + 1)2; 6.8 Metaphysicalistic and Reversely Metaphysicalistic Gestalts Metaphysicalistic formulas concern the interior (internal states) of an informing entity and, in this way, replace the reductionistic and rigidly (algori­thmically) determined propositions and predica­tes. In such a context, metaphysicalistic formulas can behave informationally, tha t is, as circularly and intentionally spontaneous entities. One can imagine how a metaphysicalistic formula—on the level of the conscious informational phenomena­lism in the living brain (mind)—models the ner­vous processes constituting the essential conscious entity. In neural systems, the one-way (direction) propagation of information—from synapses, den­drites, neuronal somata, axons to the synapses, and so forth—is a commonly recognized pheno­menon (scientific philosophy). However, on the conscious or artificial (constructionist) level, the direction of information propagation can be rever­sed in the form of the thought flow or a machine processing2 . Such a reverse metaphysicalistic pro­cess can represent an essentially different interpre­tation of the original process, particularly in the sense of specifically changed causalism where, in a cycle, causes and effects interchange their roles. One has to remind tha t the original and the rever­sed process take plače in a circularly interweaved environment where a strict distinction of causes and their effects (consequences) is no longer pos­sible. A metaphysicalistic direct and reversal struc­ture is graphically presented in Fig. 8. The bi­directional metaphysicalistic system is much more cycled than a standard metaphysicalistic system of an entity, represented in Fig. 10. While the original system has only 6 loops, the additional reversing causes a 30-loop structure, according to the formula \n{n + l) and Fig. 8, \n{n + l)+A = 32 for n = 7. The reader can calculate the en­tire complexity of the circularly structured me- See, for instance, the thermodynamic theory of thou­ght processes [11]. a)^=3Q^=^^ ^ entity's metaphysicalistic metaphysicalistic metaphysicalistic metaphysicalism informing counterinforming embedding Figure 8: A graphical interpretation of the circularly (32-tuple-loop) structured basic metaphysicalistic system of informational entity a being informationally impacted by the exterior entity /3. taphysicalism and its reverse in the form of par­ticular informational gestalts by himself/herself. 6.9 Parallel Gestalts Parallel gestalts are nothing other than gestalts of gestalts. One can imagine in which way such a si­tuation appears when examining parallel systems of arbitrarily structured (e.g., circularly circular) serial formulas. Parallel gestalts concern systems of serial for­mulas where each formula has its gestalt and the possibilities of different serial formulas included in different gestalts has to be considered. Evidently, in such a čase, the sum of the gestalt possibilities can be taken into account. For a gestalt of a gestalt as a parallel system and a gestalt of parallel systems of basic transiti­ons, there is evidently, r(r(^)) = r(^); r(r(^) ) = r(^) ; 7 Inference Gestalts Besides axioms, inference rules build up the skele­ton of logical inferetialism, consisting of processes of deriving, deducing, inducing, abducing, etc. in the framework of logical reasoning and theories constructing. Modi informationis (modes of informational in­ference) can be used in different informational de­duction processes. The basic rules (principles) of inference in the propositional and predicate logic are, for instance, substitution, modus ponens and modus tollens, which perform in the framework of truth and falseness (the principle of tertium non datur) . The reader can imagine how these rules can be transferred in the realm of informational entities (a kind of informational logic). Beside these principles, other inference rules can be in­troduced, known in the common speech as mo­dus agendi, essendi, rectus, operandi, obliquus, vivendi,' possibilitatis, etc , as already constitu­ted in the Latin Ianguage. Informational modus Figure 9: Informational graphs for informational modus agendi (1), modus ponens (2), and modus tollens (3). agendi (informational phenomenalism) of an en­tity, marked by operand a, is shown by graph (1) in Fig. 9 and represents the mode in which an in­formational entity (operand, phenomenon, thing, process) acts or operates (informs and is infor­med), tha t is, ^\=a; \ input (internalism) a ^ a \=; output (externalism) yQf |= a J interior (metaphysicalism) The 'feedback' arrow (vector) hides the potentia­lity of a' s decomposition (the a' s metaphysicalism a (= (x), the input arrow presents the a' s interna­lism |= a, and the output arrow the o;'s externa­lism a (=. • Informational modus ponens represents the in­terpretation of the logical (philosophical) modus •> ponens into informational realm. Similarh/, the informational fnodus tollens represents the inter­pretation of the logical (philosophical) modus tol­lens into informational realm. The detachment (modus ponendo ponens and modus tollendo tol­lens) formulas are, respectively, 13 (MP ) a: a and (MT ) P Their graphs (2) and (3) are drawn in Fig. 9. The arrows are marked by ';' (informational paral­lelism), =>• (informational implication), ^ (in­formational detachment), and \fi (particular no­ninforming). It is to stress tha t (L \f^ L) =>• (L 'l^j b^ 0 expresses the modus existendi of a cer­tain noninforming of the entity presented by the informational operand L. Evidently, the gestalts for MP (rule marker pMP) and MT (rule marker pMT) are, respectively, equal to the rules themselves, that is, and T(pMT) = r(pMP) pUT because the parallel structured premises a; a =4> /3 and a = > /3;/3\fif3 have to be treated as indi­visible entities. Several other forms of modi informationis can be discussed. The one concerning the intention of an informational entity is the so-called modus rectus, by which the detachment of the intention of an informing entity would be possible. 8 Intelligent Gestalt s Intelligent gestalts are a consequence of intelligent informational formulas. Which kind of a formula could be comprehended as informing in an intel­ligent way? What is intelligent informing? Definitio n 28 [Intelligent Informational Formula] An intelligent informational formula possesses its specific intelligent metaphvsicalism [13]. The ro­ugh structure of an intelligent formula, not being decomposed to the necessarg details yet, is given bu Fig. 10 and expressed as intelligence t concer­ning a (functionallu) bu the parallel sgstem of ba­sic transitions. • How can the graph in Fig. 10 be described in a form embracing ali possible formalistic pheno­menalism? The answer is: by a parallel system of A.P. Zeleznikar basic transition formulas determined by the pa­ths between two entities. So, let us construct the parallel system (Fig. 10) in the form /a (= i; <• t = 3 t ; • 3* h it; u N L; '<(<*) L t f = CM ct h )^(r(^);r<(^)) where T<(9_+) is a system of ali gestalts for for­mulas (pSy (subformulas) obtained from formula (p^ as possible, in respect to <^_>. shorter formu­las. Formula ipS> is any formula which follows (is constructed) from the graph of formula ., star­ting from an arbitrary operand in the direction of intending sensmg observing being conscious under­standing understanding informing Cv V (., in­cludes an infinite number of formulas obtained in the follotving manner: (a) Let (5 (ip^) denote the graph of circular for­mula (p^, that is a circular informational graph. (b ) Let the construction of a formula begin at the plače of any graph operand (circle, oval). (c) A formula ip, serial or circular, of the star gestalt r*(<^.) , is any formula obtained by moving from a starting circle (operand) along the graph arrouis (operators) and ending at the arbitrary circle or arrow. D The example which follows shows how formu­las can come to existence within a star gestalt, using an informational graph. Let us show how, according to the given informational graph, an in­formational formula can arise. In [14], p. 63, For­mula (63), the following formal situation of the understanding i ! and metaphysicalism /xt within an intelligent i entity is discussed: il—loop tli—loop (*H((u|=/i t )M)|=((M.hii)l=/'.)J)l= * t—loop This formula is a step towards the decomposition of entity i. The underlying graph for this formula is shown in Fig. 12. The il-, /it-, and t-loop are not meant to be informationally (causally) isolated loops. Markers il , /xt, and L can appear in other formulas of a system. In this way, they can stay open in respect to any actual environment, being changed by the exterior influences. The reader can see how different informational formulas can arise from the informational graph 0 Figure 12: A graphical interpretation of the circu­larly (double-loop) structured parallel transitional system (L (=il;i l |= /v,^4 (= t; M«. (=-^)­ in Fig. 12. To obtain regular informational for­mulas, arbitrary transitions, for example, from a beginning entity in the graph, say t, to an entity are necessary. The following formula system (a part of the star gestalt) shows the emerging of formulas using the graph in Fig. 12: ^(((iih/^hii)^); t|=(((il|=/xt)|=il)l=((^r=il)|=^); (t \= (((ii N /O Nil) h ((M* M ) h M J h *; In such a transition through the graph, the cho­osing of parenthesis pairs is spontaneous and, in this respect, different informational formulas can come into existence. Looking causally, each of the listed formula represents a special čase and can deliver, for example, different semantics con­cerning the involved entities of a formula. 11 Conclusion The introduced informational frame and informa­tional gestalt view have brought new insight into the possibilities of a general informational theory [13, 14, 15, 16, 17, 18, 19, 20]. The informer-observer problem can be exhau­stively analyzed through the study of framing of the transition a |=o|= (3, for instance, in the form Informatica 20 (1996) 65-94 93 \= au • |= «12 • h • •Oilm . \= °1 |= «21 . |= «22 • (= ' ••«2m -\= °2 a t= otki • | = ak2 • (= • • • p,k What is a's informing and what is /?'s observing? How does phenomenon a inform its reality and how does phenomenon /3 observe the a's reality? Is the informing of a nevertheless observingly pre-understood by (3 and, in this way,is a's reality comprehended in advance by /?? The last en-framed and formalized formula system offers not only various possibilities for such interpretations, but also much more than it can be said by words. The reader can observe that both 4>a and @ are harmonious frames, that is well-formed formulas. In this respect, a is an independent informer and /3 is an independent observer. Phenomenon a in­forms its reality by k different phenomenalisms, so phenomenon (3 can observe these k phenome­nalisms of a by the k properly chosen observatio­nal phenomenalisms, when frames a and p are concatenated by the separator frame 0 (perfor­ming as a concatenation operator) into the resul­ting transition of the form a \=o (= /3. Let us suppose that a informs its reality by k different phenomenalities and that (3 has chosen k adequate phenomenalities for observing of a. The number of different phenomenalities is not limited and, in some way, it depends on the observing capabilities of/3, so, in principle, k —} oo. What are the observing phenomenalisms of /3? Let a represent a physical phenomenon in the sense of the contemporary physical scien­ces. The observing entity (observer, appara­tus) (3 can use different theoretical and expe­rimental concepts, methodologies, and methods Informatica 20 (1996) 65-94 for observation of a, for example, mathema­tical (formalistic, recursive), Euclidean (geo­metrical), Newtonian (gravitational), Hamilto­nian (mechanical), Maxwellian (electromagne-tical), Lorenzian (particle-motional), Einstei­nian (relativistic), quantum, Hilbertian (axioma­tic, spatial), Schroderian, cosmological, Weylian (quantum-gravitational), computational, mind-informational [9], Slechtanian (informationally thermodynamical [10, 11]), etc. AH these theo­ries and methods belong to scientifically accepted forms of informationalism (artificialisms, metho­dologisms, scientisms, cybernetism) and depend on human consciousness, intuition, and 'reality­adequateness'. By development of science, theo­ries and methods will improve, change, and their number will increase. But, the observer entity voill observe only the phenomenalisms for which it is theoretically, methodologically, and experimen­tally capable [4]. In this sense, the preunderstan­ding of ct's phenomenalism will be, in a way, pre-understood by the observer. References [1] Patricia Smith Churchland: Neurophiloso­phy. Towards a Unified Science of the Mind-Brain. A Bradford Book. The MIT Press, Cambridge, MA, 1986. [2] J.A. Fodor: The Language of Thought. Harvard University Press, Cambridge, MA, 1975. [3] J.A. Fodor: Representations. MIT Press, Cambridge, MA, 1981. [4] D. Gernert: What Can We Learn from Internal Observers? In H. Atmanspacher and G.J. Dalenoort (Eds.): Inside versus Outside, Springer Series in Synergetics 63 (1994), 121-133. [5] M. Heidegger: Sein und Zeit. Sechzehnte Auflage. Max Niemeyer Verlag, Tiibingen, 1986. [6] M. Heidegger: Being and Time. Translated by J. Macquarrie & E. Robinson. Harper & Row, Publishers, New York, 1962. [7] D. Hilbert und P. Bernays: Grundlagen der Mathematik. Erster Band. Die Grundlagen A.P. Železnikar der mathematischen VVissenschaften in Ein­zeldarstellungen, Band XL. Verlag von Julius Springer, Berlin, 1934. [8] n.C . HOBHKOB: BneMenmu MameMamuHecKOu AOZUKU. rOCyaapCTBeHHO e H3.HaTeJIbCTBO CJ)H­SHKo-MaTeMaTiraecKOH JiHTepaTypM ($JGMaT­rira), MocKBa, 1959. [9] R. Penrose: The Emperor's New Mind. Ox­ford University Press-Vintage, New York, 1990. [10] J. Slechta: The Brain as the 'Hot' Cellu­lar Automaton. Proceedings of the 12th Int. Congress of Cybernetics, pp. 862-869, Na­mur, Belgium, 1989. [11] J. Slechta: On Thermodynamics ofthe Tho­ught Processes within a Living Body in a Changing Environment. Proceedings of the 12th Int. Congress of Cybernetics, pp. 878­885, Namur, Belgium, 1989. [12] A.N. Whitehead and B. Russel: Principia Mathematica. Cambridge University Press, Cambridge, 1950 (in three volumes). [13] A.P. Železnikar: Metaphysicalism of Infor­ming. Informatica 17 (1993) No. 1, 65-80. [14] A.P. Železnikar: Formal Informational Prin­ciples. Cybernetica 36 (1993) No. 1, 43-64. [15] A.P. Železnikar: Verbai and Formal Deduc­tion of Informational Axioms. Cybernetica 37 (1994) No. 1, 5-32. [16] A.P. Železnikar: Informational Being-in. In­formatica 18 (1994) No. 2, 149-173. [17] A.P. Železnikar: Informational Being-of. In­formatica 18 (1994) No. 3, 277-298. [18] A.P. Železnikar: Principles of a Formal Axi­omatic Structure ofthe Informational. Infor­matica 19 (1995) No. 1, 133-158. [19] A.P. Železnikar: A Concept of Informational Machine. Cybernetica 38 (1995) No. 1, 7-36. [20] A.P. Železnikar: Organization of Informatio­nal Metaphysicalism. Cybernetica 39 (1996) (to be published).