
Alexander Ionov, alexander_ionov@mail.ru Velikiy Novgorod, Russia 1988 Complex Logic for the Identification of Systems Considering Possible Errors 1. Establishing the Task The development of modern methods of computer technology created conditions for modelling of the objects, that have traditionally been considered nonmaterial such as reflections [3,4] and other functions of artificial intellect. There's a need to look at the active interaction of the elements in the IntellectComputer system (ill.1). ill.1 On the other hand, the apparatus of traditional binary logic to describe such interactions is not enough. Let's take the situations inside the system "User  Educational Computer Software", when a simple question: 2x2=? needs an answer. Situation 1. The software contains the right answer <4>, the user types in another number, for example <5>. The answer would naturally be considered false by the computer. Situation 2. The user gives the right answer <4>, but the program has the wrong number <5>, so the answer would again be considered false. Situation 3. Both the user <3> and the computer <5> are wrong, and the latter delivers the message to the former that he was wrong. Nondistinguishability of the above situations proves the known paradox of classical algebra of logic: "true results from false". Similarnondistinguishability will be true with models of the images resulted from identification, and inevitably containing errors. Clearly, there's some incorrectness in the solution of a logical task. The nature of such incorrectness and methods of its solution are the subject of this thesis. ^ Let's analyze two of possible user's answers: 2 x 2=4 (1) 2 x 2=5 (2) assuming there could be a chance of an error when they were formed. We'll use the symbolism of complex numbers' theory and enter co efficiencies a, b, c, d in (1) and (2) with complex parts of comultipliers (2+ia) x (2+ib) = 4 (3) (2+ic) x (2+id) = 5 (4) Let's do the equation (3), choosing such a correlation of co efficiencies a and b, that it will make it impossible to bring an error to zero, such as when b = 1/a. Then, opening the brackets in (3) we'll get the equation 2ai +2i/a = 1, from which we'll determine two symmetrical roots a = 0,780i; b = 1,280i. After the similar solution for the equation (4), we'll get c = 0,618i; d = 1,618i. This consideration of possible errors leads to following conclusions: a). Real error (a = 0, b = 0) can be only in the true statement (1); b). Both true (1) and false (2) statements may contain imaginary errors. Proceeding from this let's analyze the right parts (1) and (2) and the logic of their interaction with errors in the left parts. ^ We'll call "a structure" any controlling object containing an error that may be expressed in a complex way O = Re + Im, (5) where Re is a real (obvious) and Im  imaginary (hidden) components of the error O. As a logic structure let's view the description of the sentences (1) and (2) in the "IntellectComputer" system. So far we used two obvious logical terms true (1) and false (0). Considering (5) let's enter the following classification: 0: (2x2=4) true' (6) 1: (2x2=4) false' (7) 0': (2x2=5) 'true' (8) 1': (2x2=5) 'false' (9), where these sentences mean: 0: or true' or HET'/NO'  denial of the lack of error; 1: or false' or DA'/Yes'  denial of the presence of error; 0': or ‘true’ or HET/NO  confirmation of the lack of error; 1': or ‘false’ or DA/Yes  confirmation of the presence of error; For the introduced logical terms the following equations are fulfilled DA ='false', false' = DA', HET = 'true', true' = HET'. While a logical structure may be described as a point in plattitude, consisting of complex coordinates DA + i HET, true' + i false'. Let's analyze some characteristics of such structures. a). A structure will be called adequate if the following equation is fulfilled: DA + i HET = true' + i false' (10) and nonadequate in the opposite case. It's easy to see that for it to be adequate it's sufficient to have equality of real and imaginary parts in (10). Whereas the structure is brought to usual Boolean variables, without the imaginary part (true'=DA, HET = false'). b). The structure will be called extraadequate in case of the equality of real parts in (10), where DA = true', and intraadequate, when only imaginary parts are the same HET = false'. c). The case of nonadequacy, where DA =false', and HET = true' we'll call reverse synchronicity or readequacy of the structure. It's clear that the most general characteristic of logical structures is nonadequacy, i.e. presence or possibility of an error. Taking it into consideration let's form a description of the system as a sum total (DA + i HET) + i (true' + i false') (11) Moving toward the abbreviated symbolic marks and putting the imaginary and substantial parts of the first component (11) one under another, and similarly for the second component on the right we'll get: 0' 1 1' 0 (12) The type's form (12) is a complex analogue of a traditional bit. And may be called Nata (from Latin naturalis  natural). Let's form the type of Nata for the particular structural cases: adequate 0' 1 1' 0 readequate 0' 1 1' 1 extraadequate x x 1' 0 intraadequate 0' 1 x x. Whereas "x" means any form out of (0' , 0, 1', 1). The rest of the structures may be done with the help of the following arithmetical operations. ^ First let's define more precisely logical sense of the introduced complex statements: (0) true'  real truth (TRUTH) (0') false'  real lie (LIE) (1) HET(NO)  imaginary lie (ERROR) (1') DA(YES)  imaginary truth (PLAUSIBILITY) At the same time the above mentioned statements may be defined using the terms true and false (1 and 0) as follows: 1 ^ (PRAVDOPODOBIE)  Russian term 1 LIE 0' = 0 + i1 = 0 (14) (LOZH) 0 TRUTH 0 = 0 + i0 = 0 (15) (ISTINA) 0 ERROR 1 = 1 + i0 = 1 (16) (OSHIBKA) From here one can see the possible meaning of the terms: false – pure imaginary TRUTH (absence in (15) of the real part); true – pure imaginary LIE (follows out of (14) with the absence of the real part). Thus, traditional binary logic is included in complex one, similar to actual numbers being included in the sum total of all complex numbers. Let's use the symbols (13)(16) to reflect such a connection. Let's enter the logical variables A, B, C and so on. Then the structure could be built as follows: A B C D , (17) where variables A, B, C, D may adopt meanings from the multitude {0, 1, 0’, 1’, 0, 1) (18) Let's define logical operations as means of combining variables in structures and their transformations. Let’s illustrate addition it with ill.2 ill.2 Universal formula: B A+B = A (19) Examples: 1 1 1 0 0 + 1 = 0, 1 + 0 =  1 The multiplication may be shown by ill.3 ill.3 Universal formula: A x B = A B (20) Examples: 0 1 0 1 0 x 1 = 0 1, 1 x 0 = 1 0 Subtraction is shown on ill.4 ill.4 Universal formula: B A – B = A (21) Examples: 0 1 1 – 0 = 1, 1  (1) = 1 Here an operation of negativity is used: (A) = A (22) Diagram shows the division (ill.5): ill.5 Universal formula: A : B = B A (23) Examples: 1 1 1 : 0 = 0 1 , 1 : 0 = 0 1 The above introduced operations, set with the standard ratio of sum total, follow commutative, distributive and other laws for adequate structures only. But, as shown in (10), such structures don't need complex logic, as they are described with Boolean variables. Let's see the operations of ratios, concerning nonadequate structures. ^ Using the characteristics of readequate structures, enter new ratios of adequacy on the basis of an operation of conversion. Conversion of the structures can be shown with the universal equation Y ' X X = Y (24) and the sign " ' ", as an exchange of positions between the real and imaginary parts of the structure. Then the ratio A = = B (25) will be called ratio of adequacy, and the structures A and B will be adequate, under condition that B=A'. From here let's formulate algebraic principles of complex logic (we'll call it relogic) for nonadequate ( nonBoolean) structures.
A = = A' (26) relogic: It's easy to see that the reverse (26) correlation A' = = A is also fulfilled. It's clear, that the principle of identity is kept in Y Y X = X, if their components are mutually identical, X = X and Y=Y.
A = = B = = C A = = C (27)
A = = B = C A = = C (28) 3. The principle of noncontradiction follows from (26): "Two different structures are not identical but may be adequate". Thus out of (10) follows DA + i HET == false' + i true' (29) Let's use the principle of adequacy to Boolean (purely real) variables true and false. With (28) we have: true = = (true)' = false or true = = false , (30) which shows that relogic is a peculiar conversion of binary logic, including the latter in it. Following the above defined principles let's show the justice of the commutative law of addition. As follows from (19) A A + B = = B , (31) from here with (24) and (26), ( A + B ) = = ( A + B )' = B + A, and according to (28) ( A + B ) = = ( B + A ) (32) Q.E.D. The result of (32) is the invariation of the operation of addition as related to the correlations of identity and adequacy. Let's show that it's true also for the multiplication structures. According to (20) and (24) A x B = = B A (33) Then A x B = = (A x B’' = B x A, as needed. Similarly one can prove the justice of associated and distributive laws in relogic. According to (21), (23) let's define the operations of subtraction A A  B = =  B (34) and division A : B = = A B , (35) thus the multitude of viewed structures makes a closed field. ^ To analyze the situations, described when the task was established with its noncorrectness in mind, additional information is needed. Let's assume that the described dialogue between the user and the computer is going on, and the latter suggests the user should verify the result introducing the message "1" (true) or "0" (false). We shall seek the structure X, corresponding to the model of the situation according to the following correlation of adequacy: B Y A = = X, (36) where the structures A, B, Y adopt the meaning of true or false, depending on:  the imaginary confirmation of the result by the user (Y);  the correctness of the computer equivalent (B);  evaluation of the user's reply by the computer (A). Let's assume that when Y=0, the user gets the right for the new answer on the raised question, which may lead to a new situation. That’s why it will be natural to analyze the situations where Y=1. Situation 1 leads to the following correlation of adequacy: 1 1 0 = = X (37) X 1 from here, according to (27) follows the equation 1 = 0, or in a usual complex form: 1 + i X = 0 + i 1 (38) 1 Doing the equation (38), we get X = 1 + i1, or X = 1, which according to (13) is defined by the term PRAVDOPODOBIE/PLAUSIBILITY. Underline, that such a situation is corresponding to the well known liar's paradox. Situation 2 is described by the correlation: 0 1 0 = = X (39) from here, after the similar solution, X = 1, which corresponds to the term OSHIBKA/ERROR. It's easy to notice that the latter term is invariant in relation to the situations 2 and 3, accordingly for the next cases we'll consider Y=0. Situation 3, after all the above statements is described by the correlation: 0 0 0 = = X , (40) which is followed to X = 0, which corresponds to the term false. 7. Conclusion The above viewed principles of complex approach toward the identification of the logical structures, considering not only obvious but also imaginary (hidden) forms, allow for the noncorrect (from the traditional logic point of view) solutions of the existing problems. The apparatus of analysis and synthesis, called relogic, developed for such purposes, is operating with the complex statements and has the following characteristics: Erroneous Statement (ERROR) denies its erronicity; True Statement (TRUTH) denies its lack of erronicity; Plausible Statement (PLAUSIBILITY) confirms its erronicity; False Statement (LIE) confirms its lack of erronicity. Also with imaginary statements: true  pseudo false statement; false  pseudo true statement. Closeness of the described logical system suggests inclusion of the new terms, and the possibility of its practical usage in the solutions for the identification and management of systems with intellect. 8. List of Literature 1. L. Kutyura "Algebra of Logic" Odessa, "Mathesis", 1909. 2. A. Ionov "Automatization of the modelling of nonlinear dynamic objects using the soft are package" Moscow Institute of Energy, 1982. 3 . A. Ionov "Analysis and Synthesis of Images as nonlinear dynamic objects " Novgorod, 1986. 4. A. Ionov "Image Identification with Computers" Tula, 1987. 5. L. Pontryagin "Algebra" Moscow, “Nauka”, 1987. 