 1988 Complex Logic for the Identification of Systems Considering Possible Errors

 Название 1988 Complex Logic for the Identification of Systems Considering Possible Errors Дата конвертации 15.07.2012 Размер 45.79 Kb. Тип Документы
 Alexander Ionov, alexander_ionov@mail.ru Velikiy Novgorod, Russia 1988Complex Logic for the Identification of Systems Considering Possible Errors1. Establishing the TaskThe development of modern methods of computer technology created conditions for modelling of the objects, that have traditionally been considered non-material 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 Intellect-Computer system (ill.1). ill.1On 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. Non-distinguishability of the above situations proves the known paradox of classical algebra of logic: "true results from false". Similarnon-distinguishability 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.^ 2. Considering Possible ErrorsLet'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 co-multipliers(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.^ 3. Logical Structures, considering errorsWe'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 "Intellect-Computer" 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 non-adequate 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 extra-adequate in case of the equality of real parts in (10), where DA = true', and intra-adequate, when only imaginary parts are the same HET = false'. c). The case of non-adequacy, where DA =false', and HET = true' we'll call reverse synchronicity or re-adequacy of the structure. It's clear that the most general characteristic of logical structures is non-adequacy, 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 re-adequate 0' -1 1' -1 extra-adequate x x 1' -0 intra-adequate 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.^ 4. Operations with the StructuresFirst 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 ^ PLAUSIBILITY 1' = 1 + i1 = 1 (13) (PRAVDOPODOBIE) - Russian term1 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.2Universal formula: B A+B = A (19)Examples: 1 1 1 0 0 + 1 = 0, -1 + 0 = - 1The 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 0Subtraction is shown on ill.4 ill.4 Universal formula: -B A – B = A (21)Examples: -0 1 1 – 0 = 1, 1 - (-1) = 1Here 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 1The 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 non-adequate structures.^ 5. Ratios of Logic StructuresUsing the characteristics of re-adequate structures, enter new ratios of adequacy on the basis of an operation of conversion. Conversion of the structures can be shown with the universal equationY ' X X = Y (24)and the sign " ' ", as an exchange of positions between the real and imaginary parts of the structure. Then the ratioA = = 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 non-adequate ( non-Boolean) structures. The principle of adequacy: "Any structure is adequate to its conversion", i.e. 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 inY Y X = X, if their components are mutually identical, X = X and Y=Y. The principle of syllogism, expressed dually: A = = B = = C A = = C (27) "out of three adequate structures two are identical", or in another form: A = = B = C A = = C (28)3. The principle of non-contradiction follows from (26): "Two different structures are not identical but may be adequate". Thus out of (10) followsDA + 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 subtractionA A - B = = - B (34) and divisionA : B = = A B , (35)thus the multitude of viewed structures makes a closed field.^ 6. The Interpretation of the ResultsTo analyze the situations, described when the task was established with its non-correctness 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. ConclusionThe 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 non-correct (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 Literature1. L. Kutyura "Algebra of Logic" Odessa, "Mathesis", 1909. 2. A. Ionov "Automatization of the modelling of non-linear dynamic objects using the soft are package" Moscow Institute of Energy, 1982. 3 . A. Ionov "Analysis and Synthesis of Images as non-linear dynamic objects " Novgorod, 1986. 4. A. Ionov "Image Identification with Computers" Tula, 1987. 5. L. Pontryagin "Algebra" Moscow, “Nauka”, 1987.

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