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Symmetry Festival 2009. Abstracts. Budapest 2009. MATRIX OF COMPLEMENTARITY FOR METAGENETIC Kamil Bakhtiyarov Professor of Mathematics, Doctor of Philosophy., Logic (Moscow, RUSSIA). Email: [email protected] Abstract. The letters of a genetic code are taken for a sample: A (adenine) as a maximum, u (uracil) as a weak minimum, and also the inverted letters: V as a minimum, n as a weak maximum. The Matrix of Complementarity, columns of which represent complementary pairs, has been constructed under laws of creation of a living organism. The complete block Ψ characterizes dependence on Last Name-dominants A and V; and the split block (↓↑) – dependence on First Name-dominants columns ↓ and ↑. Isomorphism of matrices of a genetic code and mental types shows predisposition of characters, in many respects determined by their genetic code. Keywords: complementarity, dominants, isomorphism, genetic code, mental types. Stabilité – première condition du bonheur public. Comment s’accomode-t-elle avec la perfectibilité indefinite. А.S.Pouchkin The author offers the Matrix of Complementarity which uses the complementary pairs instead of contrasts of the traditional Square Of Opposition to overcome the infinite discontinuity (of second kind) for one of its values. Four values underlie the Square of Opposition which has a point of an infinite break point between + ∞ and – ∞. Belnap interpreted the uncertainty 0/1 as None (neither truth, nor false), and the infinity 1/0 as a nonsense, Both (both truth and false). (Belnap N.D. 1976). The author thinks if to use the value 1 as the true label it would be natural to use the opposite magnitude as the false label, a negative number –1. Then the logic zero can be used for the uncertainty, and inverse magnitude would be the logic infinity, the nonsense (Bakhtiyarov K.I. 2003, 2005). Thus, we receive two pairs of the oppositions: by the sign the values are +1 and –1, and by the magnitude they are zero 0 and infinity 1/0 = 2 ± ∞. Then use of the zero by computer by default will correspond to uncertainty. It is better than false presumptions. Under the approach based on the Square of Opposition and a principle of antecedence of the outgoing the cycles arise causing «the epicycles of the sixes and nines» (for numbers of hydrogen bonds) in the codon table (Petoukhov S.V. 2001, 2008). To overcome these artificial artefacts it was necessary to introduce a natural disposition of the complementary pair elements opposite to each other and to introduce a principle of autecedence of the incoming. The Matrix of Complementary is offered to be constructed under the laws of living organism creation, using a natural disposition of the complementary pair elements. As a complementary pair we consider letter A (adenine) as a maximum sign, and the letter U (uracyl) as a weak minimum sign. For another complementary pair we accept the similar inverse signs: the letter V as a minimum sign (cytosine), and the letter n as a weak maximum sign (guanine) (Bakhtiyarov K.I. 2007, 2008). We have The Matrix of Complementary for the cardinal points (on the left) and the Square of Opposition for properties (on the right) which differ by the disposition of elements. n A n A V u u V The Square of Opposition is an artificial logic design generating paranoiac infinite amplitude of oscillations."castling" of its elements makes the bridge between artificial intelligence and natural intelligence (NI). It suggests a way to create the artificial intelligence (AI). The Square of Opposition (on the right) is the artificial logic construction causing the discontinuity of second kind. In case of transposition of elements or "castling" in the lower row we have The Matrix of Complementary (on the left) that is free of this nonsense. It allows to restore the course of nature that ensures the continuous development, similar to monthly or annual cycle, characteristic for biological and Mental processes. The Square of Opposition looks like a table but comes to the linear scale what becomes obvious if we perform projecting operation. 3 The matrix consists of 4 units, each of which is it a miniature, operating as the fractal multiplier. Considering the type of pairs (x1, x2) = (First Name, Second Name) we accept the primacy of the surname (the second name). We have two dominant surnames: vowel *A (liberalism top is polygamous, opened) and consonant *V (the conservatism crater is monogamous, closed). Non-dominant surnames n, u: with vowel names A *, u* aura of gold pair - adjoin the block of vowels and with consonant names V *, n* - volensnolens to the block of consonants. Complemtary pairs organise combinations of the strong character with the gentle one. The matrix Ψ = (↓↑)are columns of complementary pairs ↓ and ↑: n A – monogamous consonants, ↑= ↓= V – polygamous vowels. u n A V u Non-Kronecker (left) tensor matrix square Ψ = (↓↑) n nn An n A AA gly STOP, arg/ser glu/asp Vn un arg STOP, trp/cys gln/his STOP/tyr nV AV nu lys/asn ΨA = ΨV results in Ψ2 = VA uA Au ala thr val ile, met/ile VV uV Vu uu pro ser leu leu/phe (↓↑)u The bold letters mean "polygamous" cells (the left table), generating the codones rich with amino acids (the right table). The problem on structural isomorphism of mental, physiological and natural phenomena was considered in antiquity. Since Hippocrates four temperaments were selected similar to aggregate state of substance. 01) Fire-passion, "summer", fire. Actors histeroids (A). 02) Earth-power, "winter", ice. Power epileptids (V). 03) Water-potion, "autumn", water. Scientists schizoids (u). 04) Air-media, "spring", steam. Media cycloids (n). n x’y V x’y’ A xy u xy’ Actually the table is a two-letter logic diagram by L. Carroll. For offered matrix classification the indexing of columns is from the left to the right, what means xy in the first 4 quadrant, x’y in the second quadrant, xy ’ in the third quadrant, x’y ’ in the fourth quadrant of the Cartesian coordinate plane. Isomorphism of matrices of a genetic codes and mental types shows the propensity of the characters in many respects defined by their genetic code. The Matrix of Complementary for psychological types nn.Manipulator Goebbels Vn.Informant Pavlic Morozov nV.Politican Lenin VV.Leader Ceasar An.Showman Chaplin un.Commentator Engels AV.ArtLeader Nero uV. Critic Iudushka Golovlev nA.Artist Shalyapin VA.Producer Meierkhold nu. Fantast Wells Vu.Methodologist L'Hôpital AA. Lover Don Juan uA. Art Critic Bakhrushin Au.Teacher Socrates uu.Scientist Leibniz : The microlevel of kinds (subdivisiona) resembles the macrolevel of types (units) that generates the fractality (the self-similarity) of the structure of the basic table. The matrix consists of 4 units, each of which is it a miniature, operating as the fractal multiplier. Logic positional dominance facilitates the qualitative leap as the positional notation at highest levels does not need new letters. The genetic code allowed us to glance at knowhow of the Nature and to learn the formation principles for the new things. The Matrix of Complementary will allow to overcome the discontinuity (of second kind) between natural and an artificial intelligence. REFERENCES Bakhtiyarov K.I. (2003) Algebra of logic for the computer// Tezisy 4-oi Mejdunarodnoi konferencii "Smirnovskie chteniya". Moscow,. pp. 25-27. (in Russian). Bakhtiyarov K.I. (2005) Logical Multidimensionality (vectorial and matrix models) // The Bulletin of the Moscow State University, series 7, Philosophy, № .5, p. 85-95 (in Russian). Bakhtiyarov K.I. (2007) Alphabet of Universal Language // Smirnov’s Readings on Logic. Proseedings 5-th Conference. Мoscow, p. 21-23 (in Russian). Bakhtiyarov K.I. (2008) Principles of Universal Language (positional dominance and fractality) // The Oppositions and Paradoxes. Мoscow, p. 216-233 (in Russian). Belnap N.D. (1976) How a Computer Should Think. Contemporary Aspects of Philosophy Proceedings of the Oxford International Symposium. Petoukhov S.V. (2001) The Biperiodic Table of Genetic Code and Number of Protons. Moscow (in Russian). Petoukhov S.V. (2008) The Matrix Genetics, Algebrases of Geneic Code, Noise Immunity. Moscow (in Russian).