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Metrics of Vector Logic Algebra for Cyber Space

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Metrics of Vector Logic Algebra for Cyber Space Metrics of Vector Logic Algebra for Cyber Space Vladimir Hahanov, Senior Member, IEEE , Svetlana Chumachenko, Member, IEEE, Karina Mostovaya II. B-METIC OF THE VECTOR DIMENSION Abstract - The algebraic structure determining the vector- matrix transformation in the discrete vector Boolean space for the Vector discrete logic (Boolean) space determines the interac- analyzing information based on logical operations on associative tion of objects through the use of three axioms (identity, sym- data. metry and triangle) forming a nonarithmetic B-metric of vector Keywords - vector-matrix transformation, discrete vector Boo- dimension: lean space, information analysis. ⎧d(a, b) = a ⊕ b = (a i ⊕ bi ),i = 1, n; I. INTRODUCTION ⎪d(a, b) = [0 ← ∀i(d = 0)] ↔ a = b; ⎪ i HE purpose of this article is significant decreasing the analy- B = ⎨d(a, b) = d(b, a); sis time of associative data structures through the develop- ⎪d(a, b) ⊕ d(b, c) = d(a, c), T ⎪ ing metrics of vector logic algebra for parallel implementa- ⎩⊕ = [d(a, b) ∧ d(b, c)]∨[d(a, b) ∧ d(b, c)]. tion of vector operations on dedicated multiprocessor device. Vertices of the transitive triangle (a,b,c) are vectors (Fig. 1), The problems are: 1. Develop a signature, satisfying a system of which identify the objects in the n-dimensional Boolean B- axioms, identities and laws for the carrier, which is represented Space; the sides of triangle d(a,b), d(b,c), d(a,c) are the distances by a set of associative vectors of equal length in the logic vector between vertices, which are also represented by vectors of the space. 2. Create a signature of the relations for the carrier, repre- length n, where each bit is defined in the same alphabet as the sented by a pair: an associative vector – an associative matrix. 3. coordinates of the vectors-vertices. Develop a signature of the transformations for the carrier, repre- sented by a pair of associative matrices of equal length. The research subject is the algebraic structures and logic spaces, focused to creating the mathematical foundations of ef- fective parallel computing processes, implemented in a multi- processor dedicated product. Fig. 1. Triangle of the vector transitive closure References: 1. Technologies for parallel computing by dedi- Vector transitive triangle is characterized by complete anal- cated multiprocessor systems [1-2, 10, 11, 15]. 2. Algebraic ogy with the numerical measurement of the distance in the met- structures, focused to creating a mathematical apparatus for par- ric M-space, which is specified by the system of axioms, deter- allel computing [3-4, 7-10]. 3. Process models for the solving mining the interaction between one, two and three points of any real-world problems on the basis of effective parallel computing space: [5, 6, 11, 13]. ⎧d(a, b) = 0 ↔ a = b; ⎪ M = d(a, b) = d(b,a); ⎨ ⎩⎪d(a, b) + d(b,c) ≥ d(a,c). The specific of metric triangle axiom lies in numerical (sca- Manuscript received September 23, 2011. lar) comparison the distances of three objects, where the interval Vladimir Hahanov is with the Kharkov National University of Radioelectron- ics, Ukraine, 61166, Kharkov, Lenin Prosp., 14, room 321 (corresponding au- uncertainty of the result – two sides of a triangle can be greater thor to provide phone: (057)7021326; fax: (057)7021326; e-mail: hahanov@ or equal to a third one – not really suitable for determining the kture.kharkov.ua). exact length of the last side. Removal of this disadvantage is Chumachenko Svetlana is with the Kharkov National University of Radio- electronics, Ukraine, 61166, Kharkov, Lenin Prosp., 14, room 321 (phone: possible only in a logical vector space, which can form a deter- (057)7021326; fax: (057)7021326; e-mail: ri@ kture.kharkov.ua). ministic view for each characteristic of the process or phenome- Karina Mostovaya is with the Kharkov National University of Radioelectron- non state. Then the numerical uncertainty of the third triangle ics, Ukraine, 61166, Kharkov, Lenin Prosp., 14, room 321 (phone: (057)7021326; fax: (057)7021326; e-mail: [email protected]). side in a vector logical space takes the form of the exact binary vector, which characterizes the distance between two objects R&I, 2011, No 4 11 and is calculated on the basis of knowledge of the distances for a 1 0 0 0 1 0 0 1 the other two triangle sides: b 1 1 0 0 1 1 0 0 d(a,b) ⊕ d(b,c) = d(a,c) . c 1 1 1 1 0 0 0 0 B = d(a,b) 0 1 0 0 0 1 0 1 The three axioms of the determining a metric are redundant, d(b,c) 0 0 1 1 1 1 0 0 at least for the vector space, where a single axiom can be used – d(a,c) 0 1 1 1 1 0 0 1 the interaction between three points: d(a, b) ⊕ d(b, c) ⊕ d(a, c) = 0 . Two identities are followed from Vector equal to zero (empty set) for all coordinates means a this law, which determine the relations between one and two full match the response and query. As well as the vector equal to points in a space: 1 (symbol x) for all digits indicating complete contradictoriness ⎧d(a,b) =d(b,a) =0→c =∅; the response and query. Number of gradations for a variable can d(a,b)⊕d(b,c)⊕d(a,c) =0→⎨ be a finite number that multiple of a power 2 ⎩d(a,a) =0→{b,c}=∅. n 2 4 The following fact is interesting. Having regard to the cycli- α = 2 → {2 = 4, 2 = 16} , which is determined by the power cal nature of the triangle, for any two known adjacent (incident) of the Boolean on the universe of n primitive. Otherwise, the components the third one can be calculated. This concerns both symmetric difference can exist only in closed concerning the to states (codes) of vertices and to the distances between them: set-theoretic operations alphabet. Thus, the interaction of two objects in a vector logical space can have either binary or multi- ⎧d(a,b) = d(a,c) ⊕ d(b,c) ⎧d(b,c) = b ⊕ c ⎧a = d(a,b) ⊕ b ⎪ ⎪ ⎪ valued deterministic scale of measuring interaction. Hasse dia- ⎨d(b,c) = d(a,b) ⊕ d(a,c) ⎨d(a,с) = a ⊕ c ⎨b = d(b,c) ⊕ c gram of any finite number of primitives (1,2,3,4, ...) can be ⎪d(a,с) = d(a,b) ⊕ d(b,c) ⎪d(a,b) = a ⊕ b ⎪c = d(c,a) ⊕ a ⎩ ⎩ ⎩ packed to a variable of logical vector. Moreover, 16 gradations Isomorphism of the set theory concerning the algebra of logic (for instance) of vector interaction by the four primitives exactly allows determining the vector set-theoretic S-space, where the indicate not only the degree of proximity by the variable, but in triangle axiom is defined by symmetric difference ∆ , which is what way they differ – by some primitives, or their combination. analogous to the operation XOR in Boolean algebra: Vector operation XOR actually smooths out the changes in the two codes or vectors, that is of interest for the creating digital ⎧d(a,b) = a∆b = (a ∆b ),i =1,n; i i filters. If it is applied many times, we can get a binary pyramid, ⎪d(a,b) =[∅ ←∀i(d = ∅)] ↔ a = b; ⎪ i where the last vertex is always the zero vector. Thus, the ob- S = d(a,b) = d(b,a); ⎨ tained pyramid makes it possible using some redundancy to cor- ⎪d(a,b)∆d(b,c) = d(a,c), ⎪ ~ ~ rect errors in the process of information transferring. The proce- ⎩∆ = [d(a,b) ∩ d(b,c)]∪[d(a,b) ∩ d(b,c)]. dure of convolution distances in order to verify the errors of data Here ∆ is the symmetric difference operation on the four-digit transferring for the number of vectors equal degree 2 is pre- set-theoretic alfabet α = {0,1, x = {0,1},∅}, represented by the sented below. 1) Compute all the distances between the binary codes, including the last and first vectors, resulting in a closed following table: geometric figure ci = ai ⊕ ai+1(i = n → i +1 = 0) . 2) Compute ∆ 0 1 x ∅ all distances between non-overlapping pairs of obtained in the 0 ∅ x 1 0 first stage codes c = a ⊕ a (i = 1,2,3,...,n) . 3) Repeat pro- 1 x ∅ 0 1 i 2i-1 2i x 1 0 ∅ x cedure 2 to obtain a package equal to zero in all coordinates. ∅ 0 1 x ∅ The procedure is illustrated by the following calculations: When determining the distance between two vectors in the S- 0 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 space the symmetric difference is used, which is isomorphic to 1 1 1 1 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 1 → → → → 0 0 0 0 the XOR-operation in the Boolean B-space. Examples of calcu- 1 0 1 0 1 1 1 1 0 1 0 1 1 1 1 1 lating the distances between vectors in both spaces (S, B) are 0 1 0 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 given below: 0 0 0 1 0 1 0 0 a 1 0 0 0 1 0 0 1 Similar actions can be performed and for multivalued vectors, b x x 0 0 1 1 0 0 where, for instance, every coordinate is defined in four-digit set- c x x x x 0 0 0 0 theoretical alphabet, and the procedure is reduced to the obtain- S = d(a,b) 0 1 ∅ ∅ ∅ x ∅ x ing a vector of empty values coordinates: d(b,c) ∅ ∅ 1 1 x x ∅ ∅ d(a,c) 0 1 1 1 x ∅ ∅ x 12 R&I, 2011, No 4 0 x 0 1 x ∅ 1 ∅ according to all the textbooks, is not a triangle, because three 1 x x 1 ∅ ∅ 0 0 1 x 1 x x 0 0 1 x ∅ x 0 points are located in line, Fig.
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