BOOLEAN DIFFERENTIATION EQUATIONS APPLICABLE IN RECONFIGURABLE COMPUTATIONAL MEDIUM

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1 MATEC Web of Cofereces 79, (016) DOI: / mateccof/ T 016 BOOLEAN DIFFERENTIATION EQUATIONS APPLICABLE IN RECONFIGURABLE COMPUTATIONAL MEDIUM Staislav Shidlovskiy 1, 1 Natioal Research Tomsk State Uiversity, , Tomsk, Russia Natioal Research Tomsk Polytechic Uiversity, , Tomsk, Russia Abstract. High performace computig eviromet sythesis with parallel architecture recostructig throughout the process itself is described. Sythesized computatioal medium ivolvig Boolea differetial equatio calculatios so as to fuctio i real-time image processig. Automato imagig was illustrated ivolvig the rearragemet of every processig medium elemet to calculate the partial differetials of -th order i respect to Boolea fuctio variables. The method of obtaiig settig codes for each elemet was also described. A example i calculatig d -order Boolea derivative to two differetials i respect to Boolea fuctios, depedig o three argumets withi the recostructible computatioal medium of 8x8 processig elemets was give. 1 Itroductio Architectig updated robotic systems fuctioig autoomously i a-priori ukow medium ivolves directly positioed i- robot computer systems i order to process a cosiderable body of sematic iformatio. Real-time processig of this iformatio attributable to classical computig architecture results i icreasig robot weight ad structure desig. Oe possible solutio of above-metioed problem could be applyig the recofigurable computig medium (RCM) [1 6]. Hardware-i-the-loop RCM desig ad its fuctioig is based o grouped evaluator model: S =< C, G, A( P( D)) >, where, С = {c i } set of processig elemets c i, i = 0, 1,, N 1; G itercomputig likage matrix structure (grid); A processig elemet algorithm set C iterrelated with G, i hardware parallel support program P of data processig D. Grouped evaluator costruct H =< C, G > reflects the followig architectural cocept: 1) parallelism i processig (parallel processig elemet algorithm set C itercomputig with G); ) programmability structure; 3) homogeeity of structure H (homogeeity of processig elemets c i C ad macrostructure G). The Authors, published by EDP Scieces. This is a ope access article distributed uder the terms of the Creative Commos Attributio Licese 4.0 (

2 MATEC Web of Cofereces 79, (016) DOI: / mateccof/ T 016 Let's cosider the case whe fuctioig processig elemet algorithm c i is the basis {AND, OR, NOT}, thus acquirig the hardware ivariace property. Algorithm desig i terms of solvig oe specific problem-orieted task via RCM, bearig o this limitatio is associated with sigificat peformace. This paper describes the sythesis of such RCM for Boolea differetiatio implemetatio ad itegratio. Problem statemet Usually, biary data is represeted as {0, 1} ad would look like arbitrary fiite dimesio matrix. By zero extesio it will be represetable as matrix R, where positive iteger umber. This matrix R could be iterpreted as microsequecig or code of Boolea fuctio system, biary image (aimated obect frame), dyamic biary system state at the preset samplig time, operatig techical system compoet iteractio graph. Further, this matrix will be cosidered as a system of Boolea fuctios (BF) f ( X ), = 0, 1, variables x 1, x,, x, Each BF has its ow vector value Τ (0) (1) ( 1) x x x X = K, where Τ traspositio symbol; elemet () t x (0,1) -value t1 t of BF for set of variables x, x,, ; t 1, t,, t biary represetatio parameter 1 t x 0 1 t = 0, 1; x = x, x = x. The system of BF f (Х), havig its ow vector value X, ca be writte i the form of the matrix R = X X 1 1 X0. I [7] the algorithm of Boolea derivative calculatios by private variable is: ( k ) X ( k ) () (1) = D D D X (mod ), (1) x1 x x k () i D = I i 1 i I. () where I idetity matrix. The aalysis of the algorithm shows that RCM processig elemet should implemet oe of the followig automato-based display image (Figure 1). Figure 1. Automato-based display images.

3 MATEC Web of Cofereces 79, (016) DOI: / mateccof/ T 016 Applyig automato-structured method [5] RCM fuctioig processig elemet algorithm could be sythesized: f1 = ( xyz ) + ( xyz 1 ) + ( xz 1 3) + ( xyz z3) + ( xyz 1 z3), f = x, f = xzzz + xzzz + xzzz + yzzz + yzz + yzz + yzzz f = xzzz + xzz + yzz + yzz + yzz. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) where, x 1, x, y 1, y data iputs; f 1,, f 4 outputs; z 1,, z 3 setup iputs of RCM processig elemet [3, 5]. Realizatio of algorithm (1) ito RCM via evaluators, implemetig automato images (fig.1) are plotted as the matrix structure of computig-based system. Determiig the matrix Z cofiguratio is based o the settig up each processig elemet withi oe possible automato image: Z= M i (3) i = ( Ao I ) k, M 1 1 = ( ) k T 3 = ( ( ) ) k3 M AoI o H, M Ao I o H, M = A k, A = D... D D (mod ) (4) 4 4 ( k ) () (1) where, o Hadamard product; I x idetity matrix; H x upper triagle matrix of oe; vertical overlie ( ) meas x-elemet iversio i biary matrix; Т traspositio matrix symbol; k i automato image code. 3 Example It is required to calculate Boolea secod-order derivative with respect to the BF variables x 1 ad x, as i the followig expressio fi ( X ) = x 1 x x 3. (5) To implemet Boolea derivative calculatio by private variable four basic evaluator elemets are ecessary (fig. 1). Each elemet is deoted by k i (i = 1,,, 4). Value k i correspods to each automato code of its cofiguratio, itroduced as decimal umeratio (as follows k 1 = ; k = 3, k 3 = 4, k 4 = 5). We apply i (5) all possible argumet values: X = [ ] T Accordig to (3) ad (4) the followig ca be obtaied: A = =., 3

4 MATEC Web of Cofereces 79, (016) DOI: / mateccof/ T Z = Obtaied matrix of Z cofiguratio correspods to RCM, depicted i Figure. Figure. RCM cofiguratio. Havig geerated the sigals correspodig to (6) the result providig the BF calculatio by private variables (5) x 1 ad x is recorded o data iputs of obtaied RCM ad o medium retur: X T = [ ]. x x 1 4 Coclusio RCM algorithms were sythesized to provide the calculatio of BF private variables. These algorithms meet the specified requiremets applicable for VLSI process. I this case, such algorithms ivolve regular ad localized computatio commuicatio, orieted o pipelie computatios. Proposed algorithm cofiguratio (3), ivolvig specific settig code iputs for each RCM processig elemet, is resposible of calculatig Boolea derivatives to differet argumats withi oe ad the same RCM. Accordig to the referece descriptio, the sythesis method of specific Boolea fuctio evaluators is based o the operatig diagram of the previously selected Boolea 4

5 MATEC Web of Cofereces 79, (016) DOI: / mateccof/ T 016 base (AND, OR, NOT) ad is housed i sythesized-based medium. It should be oted that the prescribed fuctioal diagram or logical etwork withi selected medium is 'fitted'. Obtaied algorithms of operatig RCM (recostructible computatioal medium) could be the basis of developig special-purpose evaluators producig basic operatios of morphologically processig digital images. Ackowledgmet The followig research has bee fiacially supported by Russia Foudatio for Basic Research (RFBR) А Itelliget Recofigurable Cotrol Systems, Navigatio ad Image Processig for Autoomous Mobile Robots. Refereces [1] I. Kulagi, A. Pazikov, M. Kurosov, Lecture Notes i Computer Sciece 951, 405 (015) doi: / _39 [] N. Maa, Sh. Wag, Ali S. Mohsi, X. Cui, Yu. Peg, MATEC Web of Cofereces 45, (016) doi: /mateccof/ [3] S.V. Shidlovskii, J. Comp. Sys. Sci. Iter. 45, 8 (006) doi: /S [4] A.I. Kalyaev, I.A. Kalyaev, J. Comp. Sys. Sci. Iter. 55, 11 (016) doi: /S [5] I.A. Kalyaev, I.I. Levi, A.I. Dordopulo, L.M. Slaste, IFAC Proceedigs Volumes 46, 10 (013) doi: / CZ [6] D.V. Shashev, S.V. Shidlovskiy, Optoelectroics, Istrumetatio ad Data Processig 51, 7 (015) doi: /S [7] S. Jaushkevich, D. Bokhma, R. Stakovich, G. Tosic, V. Shmerko, Avtomatika I Telemekaika (Joural of Automatio & Remote Cotrol) 6, 155 (004). 5