A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System

A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System

A new minimization method of the logic functions system of n variables in polynomial set-theoretical format is considered. The method is based on the splitting procedure of minterms system and on the generalizing the conjuncterms set of the different ranks by the settheoretical simplifing rules. The...

Повний опис

Збережено в:
Бібліографічні деталі
Дата:2015
Автор: Rytsar, B.Ye.
Формат: Стаття
Мова:English
Опубліковано: Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України 2015
Назва видання:Управляющие системы и машины
Теми:
Новые методы в информатике
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/112536
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System / B.Ye. Rytsar // Управляющие системы и машины. — 2015. — № 5. — С. 13–21. — Бібліогр.: 40 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-112536
record_format dspace
spelling irk-123456789-1125362017-01-23T03:03:39Z A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System Rytsar, B.Ye. Новые методы в информатике A new minimization method of the logic functions system of n variables in polynomial set-theoretical format is considered. The method is based on the splitting procedure of minterms system and on the generalizing the conjuncterms set of the different ranks by the settheoretical simplifing rules. The advantages of the suggested method are illustrated by the examples. Розглянуто новий метод мінімізації системи логікових функцій від n змінних у поліноміальному теоретико-множинному форматі, що ґрунтується на процедурі розчеплення системних мінтермів та узагальнених теоретико-множинних правилах спрощення множин кон’юнктермів різних рангів. Переваги методу ілюструються прикладами. Рассмотрен новый метод минимизации системы логических функций от n переменных в полиномиальном теоретико-множественном формате, основанный на процедуре расцепления системных минтермов и обобщенных теоретико-множественных правилах упрощения множества конъюнктермов разных рангов. Преимущества метода иллюстрируются примерами. 2015 Article A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System / B.Ye. Rytsar // Управляющие системы и машины. — 2015. — № 5. — С. 13–21. — Бібліогр.: 40 назв. — англ. 0130-5395 http://dspace.nbuv.gov.ua/handle/123456789/112536 519.718 en Управляющие системы и машины Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Новые методы в информатике
Новые методы в информатике
spellingShingle Новые методы в информатике
Новые методы в информатике
Rytsar, B.Ye.
A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System
Управляющие системы и машины
description A new minimization method of the logic functions system of n variables in polynomial set-theoretical format is considered. The method is based on the splitting procedure of minterms system and on the generalizing the conjuncterms set of the different ranks by the settheoretical simplifing rules. The advantages of the suggested method are illustrated by the examples.
format Article
author Rytsar, B.Ye.
author_facet Rytsar, B.Ye.
author_sort Rytsar, B.Ye.
title A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System
title_short A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System
title_full A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System
title_fullStr A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System
title_full_unstemmed A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System
title_sort new method of the logical functions minimization in the polynomial set-theoretical format. 3. minimization of function system
publisher Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
publishDate 2015
topic_facet Новые методы в информатике
url http://dspace.nbuv.gov.ua/handle/123456789/112536
citation_txt A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System / B.Ye. Rytsar // Управляющие системы и машины. — 2015. — № 5. — С. 13–21. — Бібліогр.: 40 назв. — англ.
series Управляющие системы и машины
work_keys_str_mv AT rytsarbye anewmethodofthelogicalfunctionsminimizationinthepolynomialsettheoreticalformat3minimizationoffunctionsystem
AT rytsarbye newmethodofthelogicalfunctionsminimizationinthepolynomialsettheoreticalformat3minimizationoffunctionsystem
first_indexed 2025-07-08T04:05:03Z
last_indexed 2025-07-08T04:05:03Z
_version_ 1837050108083961856
fulltext УСиМ, 2015, № 5 13 Новые методы в информатике UDC 519.718 B.Ye. Rytsar A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. 3. Minimization of Function System Рассмотрен новый метод минимизации системы логических функций от n переменных в полиномиальном теоретико-множест- венном формате, основанный на процедуре расцепления системных минтермов и обобщенных теоретико-множественных пра- вилах упрощения множества конъюнктермов разных рангов. Преимущества метода иллюстрируются примерами. A new minimization method of the logic functions system of n variables in polynomial set-theoretical format is considered. The method is based on the splitting procedure of minterms system and on the generalizing the conjuncterms set of the different ranks by the set- theoretical simplifing rules. The advantages of the suggested method are illustrated by the examples. Розглянуто новий метод мінімізації системи логікових функцій від n змінних у поліноміальному теоретико-множинному фор- маті, що ґрунтується на процедурі розчеплення системних мінтермів та узагальнених теоретико-множинних правилах спро- щення множин кон’юнктермів різних рангів. Переваги методу ілюструються прикладами. As it is known [27–29], in general case the system F( X ) of the functions fi(x1, x2 ,..., xn ) , i 1,2,..., s , in the disjunctive set-theoretical format is reflected by the PSTF {Yi 1,Yi *} of the following expression: F( X )  Y1 1 {11,12,...,1k1 }1, Y1 * {k1 1,k1 2,...,2n k1 v1 }* Y2 1 {21,22,...,2k2 }1, Y2 * {k2 1,k2 2,...,2n k2 v2 }* .................................................................................. Ys 1 {s1,s2,...,sks }1, Ys * {ks 1,ks 2,...,2n ks vs }*        , vi  2n  ki, (27) where ij are conjuncterms of the ranks },...,2,1{ nr ; the mark * changes the symbol ~ or 0 depend- ing on the fact which part of п-dimension space of the given system F( X ) belongs to indefinite values of the functions fi for all i 1,2,..., s , namely: if Yi * Yi ~ , here | Yi ~ || Yi 0 Yi 1 |, then (27) is the sys- tem of inpredeterminated (incomplete) functions reflected by the STF {Yi 1,Yi ~}, if Yi * Yi 0 , here | Yi ~ || Yi 0 Yi 1 |, then (27) then (27) is the system of weakly determinated (incomplete) functions re- flected by the STF {Yi 1,Yi 0}, and if Yi * , then (27) is the system of complete functions reflected by the STF {Yi 1}. According to the suggested method of minimization in the polynomial set-theoretical format the sys- tem F( X ) of the functions fi(x1, x2 ,..., xn ) is given by the minterms mij (i.e. conjuncterms of n-rank) of the perfect PSTF {Yi  ,Yi *}, where 1 ii YY  and ** ii YY  in case of the perfect STF {Yi 1,Yi *} (27) , i 1,2,..., s: F( X )  Y1  {m11, m12,..., m1k1 } , Y1 * {mk1 1, mk1 2,..., m2n k1 v1 }* Y2  {m21, m22,..., m2k2 } , Y2 * {mk2 1, mk2 2,..., m2n k2 v2 }* ........................................................................................... Ys  {ms1, ms2 ,..., msks } , Ys * {mks 1, mks 2 ,..., m2n ks vs }*        , vi  2n  ki. (28) 14 УСиМ, 2015, № 5 The minimization of the complete and incomplete functions system in the polynomial set-theoretical format will be considered on the basis of compatible realization when the search of the minimal PSTF {Yi  ,Yi *} of the system F( X ) (28) is done with maximal use of equal conjuncterms of its function. For this similarly as in the case of compatible minimization of the system in the disjunctive format [27– 29] with minterms of the perfect PSTF {Yi  ,Yi *} (28) a set of so called system minterms (m)1,2,..., s , s {1,2,..., s}, is formed the indices of which 1,2,..., s mark belonging of equal minterms of the system (28) to its certain functions. The algorithm of the system minimization F( X ) (28) is realized in the similar way as for one func- tion in two stages (see p. 2.1). On the first stage we do the splitting procedure of the system minterms with the help of splitting matrix, covering of which results in formation of a set of system conjuncterms of different ranks. On the second stage after distribution of these conjuncterms in functions of the system F( X ) the procedure of iterative simplification, as a result of which the minimal PSTF },{ * ii YY is formed taking into account compatible realization of the system. 3.1 Algorithm of minimization of a complete functions system. Examples of minimization The algorithm of the compatible minimization system F( X ) of complete functions given by the perfect PSTF }{  iY , Yi * , is realized in the following way. On the first stage the set }{  IY , },...,2,1{ sI  , of the system minterms (m)1,2,..., s , which are split with the help of the matrix r nM , is formed from the given minterms. The minimal covering of the r nM differs from the analogical proce- dure in case of one function (p. 2.1). It is connected with the fact that generative elements of the matrix Mn n1 are the system minterms sm ,...,2,1)( and consequently every element of the Mn n1 is a splitting system conjuncterm of (n 1)-rank (i n1)1,2,..., s will have the same index with subsets of functions of the given system as its generative element. The elements of minimal covering of the matrix Mn n1 in the same way as in case of one function will be identical to the system conjuncterms-copies of (n 1)-rank. But among them a decisive role for realization of compatible minimization of the system will play those ones the indices of which contain the greatest quantity of numbers. So, if (i r1)1,2,..., s and ( j r1)1,2,...,s", s , s {1,2,..., s}, these are identical system conjuncterms-copies of (r 1)-rank of the matrix M n r, r 1,2,...,n 1, then they can be elements of its covering if the indices of their generative form (i r )1,2,..., s and ( j r )1,2,...,s" form a not empty intersection, i.e. {1,2,..., s }{1,2,...,s"}. For example, let the system minterms (100)1,2,4, (110)1,3, (010)1,2,3 be generative elements of the matrix Mn n1. For the mask {l  l} the identical system conjuncterms-copies will be (1 0)1 and (1 0)1, the index of which determines the intersection {1,2,4}{1,3} {1}, and for the mask {ll} will be (10)1,3 and (10)1,3 because {1,3}{1,2,3} {1,3}. So, in this case for covering of the matrix Mn n1 it is advisable to choose the second pair of the mask {ll}. The system conjuncterms-copies of the matrix Mn r as well as for one function will be underlined We should note that it is unadvisable to use a compatible minimization of a system with the complete functions in the case if the given the perfect PSTF (28) contains less than two minterms or it does not contain any common minterms at all. Then, instead of formation of a set of system minterms, independ- ent minimization of system is used when every its function is minimized separately. After this a possibil- ity to form the more the better common system conjuncterms is searched. In this sense the splitting pro- cedure is quite convenient as the matrices of splitting can be covered for identical masks of literals. УСиМ, 2015, № 5 15 The cost of the compatibly minimized system realization F( X ), similarly as in the case of minimiza- tion of one function, will be determined on the basis of the interrelation k * / kl * / kin * , where numeric pa- rameters k, kl and kin have something to do with different system conjuncterms of the minimal sys- tem of the PSTF . The described algorithm of the system minimization of the complete functions with the help of split- ting method in the polynomial set-theoretical format will be considered for the system F( X ) , f i (x1, x2 , x3 ), i 1,2, of the perfect STF      11 2 11 1 )}111(),010{( )}111(),101(),010(),001{( Y Y , on the example of which the authors [20, example10, p. 391] illustrate efficiency of their own exorlink method. First of all, having changed the upper indices 1 on  in the given system of the perfect STF {Y1,2 1 } we get the system of the perfect PSTF . The set of the given system minterms will be formed from the minterms of the last one and the splitting procedure with the help of the matrix M3 1 will be done: Y1,2  {(001)1,(010)1,2,(101)1,(111)1,2}  S l   l    l         0   0   1  1  0  11,2 0  11,2   11   0   11  11,2           0011 0101, 2 1011 1111, 2 . This procedure corresponds to the first step on the first stage of the algorithm (see p. 2.1). In the sec- ond step we do covering of the matrix by singled out identical conjuncterms-copies M3 1:      2,12,12,1111 )110(,)011(,)1(,)111(,)011(,)1( C . Having distributed system conjuncterms in the function, we get the system of the PSTF , with the underlined elements of which we will do for every system function. The procedures of simplification according to the rules (2), (3) and (6) of the respective theorems (p. 2.1): Y1  {( 1),(111),(1),(110)} {( 1),(1),(11)} {( 1),(01)} Y2  {(1),(011),(110)} {(1),(11),(11)} {(11),(01)}    . So, the minimal system of the PSTF Y1  {( 1),(01)} Y2  {(11),(01)}    , cost of realization, which is reflected by the interrelation k * / kl * / kin *  3/ 5/1, and corresponds to [20]. Example 13. To minimize the system F( X ) of complete functions f i (x1, x2 , x3 ), i 1,2,3, given by the perfect STF Y1 1 {0,2,5,6}1 Y2 1 {1,3,5}1 Y3 1 {0,1,2,3}1      , in the polynomial set-theoretical format with the help of splitting method [21, p. 35] on the example of this system the author illustrates efficiency of xlinking method. Solution. Having transformed the given system of the perfect STF {Y1,2,3 1 } in to the system of the perfect PSTF and having formed from it a set of system minterms we will do the splitting pro- cedure with the help of the matrix and the coveting procedure of the last one M3 1: Y1,2,3  {(000)1,3 ,(001)2,3 ,(010)1,3 ,(011)2,3 ,(101)1,2 ,(110)1}  S 16 УСиМ, 2015, № 5  S l   l    l          0  1,3 0  2,3 0  1,3 0  2,3 1 1,2 1 1 0  0  1 1 0  1   0  1   0  1  1   0            C      12,13,23,23,23,13,13,1 )110(,)101(,)010(,)000(,)0(,)011(,)001(,)0( C . We will do the splitting procedure with system minterms of the formed set with the help of the ma- trix M3 2 and the covering procedure of the last one: (001)1,3 ,(011)1,3 ,(000)2,3 ,(010)2,3 ,(101)1,2 ,(110)1 S  S ll  l  l ll         00  01 00  01 10  11 0 11,3 0 11,3 0 02,3 0 02,3 11 10 01 11 00 10 01 10            C {(0 1)1,3 ,(0 0)2,3 ,(101)1,2,(110)1} . Having distributed system conjuncterms in the functions we get the system of the PSTF , with the underlined elements, which we transformation according to the rules (2), (3) and (7) of the corre- sponding theorems: Y1  {(0  ),(0 1),(101),(110)} {(0  0),(11),(11)} {(0  ),( 1),(11)} Y2  {(0  ),(0  0),(101))} {(0 1),(101)} {( 1),(111)} Y3  {(0 1),(0  0)} {(0  )}      . Answer. Minimal system of the PSTF looks like: Y1  {(0  ),( 1),(11)} {(0,1,2,3),(1,3,5,7),(6,7)} {0,2,5,6} Y2  {( 1),(111)} {(1,3,5,7),(7)} {1,3,5} Y3  {(0  )} {0,1,2,3}      , A cost of realization reflects the interrelation k * / kl * / kin *  4/7/1. If compared with [21] it is a better result, where this system is minimized in a compatible way by xlinking method with the interrelation k * / kl * / kin *  4/9/4, namely: Y1  {(0  ),(0 1),(110),(101)} {(0,1,2,3),(1,3),(6),(5)} {0,2,5,6} Y2  {(0 1),(101)} {(1,3),(5)} {1,3,5} Y3  {(0  )} {0,1,2,3}      . Example 14. To minimize the system F( X ) of complete functions f i (x1, x2 , x3, x4 ), i 1,2, given by the perfect PSTF Y1  {3,5,6,8,9,12,15} Y2  {1,2,4,7,10,11,12,13}    , in the polynomial set-theoretical format with the help of splitting method , the author [33, p. 223] illustrates a conventional K-maps method. Solution. We apply independent minimization to the given system as its the perfect PSTF }{ 2,1 Y contains only one common minterm (1100). So, for minimization of the function f1 we do the splitting procedure of the minterms of the perfect PSTF }{ 1 Y with the help of the matrix M 4 2 and covering of the last one which is highlighted in bold font: Y1  {(0011),(0101),(0110),(1000),(1001),(1100),(1111)} S УСиМ, 2015, № 5 17  S ll   l  l  l  l ll  l  l   ll                  00   01  01  10   10   11  11  0 1 0 0  0 1 10  10  10  11 0  1 0  1 0  0 1 0 1 1 1 0 1 1 01 10  11 00  00  10  11 0 1 11 10 0 0 0 1 10 11  11  01  10  00  01  00  11                    C         )1101(),0111(),11(,)1101(),01(,)0111(),0010(),10( C {(0 1),(1 0 ),(11),(0010)} . Having applied to the underlined elements the rule (3) of theorem 1, we get the PSTF Y1  {(1 ),( 1),(11),(0010)} . We apply the analogical transformation procedures to minimization of the function f2: Y2  {(0001),(0010),(0100),(0111),(1010),(1011),(1100),(1101)} S  S ll   l  l  l  l ll  l  l   ll                  00   00   01  01  10   10   11  11  0 0  0 1 0 0  0 1 11 11 10  10  0  1 0  0 0  0 0  1 1 0 1 1 1 0 1 1 00  01 10  11 01 01 10  10  0 1 0 0 10 11 0 0 0 1 10 11  01  10  00  11  10  11  00  01                    C          )}10(),01(),10{()0101(),10(,)0011(),01(,)0101(),0011(),10( C . So, the minimal PSTF Y2  {(0  1),( 1),(1 )} . Answer. Minimal system PSTF }{ 2,1 Y looks like: Y1  {(1 ),( 1),(11),(0010)} {(8,9,10,11,12,13,14,15), (2,3,6,7,10,11,14,15),(5,7,13,15),(2)} {3,5,6,8,9,12,15} Y2  {(0  1),( 1),(1 )} {(1,3,5,7),(2,3,6,7,10,11,14,15), (4,5,6,7,12,13,14,15)} {1,2,4,7,10,11,12,13}       . The cost of realization of the minimized system is equal to k * / kl * / kin * 5/11/1, which corresponds to [33]. 3.2. Algorithm of minimization of incomplete functions system. The examples of minimization In case of the system of incomplete (inpredeterminated and weakly determinated) functions (27), given by the system (28) of the perfect PSTF },{ * ii YY , the algorithm of minimization by the method of conjuncterms splitting in the polynomial set-theoretical format combines realization of two algorithms: of system minimization of a complete functions (p. 3.1) taking into account the form of the giving sys- tem of incomplete functions (28) and minimization of an incomplete function (p. 2.2). Respectively, in this case the set of system minterms will consist of two subsets which will be separated by the symbol  IY and reflected as }{ * II YY  , },...,2,1{ sI  , where  IY is the set of system minterms of the perfect PSTF }{ * IY , and * IY is the set of system minterms of the perfect PSTF }{ * IY , here the mark * in particular case will be replaced by the symbol ~ or 0 in an analogical way to (27). Further, the minterms system undergo the splitting procedure with the help of the splitting matrix r nM and its covering, as a 18 УСиМ, 2015, № 5 result of which some set of system conjuncterms of different ranks }{ * II YY  is formed. We should note that in the course of covering the matrix r nM two procedures are realized at a time: making the matrix compatible as its elements are used to maximum extent with higher capacity of the set I, and making it more definite (predeterminate) of the system due to use of the elements of the submatrix * IY . After distribution of the last ones in the functions of the system we get PSTF },{ * II YY , the elements of which for every function further undergo simplification procedure according to the rules of the respec- tive theorems (p. 1.2), selecting out of possible variants of transformation those which will secure the compatible minimization of the given system in the best way. Given further examples illustrate the peculiarities of minimization of the system of incomplete func- tions by the suggested method. Example 15 [39, p. 228, example 5.1]. To minimize the system F( X ) of incomplete functions f1(a,b,c) and f2(a,b,c), given by the truth table with the help of splitting method in the set-theoretical format. Solution. The given system F( X ) has the perfect PSTF         ~~ 22 ~~ 11 }4,3,1{;}7,2{ }5,4,1{;}6,3,0{ YY YY . We will form a set of system minterms of its minterms doing the splitting procedure with the help of the ma- trix  ~ 2,12,1 YY  , and the procedure of its covering the matrix M3 1, for ex- ample, for the mask }{ l : S YY   })101(,)100(,)011(,)001()111(,)110(,)011(,)010(,)000{( 12,122,121121 ~ 2,12,1   S l   l    l         0   0   0   1  1  0   0   1  1  0 1 12 11 11 12 0 1,2 12 0 1,2 0 1   0   0  1   0  1  1  1   0  1            C      12,122,11112221 )101(,)100(,)011(,)001()111(,)010(,)1(,)110(,)011(,)1(,)000(}{ l C . After removal of the system minterm (011)2 the set of covering will look like       12,122,1111221 ~ 2,12,1 )101(,)100(,)011(,)001()111(,)010(,)1(,)110(,)1(,)000( YY . Having distributed the system conjuncterms in the functions, we get the system PSTF }{ ~ 2,12,1  YY          )}100(),011(),001()110(),1{( )}101(),100(),001()111(),010(),000(),1{( ~ 22 ~ 11   YY YY . We will do the splitting procedure with the minterms of the PSTF }{ ~ 11  YY  of the function f1 with the help of the matrix 1 3M and the procedure of its covering: CS l l l                              101100 000110 110100 )}101(),100(),001()111(),010(),000{(         )}1(),0{()011(),1(,)011(),0( C . I a b c f1 f2 0 1 2 3 4 5 6 7 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 ~ 0 1 ~ ~ 1 0 0 ~ 1 ~ ~ 0 0 1 УСиМ, 2015, № 5 19 Having taken in to account (3) 0    1               1    0              we will get two solutions of the minimal PSTF Y1   (1), 1. (0  ),( 1) 2. (1 ),(  0)              . We will do the analogical procedures for the minterms of the PSTF }{ ~ 22  YY  of the function f2, having applied the matrix M3 2 for their splitting:                               )}10(),01{( 00110110 01101001 10010011 )}100(),011(),001()110{( CS ll ll ll  . After the transformation according to the rule (3) 1 0 0 1        1   1      ,   0 0               , we get the minimal PSTF Y2   (1), 1. (1 ),( 1) 2. (  0),(0  )           . Answer. The given system of functions has two solutions of minimization which reflect the PSTF 1. Y1  {(1),( 1),(0  )} {(2,3,6,7),(1,3,5,7),(0,1,2,3)} {0,3,5,6} Y2  {(1),( 1),(1 )} {(2,3,6,7),(1,3,5,7),(4,5,6,7)} {1,2,4,7}    2. Y1  {(1),( 0),(1 )} {(2,3,6,7),(0,2,4,6),(4,5,6,7)} {0,3,5,6} Y2  {(1),( 0),(0  )} {(2,3,6,7),(0,2,4,6),(0,1,2,3)} {1,2,4,7}    , where the minterms of the sets ~ 1Y and ~ 2Y introduced as a result of additional predetermination are in bold font. The analytical expressions correspond to these solutions: 1. f1(a, b,c)  b c a f2(a, b,c)  b c a    ; 2. f1(a, b,c)  b c  a f2(a, b,c)  b c  a    . Cost of realization of the system for the solution 1 is equal to k * /kl * / kin *  4/4/1, and for the solution 2 is k * / kl * / kin *  4/4/2. Both solutions are a better result if compared with [39], where cost of realization of the system is equal to k * / kl * / kin *  4/7/2, namely: f1(a, b,c)  b c  ab f2(a, b,c)  b abc    . Example 16. To minimize the system of incomplete functions given by the perfect STF Y1 1 {1,4,6}1;Y1 ~ {3,5,7}~ Y2 1 {0,2,4,7}1;Y2 ~ {1,6}~    in the polynomial set-theoretical format with the help of splitting method (borrowed from [40, p.4, example 3]). Solution. We form a set of the minterms system of the perfect PSTF         ~~ 22 ~~ 11 )}110(),001{(;)}111(),100(),010(),000{( )}111(),101(),011{(;)}110(),100(),001{( YY YY doing the splitting procedure with the help of the matrix M3 2 and the procedure of its covering. For example for the mask }{ ll  we have: S YY   })111(,)110(,)101(,)011(,)001()111(,)110(,)100(,)010(,)001(,)000{( 12112212,1212 ~ 2,12,1  20 УСиМ, 2015, № 5 CS ll ll ll                              12112212,1212 12112212,1212 12112212,1212 1101011101110100010100 11011110110101 1111100100111110010000 10001000  })111(,)110(,)100(,)10(,)00{(}{ 212,112ll C . Having distributed the system conjuncterms in the functions, we get the system PSTF  2,1Y , doing step by step simplification for  1Y according to the rules (2) and (3), and for  2Y (7) and (2):                   )}11(),0{()}111(),110(),0{()}111(),100(),00{( )1(),1(.2 )0(),0(.1 )}01(),10{()}110(),100(),10{( 2 1 Y Y . So, having taken into account the realization of the compatible minimization, the given system has the minimal PSTF         }7,4,2,0{)}7,6(),6,4,2,0{()}11(),0{( }6,4,,1{)}6,4,2,0(),3,2,1,0{()}0(),0{( 2 1 Y Y 3 . Answer. Cost of realization of the minimized system is equal to k * /kl * / kin *  3/4/2 and is better than all seven results of additional predetermination given in [40, table 4, p. 5]. Conclusion This article that is a logical continuation of the previous ones (see 1. Generalized of Set-Theoretical Simplify Rules of Conjuncterms; 2. Minimization of Complete and Incomplete Functions) describes the algorithm of the system minimization method of the complete and incomplete functions with n variables on the basis of conjuncterms splitting in the polynomial set-theoretical format. Efficiency of the suggested method is illustrated on the examples of minimization in the polynomial format of functions system bor- rowed from the well-known publications, the authors of which demonstrate their own methods. On the ba- sis of the results comparison one can note that the described in the mentioned above papers advantages of the suggested method can be proved also in the case of application to the systems minimization of the functions. 1. Besslish P.W. Efficient computer method for EXOR logic design // IEE Proc. Pt. E. – 1983. – 130, – P. 203–206. 2. Sasao T. Switching Theory for Logic Synthesis. Kluwer Acad. Publ. – 1999. – 361 p. 3. Papakonstantinou G. A Parallel algorithm for minimizing ESOP expressions // J. Circuits Syst. Comp. – 2014. – 23, issue 01. 1450015. – 17 p. 4. Saul J. Logic synthesis for arithmetic circuits using the Reed-Muller representation // Proc. of Europ. Conf. on Design Automation, IEEE Comp. Society Press, March 1992. – P. 109–113. 5. Perkowski M., Chrzanowska-Jeske M. An exact algorithm to minimize mixed-radix exclusive sums of products for in- completely specified boolean functions // Proc. Int. Symp. Circuits Syst., New Orleans, LA, May 1990. – P. 1652–1655. 6. Tsai C., Marek-Sadowska M. Multilevel Logic Synthesis for Arithmetic Functions // Proc. DAC’96. – June 1996. – P. 242–247. 7. Takashi Hirayama, Yasuaki Nishitani. Exact minimization of AND-EXOR expressions of practical benchmark func- tions // J. of Circuits, Syst. and Comp. – 2009. – 18, N 3. – – P. 465–486. 8. Debnath D., Sasao T. Output phase optimization for AND-OR-EXOR PLAs with decoders and its application to design of adders // IEICE Trans. Inf. & Syst. – July 2005. – E88-D, N 7. – P. 1492–1500. 9. Fujiwara H., Logic testing and design for testability // in Comp. Syst. Series. Cambridge, MA: Mass. Inst. Tech., 1986. – P. 821–826. 10. Sasao T. Easily testable realizations for generalized Reed-Muller expressions // IEEE Trans. On Comp. – 1997. – 46, N 6. – P. 709–716. 11. Faraj Khalid. Design Error Detection and Correction System based on Reed-Muller Matrix for Memory Protection // Int. J. of Comp. Appl. (0975-8887). – Nov. 2011. – 34, N 8. – P. 42–55. УСиМ, 2015, № 5 21 12. . Exact ESOP expressions for incompletely specified functions / M. Sampson, M. Kalathas, D. Voudouris et al. // VLSI J. – 2012. – 45, issue 2. – 197–204. 13. Stergiou S., Papakonstantinou G. Exact minimization of ESOP expressions with less than eight product terms // J. of Circuits, Syst. and Comp. – 2004. – 13, N 1. – P. 1–15. 14. Debnath D., Sasao T. A New Relation of Logic Functions and Its Application in the Desing of AND-OR-EXOR Net- works // IEICE Trans. Fundamentals. – May 2007. – E90-A, N 5. – P. 932–939. 15. Mishchenko A., Perkowski M. Fast Heuristic Minimization of Exclusive-Sums-of-Products // Proc. Reed-Muller Inter. Workshop’01. – 2001. – P. 242-250. eecs.berkeley.edu 16. Wu X., Chen X., Hurst S.L. Mapping of Reed-Muller coefficients and the minimization of exclusive-OR switching func- tion // IEEE Proc. Pt. E. – Jan. 1982. – 129. – P. 5–20. 17. Fleisher H., Tavel M., Yager J. A computer algorithm for minimizing Reed-Muller cannonical forms // IEEE Trans. Comp. – Feb. 1987. – C-36.– P. 247–250. 18. Even S., Kohavi I., Paz A. On minimal modulo-2-sum of products for switching functions // IEEE Trans. Electr. Comp. – Oct. 1967. – EC-16. – P. 671–674. 19. Helliwell M., Perkowski M. A fast algorithm to minimize mixed polarity generalized Reed-Muller forms // Proc. of the 25th ACM/IEEE Design Automat. Conf. – 1988. – P. 427–432. – IEEE Comp. Society Press. 20. Song N., Perkowski M. Minimization of Exclusive Sum-of-Products Expressions for Multiple-Valued Input, Incom- pletely Specified Functions // IEEE Trans. Comput.-Aided Design of Integrated Circuits and Sys. – Apr. 1996. – 15, N 4. – P. 385–395. 21. Saul J. Logic synthesis based on the Reed-Muller representation, 1991. – http://citeseer.uark.edu:8080/citeseerx/ viewdoc/ summary;jsessionid=A765F8A29F4FB9D2143A1DB7CDC91593?doi=10.1.1.45.8570 22. Zakrevski A. Minimum Polynomial Implementation of Systems of Incompletely Specified Functions // Proc. of IFIP WG 10.5 Workshop on Appl. of Reed-Muller Expansion in Circuits Design 1995, Japan.– P. 250–256. 23. Brand D., Sasao T. Minimization of AND-EXOR Using Rewrite Rules // IEEE Trans. on Comp. – May 1993. – 42, N 5. – P. 568–576. 24. Knysh D., Dubrova E. Rule-Based Optimization of AND-XOR Expressions // Facta Universitatis (Nis), Ser.: Elec. En- erg. – Dec. 2011. – 24, N 3. – P. 437–449. 25. Wang L. Automated Synthesis and Optimization of Multilevel Logic Circuits, 2000. – http://researchrepository.napier. ac.uk/4342/1/Wang.pdf 26. Stergiou S., Daskalakis K., Papakonstantinou G. A Fast and Efficient Heuristic ESOP Minimization Algorithm // GLSVLSI’04. – Boston, Mass., USA. – Apr. 26–28, 2004. 27. Рицар Б.Є. Мінімізація булових функцій методом розчеплення кон’юнктермів // УСиМ. – 1998. – № 5. – С. 14–22. 28. Рицар Б.Є. Теоретико-множинні оптимізаційні методи логікового синтезу комбінаційних мереж: Дис. докт. техн. наук. – Львів, 2004. – 348 с. 29. Рицар Б.Є. Мінімізація системи логікових функцій методом паралельного розчеплення кон’юнктермів // Вісник Нац. ун-ту «Львівська політехніка» «Радіоелектроніка та телекомунікації». – 2013. – № 766. – С. 18–27. 30. Рицар Б.Є. Числова теоретико-множинна інтерпретація полінома Жеґалкіна // УСиМ. – 2013. – № 1. – С. 11–26. 31. Рицар Б.Є. Візерунки булових функцій: метод мінімізації // УСиМ. – 2007. – № 3. – С. 34–51. 32. Tran A. Graphical method for the conversion of minterms to Reed-Muller coefficients and the minimization of exclu- sive-OR switching functions // IEE Proc. – March 1987. – 134, Pt.E, N 2. – P. 93–99. 33. Tinder R.F. Engineering digital design. – Acad. Press, 2000. – 884 p. 34. Закревский А.Д., Поттосин Ю.В., Черемисинова Л.Д. Логические основы проектирования дискретных устройств. – М.: Физматлит, 2007. – 592 с. 35. Sasao T. A Design Method for AND-OR-EXOR Three-Level Networks. – http://www.researchgate.net/publication/2362534 36. Tran A. Tri-state map for the minimization of exclusive-OR switching functions // IEE Proc. – Jan. 1989. – 136, Pt.E., N 1. – P. 16–21. 37. Optimisation of Reed-Muller logic functions / L. McKenzie, A.E. Almaini, J.F. Miller et al. // Int. J. of Electronics. – Sept. 1993. – 75, N 3.– P. 451–466. 38. Majewski W. Uklady logiczne. Wzbrane zagadnienia. – Warszawa: Wydawnictwo Politechniki Warszawskiej, 1997. – 180 s. 39. Zeng X., Perkowski M., Dill K. Approximate Minimization Of Generalized Reed-Muller Forms // Proc. Reed-Muller’95, 1995. – P. 221-230. 40. Al Jassani B.A., Urquhart N., Almaini A.E.A. Minimization of Incompletely Specified Mixed Polarity Reed-Muller Functions using Genetic Algorithm. – http://reasearchrepository.napier.ac.uk/3500/1/SCS09-CE-24[1].pdf Поступила 09.04.2015 E-mail: bohdanrytsar@gmail.com © Б.Е. Рыцар, 2015  << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description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> /CHS <FEFF4f7f75288fd94e9b8bbe5b9a521b5efa7684002000410064006f006200650020005000440046002065876863900275284e8e9ad88d2891cf76845370524d53705237300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c676562535f00521b5efa768400200050004400460020658768633002> /CHT <FEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef69069752865bc9ad854c18cea76845370524d5370523786557406300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002> /CZE <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> /DAN <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> /DEU <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> /ESP <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> /ETI <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> /FRA <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> /GRE <FEFF03a703c103b703c303b903bc03bf03c003bf03b903ae03c303c403b5002003b103c503c403ad03c2002003c403b903c2002003c103c503b803bc03af03c303b503b903c2002003b303b903b1002003bd03b1002003b403b703bc03b903bf03c503c103b303ae03c303b503c403b5002003ad03b303b303c103b103c603b1002000410064006f006200650020005000440046002003c003bf03c5002003b503af03bd03b103b9002003ba03b103c42019002003b503be03bf03c703ae03bd002003ba03b103c403ac03bb03bb03b703bb03b1002003b303b903b1002003c003c103bf002d03b503ba03c403c503c003c903c403b903ba03ad03c2002003b503c103b303b103c303af03b503c2002003c503c803b703bb03ae03c2002003c003bf03b903cc03c403b703c403b103c2002e0020002003a403b10020005000440046002003ad03b303b303c103b103c603b1002003c003bf03c5002003ad03c703b503c403b5002003b403b703bc03b903bf03c503c103b303ae03c303b503b9002003bc03c003bf03c103bf03cd03bd002003bd03b1002003b103bd03bf03b903c703c403bf03cd03bd002003bc03b5002003c403bf0020004100630072006f006200610074002c002003c403bf002000410064006f00620065002000520065006100640065007200200035002e0030002003ba03b103b9002003bc03b503c403b103b303b503bd03ad03c303c403b503c103b503c2002003b503ba03b403cc03c303b503b903c2002e> /HEB <FEFF05D405E905EA05DE05E905D5002005D105D405D205D305E805D505EA002005D005DC05D4002005DB05D305D9002005DC05D905E605D505E8002005DE05E105DE05DB05D9002000410064006F006200650020005000440046002005D405DE05D505EA05D005DE05D905DD002005DC05D405D305E405E105EA002005E705D305DD002D05D305E405D505E1002005D005D905DB05D505EA05D905EA002E002005DE05E105DE05DB05D90020005000440046002005E905E005D505E605E805D5002005E005D905EA05E005D905DD002005DC05E405EA05D905D705D4002005D105D005DE05E605E205D505EA0020004100630072006F006200610074002005D5002D00410064006F00620065002000520065006100640065007200200035002E0030002005D505D205E805E105D005D505EA002005DE05EA05E705D305DE05D505EA002005D905D505EA05E8002E05D005DE05D905DD002005DC002D005000440046002F0058002D0033002C002005E205D905D905E005D5002005D105DE05D305E805D905DA002005DC05DE05E905EA05DE05E9002005E905DC0020004100630072006F006200610074002E002005DE05E105DE05DB05D90020005000440046002005E905E005D505E605E805D5002005E005D905EA05E005D905DD002005DC05E405EA05D905D705D4002005D105D005DE05E605E205D505EA0020004100630072006F006200610074002005D5002D00410064006F00620065002000520065006100640065007200200035002E0030002005D505D205E805E105D005D505EA002005DE05EA05E705D305DE05D505EA002005D905D505EA05E8002E> /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.) /HUN <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> /ITA <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> /JPN <FEFF9ad854c18cea306a30d730ea30d730ec30b951fa529b7528002000410064006f0062006500200050004400460020658766f8306e4f5c6210306b4f7f75283057307e305930023053306e8a2d5b9a30674f5c62103055308c305f0020005000440046002030d530a130a430eb306f3001004100630072006f0062006100740020304a30883073002000410064006f00620065002000520065006100640065007200200035002e003000204ee5964d3067958b304f30533068304c3067304d307e305930023053306e8a2d5b9a306b306f30d530a930f330c8306e57cb30818fbc307f304c5fc59808306730593002> /KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020ace0d488c9c80020c2dcd5d80020c778c1c4c5d00020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e> /LTH <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> /LVI <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> /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR <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> /POL <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> /PTB <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> /RUM <FEFF005500740069006c0069007a00610163006900200061006300650073007400650020007300650074010300720069002000700065006e007400720075002000610020006300720065006100200064006f00630075006d0065006e00740065002000410064006f006200650020005000440046002000610064006500630076006100740065002000700065006e0074007200750020007400690070010300720069007200650061002000700072006500700072006500730073002000640065002000630061006c006900740061007400650020007300750070006500720069006f006100720103002e002000200044006f00630075006d0065006e00740065006c00650020005000440046002000630072006500610074006500200070006f00740020006600690020006400650073006300680069007300650020006300750020004100630072006f006200610074002c002000410064006f00620065002000520065006100640065007200200035002e00300020015f00690020007600650072007300690075006e0069006c006500200075006c0074006500720069006f006100720065002e> /RUS <FEFF04180441043f043e043b044c04370443043904420435002004340430043d043d044b04350020043d0430044104420440043e0439043a043800200434043b044f00200441043e043704340430043d0438044f00200434043e043a0443043c0435043d0442043e0432002000410064006f006200650020005000440046002c0020043c0430043a04410438043c0430043b044c043d043e0020043f043e04340445043e0434044f04490438044500200434043b044f00200432044b0441043e043a043e043a0430044704350441044204320435043d043d043e0433043e00200434043e043f0435044704300442043d043e0433043e00200432044b0432043e04340430002e002000200421043e043704340430043d043d044b04350020005000440046002d0434043e043a0443043c0435043d0442044b0020043c043e0436043d043e0020043e0442043a0440044b043204300442044c002004410020043f043e043c043e0449044c044e0020004100630072006f00620061007400200438002000410064006f00620065002000520065006100640065007200200035002e00300020043800200431043e043b043504350020043f043e04370434043d043804450020043204350440044104380439002e> /SKY <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> /SLV <FEFF005400650020006e006100730074006100760069007400760065002000750070006f0072006100620069007400650020007a00610020007500730074007600610072006a0061006e006a006500200064006f006b0075006d0065006e0074006f0076002000410064006f006200650020005000440046002c0020006b006900200073006f0020006e0061006a007000720069006d00650072006e0065006a016100690020007a00610020006b0061006b006f0076006f00730074006e006f0020007400690073006b0061006e006a00650020007300200070007200690070007200610076006f0020006e00610020007400690073006b002e00200020005500730074007600610072006a0065006e006500200064006f006b0075006d0065006e0074006500200050004400460020006a00650020006d006f0067006f010d00650020006f0064007000720065007400690020007a0020004100630072006f00620061007400200069006e002000410064006f00620065002000520065006100640065007200200035002e003000200069006e0020006e006f00760065006a01610069006d002e> /SUO <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> /SVE <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> /TUR <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> /UKR <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> /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice

Схожі ресурси