SU1388890A1 - Functional converter - Google Patents

Abstract

The invention relates to automation and computing and is intended to design logic circuits. The purpose of the invention is to enhance the functionality by finding any of the 2 polynomial representations of the Boolean function. The converter contains / G information inputs 1, outputs 2, counter 3, switch 4, block of UNEMPLARABILITY 5, trigger block 6, synchronization input 7, set to local state 8, configuration inputs 9. The converter calculates the coefficients of such polynomial representations of boolean functions in which variables are included not only directly, but also with an inversion sign. Depending on the chosen method of inverting the variables specified by the counter code, the switch performs a permutation of the components in the truth vector of the Boolean function applied to the information inputs. The block of UNEQUALITY elements forms the polynomial coefficients of the selected type, which at the end of the cycle are entered into the block of triggers, from which the result is taken. 2 hp f-ly, 3 ill. J (L to 00 oo 00 co o

Description

ten

15

The invention relates to automation and computing and is intended for the mechanization of manual methods for the synthesis of logic circuits in automated design systems.

. The purpose of the invention is to expand the functionality of the converter by finding any of the 2 polynomial representations of the Boolean function.

FIG. 1 shows a functional diagram of the converter; in fig. 2 and 3-: block diagrams; UNEQUALITY and switch with.

The functional converter contains information inputs 1, outputs 2, counter 3, switch A, containing 2 multiplexers 4) -4jh, 20 block 5 elements UNIMPLANITY, containing groups of 2 elements UNACT UNIVERSAL, block of trigger 6, initial synchronization state and set-up inputs 9.

Consider the algorithm implemented by the proposed device for calculating the coefficients of the Zhegalkin polynomial,, where each variable is included either directly, or with inversion. Let 9adan be a boolean function with its truth vector f (fo, ... jfjo.,), Where f; may be either zero or one (j “0,1, ...,). An arbitrary Boolean function is represented by a Zhegal polynomial of the following form: 1.

1388890

f (x ,,:

itX

where x; x

Veli or zero

So that 3; s F (k f multiplied (crown

rassmat changes in

From the best you can in the course

Np

35

where n

functions

writing down

,) --- ZF (4k,)

 K, 0

(one)

where the sum sign is modulo two; x; or x ;, or x (, ..., n); x 1 if k; 0;

The values of F (kni ..., k,) can be either null or units.

To obtain the vector of the coefficient 3; ov F (k, ..., k,), it is necessary to multiply the vector f by the n-fold direct (Kronecker) product of the matrix

ГЬ; ЬП Ь J j

(2)

considered at, ..., p. If the variable xj is not inverted, then otherwise

In order to find the n-fold direct product of the Hn matrix (2), one can use the following recurrence relation:

ь.н „., h, н„.,

Mon „., 1

Hh-, J)

where H, is a matrix of size 2 2, which is (n-1) -fold Kronecker's product of matrix (2). When multiplying a vector of values

functions Г (х „, ..., х) on the matrix Н„

is written like this:

This relationship can be represented

as a system of the following equalities:

F (0,0,0) h, hih, fo®hihah, f, ®h, hih, fj®h, h ,, h, f ,, f4 © h ,, fseh ,, fg €

®h, hjh, f, j „

(0,0,1) h ,, ®h, h, if, ®h, hifj®h, hj.fs®h, hif4®h ,, hjf5 © h3hjfsfflh, hj.f-,;

F (0, l, 0) hjh, fjShjh. f, ®h ,, h, fi®h, h, f., © h, h, f4®h, h, Gu i b, b, f; f;

: r (0,1J1) h, fo®hsf, © h, f2®l, f, ®h, f ®h ify © h, f «®h, f,; .34

F (l, 0,0) h5li, fo®hjh, f, .fi®liih, f, ®h, h, f4®h ,, h, fy © hjh, f, ®hih, f;

(1, 0.1) H2fo®hjf, ®hi: ra®h, .f, ®hif4a) E fs®hjf6®hzf7 i

Fd, l, 0) -h, fo € h, f, © h, f4 © h, f, ®h, f4 © h.fs®h, fe®h.f-,;

F (I, 1, 1) fo® f, ffif j®f, ®f ®f: f6 © f7 .-

For the algorithm, the operation is determined by the system of equalities (3).

Function Converter 2

keeps a group of

inputs, and

as a system of the following equalities:

Functional Converter co 2

keeps a group of

inputs, and

31388890

the order of connecting the inputs Nj (Ng. and the numbers N, (. .NaL) of the input J-ro 0,) of this group to the informational (, 2-O multiplexer, where

p-1 (n1 {-It (O

 N, + ... + 2-N + К „.

(five)

© N;:, i (r;)

the switch inputs are described by the expression

Obtaining the binary representation (4) of the numbers of the inputs of the functional converter from the binary representation. Here the N and N values are the multiplexer number and the DM number are the components of the binary representation Q of the multiplexer inputs with the order number N (,. К). illustrated by a table.

000001010011100101110111

001000011010101100111po

010011000001po111100101

011010011000111po101100

1001011 111.000001010011

101100111po001000011010 to 111100101010011000001 111po101100011010001000

The proposed device allows calculating the coefficients of the Zhegalkin polynomial in one operation cycle by multiplying the vector of values of the function T (x, ..., x,) by the matrix Hn.

The required form of the Zhegalkin polynomial is given by the values hr ,,. .., h ,, stored in counter 3. The value of h; (i 1, n) is equal to one if the corresponding variable x; inverted, and identically zero otherwise.

After entering into the counter 3 a certain combination of the quantities hn, ..., h, from the inputs 9, the counting of the clock pulses begins, and at the outputs of the counter 3 successively different combinations of the quantities bn ... b, appear. This allows the calculation of any of the 2 polynomial representations. Boolean function, starting from the given

I, h, h2h, fo®h5h2h, f, ®h3h2h, f, j®h, hjh, 1 HcEgK, f, .- fo © ii, hjh, f, @ E, hjh,. 15 th zy 2.y, G5 (© Y, Y. gb, f ,, fo @ li, hili,, hih, f, ®h, hih ,, h-, f, ®ff, h5, h,, f4 @ h, hj.h, fy © h, hjii, GB © th, b., b, g5®Y5Kgb,, hih, f, ®fi, hzh,

p-1 (n1 {-It (O

 N, + ... + 2-N + К „.

(five)

35

40

45

which is determined by the recording combination hh, ..., h ,. The obtained results are compared with the aim of choosing the best one according to the given criteria (for example, the criterion of the minimum number of units in the Zhegalkin polynomial providing the minimum of hardware).

Consider the operation of the proposed device for described by the system of equations (K).

The inputs IQ-I, of the functional converter, respectively, p €) are given the values f ;,, f,, f jf,, f jfj-j. fg jfy boolean function.

The combination of values hj, h, h, from the output of counter 3 is fed through the control inputs of switch 4 to the address inputs of multiplexers 4 (-4g The signals at the outputs from multiplexers 4, -4g are given by the following expressions respectively:

f3®h, h2h ,,, f5®h, h3, h, f6 © h,; h2h, f7; f2 © h, hjfi, fs®hJЯ2h, fd®hıh, JH, f, © h ,, fв; f, © h, hih, f6 © h ,,, f4 © h3hih / f5. ; , f, ®h, fi., h, f6®h, h ,, h, f6 @ h, hjh, f4; f7 © h, hih, fo © h, hih, f, © h, hih, f Shjhjh, f,; f6®h, hj.S, f, ®h, li2h, fo®h, lizh, f, ®h ,, fi;

50

513888906

,, 1t, fpffihjh.h, ,, f4®h3h, h, f; ®h, h, h, fi®h, h, h, f, ®h, hih, fo®h, h, h, f, ; .ih, f, © h, h ,,, h, fe © hjhjh fsfflh.,, f, ®h, h, h, fi®h, h, h, f ,, f;

where the logical OR operation is replaced by the sum modulo two, thanks to the following.

In the Zhegalkin algebra, the identity xvy x®y®xy exists, which turns into if x and y of the UNEQUAL VALUE there are magnitudes of conjunctions containing

(hjh2h ,, hih, f, ®h ,, f ,, f, © h, hih, Gf®Ь ,,, Gb © b, b ji, f,), (h, h2fo®h, ijf, © h ., hzf ,, h jfg®h, h2f6®h,); (h, h2h, fj h h ,,, fo h h, hji, f, h h, h, h, gb з hb, f, h, h, h, f4. h h, h., h, f); (h, h.jf2®h, h f3 © h-5b ,, fo®h5hzfi®h, h, fg © h ,, h, tf4®h3hj, f5-); (h, h5h ,, h2h ,, hih, f6 © h, hjh.fr © h, h., h, fo®h ,, f, ® h, h ,, h, fj © h, h, h, f3 ); (h, h2f4®h, h, f5.0h, hjfg®h2h f7®h, hjf (, ffih, hif, © h, hif2®h, h.if,); (h, hih, f6®h, hih , f7®h, h2b,,, ,, f, ®h, h., h, fo © h, h., h, f,); (h, h f6®h5hifT®h ,, h ,,,, h.jf ,, fflh3h2f, ®h, hjfo®h, h ,, f,)

respectively.

This allows for the outputs of the second group of elements of UNIQUENESS and

(h, hah,, hih, f, © h ,,, f, ®h, h ,, h,, hjh, fy®h, hih, fg®h ,, f);

(h; hifoeh, hif, eh ,, hifi®h, hifj © h, h,. f4®h, hify®h, hzf6®h, hif);

(hjh, fj © hjh, f, ®h, -h, Ge © b, b, f4®h, h, fgffihjh, fsfflhjh, f);

(h ,, f, ®h, f2®h9: f5®h, f4 © h,: fs®h., fg®h3f;);

(h, bih; f4 0h ,, ®h ,, f6 © h ,, f, ®h ,, 4®h, h.jh, f, ®h, hjh, f, jiBh, h., h, f, );

(h, h5f4 @ h, hjfs®h.jhzfg®h, hjfr®h, h, fo®h, hif, ®h ,, hifs) .;

(h.jh ,, ,, ih, fs®h, h, h, fo © h, h, fj®h, h, f, ®h, h, f ,,);

(h, f4®h5fs © h, f6 © h5f ,, ®h, fj, ®h, f3).

Ha inputs of flip-flops 6 (-69 from the outputs of KNOB, the values of

the third group of elements of the UNEQUALITIES ,, -

(hjh-ih, fo @ h ,, ®h, h2h, f2®h, h.ih,, f4 @ h, h.ih, f5-fflh, h5, h,, f); (hjhjfp®hjhif, a) hjh54 © h3h f,., hjf4 © h, hi.fs © h5hife®h, h2f7); (h, h,, f ,,, f, ®h, h, f.®h5h, fy @ h, h, fg®h, h, f,); (h, fo® h, f, © h, f.j © h, f ,, f, “h3f6 © h f,); (hjh, f ,, fi®hjh,, ih, f4 © hih,, fgShjh, f); (h ,, fo®hifi © hifi © hzf, 0h, f ®hify © hif6®hjf); . (h, fo®h, f, ®h, fz®h, f5®h, f4 @ h, f5eh, f6®h; -f,);

(f, & f j® f, © f C. fy® f 6 ® f 7)

respectively.

As a result, at the end of the cycle at the outputs 2, -29 of the converter, the values are determined by the system of equalities (3),

Claims (3)

1. Functional converter, containing a counter, a switch, a block of triggers, and a block of elements of an UNEVEN and inverse values of some variable, for example h, f (, ®h, f, -h, .f,. Ha outputs of the first group, as well as at the inputs of the second group of elements 25 at the inputs of the third group of elements NECKNESS, get respectively the values: VALUE, the information inputs of the converter are connected to the information inputs of the switch, the control inputs of which are connected to the outputs of the counter, the synchronization input of which is connected to the synchronization input of the converter and clock inputs of the trigger block, the outputs of which are set to the initial state connected to the entrance reset block triggers, information inputs of which are connected to the output. the house of the UNEQUALITY block, the inputs of which are connected to the switch outputs, which are so that, in order to extend the functionality by finding any of the 2 polynomial representations of the Boolean function, the tuning inputs of the converter are connected to the information inputs of the counter, the enable input whose records are connected to the setup input to the initial state of the converter. 2. The converter according to claim 1, which is characterized by the fact that the block of UNEMPLARABILITY elements contains n groups of UNIMQUENCY elements in each, with the inputs of the j-ro element UNCHARTERNESS of the first group (,) connected to (2j-l) -M five 0 and 2J-M inputs of the block, the inputs of the J-ro element of the K-th group (, p) are connected to the corresponding outputs of the elements UNEMBLEMENT (K-1) -th group and the inputs of the block, from the first to the 2nd outputs of the block are connected respectively to the first the input of the block, the outputs of the elements of the UNIFORMITY of the first through (n-1) -th group, the outputs of the elements of the UNEQUAL DETAIL of the n-th group are connected to (+ 1) -th of the outputs of the block 3. The converter according to claim 1, characterized in that the switchboard contains 2 multiplexers, the control inputs of the switch connected to the control inputs of the multiplexers, whose outputs are connected to the switch outputs, the information inputs of which are connected to the corresponding information inputs of the multiplexers.