SU1550507A1 - Device for polynominal decomposition of logic functions - Google Patents

Abstract

The invention relates to computing and can be used in computers interpreting high-level programs, as well as in processors oriented towards the effective solution of certain problems. The purpose of the invention is to expand the functionality of the device due to conjunctive-polynomial decomposition of logical functions in arbitrary K≤N variables. This is achieved by the fact that a device for polynomial decomposition of logical functions contains N (N-number of variables of a decomposable logical function) groups of logical cells with 2 ^N-1 cells each, N - 1 control nodes and N - 1 switches, and each the cell contains And elements and the Element CONSTRUCTION BY MODULE 2, 2 ^N information inputs, N configuration inputs and 2 ^N outputs. The information inputs of the device are supplied with a truth table of the decomposable logic function, the binary inputs of the device receive the binary vector settings U = (U ₁ , U ₂ , ..., U _N ), which determine the variables according to which the vector components are decomposed U define the variables on which the decomposition is performed. At the outputs of the device, successively formed truth tables of logical functions = _S (S = 0.1, ..., 2 ^to -1), into which the initial logical function of N variables is decomposed. 1 hp f-ly, 2 tab. 7 il.

Description

switch; in fig. 6 - functional scheme of the second switch; 7, state graphs of the second switch for the considered example,

A device for polynomial decomposition of logical functions (Figs. 1-) contains 2-4 logical cells of the first group If-1 4, 2B 4 logical cells of the second group; 4 logical cells of the third group

4V3.

5 1 s, 6, -6b,

V7,

 vyhop 1 2 control nodes 41 and, 4, switch 5 and 5, informational: the inputs of the configuration inputs are ids. h

Each logic cell 1, 2 and 3 (form 2) contains two informational lines 9 and 10, setup input 11, output 12, and element 13 “element 14 MODULE 2 CONDITION.

The control unit 4 contains two elements NOT 15 and 1 b, element OR-NOT 7, element OR 18, element AND 19, station 20 COMPONENT ON MODULE 2 20 five exits 21-25. The control unit 4a contains the element ij-seHT OR 26, the element NOT 27, two elements AND 28 and 29 and three outputs 30-32.

Switch 51 form the four Ele Yenta AND-OR 33-36.

Switch 5 Ј consists of six (with the element AND-OR 37-42.

The device worked as follows,

On the j-th information input ("2 ,,,., 2П) of the device, the value y is applied: decomposable logic function f" f (x ,, x, ..., x) on the (j-lX-th set

x

xn -

Ј (x1sx and $ ot

X "to

that K 5, x.

about ", x g

2k-4 F K

50

s (xr.

(ABOUT

many possible

  1 "l c-nca J L

variables x1, ha ,. “. , x n “The binary input vector components are supplied to the configuration inputs 7 of the device

U "(Uf, 1,. ,,, Uh), defines the variables that can be decomposed by. In this case, if U 1 (, 2,.,., P) 9 then the decomposition in the variable x is performed; if U.0, x decomposition is not performed. At outputs 8 of the device, successively the truth tables of the logical function s (, o, 2-1) are formed, into which the original function Ј раз х (x15x

five

pairs of different conjunctions of rank r (, 1, .. „, k) of variables x, x. , at,, .., x, on which the smallpox {ate-ats conjunctive-polynomial decomposition.

The value of the function vs (xf "-t" x. "

CG "(

S

(2 S + v + l) -M output device.

Table 1 presents the types of 2–8 possible conjunctive-polynomial expansions of the logical function of three-variables (xf, x, x)

 S tsp (1

on the vth variable set

(.,, 0, o, x f p (, 1, ..., 2 -1; 0,1,0 .., 2k-1) is formed on

hg „

It is known that an arbitrary logical function of variables (x1, x,..., Xh) admits a conjunctive-polynomial expansion in the variable x; () of the form:

five

,,, .., xn) V0 (x1, | ... xn) @ x. V, (xf,.,. .X

i-f

where (x ..., xlVl,

X1-1

, hp) (2)

 x

X

   whether

Xh)

 "Xn)

AND

f (x ,, ... jXjH, OjX, i

iv, (x 1t ..,, x j. ,, x fH, ...., xn) in

- (- (h.i) g / „„

(j L x i-i

0, xi +1

X

() ® f (x,, ..., x,. ,, i,

five

x) ",)" "HP

Execution x f, x

five

d

0

LP)

expansion (2) with respect to per- {f, xfi, ..., XfK makes it possible to represent an arbitrary logical function f (x ,, x1, .., xn) as a polynomial sum (1) of logical functions n-k variables vs (xj

x | ...

xp ") and X |

and"

R

mz

f and i k. i

G.-1 1 GNG

  x txi xnJ / lx (1 x g

.xt, "

The source for constructing the truth tables of functions is the vector of knowledge (U, U, ..., y Ja (y

functions

chenii w ..., y °

about i decomposable

i

f "f (x"

xd, ..., xb) „We believe that decomposition

but

4 modules are produced sequentially in variables K (, .., X (. Next, a sequence of vectors w, Wj, .. ,, wk is formed, the components of which are calculated according to the following recurrence relations:

v h-

(3)

 FV © V () ijm «t (inlm

where i is 0,1,. , ,

t-l, 2, ..., m; , 2, ..., k.

In accordance with the above decomposition algorithm (3), the device contains logic cell sequences (Fig. 2), whose operation is described by the expression:

 & ftit

where / 3 ,, and / Jj - respectively, the signals on the first and second information inputs of the logic cell; U. (.,., ...

.. p) - signal on the tuning

the input of the logic cell (the index i indicates that the cell is located in the i-th language and the ith component of the us-three U vector (U, p ..., nn) is fed to it; ai is the signal at the output of the logic cell.

As noted above, if U, 1, then the decomposition is carried out in the variable x. (Lii, 4n), i.e. ,

NEW BINARY CODE (Z Y, 2, ..., 7.

which is intended to switch the input signals of the switch in q directions.

Signal conditioning

 , z V 7 slrll to 1 hour:

ten

(41

 1 if U, Uj, f.,. , and,., or z, if I U q-l; ;

/} J o

one ; , 3,.,. ,P; , 3q.

Control nodes 4 can be built on the basis of a fundamental symmetric multi-popp

Then (A) can be represented in pn-de:

rov

25 (5)

, 3, ..., P;

i where F., F., (1, DGG 9, .., U. Л - Аук20

 J “Therefore, rf p, / / 14 3Q is a damentalist (elementary) symmetric of the 1st rus of logical cells, the provisional Boolean functions depending

on variables t, U4,. , , 11., where, 1. about. , q-l. In this case, F, converts the input vector of values according to (3). When variable x -, belongs to the set

p, Qt 1,. about. , q-l. In this case, G, -1 then and only when. . , +

.х1к + а х Ь рус L0 35 /

of the logic cells provides the Transit-Functions Fp (, 1, 3 .., P) to the second inputs of the P-th control node (, 3, ... n-2) from the second outputs (Pl) -ro of the control node and g0 is used to generate signals. (5). In turn, on their

Well, transfer the input vector of values to the output.

To extract from the vectors w ,,;,.,, .., WH, consecutive tuples of the values of the functions s serve switches 5 (their total number is n-1).

If the sum of the values of the first q components is U1, U7, ..., 11 (, 3, ..., n)

the tuning vector U is 5. and, Cl,

ffl "t

then (ql) -ft switch 5 ensures the formation at its outputs of an ordered vector of values, in which truth tables of intermediate decomposition functions (, 1, ....), depending on kd variables, are arranged sequentially in order of increasing measures functions, i.e. V. vV,,.

Control nodes are used to control the switches. Moreover, each (q-O-th switch (, 3, ««., P) corresponds to (q-l) -ft node

45

the second outputs of the p-th control unit implements the outcome of the F p puncture

6

F P. | (I about which arrive at the second inputs (P +) of the control unit:

B Fp if

6 - 6 if

50

if b P + 1.

(6)

55

 F p U p + 1 Fp

P F6 1 IR ", tf,

For the considered example (), taking into account (5), (6), it is possible to write the conditions of the formation of signals by the control nodes 4:

z; -u, vujj J

zJ-U,

504076

control, which at the main outputs forms the q-pa unplaced :-p "

NEW BINARY CODE (Z Y, 2, ..., 7.).

which is intended to switch the input signals of the switch in q directions.

Signal conditioning

 , z V 7 slrll to 1 hour:

(41

 1 if U, Uj, f.,. , and,., or z, if I U q-l; ;

/} J o

, 3,.,. ,P; , 3q.

Control nodes 4 can be built on the basis of a fundamental symmetric multi-popp

Then (A) can be represented in pn-de:

rov

(five)

, 3, ..., P;

i where F., F., (1, DGG 9, .., U. L - Auk

damentalish (elementary) symmetric Boolean functions that depend on

five

the second outputs of the p-th control unit implements the outcome of the F p puncture

6

F P. | (I about which arrive at the second inputs (P +) of the control unit:

B Fp if

6 - 6 if

0

if b P + 1.

(6)

 F p U p + 1 Fp

P F6 1 IR ", tf,

For the considered example (), taking into account (5), (6), it is possible to write the conditions of the formation of signals by the control nodes 4:

z; -u, vujj J

zJ-U,

u, vu2;

, F U4;

Fj-U, U;

z -Fjvfis;

"Sl-UjF1 ,;

.

The first 4 and second 4 node-XJOB control circuits are constructed according to (7) n (8) in the device () ft the switch contains 2 p-2 elements 1, (q "2,3 ,, .., n) s where, , .., q is the number of two-input AND elements, the outputs of which are connected to the inputs of the corresponding element OR.

As an explanation in FIG. 7 (a,) „c), the graphs of the second

(7) Switch 5g for the example in question, the directions of information transfer are indicated for various

e vectors settings. Such graphs can be easily constructed for all switches for an arbitrary value of n.

(8) | On the basis of graphs, uniquely | Goiivoda functional diagram of the corresponding switches.

As an illustration, Table 2 presents eight types of conjunctive-polynomial expansions of the logical function 15 Din implemented by the device in three variables:

f (x, x7, x3) x1xtx3vx, x3.

In Table 2, a bold line highlights the tuples of the values of logical functions.

4V (, l, ...,.

20

As follows from table 2, takes place

As follows from table 2, takes place

Ј (x, xl, x3) x1xax3vx1Xj

xtx3 ©, YH s)

 (x, ® x 3) F xgh, he

 © xi (x1vx-t)

 © x, he f x, f x, hghe

xg f x} xr © x, xg f x, x3x,

x, © he @ hgh1 © x7x3x,

 1 © he © x5®: CGF3, ®

Thus, the device allows 2 conjunctive-polynomial decompositions to be performed.

Claims (1)

Invention Formula 1 A device for polynomial decomposition of logical functions, containing n (n is the number of variables of a decomposable logical function) of groups of logical cells in each cell, the first information input of the 1st logical cell (, ..., 2nd 1) of the first group connected to the i-th information input of the device, () whose information code is connected to the second information input of the 1st logical cell of the first group, the first information input of the j-th logical cell of the q-th group (, 2, .., fc ;, 3 ,,,, n;) is connected to the jth information input of the device, (2n + j} -Pi infor Discount koto- cerned input coupled to a second data input of the j-th logic cell q-th group, the first data input,) -ft logic cell q-th five 0 5 " the group is connected to the output (2 -2t + + j) -ft of the logical cell (qO-th group, the output (-t + j) -8 of the logical cell of which is connected to the second information input (2 n -t + j) -ft of the logical cell q-th group, the first information input of the device is connected to the first output of the device, the 2nd output of which is connected to the output of the 2nd logical cell of the p-th group, characterized in that, in order to extend the functionality by means of conjunctive polynomial expansion of logical functions in arbitrary variables, contains nl control nodes and n-commutation oors, the information inputs of the first of which are connected respectively to the outputs of the first 2nd logical cells of the first group and the outputs of the first 2 logical cells of the second group, the information inputs of the 1st switch (3, ..., n-1) are connected respectively to the outputs vx of logic cells 1st group -2, „. ,, h -2P - M), the outputs of the first h logical (1-Н) -th group of cells, the outputs and the first information inputs wx of logical cells of which (,, ...., -h), the outputs (il) -ro of the switch are connected to the first and second information inputs wx, respectively logic cells of the () -th group, m-th output (go "1,2,.,.,) (nl) -ro switch is connected to (m +) - m device output, the z-th tuning input of which (i1, p ) is connected to the configuration input of the i-th logical cell of the z-th group, the first and second configuration inputs of the device are connected respectively to the first and second inputs of the first control node, the first outputs of which ineny respectively with the control inputs of the first switch (1 + 1) -th adjusting ustroy--OPERATION input is connected to the first input of the 1st Table 1 Types of conjunctive polynomial expansions in arbitrary kin variables with Table 2 Truth table of logical functions ys (, r ,, .., 2 -I; kin-3) for the example in question control unit, the first outputs are connected to the control inputs of the 1st switch, the second inputs of the 1st control unit are connected to the second outputs (1-1) of the control unit. 2, the device according to claim I, characterized in that each logical cell contains an element COMPLEX ON MODULE 2 and element AND, the first input of which is connected to the first information input of the logical cell, the configuration input of which is connected to the second input of element AND, the output of which is connected to the first input of the ADDITION ON MODULE 2 element, the second input of which is connected to the second information input of the logic cell, the output of which is connected to the output of the ADDITION ON MODULE 2 element. // Phie.1 13 t: No 5ifJ R 7g  Fig.Z D) Js W / h 20 No 17 25 21 yy J L.J fig a f /// Fiyo.5 32 31 30 Phage. 6 tf. 1S2 + V " V “F is, + f6, , - “t% % °