Algebraic models and methods for combinational circuits identification Текст научной статьи по специальности «Математика»

ALGEBRAIC MODELS AND METHODS FOR COMBINATIONAL CIRCUITS IDENTIFICATION

VOLODYMYR G. SKOBELEV

Institute of Applied Mathematics and Mechanics of National Academy of Science of Ukraine, Rose Luxemburg Street, 74, Donetsk, 83114, Ukraine. E-mail: skbv@iamm.ac.donetsk.ua

Abstract. This paper is a survey of some of the author’s results in resolving via algebraic models and methods of some problems of discrete systems’ identification, with origins in technical diagnostics. Analyzed class of devices is formed by combinational circuits, i.e. by devices described via Boolean functions. In terms of this mathematical model resolving of the following two basic problems is presented schematically. The first one is elaboration of controllability/observability analysis technique for Boolean functions, intended for tests’ design, via disjunctive normal forms. The second one is elaboration of technique for identification of Boolean vector-functions, intended for design of on-line control as well as of off-line checking, via methods of finite fields theory.

Key words: Boolean functions, controllability/observability, fault detection, finite fields, identification, minterms.

1. Introduction

For the last past years an intensive evolution of Discrete Systems Theory is characterized by springing up a series of weakly interacted and insufficiently studied sections, in majority with origins in applied (to a considerable extent, engineering) problems. Although it is developed sufficiently wide class of mathematical models (of non-numerical nature, as a rule) nowadays searching go on to be the principal method of investigation of discrete systems. This results in the lack of effective theory of discrete systems’ analysis and, in particular, in the lack of the theory of discrete systems identification. And, at last, it is a very weak tendency in decreasing the gap between theoretic and engineering, heuristic by its nature, methods of discrete systems analysis. It is evident that resolving of a wide class of applied problems is naturally reduced to resolving of some problems, which deal with different aspects of discrete systems identification. It is also evident that the effectiveness of methods, intended for resolving of discrete problems to a considerable extent depends on replacing searching by algebraic operations. Thus the elaboration of technique, based on interaction of searching with algebraic methods and intended to resolve problems of discrete systems identification is actual, fruitful and promising.

Some attempts to develop the above mentioned technique for combinational circuits identification are presented in this paper. The choice of the class of investigated discrete systems is justified by the following two factors. Firstly, effective methods of combinational circuits identification form the strong base for effective resolving of a sufficiently wide class of problems of technical diagnostics. Secondly, combinational circuits are presented adequately via B oolean functions. Different aspects of Boolean functions were

investigated deeply and some powerful methods of their analysis were developed.

The rest of the paper is organized as follows. Section 2 presents the technique for controllability/observability analysis of Boolean functions. Sets and parameters of the controllability and the observability are determined via the minterms, algorithms for their computations are proposed and extended for compositions of Boolean functions. Applications for tests design are outlined. Section 3 presents the technique for identification of Boolean vector-functions, intended to resolve the problems of on-line control as well as off-line checking. Proposed approach is based on applying the methods of the theory of vector spaces over finite fields for design for the given Boolean vector-function some characteristic function in the form of the set of special Boolean matrices. Final remarks are given in the concluding Section 4.

2. The Controllability/Observability of Boolean Functions

It is well known that a brief analysis of the testability of a combinational circuit C is often reduced to an estimation of the controllability and the observability parameters for its nodes (see, for example, [1,2]). As a rule, these parameters are determined as follows. The controllability of a node v of C is the least number of external input channels of C , such that by fixing the values of these channels we guarantee the prescribed value in v, regardless the values of other external input channels. Similarly, the observability of a node v of C is the least number of external input channels of C , such that by fixing the values of these channels we would observe via some external output channels the value of v, regardless the values of other external input channels.

Generally accepted approach for the controllability/ observability analysis deals exceptionally with estimations of parameters and is based on simple stochastic models (see, for example, [2]). Insufficient workability of this approach results in preliminary groundless conclusions in the presence of gaps between estimations and the ways of attainment of these estimations. To avoid these bottle-necks the following approach for the controllability/observability analysis of combinational circuits is proposed (see [3,4], for details).

It is natural to refer to a set of fixed values of external input channels of C, that guarantee the prescribed value inv regardless the values of other external input channels as to a set of controllability of v. Similarly, we would refer to a set offixed values ofexternal input channels of C, that guarantee the observation of the value of v via some external output channels regardless the values of other external input channels as to a set of observability of v. It is evident that sets and parameters of the controllability and the observability of a node v are completely determined by the Boolean function realized by the fragment of C , generated by v , and do not depend on the structure of this fragment at all. Thus, the above pointed sets and parameters could be determined and investigated via Boolean functions. This leads to unified design and analysis of algorithms for computation of these sets and parameters. Moreover, some means to automate the process of the controllability/observability analysis for combinational circuits may be developed. Now we present the proposed approach schematically.

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Let E = {0,1} and P2 (n) be the set ofall Boolean functions f: En ^E . If f e P2(n), 1<i1 <...<ir <n and

01, ..., c r e E then it is denoted byfj}’’the Boolean

function obtained by substituting inf constants cj (j = 1,...,r) instead of the variables xi. . For any f e P2 (n) a set s = {(ij,cj) | j = 1,.. .,r} is called to be a set of:

1) a -controllability (a e E) iff = a ;

2) (i,a) -observability (1 ^ i ^ n;i ^ ij(j = 1,...,r); ae E) iff f li,'",ir = x“.

Words combination a minimal set denotes any minimal by cardinality set with the given property and words combination an irreducible set denotes any set with the given property which contains no proper subset with the same property.

For any f e P2(n) and ae E by (f, a) and

Smm (f, a) there are denoted the sets ofall, correspondingly, irreducible and minimal sets of a -controllability. Similarly, by Sf a) and S^Tf,!,«) (ie{1,...,n}\{i1,...,ir})

there are denoted the sets of all, correspondingly, irreducible and minimal sets of (i, a) -observability. The parameters of a -controllability v c(a), (i, a) -observability v o (i, a) and i-observability v o(i) are determined in the following way

vc(a) = || s | (s e Smin (f, a)), if Scr (f, a) ^0

f I », if Scr (f, a) = 0

vo(i a) = j| s | (s e Smf a)),if Sjf (f,i, a) ^0

f 1 », if Sjr (f, i, a) = 0

v o (i) = min{v ° (i,0), v o(i,1)}.

Now the technique for computing of the sets

Sjr(f,a), Sior(f,i,a), Smin(f,a), STfa) as well as of the parameters v c (a), v o (i, a), v o (i) would be outlined. Let f e P2 (n). We set Nf = {e e En | f(e) = 1}.

By D aim it is denoted the disjunctive normal form (DNF) consisting ofall irreducible minterms of Boolean function f . We associate with any set

s = {(ij, a j)| j = 1,...,r}

(either of a -controllability, or of (i, a) -observability) the

minterm K(s) = x^1 •...xCTr.

i1 h

Theorem 1 [3,4]. For any f e P2 (n) there hold the following two statements:

1) s e Scr(f,a) « K(s) e Dai,"1;

f

2) s e So(f,i, a) ^ x“ • K(s) e Daim, where fia is

i fi,a ’

the Boolean function, such that the set Nf. is the set ofall

128

vectors (P1,..., p i_1, a, P i+1, ..., P n) e Nf, such that (P1, • • •, P i-1, a, P i+1, • • •, P n) ^ Nf.

Theorem 1 implies that: design of irreducible and of minimal sets of either of the controllability, or of the observability

1) is equivalent to design of, correspondingly, irreducible minterms and irreducible minterms of minimal rank;

2) design of all irreducible sets of either of the controllability, or of the observability is equivalent to design of the DNF consisting of all irreducible minterms;

3) computing of the parameters v c(a), v o(i, a) and v o(i) is equivalent to computing the minimal rank of irreducible minterms for f .

Corresponding algorithms are developed and investigated systematically in [3,4].

It is worth to note that the suggested approach provides us with powerful tools applied in DNF analysis (see [5], for example). Indeed, let A = log log n . Typical values of parameters imply that for almost all Boolean functions f e P2 (n), for all 1 < i < n and a e E there hold the following estimations:

1) for S e {Sjr (f, a),Sior(f,i, a)}

n(1_Sn >A _ 2n < | s | < n(1+^n ) A • 2n, where 5 n ^ 0, 5 n ^ 0, when n ;

2) |Smin(f, a)| = o(|Scr(f, a) |) (n -+cc),

|Smn(f,i, a)| = o(|Sior(f,i, a) |) (n ^»);

3) for almost all sets s e S^ (f, a) u Sjr (f, i, a)

n - A - log((1 + s) • A) < |s| < n - A;

4) for ve{vc(a), vo(i, a)}

n - log n + 2 <v< n - A +1.

The last inequalities provide us with approximation method for computing the values ofthe parameters ofthe controllability and the observability. Indeed, to compute vc(a) (correspondingly, v o (i, a)) it is sufficient to compute any s e Sjr (f, a) (correspondingly, s e Sior (f, i, a)) and set v c (a) = | s | (correspondingly, vo (i, a) = | s |). In almost all cases the mistake of approximation does not exceed the value

A = 0(log - ) (n —^ tx>).

log n v ’

Developed approach is naturally extended for composition g^ of Boolean functions f(x1, ..., xn) and

gj(yj1,•••,yjmj) (j = 1, — ,k). Indeed, let the sets of variables of the given Boolean functions be pairwise disjoint. Then:

1) to design some set s e Slf (fg1,”'gk , a) it is sufficient

&1?• • •&k

to substitute in any set

sj e scr(gj, a j) (j = 1,... ,k)

(ij,c j);

s'e Scr(f, a) some set instead of every pair

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2) to design some set s e ' 'gk , i, a) it is sufficient:

a) in the case, when i ^ ij (j = 1, ...,k) to substitute in any set s'e Sjo'(f,i,a) some set

sj e SC(gj, a j) (j = 1,... ,k) instead of every pair

(i j, c j); b) in the case, when i = hlh to substitute in any

set s'e Sf a) (correspondingly, in any set s'e Sf a)) some set sj e S* (gj, a j) (j = 1, — ,k) instead of every pair (i j, a j) and to join the obtained set with some set sh e Sor(f,hlh,1) (correspondingly, with some set sh e S^T (f,hlh,0));

3) to design the sets

SjF(fl1”"gk,a) So'(fi1”-,ik ,i,a)

c v g1,---gk o v g1,-gk

it is sufficient to accomplish described above substitutions in all possible ways.

Computing the values of the parameters

V-* ...ik(ia)

g1,-,gk g1,-,gk

is naturallyc reduced to simple operations over the values vc(a), vgj(cj), vo (i,a) and vo (hlh,a).Indeed, let

, ox lv g- (c),ifi e {iL •••,ik} and i = ij ^(xi ) =\ gij

[ 1, if i e {1,...,n}\{i1,...,ik}

For a minterm K

r

A

l=1

X

Ol

r

we set p.(K) = £ p(x l=1

©),

jl

taking into account that a + <x> = <x> + a = <x> + <x> = <x>.

Then v c- i (a)

fi1,-,ik v 7

g1,-,gk

min p(K(s))

seS{;r(f,a)

v o- i (i, a) = <

fl1,---,lk v ’ 7

g1,-,gk

min |a(K(s)),

seSjJ' (f ,i,a)

if i e{1, .^,n}\{i1,_,ik} min{v1,v2}, if i = hl

(h = ij ,l = 1,... ,m)

Let us compute 0-controllability of the node Z1 and 0-observability of the node X3 in the external output Z1. We

get Dy™ = X1 • X2 v X1 • X2, D-1™ =

Y1

X1 • X2 v X1 • X2:

Daim. t-\ aim

y3 = X3 • X4 , D— = X3 v X4 ,

Daim _ T-.aim

z1 - y1v y3, D- = y1 • — .

The 0-controllability of the node Z1 is computed in the following way

v Z1 (0) = ^(y1 • — ) = ^(y1) + ^(—) =

= min{p(X1 • X2), ^(X1 • X2)} + min{p(x3), 1^x4)} =

= min{p(X1) + p(X2), |a(x1 ) + p(x4)} + min{1,1} =

= min{1 +1,1 +1} +1 = min{2,2} +1 = 2 +1 = 3.

The 0-observability ofthe node X3 in the eXternal output Z1 is computed in the following way

V°1 (X3,0) = min{v1, v 2}.

Since V1 = min p(K(s))+v° (x3,1) = <x>+<x> = <x> ,

seSi0r(z1,Y3,0) Y3

v 2 = min |a(K(s)) + v -_(x3,0) =

seSor(z1,y3,1^ y3

= min{p(y1)} + ^4) =

= min{p(x1 • X2), ^(X1 • X2)} +1 =

= min{p(x1) + p(x2), |a(x1) + p(x2)} +1 =

= min{1 +1,1 +1} +1 = min{2,2} +1 = 2 +1 = 3,

then v° (x3,0) = min{<x>,3} = 3 .

1

If we substitute in the DNF D Z-™

ofthe literal yi and the DNF D^1?

y3

we get 3

the DNF D^ instead Y1 —

instead ofthe literal y3 ,

DZ^ = (X1 • X2 v X1 • X2) • (X3 v X4) =

Z1y1,y3

where V1 = V1

min |a(K(s)) + v° (hl,1),

seSo'(f,h,a)

= min M(K(s)) + v° (hl,0)

seSor(f,h,a) sh

Example. Letusconsidercombinational circuit C (see Fig. 1), where

y1 = X1 © x2, y2 = x2 © x3, — = X3 • X4,

= X1 • X2 • X3 v X1 • X2 • X4 v X1 • X2 • X3 v X1 • X2 • X4 .

Thus, the set of all irreducible sets of 0 -controllability of the node Z1 consists of the following four elements

s1 = {(X1,1),(X2,1),(X3,1)}, s2 = {(X1,1),(X2,1),(X4,0)}, s3 = {(X1,0),(X2,0),(X3,1)}, s4 = {(X1,0),(X2,0),(X4,0)}.

These sets are also the minimal ones for 0-controllability of the node Z1 . It is worth to note that any combination of values of input variables, consistent with these sets (such as the combination (0,0,1,0)) detects the fault = 1 at the node

Z1 . It is also evident that any set of four combinations {x1, X2, X3, X4} of values of input variables, such that the combination x- (i = 1,2,3,4}) is consistent with the set s- is a test for identification the fault = 1 at the node Z1.

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If the sets of variables of the given Boolean functions f(x1;...,xn) and gj(yj1,••• ,yjmj)(j = 1, — ,k) are not pairwise disjoint then applying of above described technique guarantee the design of some approximate solutions only. Design of the exact solutions may be provided in the

Daim CT j

gj

instead of every literal x °j in Dfm . Secondly, the obtained

ij

formula must be transformed by applying identities a • (b v c) = a • b v a • c, a • a = 0, a • a = a. Thirdly, the obtained DNF must be transformed into the DNF

Daim

fh-A . gb-.gk

Developed approach forms the base for design of a subsystem intended for controllability/observability analysis of combinational circuits as well as for tests’ design, which is compatible with a simulating system. Thus, some means intended to automate the process of tests ’ design are proposed. The structure of this subsystem is shown in Fig,2.

Fig-2. The structure of subsystem, intended for controllability/observability analysis.

Block 1 is intended to construct the description of the given circuit via a net constructed by applying some basic elements. For every basic element the Library, realized via Block 2, consists the set of all irreducible sets of the controllability and of the observability as well as the values of the parameters of the controllability and of the observability. The Library permits an extension, realized via Blocks 3 and 4. Block 3 via the interaction with the simulation system constructs input data for Block 4. The base of the last is formed by some algorithms for design if irreducible minterms and DNF consisting of these minterms (see [4], for details). Block 5 computes the sets and the parameters of the controllability and of the observability for compositions. If the obtained results are unsatisfactory then the investigated fragment is declared as a basic one, its characteristics are computed and inserted into The Library. The scheme description is revised and new description is analyzed.

3. Identification of Boolean Vector-Functions via Finite Fields

One of distinguishing features of on-line control of discrete devices lies in the necessity to provide real-time analysis of corresponding input-output pairs in order to prevent the transmission of incorrect information (see Fig. 3).

It is worth to note that the real-time analysis of input-output pairs is also preferable and, moreover, in some cases it is essential for off-line checking of discrete devices. If the analyzed device is a combinational circuit then it, being fault-free, realizes the prescribed Boolean vector-function f : Em ^ En. Thus, it naturally arises the following Problem: for the given Boolean functionf and a set Q c Em +n it is necessary to check if the inclusion Q c graph(f) holds. It is evident that this Problem is a version ofthe classic Problem of Boolean functions’ identification. Thus, classic methods based on searching may be applied directly. The main disadvantage of this approach is its high complexity, that leads to impossibility of the real-time analysis. It is well known that to reduce the complexity, it is sufficient to replace searching by algebraic operations, wherever it is possible. Just this approach for solving of the investigated Problem is proposed in what follows (see [4,6,7] for details). The main idea of the proposed approach is to explore effectively the factor that the set Em+n is the vector space GF m+n(2).

Let Pm,n be the set of all Boolean vector-functions f : Em ^ En . Since any Boolean vector-function f e Pm,n may be presented in the form f = (fj,..., fn), where fi e P2(m) for all i = 1,...,n, then

Pm,n = (P2(m))n . We set

To(m) = {f e P2(m)|f(0,_0) = 0}

m times

Theorem 2 [4,6]. For any Boolean vector-function f e (To (m))n it holds the identity

graph(f) = U V

VeLingraph(f) ’

where Lingraph(f) is the set of all maximal (relatively to the inclusion relation) subspaces of GFm +n (2), which are subsets of the set graph(f).

Since g(x) = f (x) + f (0) e (T0(m))n for any Boolean vector-function f e Pm,n , it is sufficient to deal only with the set (T)(m))n of Boolean vector-functions. Theorem 2 justifies the following structure. A set

X f = {M i|i = 1,... ,l}

ofmatrices over the field GF (2) is called to be a characteristic function of the set graph(f) (f e (T (m))n), if it holds the following condition: forany a e Em+n there exists M i ex f, such that a • M i = 0 iff a e graph(f). Theorem 2 provides us withthe following effective method fordesignofacharacteristic function of the set graph(f) (f e (T0(m))n).

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Theorem 3 [4,6]. For every Boolean vector-function f e (To(m))n there exists a characteristic function

1f = {M v | V e Lin graph(f)}

of the set graph(f), where every matrix

MV (V e Lin graph(f)) is a (m + n) x (m + n - DimV) -matrix, whose columns m1;..., mm+n_DimV form some basis of orthogonal complement V1 of the subspace V .

Assembling the obtained results altogether, the following algorithm for resolving of the investigated Problem is proposed

Step 1. If f g (L0 (m))n , then f := f + f (0), Q :=Q + f (0). Step 2. Design the characteristic function

Xt = {Mv | V e Lin graph(f)}.

Step 3. Compute (a) = {a • MV | V e Lin graph(f)}

for all a eQ .

Step 4. If 0 £ xf (a) for all a eQ, then Q c graph(f), else Q graph(f) .

Software and/or hardware implementation of the proposed algorithm forms the base for unified design of means for online control as well as for off-line checking of combinational circuits.

4. Conclusion

Presented in the given paper approach for resolving of some problems of Boolean functions identification is based on joining together of searching with algebraic methods and is intended for design of some technique for unified analysis of combinational circuits. Software/hardware implementation of proposed algorithms form a strong base for design of means for automation of resolving of some problems of technical diagnostics as well as on-line control of discrete devices.

Possible ways of further working out of the proposed approach of controllability/observability analysis of Boolean functions lie in extraction and investigation of specific subclasses of Boolean functions. Proposed methods of

identification of Boolean vector functions may be easily revised into the methods of identification of discrete vector functions f :{0,1,...,k}m ^{{0,1,...,l}n, where k and l, both, are powers of the number 2. If either k, or l is not a power of the number 2 then the set graph(f) cannot be presented in the form of the union of some subspaces. In this case the proposed approach may lead to design of approximate solutions, only. Thus, it naturally arises the problem of

extracting of sets of subspaces Lin1 graph(f) and Lin 2graph(f), such that the following inclusions hold

U V c graphf) c y V

V&Lini graph(f) VsLin2 graph(f)

Thus, some technique, intended for effective analysis of sets graph(f )\ U V and U V \ graph(f),

VeLini graph(f) VsLin2 graph(f)

both, must be developed.

It is also worth to note that the proposed approach for identification ofBoolean vector functions as well as discussed above its generalizations to the identification of discrete vector functions are directly applied for resolving of some problems, connected with preventing any unauthorized access to algorithms as well as to data implemented via software.

The author is grateful to Prof. D.V. Speranskij (Saratov State Univ., Russia) for the numerous discussions, interesting comments and suggestions.

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S.V. Yablonskij and O.B. Lupanov. Moskow: Nauka, 1984. 312p. 48 (in Russian). 6. Speranskij D. V., Skobelev V.G. Identification ofboolean functions via linear algebra methods // Ukrainskij Mathematicheskij Zhoornal. 1995. Vol. 47, N° 2. P. 260-268 (in Russian). 7. Skobelev V.G. Algebraic models for discrete systems’ analysis // Mathematical Notes, 2001. Vol. 2, N° 1. P. 61-68.

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