- Email: [email protected]
Disjunctive normal forms of Boolean functions with a small number of zeros
185 7. ZHURAVLEV YU.I., Well-posed algebras over sets of improper (heuristic)algorithms, I, Kibernetika, 4, 14-21, 1977. 8. AIZENBERG N.N., Spectrum of the convolution of arbitrary signals in an arbitrary basis, Dokl. Akad. Nauk SSSR, 241, 3, 551-554, 1978. 9. AIZENBERG N.N. and TROFIMLYUK O.T., Shift, convolution and correlation function in an arbitrary basis, Dokl. Akad. Nauk SSSR, 250, 1, 47-51, 1980. 10. AIZENBERG N.N. and TROFIMLYUK O.T., Conjunctive transformation of discrete signals and their application in the search for criteria and the recognition of the monotonicity of the functions of the algebra of logic, Kibernetika, 5, 138-139, 1981.
Translated by E.L.S.
U.S.S.R. Comput.Maths.Math.Phys.,Vo1.27,No.3,pp.185-190,1987 Printed in Great Britain
0041-5553/87 $lo.00+0.00 01988 Pergamon Press plc
DISJUNCTIVE NORMAL FORMS OF BOOLEAN FUNCTIONS WITH A SMALL NUMBER OF ZEROS* A.YU. KOGAN
An algorithm is proposed for constructing reduced disjunctive normal forms (DNF) of Boolean functions given with zero sets wthat is most effective for a small number of zeros. Sufficiently accurate constructiveupperestimates are obtained for the length of the shortest DNFs and the complexity of the minimum DNFs of almost all Boolean functions with a small number of zeros. In diverse applied problems Boolean functions are used that have a small number of zero sets /l/. Such functions f&‘*” are defined by the enumeration of a set of zeros m', (the matrix of zeros
d'Eiif,f). M,-llu,'ll,'r,'=:::;~, Both the reduced disjunctive normal form (RDNF)
of f and a realization of f, that is sufficiently short both in length (the number of elementary conjunctions (EC)) and in complexity (the number of symbols), must be constructed in the class of DNFs. These problems are also considered in this paper. The concepts and notation used below without definition can be found in /l/ to /3/. Sect.1 The S-algorithm for constructing RRNFs of Boolean functions. The algorithm for constructing the RDNF (S-algorithm),that is oriented towards the method under consideration of the definition of a function /=Ph" by a matrix of zeros Ml, is based on the following assertion. Theorem 1.
For the elementary conjunction
to be a simple implicantoff it is necessary, and for the condition of admissibility of the EC K also sufficient, that points d',....d'=fl,exist such that a,,'=@, Ve{l, 2....(t),aj,'=o, VsE{i, Z,..., t),s+r. Proof.
If K is a simple implicant of f then for any K, =
s=(1,2,...,t}
i z;; t=l.I+8
is not an implicant of f i.e. there exists a point $OEiti such that K,(a')=l and thus Ctj,'=O, VSE{i, 2,...,t), S+r. And since K(?")=O then a,JLa',.The necessity is proved. If K is an implicant of f and there exist points B’, . .,d’ that satisfy the condition of the theorem, then for any s=(1,2,...,t)and the EC defined above, K.(a")=l i.e. K. is not an implicant of f and thus the EC K is a simple implicant of f. The theorem is proved. Corollary 1. The rank of the simple implicants of f&,” does not exceed min{k,n}. The following S-algorithm for constructing all simple implicantsof rank t,3
For each set of t rows of the matrix M, in the submatrix of dimensions
*Zh.vychisl.Mat.mat.Fir..,27,6,924-931,1987
txn
186
that corresponds to this set, separate all the subcolumns of weight 1 and t-l them into groups of identical subcolumns (taking into account opposites).
and divide
Step 2. If a group is less than t then the required set of rows of simple implicants of rank t is not defined. If a group is exactly t then, by separating out one subcolumn from each group, sets of t subcolumns must be formed in all possible ways. Step 3. Put the set of subcolumns (O+l'E(1, 2,. . ..t}. Uj,=(u,’i,,..-1 Uj,‘) ; qrr=CT7; uj,“=u,, sfr) correspondence with the EC (1) and verify that this EC is allowable i.e. that the columns with numbers ]I,..., jl in each of the remaining k-t rows of M, separate a subrow that does not correspond to a non-allowable subrow (al, . . ..a[). The set of constructed allowable ECs is the set of all simple implicants of f of rank t. The simple implicants of rank 1 and 2 are formed in an obvious way for single scanning of, respectively, all columns and all pairs of columns from iV1. From /l, Theorem 2/ it follows that any algorithm for constructing the RDNF of the function f=Phn can be applied to constructing the RDNF of the function cp,the matrix of zeros MO of which contains one of each group of identical columns (takingintoaccountopposites)of The RDNF of f can be written down directly from the RDNF of cpand the numericity of the ,M,. groups in Mt. Therefore, the effectiveness of any algorithm for constructing a RDNF can be evaluated on functions whose matrices of zeros don not contain like columns (taking into (n-+m) there corresponds account opposites). Since for almost all functions with k zeros one and the same functions 90 (called complete) whose matrix of zeros contains all exp,(k--I) different columns (taking account of opposites) /l, assertion 3/then the effectiveness of algorithms for constructing the RDNF of functions with a small number of zeros canbeevaluated on the complete function 'pa. in
Theorem
2.
The FCDNF of 'pahas length
e(k)+0
N(k)=exp*{~~l+E~k)l},
as
k--t=.
Proof. Using Theorem 1 we shall calculate the number N,(k) of simple implicants of cpO of rank t,3GtGk. Fix t rows of the matrix of zeros ((f) methods of fixation). In the matrix obtained each group of identical subcolumns (taking into account opposites) contains terms. Separating one subcolumn in each of the t different groups that contain exp,(k--t) different sets of subcolumns of weight 1 and t-1 in all possible ways we obtain exp,[t(k--t)] columns. Add up the number of sets of columns that at least in the selected i rows of the remaining k--t rows ((“c’) ways of choosing) separate a non-allowable subcolumn. In each members. Therefore the required number group in such sets there are e&ctly expz(k-t--i) Since the simple implicants define those sets of sets of columns is equal to exp,[t(k--t-i)]. rows, that do not correspond to of columns that separate the subrows in the remaining k-t a non-allowable subrow, then, according to the inclusion-exclusionformula, the fixed t rows define the following number of simple implicants:
From which [exPx(t)--ll’-‘~N,(k)i(
t ) [exp,(t)-i]“-‘G(
: ) exp,(G),
Note that N,(k)<
N,(k)=&
2 ( expr(;-i)
;) .
) =z ( ; ) erp,(
Thus we have
N(k)=iN,(k)Gi(
1-I
F ) exp,(:)=exp,($+k)s
1-o
exPz[~[l+e,(k)l),
e,(k)+O,
k-t-,
N(k) > Ntkjaj W > (exp, LWI - lYkbl = sxp,($[i +s, Wl),
e*(k)-+O,
kdW.
Hence
N(k)=exp,(%L1+s(k)l},
e(k)+O,
k+-.
187 The theorem is proved. Corollary 2. The complexity of the S-algroithm on the complete function cp~is equal to the S-algorithm is asymptotically linear. LT'+*'~)(~), e(k)*0 as k-m, i.e. In fact, as follows from the proof of Theorem 2, the S-algorithm for constructing all simple imp&cants of spoofrank t uses not more than (
t
)
(texp*(k---l)-W(k-t)exp,[t(k--t)]}
operations, i.e. the general complexity of the S-algorithm does not exceed
{texpz(k--l)+tJ(k-t)exp,[t(k-t)l)C kexpz(2k-l)+:exp,(c+k)= I+C(A)(k), e,,(k),
exp, (;[2+e,(k)]}=N
c(k)+0
as k+m.
Note that earlier well-known algorithms for constructing RDNFs of Boolean functions given with zero sets, such as the method in /4/, that consist in the case quoted of the transformationofareal CNF of a function into a DNF multiplying the brackets and applying an absorption operation and the method for constructing implicants of order t by considering all sets from t columns on the full function q0 have the respective complexities exp2[k(k-I)]=expz(k2[
l+~,(k)]}=NL+eP)(k),
L
expz~-l)
)a.&-
[esp,(k-l)-kJk=
(=‘I
where e,(k), . . . , El(k)+0 as k-t=. Thus the complexity of well-known algorithms for constructing the RDNF on the full function Q is asymptotically equal to the fourth degree of the length of the RDNF. Therefore the S-algorithm that has an asymptotically linear complexity is the most effective algorithm for constructing‘the RDNF of functions with a small number of zeros.
2. Short DNFs of Boolean functions with a small number of zeros. As is well-known /l, Theorem 1, 4/, the length of the shortest DNF of a function from PA" is greater than min{[kn/2], n+m(k)) where :\a]is the integer part of a (Tal=-l-al),m(k)= m(3)=1, m(4)=4, m(k)
3.
Almost all functions f=P,“,
log,n
of length ]D,(: k (log? k -
ID,I<=-
4
Proof.
II
log, n
log,log, n)
log, n
For each variable x,of a function j&," &(f)=nlax(l{iliE{l,
2, . .
we introduce the notation
. , k), a’=.V., a,‘=l)~,
9 I{ili=(l, _,...
, k}, d’~iV,, at=O} I). Expanding f in the variable
XJ~,
where jo=arg min I=,*.*,..
we
.n,
4(f)
obtain f=~J~fJ.“VxJofJ.‘~
and functions fh",h' depend on n-i variables and have the zero sets, Lj.(f) k-Lo(f). The expansion process described above can be applied to each of the Boolean functions obtained that have more than log,n zeros. As a result we obtain the expansion The
f where
the EC A,
has rank a,,f“IEP;-%
m
’
=m<,Amfm,
we shall
show that for almost all fEPHn, log,ntk~0(~~~~(~/2)), k,,,+og.n
n+m,
vm~{l,2,...,iCi)
is satisfied. For this it is sufficient for almost all f to show that if in the expansion of f the term A,f,, k,>logz n appears, then min L,(f7) 4’/,,k,. We will obtain an upper estimate of the number of functions in PA” for which this is not so. For each such function f in hi’, it is possible to produce such k,>logzn rows, that separate This enables us or not less than ‘l,,k,. subcolumns of weight either not greater than 'i,,k,, to obtain an upper estimate of the number N of different columns that arise from M,:
~:t**, Ndew(k-k,)
x (‘i)+ k
[
( t: )] -c
j=:/,&,
j-0
exp,(k-k,)exp,(k,).2esp
here the following ineuality /4/ is used:
n
cc ) m>n,*+L m
2h" Cexpz(n)exp --T . ( )
The number of functions in PA" is equal to lph",=( exp;ln)) _expz($)
[1+0(l)1.
Since the matrix of zeros of functions being considered in Ph" contains not more than N different columns the number of such functions is not greater than
;<
=o(lPh”I).
*[*exp(-*)I"
Thus for almost all functions we have "/,, log, n
n,
(“/,,)“mk~k,~(‘/,,)a-k
Vm=(l, 2,. . . , M).
Thus the inequalities
M<2k
N 5 log, n ’
a,> a < !?I.
(
log* +-)log*$
> (log, k - logz log, d/2,
log, k - log, log, n + log, (1215) logz (1217)
are
satisfied. It follows from the inequalities proved andtheconstructed expansion of f that, by realising the functions fm with the shortest DNFs D f,_., we obtain the DNF D, of the function f, the length ID,/ of which for almost all f satisfies the inequality
Using
the
est&nate
for
m(k)
derived above we obtain
ID/mI~‘n-u,+exp,(k,-l)+exp,(k,-3)~’Sl,n-((logz
k-log,
log, n)/2.
Thus, for almost all f the estimates lD,I<-are correct.
12
k 13 --n-$(log,k-log,log,n)]<~5 log, n [ 8
The theorem is proved.
k (log, k - log, log, n) 2
lo& n
189
Corollary 3. The complexity of the minimal DNFs of almost all functions from P,",log,n< k90(expl(n/2)),n+? does not exeed
log, k - log, lee, n + log2(‘V& +
1
log, (“A)
log
n
P
4kn -log, n
k (log, k - log, log, n)
IL
log, n
I
.
This follows directly from the upper estimates for a, and the fact that the rank of the EC in the DNF constructed for the proof of theorem 3 is not greater than maxa, +logtn. In Note that by choosing ‘la-i/p and ‘/,-H/p instead of the numbers “1~ and 'II2respectively and using more and more accurate estimates for m(k), estimates can be obtained for the length of the shortest DNFs arbitrarily close to the expression 2k (log, k - log, log, n)
log2n and estimates of the complexity of the minimal DNFs arbitrarily close to the expression [log,k-log,log,n+l+log,n][~(i+-$-)-
2k('og'kl~~>'og'n)].
Since it is known /5/ that for almost all functions from Ph" the rank of the simple implicants is not greater than r=llog,(kn)-log,log,(kn)+51 then a stronger upper estimate for the complexity of the minimal DNFs
follows from the statement. AS L.I. Lipkin pointed out to the author, from the upper estimate obtained in /5/ of the rank of the simple implicants of almost all functions by Shannon's power method, the lower estimate of the length of the shortest DNFs can easily be obtained for almost all functions in Pa",k
kn log* n log, (kn)
’
In fact, the number of ECs of rank not greater than r does not exceed n ) expl(9G exp,(log, n[log,(kn)-
A( b.
log,log,(kn)+5l)=p,
and the number of DNFs made up from these of length not greater than 1 does not exceed P’
expa (kn)
1 -_[log,log,(kn)-5]~)=a(lP,nI).2
G-==expz k! kl
Since it is known from /5/ that for almost all functions from P,"the rank of the simple implicants is not less than Ilog,k-log,(log,klogn) 1 then, consequently, the lower estimate of the complexity of the minimal DNFs is obtained for almost all functions and is equal to Ilog,k--og,(log,klogn)l
log
a
n;&&n) 1.
Thus, for almost all functions in P*",k
min (k, n) - 2
min (k, n) + 2
2
2
I
,2n+
exp, (k + 1) - 10 + exp~(L~]+2)+exp*([~]+2)+
exP.([S])+axp,([~])]. From this, in particular, it follows thatthecomplexity of the minimal DNFs of functions in Pk",kclegan-q(n),$(n)+=, n-m, is equal to 2n[l+o(l)] (naturallythere are also functions
190
f=Pn’
where the number of zeros and single columns inM/
totals o(n)].
The author expresses his sincere gratitude to Yu.1. Zhuravlev for his help, and also thanks L.I. Lipkin for valuable advice. REFERENCES 1. ZHURAVLEV YU.1. and KOGAN A.YU., Realization of Boolean functions with a small number of zeros by disjunctive normal forms and related problems, Dokl. Akad. Nauk SSSR, 285, 4, 795-799, 1985. 2. YABLONSKII S.B. and LUPANOV O.B., Discrete mathematics and mathematical problems of cybernetics, Nauka, Moscow, 1974. 3. ERDESH P. and SPENCER J., Probability methods in combinatorics,Mir, Moscow, 1976. 4. NELSON R.J., Simplest normal truth functions, J. Symbol. Logic, 20, 2, 105-108, 1955. 5. SAPOZHENKO A.A., Values of the length and number of Tupikov DNFs for almost all not everywhere defined Boolean functions, Mat. Zam., 28, 2, 279-300, 1980.
Translated by S.R.
U.S.S.R. Comput.Maths.Math.Phys., Vo1.27,No.3,pp.190-192,1987 Printed in Great Britain
0041-5553/87 $lO.OO+O.OO @1988 Pergamon Press plc
A PROPERTY OF HURWITZ MATRICES IN THE REGULARIZATION OF A SYSTEM OF LINEAR ALGEBRAIC EQUATIONS* R.S. AYUPOV
A theorem on the existence of a scaling diagonal matrix A for a matrix D is formulated in a fairly general case that ensures that the corresponding product matrix AD is Hurwitz. This result is used in the Appendix to construct an iterational process for solving a system of linear algebraic equations. All matrices are assumed to be real. Definition. TWOprincipal minors (see /11/j of d&D are called embedded if all the diagonal elements of one of them are diagonal elements of the other one of greater dimension. Theorem. Suppose detD, where D is a matrix of order n, has a sequence, even if only one, of mutually embedded non-zero principal minors (conditionA). Then there exists a matrix . . A=diag(pi,..., P.) where b,#O, i=i,2,. . . . n such that the matrix AD is Hurwltz l.e., all its eigenvalues h, have a negative real part: Re&CO, f=i,2,..., n. 0) As is well-known, the eigenvalues hr are the solutions of the characteristic equation det(AD-hE)=qA"+q,h"-'+ . +q,_,b+q,=o, n>1, (2) with real coefficients. The proof of the theorem rests on the following lemma. Lemma. Suppose 03 in (2). If the coefficients nl(e) that depend on a small parameter a==0 satisfy the asymptotic conditions as e-+0: qi+2~+L(e)q~(e)/4,+l(e)=o(4i+z~(e)), i=o,i,. . . . n-3,
(3)
s=l,2,..., [(n-i-1)/2], where [...Idenotes the integral part of a number and if, moreover, as a+0 (4) qi(a]'o, i=o,1,. . . . n, then all the roots hi(e)of Eq.(Z) satisfy condition (1) for all e>O in a sufficiently small eo-neighbourhood of zero i.e. for O
In symmetric notation conditions (3) have the form ,im qr+.+r(a)Yr-.(a) -4 e-0 qr+.(s)qr-.+t(e)
where a-1,2,..., [n/2], k=s,s+l,..., n-s-i, and in this case (4) is taken into account. We shall show that for n=i,2 for AD to be Hurwitz it is sufficient that the inequalities of (4) are satisfied. *Zh.vychisl.Mat.mat.Fiz.,27,6,932-934,1987
(33
Translated by E.L.S.
U.S.S.R. Comput.Maths.Math.Phys.,Vo1.27,No.3,pp.185-190,1987 Printed in Great Britain
0041-5553/87 $lo.00+0.00 01988 Pergamon Press plc
DISJUNCTIVE NORMAL FORMS OF BOOLEAN FUNCTIONS WITH A SMALL NUMBER OF ZEROS* A.YU. KOGAN
An algorithm is proposed for constructing reduced disjunctive normal forms (DNF) of Boolean functions given with zero sets wthat is most effective for a small number of zeros. Sufficiently accurate constructiveupperestimates are obtained for the length of the shortest DNFs and the complexity of the minimum DNFs of almost all Boolean functions with a small number of zeros. In diverse applied problems Boolean functions are used that have a small number of zero sets /l/. Such functions f&‘*” are defined by the enumeration of a set of zeros m', (the matrix of zeros
d'Eiif,f). M,-llu,'ll,'r,'=:::;~, Both the reduced disjunctive normal form (RDNF)
of f and a realization of f, that is sufficiently short both in length (the number of elementary conjunctions (EC)) and in complexity (the number of symbols), must be constructed in the class of DNFs. These problems are also considered in this paper. The concepts and notation used below without definition can be found in /l/ to /3/. Sect.1 The S-algorithm for constructing RRNFs of Boolean functions. The algorithm for constructing the RDNF (S-algorithm),that is oriented towards the method under consideration of the definition of a function /=Ph" by a matrix of zeros Ml, is based on the following assertion. Theorem 1.
For the elementary conjunction
to be a simple implicantoff it is necessary, and for the condition of admissibility of the EC K also sufficient, that points d',....d'=fl,exist such that a,,'=@, Ve{l, 2....(t),aj,'=o, VsE{i, Z,..., t),s+r. Proof.
If K is a simple implicant of f then for any K, =
s=(1,2,...,t}
i z;; t=l.I+8
is not an implicant of f i.e. there exists a point $OEiti such that K,(a')=l and thus Ctj,'=O, VSE{i, 2,...,t), S+r. And since K(?")=O then a,JLa',.The necessity is proved. If K is an implicant of f and there exist points B’, . .,d’ that satisfy the condition of the theorem, then for any s=(1,2,...,t)and the EC defined above, K.(a")=l i.e. K. is not an implicant of f and thus the EC K is a simple implicant of f. The theorem is proved. Corollary 1. The rank of the simple implicants of f&,” does not exceed min{k,n}. The following S-algorithm for constructing all simple implicantsof rank t,3
For each set of t rows of the matrix M, in the submatrix of dimensions
*Zh.vychisl.Mat.mat.Fir..,27,6,924-931,1987
txn
186
that corresponds to this set, separate all the subcolumns of weight 1 and t-l them into groups of identical subcolumns (taking into account opposites).
and divide
Step 2. If a group is less than t then the required set of rows of simple implicants of rank t is not defined. If a group is exactly t then, by separating out one subcolumn from each group, sets of t subcolumns must be formed in all possible ways. Step 3. Put the set of subcolumns (O+l'E(1, 2,. . ..t}. Uj,=(u,’i,,..-1 Uj,‘) ; qrr=CT7; uj,“=u,, sfr) correspondence with the EC (1) and verify that this EC is allowable i.e. that the columns with numbers ]I,..., jl in each of the remaining k-t rows of M, separate a subrow that does not correspond to a non-allowable subrow (al, . . ..a[). The set of constructed allowable ECs is the set of all simple implicants of f of rank t. The simple implicants of rank 1 and 2 are formed in an obvious way for single scanning of, respectively, all columns and all pairs of columns from iV1. From /l, Theorem 2/ it follows that any algorithm for constructing the RDNF of the function f=Phn can be applied to constructing the RDNF of the function cp,the matrix of zeros MO of which contains one of each group of identical columns (takingintoaccountopposites)of The RDNF of f can be written down directly from the RDNF of cpand the numericity of the ,M,. groups in Mt. Therefore, the effectiveness of any algorithm for constructing a RDNF can be evaluated on functions whose matrices of zeros don not contain like columns (taking into (n-+m) there corresponds account opposites). Since for almost all functions with k zeros one and the same functions 90 (called complete) whose matrix of zeros contains all exp,(k--I) different columns (taking account of opposites) /l, assertion 3/then the effectiveness of algorithms for constructing the RDNF of functions with a small number of zeros canbeevaluated on the complete function 'pa. in
Theorem
2.
The FCDNF of 'pahas length
e(k)+0
N(k)=exp*{~~l+E~k)l},
as
k--t=.
Proof. Using Theorem 1 we shall calculate the number N,(k) of simple implicants of cpO of rank t,3GtGk. Fix t rows of the matrix of zeros ((f) methods of fixation). In the matrix obtained each group of identical subcolumns (taking into account opposites) contains terms. Separating one subcolumn in each of the t different groups that contain exp,(k--t) different sets of subcolumns of weight 1 and t-1 in all possible ways we obtain exp,[t(k--t)] columns. Add up the number of sets of columns that at least in the selected i rows of the remaining k--t rows ((“c’) ways of choosing) separate a non-allowable subcolumn. In each members. Therefore the required number group in such sets there are e&ctly expz(k-t--i) Since the simple implicants define those sets of sets of columns is equal to exp,[t(k--t-i)]. rows, that do not correspond to of columns that separate the subrows in the remaining k-t a non-allowable subrow, then, according to the inclusion-exclusionformula, the fixed t rows define the following number of simple implicants:
From which [exPx(t)--ll’-‘~N,(k)i(
t ) [exp,(t)-i]“-‘G(
: ) exp,(G),
Note that N,(k)<
N,(k)=&
2 ( expr(;-i)
;) .
) =z ( ; ) erp,(
Thus we have
N(k)=iN,(k)Gi(
1-I
F ) exp,(:)=exp,($+k)s
1-o
exPz[~[l+e,(k)l),
e,(k)+O,
k-t-,
N(k) > Ntkjaj W > (exp, LWI - lYkbl = sxp,($[i +s, Wl),
e*(k)-+O,
kdW.
Hence
N(k)=exp,(%L1+s(k)l},
e(k)+O,
k+-.
187 The theorem is proved. Corollary 2. The complexity of the S-algroithm on the complete function cp~is equal to the S-algorithm is asymptotically linear. LT'+*'~)(~), e(k)*0 as k-m, i.e. In fact, as follows from the proof of Theorem 2, the S-algorithm for constructing all simple imp&cants of spoofrank t uses not more than (
t
)
(texp*(k---l)-W(k-t)exp,[t(k--t)]}
operations, i.e. the general complexity of the S-algorithm does not exceed
{texpz(k--l)+tJ(k-t)exp,[t(k-t)l)C kexpz(2k-l)+:exp,(c+k)= I+C(A)(k), e,,(k),
exp, (;[2+e,(k)]}=N
c(k)+0
as k+m.
Note that earlier well-known algorithms for constructing RDNFs of Boolean functions given with zero sets, such as the method in /4/, that consist in the case quoted of the transformationofareal CNF of a function into a DNF multiplying the brackets and applying an absorption operation and the method for constructing implicants of order t by considering all sets from t columns on the full function q0 have the respective complexities exp2[k(k-I)]=expz(k2[
l+~,(k)]}=NL+eP)(k),
L
expz~-l)
)a.&-
[esp,(k-l)-kJk=
(=‘I
where e,(k), . . . , El(k)+0 as k-t=. Thus the complexity of well-known algorithms for constructing the RDNF on the full function Q is asymptotically equal to the fourth degree of the length of the RDNF. Therefore the S-algorithm that has an asymptotically linear complexity is the most effective algorithm for constructing‘the RDNF of functions with a small number of zeros.
2. Short DNFs of Boolean functions with a small number of zeros. As is well-known /l, Theorem 1, 4/, the length of the shortest DNF of a function from PA" is greater than min{[kn/2], n+m(k)) where :\a]is the integer part of a (Tal=-l-al),m(k)= m(3)=1, m(4)=4, m(k)
3.
Almost all functions f=P,“,
log,n
of length ]D,(: k (log? k -
ID,I<=-
4
Proof.
II
log, n
log,log, n)
log, n
For each variable x,of a function j&," &(f)=nlax(l{iliE{l,
2, . .
we introduce the notation
. , k), a’=.V., a,‘=l)~,
9 I{ili=(l, _,...
, k}, d’~iV,, at=O} I). Expanding f in the variable
XJ~,
where jo=arg min I=,*.*,..
we
.n,
4(f)
obtain f=~J~fJ.“VxJofJ.‘~
and functions fh",h' depend on n-i variables and have the zero sets, Lj.(f) k-Lo(f). The expansion process described above can be applied to each of the Boolean functions obtained that have more than log,n zeros. As a result we obtain the expansion The
f where
the EC A,
has rank a,,f“IEP;-%
m
’
=m<,Amfm,
we shall
show that for almost all fEPHn, log,ntk~0(~~~~(~/2)), k,,,+og.n
n+m,
vm~{l,2,...,iCi)
is satisfied. For this it is sufficient for almost all f to show that if in the expansion of f the term A,f,, k,>logz n appears, then min L,(f7) 4’/,,k,. We will obtain an upper estimate of the number of functions in PA” for which this is not so. For each such function f in hi’, it is possible to produce such k,>logzn rows, that separate This enables us or not less than ‘l,,k,. subcolumns of weight either not greater than 'i,,k,, to obtain an upper estimate of the number N of different columns that arise from M,:
~:t**, Ndew(k-k,)
x (‘i)+ k
[
( t: )] -c
j=:/,&,
j-0
exp,(k-k,)exp,(k,).2esp
here the following ineuality /4/ is used:
n
cc ) m>n,*+L m
2h" Cexpz(n)exp --T . ( )
The number of functions in PA" is equal to lph",=( exp;ln)) _expz($)
[1+0(l)1.
Since the matrix of zeros of functions being considered in Ph" contains not more than N different columns the number of such functions is not greater than
;<
=o(lPh”I).
*[*exp(-*)I"
Thus for almost all functions we have "/,, log, n
n,
(“/,,)“mk~k,~(‘/,,)a-k
Vm=(l, 2,. . . , M).
Thus the inequalities
M<2k
N 5 log, n ’
a,> a < !?I.
(
log* +-)log*$
> (log, k - logz log, d/2,
log, k - log, log, n + log, (1215) logz (1217)
are
satisfied. It follows from the inequalities proved andtheconstructed expansion of f that, by realising the functions fm with the shortest DNFs D f,_., we obtain the DNF D, of the function f, the length ID,/ of which for almost all f satisfies the inequality
Using
the
est&nate
for
m(k)
derived above we obtain
ID/mI~‘n-u,+exp,(k,-l)+exp,(k,-3)~’Sl,n-((logz
k-log,
log, n)/2.
Thus, for almost all f the estimates lD,I<-are correct.
12
k 13 --n-$(log,k-log,log,n)]<~5 log, n [ 8
The theorem is proved.
k (log, k - log, log, n) 2
lo& n
189
Corollary 3. The complexity of the minimal DNFs of almost all functions from P,",log,n< k90(expl(n/2)),n+? does not exeed
log, k - log, lee, n + log2(‘V& +
1
log, (“A)
log
n
P
4kn -log, n
k (log, k - log, log, n)
IL
log, n
I
.
This follows directly from the upper estimates for a, and the fact that the rank of the EC in the DNF constructed for the proof of theorem 3 is not greater than maxa, +logtn. In Note that by choosing ‘la-i/p and ‘/,-H/p instead of the numbers “1~ and 'II2respectively and using more and more accurate estimates for m(k), estimates can be obtained for the length of the shortest DNFs arbitrarily close to the expression 2k (log, k - log, log, n)
log2n and estimates of the complexity of the minimal DNFs arbitrarily close to the expression [log,k-log,log,n+l+log,n][~(i+-$-)-
2k('og'kl~~>'og'n)].
Since it is known /5/ that for almost all functions from Ph" the rank of the simple implicants is not greater than r=llog,(kn)-log,log,(kn)+51 then a stronger upper estimate for the complexity of the minimal DNFs
follows from the statement. AS L.I. Lipkin pointed out to the author, from the upper estimate obtained in /5/ of the rank of the simple implicants of almost all functions by Shannon's power method, the lower estimate of the length of the shortest DNFs can easily be obtained for almost all functions in Pa",k
kn log* n log, (kn)
’
In fact, the number of ECs of rank not greater than r does not exceed n ) expl(9G exp,(log, n[log,(kn)-
A( b.
log,log,(kn)+5l)=p,
and the number of DNFs made up from these of length not greater than 1 does not exceed P’
expa (kn)
1 -_[log,log,(kn)-5]~)=a(lP,nI).2
G-==expz k! kl
Since it is known from /5/ that for almost all functions from P,"the rank of the simple implicants is not less than Ilog,k-log,(log,klogn) 1 then, consequently, the lower estimate of the complexity of the minimal DNFs is obtained for almost all functions and is equal to Ilog,k--og,(log,klogn)l
log
a
n;&&n) 1.
Thus, for almost all functions in P*",k
min (k, n) - 2
min (k, n) + 2
2
2
I
,2n+
exp, (k + 1) - 10 + exp~(L~]+2)+exp*([~]+2)+
exP.([S])+axp,([~])]. From this, in particular, it follows thatthecomplexity of the minimal DNFs of functions in Pk",kclegan-q(n),$(n)+=, n-m, is equal to 2n[l+o(l)] (naturallythere are also functions
190
f=Pn’
where the number of zeros and single columns inM/
totals o(n)].
The author expresses his sincere gratitude to Yu.1. Zhuravlev for his help, and also thanks L.I. Lipkin for valuable advice. REFERENCES 1. ZHURAVLEV YU.1. and KOGAN A.YU., Realization of Boolean functions with a small number of zeros by disjunctive normal forms and related problems, Dokl. Akad. Nauk SSSR, 285, 4, 795-799, 1985. 2. YABLONSKII S.B. and LUPANOV O.B., Discrete mathematics and mathematical problems of cybernetics, Nauka, Moscow, 1974. 3. ERDESH P. and SPENCER J., Probability methods in combinatorics,Mir, Moscow, 1976. 4. NELSON R.J., Simplest normal truth functions, J. Symbol. Logic, 20, 2, 105-108, 1955. 5. SAPOZHENKO A.A., Values of the length and number of Tupikov DNFs for almost all not everywhere defined Boolean functions, Mat. Zam., 28, 2, 279-300, 1980.
Translated by S.R.
U.S.S.R. Comput.Maths.Math.Phys., Vo1.27,No.3,pp.190-192,1987 Printed in Great Britain
0041-5553/87 $lO.OO+O.OO @1988 Pergamon Press plc
A PROPERTY OF HURWITZ MATRICES IN THE REGULARIZATION OF A SYSTEM OF LINEAR ALGEBRAIC EQUATIONS* R.S. AYUPOV
A theorem on the existence of a scaling diagonal matrix A for a matrix D is formulated in a fairly general case that ensures that the corresponding product matrix AD is Hurwitz. This result is used in the Appendix to construct an iterational process for solving a system of linear algebraic equations. All matrices are assumed to be real. Definition. TWOprincipal minors (see /11/j of d&D are called embedded if all the diagonal elements of one of them are diagonal elements of the other one of greater dimension. Theorem. Suppose detD, where D is a matrix of order n, has a sequence, even if only one, of mutually embedded non-zero principal minors (conditionA). Then there exists a matrix . . A=diag(pi,..., P.) where b,#O, i=i,2,. . . . n such that the matrix AD is Hurwltz l.e., all its eigenvalues h, have a negative real part: Re&CO, f=i,2,..., n. 0) As is well-known, the eigenvalues hr are the solutions of the characteristic equation det(AD-hE)=qA"+q,h"-'+ . +q,_,b+q,=o, n>1, (2) with real coefficients. The proof of the theorem rests on the following lemma. Lemma. Suppose 03 in (2). If the coefficients nl(e) that depend on a small parameter a==0 satisfy the asymptotic conditions as e-+0: qi+2~+L(e)q~(e)/4,+l(e)=o(4i+z~(e)), i=o,i,. . . . n-3,
(3)
s=l,2,..., [(n-i-1)/2], where [...Idenotes the integral part of a number and if, moreover, as a+0 (4) qi(a]'o, i=o,1,. . . . n, then all the roots hi(e)of Eq.(Z) satisfy condition (1) for all e>O in a sufficiently small eo-neighbourhood of zero i.e. for O
In symmetric notation conditions (3) have the form ,im qr+.+r(a)Yr-.(a) -4 e-0 qr+.(s)qr-.+t(e)
where a-1,2,..., [n/2], k=s,s+l,..., n-s-i, and in this case (4) is taken into account. We shall show that for n=i,2 for AD to be Hurwitz it is sufficient that the inequalities of (4) are satisfied. *Zh.vychisl.Mat.mat.Fiz.,27,6,932-934,1987
(33