On the correctness of algebraic closures of recognition algorithms of the “tests” type

U.SS.R Comput. Mzths Math. Phys Vol. 22, No. 6, pp. 217-226, Printed in Great Britain

1982.

0041-5553/82$07.50+.00 01984. Pergamon Press Ltd.

ON THE CORRECTNESS OF ALGEBRAIC CLOSURES OF RECOGNITION ALGORITHMS OF THE “TESTS’ TYPE* A. E. DYUSEMBAEV Alma-Ata (Received 22 September 1981)

THE CORRECTNESS of algebraic closures of recognition algorithms of the “tests” type over problems with binary learning information is proved. For each of the problems considered an explicit form of correct algorithm is indicated. A dependence is revealed between systems of reference sets and problems, the correctness of the closures over which can be proved. We shall consider the algebraic closures of certain classes of algorithms for computing estimates, over the set of problems, the objects in which are binary sets of fixed length. In Section 1 we study the classes of algorithms, all the algorithms of which have as their systems of reference sets some fixed system of subsets of the set of criteria, satisfying a special condition which we introduce below. In particular, these systems may be certain sets of dead-end tests of the learning matrix. Our main result in Section 1 is to prove the correctness of the algebraic closures of the subclasses considered over the corresponding sets of problems. In Section 2 we construct extensions of the initial sets of algorithms and show that the algebraic closure of these extensions are correct over much wider sets of problems than the sets of problems considered in Section 1. We observe that, for the algorithms of the “extended” sets, the systems of reference sets may be any set of dead-end tests of the learning matrix. All the proofs given are of a constructive type: the correct algorithms in the closures are constructed in explicit form. Our main results and the proofs of our claims are based on the methods developed in [l-3]. As

in [ 1, 2] , we shall assume that the following objects are given.

1. The sef X Its elements are n-dimensional vectors, whose coordinates take the values 0 or 1. In other words, each element x of X is a finite set of binary tests, i.e. z= (;c,, . . . , 2”) , Z
*Zh. vjkhisl Mat. mat. Fiz., 22.6, 1491-1499,

1982.

217

218

A. I?.Dyusembaev

3. The set of problems (2). Each of them is a pair (10, x9), where gg= {a, . . . , z,} is a set of admissible objects, the information rows of which have to be evaluated, 10 E {I,}. 4. The set of algorithms {A} f or computing estimates, We know that every estimatecomputing algorithm A can be written as A = B oC* (see [2 ] ), where B is the operator of computing estimates, and C’*is a correct decision rule with parameters ci,‘, . . . , ciI’, czi’, . . . , czl’. We shall consider classes of estimate-computing algorithms which contain algorithms with an arbitrary (though fixed for all algorithms) system of reference sets n, which is the base of the set of tests in the following sense: the collection G?of reference sets of the estimate-computing {1,X,. . * I nl, algorithm A is a base of the set of tests {I, 2, . . . , n}, if

be admissible 5. The proximity functions. Let zl= (x1,, . . . , zin), &= (zz,, . . . ,x2,,) the reference set; then we denote by p (Gx~, I%,) the number objects, and G= (ii, . . . , it) of coincident components of the vectors as,= (xii,, . . . , s+), ox,= (.zz~~~, . . . , zzi ,). The proximity function between objects xl, x2 with respect to the reference set G is specified as follows: F (65x,, 63.~~)=

1, ecdH P(Gx~,&~)>~-E, B I-f~OTIlBHOM

CJIJWW,

where E is an integer of [0, n] . 6. Estimates and parameters of recognition operators. With each test j we associate a “weight” p(~)=p~,+. . . +pil. p,>O, j=l, 2,. . . ,TZ._ Let cjd?. 8=(i,, . . . , t,) ;thentheweight Further, let Z = be a problem. To each object Xi E 10 we assign a weight yir ri> 0, yi=y(zi), i=l, 2, . . . ) m. We denote by III’ii II9x l the matrix of the operator B. We put

Finally,

where Rij=XjnXm, CRj=X”\Rj, rim= {xi’, . . . , z,,,‘} do, P=t-F. (see PI)

We have the inequality

Algebraic closures of recognition algorithms

1. Correctnessof algebraicclosure

219

%{A 1 over the set of

problems {Zl Let Z=CZO. K*> be a problem. The initial information Zo is called R-essential for Z if, given any x1, x2 of the set zb, an object x exists of the set zrn, and also a set G exists of CI, such that p(o+, as,) +p(im, az,). We call the problem

Z=_(Zo, HP>

l)a,+a,,

. . (I, p=l,

t=1,2,.

!&normal if it satisfies the following conditions:

2,. . . ,&I,

t+i,.

. . ,z;

2) the initial information 10 is n-essential for problem Z. Since the system of reference sets (the base SZ)is fixed for all algorithms, S&essential initial information will henceforth be called simply essential, and !&normal problems will simply be called normal. The condition {x,‘, . . . , z,,,‘} ll {x,, . . . , zq} -0 will be assumed to be satisfied for every problem, since otherwise, from the sample R’*= {x1, . . . (5,) , we can remove those objects that belong to the intersection ~“‘ll8*. Any notation and definitions not given in the present paper will be found in [ 1,2]. Let (2’) be the set of problems, and let {A} ( {I?} ) be the set of estimatecomputing algorithms (operators) [3] ; also, let the algebraic closure II {I?) be quasi-complete for (2’) and let {B (i, i) } 91:: be a quasi-basis of %{B} . Theorem

1 (see [3])

A correct algorithm A for problem Z’ of {Z’} can be written as

(1.1) where IIaijIIqxl is the information matrix of problem Z’, Cil= min

Cij’,

Cz’=

max

Cpj’,

a=

1 07 PA,

max

Ir,,(i,

j)l,

(fd),(W)

i),

and k=

Theorem

lnq+ln~+lln(c,‘+c,‘)I-llnc,‘I I Ilnal

1 * +1

2

The algebraic closure ?[(A} of subclass over the set of normal problems.

{A}

of estimate-computing algorithms is correct

proof: We shall show that the linear closure L(B) is weakly complete [3] for the set of normal problems (2). In other words, we shall show that, given any problem Z of the set {Z} ,

220

A. E. Dyusembaev

there exist in L {B} operators B(i, j), i=l, 2, . . . , q, j=1,2, , . . , 1, such that, for the elements of the matrix ]]I’,, (i, j) ]Ipxl of the operator B (i, j) we have the relations

We shall assume that 4 > 1, I > 1. We shall first show that in L(B) operators Bii exist such that B(i. j)=c(i, j)Z%, c(i, j) are non-zero constants, i = 1,2, . . . , q, j = 1, 2, . . . , I We define the operator Bij=Bj+BI, where

B,=

z

Bjt,

Bij=

I#j

Bi,‘,

z

j=l,

2,. . .,I;

i=l,2,

. . . . q.

r#i

The operators RI,, j=1, 2, . . . , Z, 1= 1, 2, . . . , j-1, j+l, . . . , I, are estimate-computing operators. Each of the Bi,j (i=l, 2, . . . , q, ~=i, 2, . . , , i-l, A-1, . . . , q) , is either an estimate-computing operator, or is the difference of two such operators. Construction of the operator Bit. Let ]]I’,,]I,,,, be the matrix of the operator Bit. We specify the parameters of the operators Bit in such a way that, given any normal problem Z= O. As the set of reference sets of the operator we take the base R of the set {I, 2, . . . , n}. Further, let Z?jgR;,; then there is an object IC,EK,\R,. We put po=O, pi=l, yi=l, i=l, 2,. . . , v-1, vfl, . . . , m, E-n, Pi=. . . =p,= p. ~10. Let Sz be the set of reference sets of the operator Bit. The case zj = Et is excluded by condition 1) of the definition of a normal problem. We estimate the elements of the matrix 11 rk,, ]lqXI of the operator Bit: y(3h) =y,=o,

rkjZc,o,

c,=p(8*)+...+~‘(o:,I),

rkh<(0+772-I)c,,

h=l. 2, . . . : 1. k=l,

(1.2)

2, . . . ) 1.

(1.4)

It can be seen that (1.2) holds, from the fact that, if Kj c zr, then, by the choice of parameters if e, y1r a. * 7ym, Pit * *. , Pn, and by the fact that X’n’i’~-~g=O, we have F(Bz,, OX,) =I: KjSRrt then F(G%, Qx,) =1. Inequality (1.3) follows from the fact that, for the case Inequality (1.4) ZGg R, , we have XvERj\Rt, while forKi C Et we have z,ER,\R~. follows from (1). Consider the operators Bj, j = 1, 2, . . . , 1. By definition, B,=B,,-+ . . . +Bj* j-l+Bj, j+i+ . . . $-Bit. Let llrtPllqxl be the matrix of the operator Bj; then, from estimates (1.2) - (1.4) we obtain the following estimates for the elements of matrix I/rtP]]gX,:

(1.5) rt,qz-2)c,0+(~-f) (m-~)c,, p=l, 2, . . . , j-l,i+l,.

, . ,Z.

t=l

9 . *. , 4, f --

(1.6)

Algebraic closures of recognition algorithms

Consider the operators

Bi,‘,

i=l,

2, . . . , Q, T=!,

221

2, . . . , i-1,

ifl,

. . . , Q, i=

, 1. With respect to the normal problem 2, the operator Bij evaluates the matrix 1, 2,... such that element I’ii is much greater than element Fri. Since the initial information Zo is essential for problem 2, an object x E xrn and a reference set w”exist such that

p (Qx, BXi) Zp (ax,

6x,) ,

i.e. we have one of the inequalities: p (ax,

63x,)>p (QL, ax,),

P(QG m,)
(1.7)

(1.8)

ax,).

Construction of the operators Bi,i for (1.7). Consider the case when x E &. We put

7(x)='J, yt=y(xr) =1 p*=l,&=n-p(Gjs, Ox,),

for x,#s,

.rl~xm.

p,,=O,

As the set of reference

sets of the operator Bi, we take R. Let r be the number of the test appearing in G and not appearing in the reference sets other than G (here and henceforth, p,=l,

p,=O,

t=l,

. . n.

2?. . . . r-l.r+l..

in the proof of the theorem);

We estimate the elements of matrix

then, we put (jrvhIIr,:..,

of the operator Bi,j, generated by the chosen values of the parameters

r,,A7,

(1.9)

I-,,< (m-l), r,,
v=l,

2

(1.10) h=l,

47

,...’

2,. . . , 1.

(1.11)

Estimates (1.9) and (1.10) follow from the fact that, given the chosen values of the parameters , we

a, Yl, . . * , ym, Pl, * * * , Pn, PO, PI Inequality (1.11) follows from (1). Let x EC&.

have

F( 8z,

nz,) =I

and

F(&r,

BZ,)

=O.

We define the operator Bij as follows: j=l B,,-=B:i,-Bi,,, ~=1,2

,....

i-l,i-il,...,

,_,3 .

..,1, i=1,2,

. . . , 9,

q.

For the operator Bili, we specify the set of reference sets sl with pO=l, pr=O, E=O. The remaining parameters are the same as when constructing the operator B,i in the case when x ERj. We specify the operator Bj,i, being the same as when constructing

by the parameter values /.Q = 1,1-(1 = 0, the remaining parameters the operator B’i7 in the case when x E Ei . The set of reference

sets is LL We estimate the elements of the matrices operators

IIrYh(1) llpXlr

IlI',,(2) llP~~, llI’v~llg~~ofthe

B,:,, B,fir, B:,':

ri,(2)Gm-1,

ri,30-

20,

(1.12)

r7.i (2) 20,

(1.13)

r,,(i)

ri,(l)20,

(m-l),

]T,jIGm-1

7

(1.14)

A. E. Dyusembaev

222

r,,(2)
r,,(l)Gs+m-I, ‘v=l, 2, * . . : q,

pvh~G.7+~-11, (1.15)

h=l, 2,. . . ,1.

Inequalities (1.15) follow from (I), and inequalities (1.12) from the fact that the proximity function of the operator Blli7 ’ is such that F (BX, a)~*) =?, F(Bx, 6%) =I. Estimates (1.13) follow from the equations F(&rz, 8zi) =O, P(Bx, BX,) =I. Estimates (1.14) follow from (1.12), (1.13), and (1.15). Construction of the operators BiTJ’for (1.8). Consider the case when the object x EXj. We define the operator Bi,i as the difference between the two estimate-computing operators BL,, BZL.

We construct the operators BA,, Bii,.

’ . by the following parameter values: )&=O, k=l, a-n-p We specify the operator Blllr (as, G&). The remaining parameters are the same as when constructing Bl,i, in case (1.7). The set of reference sets of the operator Bl,i, is St.

We define the operator Bi,, as c=n-p(Bx, Bx,).

in

the same way as Bi, is, except that the parameter E is defined

It is easily verified that the elements of the matrices []I’,( 1) IlpX1,IlI’,,(2) IIqxl, IlI’,,J,,, of the operators B,f,,

Biir,

Bi,j

satisfy inequalities (1.12)-(1.15)

respectively.

For, in view of the choice of the parameters po, pl, yl, . . . , ym, p,, . . . , p,,, E, the proximity function of the operator Bi,i, is such that F ( BX, t.Gxi) =F (ax, BX,) = 1. The proximity function of the operator Bj,i, is such that, with the chosen parameter values of the operatorBj2ir,wehave F(CTIX, 3xi)=0 and F(Bx, @x1)=1. Hence(l.l2)and(1.13) follow. Inequality (1.14) follows from (1.15), (1.12), (1.13). Inequalities (1.15) follow from (1). Next consider the case when x E C’Zj. For the operator BiJ we now specify the set S2of reference sets with /.Q = 1, ~1 = 0, E =n-p (Qx, OX,), pr=l, pt4 t=l, 2, . . . , r-1, r+i, . . . ) n; 7(x)=0,

y(xt)

=I,

x~+T.

It is easily seen that relations (1.9)-(1.11) the operator Bi,i.

hold for the elements of the matrix III’y,,ll~Xlof

For, with the chosen parameter values, the proximity function F (Ox, 0x0 =0, F ( QX, 8x1)=1: hencerelations(l.9)and(l.lO)hold. Weobtain(l.ll)from(l). Consider the operator Bii. By definition, B+

c r*i

Bi,j,

i=l, 2,. . .) q,

j=i,

2, . . * ,1.

We willshowthat the elements of the matrix III’,, (i, f) llqxl of the operator Bii satisfy the relations rij(i,j)~(q-l)o-(q-l)

Irdu)

I~(q-2b+(q-l)

(m-l), (TW--I), ufi,

(1.16) (1.17)

223

Algebraic closures of recognition algotithms

p--u&j) Iqq-l>0+@.?-1) u=l,

U’l,

(m-l>,

2,. . . ) q,

(1.18)

2, . . . ,1.

In fact, relation (1.16) follows from the definition of the operator Bii and from (1.9) and (1.14); relation (1.17) follows from the definition of the operator and (1. lo), (1.14); while relation (1.18) follows from (1.1 l), (1.15) and the definition of the operator. Let us estimate the elements of the matrix Next consider the operator Bij=Bj+B(. Ilr=fi(i. i) llpXl of this operator. By definition OfBij and (1.5), (1.16), we obtain

(m-1).

rij(i,j)2(q+c,z-c,-l)o-(q-l)

By definition OfBij and (1.5) and (1.17), we obtain ,I) (m- .I>+ (Z-l) (m-l)

j) ~~(q+C~Z--C~--2)O+(q-

Ir,j(i,

c,,

afi. By definition of Bii and (1.6), (1.18), we obtain pk(i,j)

IG(q+c,i-2c,-l)cr+(q-1)

a=1,2,.

. . , i-l,

Clearly, with o>rQO,

. . , q,

i-II,.

tit (6)
rij(i,j)>lrab(4j)

I,

(m-l)+(Z-1) p=1,2,.

(m-l)c,,

. . ,j-t,j+1,.

. . ,l.

, we have

(a. P)+(i,i),

Cl!

((J), Ci2(O)
Finally, we put B (i, j) =BJij-’ (i, j) . It is easily seen that, with respect to the normal problem Z, the operator B (i, j) computes the matrix IlrlP (i, j) llgxI such that rij(i, j) =I,

I rfp (4 i) i-4,

(t, p)+

(4 i).

We shall omit the proof that a correct algorithm A for the normal problem 2 can be written in the form (1. l), since it is similar to the proof of Theorem 1 of [3]. The theorem is proved. Note 1. In order for %{B} to be quasi-complete for problem Z, it must be ((11, . . . , {n)) normal. We shall assume that Q > 1, I> 1. Assume that the problem Z=(ZO, K*) is not normal. Assume that it does not satisfy condition 1) of the definition of a normal problem, i.e. classes Kp, Kt exist such that Rp=Rt , t+p. Then, for the elements of the matrix B (2) of the operator B, we have r,(zi)=rl(Xi), i=l, 2, . . . . Q. Consider the case when the initial information Zo is not essential for problem Z. Then, there will be objects ztr zz=K, such that, for the elements of the matrix B (2) of the operator B we have the relations I’j(li)=I’,(5?), i=l, 2, . . . . 1. Hence we see that I[{B) is not quasi-complete for problem Z Consider a set of problems with disjoint classes. Denote the set by (2,). For each problem Z of this set of dead-end tests of the learning matrix Tnml (Z) (see [4] ). As the reference sets for the operator B of {B} we take the dead-end tests which form in aggregate the basis of the set {I,

A. E. Dyusembaev

224

2 ,a*-, n}. Denote by {B’} (or {A’} the subclass of operators (of algorithms) of computing estimates with the set of reference sets considered. We distinguish in (2,) a subset of problems {Z,, i}, the initial information of which is essential. Corollary1. The algebraic closure %{A’) of subclass {A’} of estimate-computing algorithms is correct over the set (2,. ,} ,

2.

Correctness of algebraic closure o[(21 the set of problems {E>

over

We shall consider estimate-computing algorithms, differing from the algorithms of set {A} in the method of computing the estimate of the object x in the class Kj, namely: I?j

(Xi)

'~i".Fj'(Zi)

+

j.Lt'rj'

(Xl)

7

pi’, ).LilE{O, I},

i=l,

27 a. * 7 q*

In addition, we shall assume that the system of reference sets for these algorithms is any set !J’ of reference sets (not necessarily a basis). Denote the set of these algorithms by {A}, and the corresponding set of operators by {B} . A problem

is called non-degenerate if the initial information 10 of Z is such

Z=(lO, %‘)

that I<, zH,,

t=1,2,.

. . ) 1,

p=l,

2,. . . , t-1, t+1,.

. . ( 1.

Denote the set of non-degenerate problems by (2). Theorem 3 The algebraic closure 9l {a} problems (2).

of the set {A} is correct over the set of non-degenerate

Proof: Let 2=X1,, _k?? be a problem of the set {Z}. We will show that the linear closure L(B) is quasi-complete for problem 2 We construct initially the set of estimate-computing operators B,ri, i=l, 2, . . . ,1, PI, 1=1,2,. . . ,i-l,i+l,. . . , I, i=l, 2 ,..., q. We specify each of the operators Bjri by the same set of parameters yl, . . . , y,,,? pi, . . . , p,, E. as for the operator Bit, used when proving Theorem 1. In addition, with object Xi of control sample 24 we associate the parameter values pi”=O, pi1=17 if Rj$R,, and the values pi’=O, pio=l, if Rj=R,. With objects of sample Tq, distinct from xi, we associate the value ~1 =@ = 0. The set of reference sets of the operator Biti is SI’. We define the operator i=l, 2, . . . , q,

j=l,

2, . . , ,1.

Algebraic closures of recognition algorithms

We can easily estimate the elements of the matrix rij(i,

I?ir(i,

r,, (i, i) =O,

III’,, ( i7 j) lipX, of the operator Bii:

(2.1)

j)>(Z-l)w,

j)<(l--2)C,c+(Z-4)

r=l,2

(m-l)C,,

,..., j-l,,r+I,...,

r=l,

225

2,. . . (2,

?=I,

(2.2)

2. 2,. . . , i-l,

i+1:.

. . , q.

(2.3)

Estimates (2.1), (2.2) are similar to (1.5), (1.6) respectively. Equation (2.3) is obvious. It is easy to find a 00 > 0 such that, with u > 00, we have

while for the elements of the matrix have relations FijCi,

IITlp(i! j) IlpXI of operator

j) =I>

I-*,

(i, i) (1,

B(i, j) =BJij-‘(i,

j)

we

(4 PI =+(4 i) *

By Theorem 1, a correct algorithm 2 for a non-degenerate problem Z can be written in the form (1. l), except that k is evaluated from the relation k=

~(lnZ+lln(c,‘+c,‘)I-llnc,‘l)]+l.

E

The theorem is proved. Let Z=(lo,

x4)

be a problem. The class Fj is called isolated in Z if RjG

Let .!?T=(Z,,H’Q

U 2,. 12)

be a problem in which every class is isolated. Call the set of such problems (z}.

Theorem 4

The linear closure L(A)

of the set {ill

is correct over the problem set

{z}

Bool: We will show that any problem Z of the set {Z} is complete with respect to L(B). We fix a pair (i, j)! l0. We construct an operator g(i, j), for the elements of the matrix IlrtP(i, j) IlqXi of which we have the relations rij(i? j) =I.

rtp(i,j) G6.

(t. p) + (L j) .

(2.4)

We initially construct the estimate-computing operator zij. As the set of reference sets of the operator Bij we take !G!‘.With object xi of the control sample FS we associate the parameter values b’=l, ktl“=O. With objects of FQ different from xi we associate parameter values ~10= ~1 = 0. There is an object xi of the set Ri such that zj$Rz, -c=l, 2. . . . ? j-l, j-i-l, . . . , 1. Then we put y>=y (Xj) =o, y,=y (t,) =I, XrfXj. Next, we put e=n, p,= . . . =p,,=p, p>u. Let us estimate the elements of the matrix of the operator Rij, generated by the chosen parameter values: fi, (4 i) 20, p=l,

2,. , . (1,

Pip (i, j) fc,m. t#i,

Ttp(i, j) =O?

226

A. E. Dyusembaev

where c,=p(Oi)+. . . +p ( B,~,). These estimates are easily found from the definition. We put B (i, j) =BijTij-’ (i, j) . Clearly, for each 6 > 0 there is a u < m such that, for the elements of the matrix of the operator 3 (i, j), we have estimates (2.4). On then using Theorem 2 of [ 11, we arrive at Theorem 4. Consider the set of problems (2,) and the set of estimate-computing algorithms {A}. Let Z be a problem of the set (2,). As the set of reference sets for algorithm 2 of set {a} we take any set of dead-end tests of the learning matrix Tnml (Z) (see [4, 51). Denote by {a,} the subclass of such algorithms. We have the corollary of Theorem 3: Theorem 5

The algebraic closure %{A,} of subclass {a,} is correct over the set of problems (2,). The author sincerely thanks Yu. 1. Zhuravlev for his guidance. Translated by D. E. Brown

REFERENCES 1. ZHURAVLEV, YU. I., Correct algebras over a set of incorrect (heuristic) algorithms, I., Kibernetika, No. 4, 14-21,1977. 2. ZHURAVLEV, YU. I., Correct algebras over a set of incorrect (heuristic) algorithms, II, Kibernetika, No. 6, 21-27,1977. 3. ZHURAVLEV, YU. I., Correct algebras over a set of incorrect (heuristic) algorithms, III, Kibernetiku, No. 2, 3%43,1978. 4. KONSTANT’INOV, R. M., KOROLEVA, Z. E., and KUDRYAVTSEV, V. B., A combinatoric-logic approach to problems of ore-content prediction, in: Prubl kibernetiki, No. 31, S-33, Nauka, Moscow, 1976. 5. CHEGIS, I. A., and YABLONSKII, S. V., Logical methods of monitoring electrical circuits, Tr. Matem. in-ta Akad Nauk SSSR, 51,270-360,1958.