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6 /o. ?t 
 
 7 ~ £6~AS Report No. UU2 
 
 JkAZ% 
 
 COO-211 8-0008 
 
 INTERVAL GENERALIZATION OF SWITCHING THEORY 
 
 by 
 R. S. Michalski, B. H. McCormick 
 
 May 3, 1971 
 
COO-2118-0008 
 
 Report No. kk2 
 
 INTERVAL GENERALIZATION OF SWITCHING THEORY 
 
 by 
 
 R. S. Michalski, B. H. McCormick 
 
 May 3, 1971 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 This work was supported by Contract AT(ll-l)-21l8 with the U. S. Atomic 
 Energy Commission. 
 
INTERVAL GENERALIZATION OE SWITCHING THEORY 
 
 R. S. MICHALSKI, B. H. MC CORMICK 
 University of Illinois 
 Urbana, Illinois 
 
 ABSTRACT 
 
 The paper considers: 
 
 (1) a Boolean algebra < 2 E , w , ^ , -, E, > of event sets E. 
 from a discrete finite vector space E, and 
 
 (2) mappings f from the set E into {[0,l], *}, where * represents 
 some unspecified value. A special case of the above is the 
 Boolean algebra and Boolean functions considered in switching 
 theory, where E is a space of binary vectors and f maps E 
 into {0,1,*}, i.e. into the endpoints of the interval [0,1] 
 and *. 
 
 A meet semi-lattice of multidimensional intervals (inter- 
 val complexes) in E is introduced and then the concepts of exact, 
 free, unordered and ordered interval covers of f are defined. 
 
 The simplest case of a cover - an unordered exact cover 
 of a set F-*-* against F - is defined as a set of interval complexes 
 whose set-theoretic union covers a given subset F- 1 $ E (defined as 
 { e f(e) >_ \ } ) and does not cover any element of another given 
 subset F°* C EXF 1 . 
 
 The concept of ordered covers was developed to accomodate 
 a preferential order in covering the set of 'mixed' events, defined 
 as { e < f(e) < 1 } (a case not considered in classical switch- 
 ing theory). 
 
 The synthesis algorithm of covers is based on the 'method 
 of disjoint stars', which has proved to be very useful for synthesis 
 of complex switching systems. Quasi-minimal covers, produced by this 
 method, are either minimal or approximately minimal. However, when 
 we cannot state that the obtained solution is minimal, an estimate 
 of its maximal possible distance to the minimum is provided. 
 
 Applications of the interval coverings concepts to pattern 
 recognition and picture filtering are delineated. 
 
 1. INTRODUCTION 
 
 Some concepts and methods, initially developed for switch- 
 ing theory purposes, seem to have more universal application, if 
 properly generalized. The generalization described in this paper 
 stems from three observations: 
 
 (l) The coverings arising in the minimization of switching circuits 
 can be viewed as a limiting case of interval coverings, intro- 
 duced in the paper. 
 
 -2- 
 
(2) The so- called method of disjoint stars, which was originally 
 developed to provide a ra I or qu mal solution of 
 the covering problem in switching theory [l,3,H], and then 
 extended to provide the quasi-minimal solution of the general 
 covering problem [2], can be applied in particular to the 
 synthesis of the above-mentioned interval coverings. 
 
 (3) The concept of a covering can be extended in yet another 
 direction to accommodate a preferential ordering of the elements 
 to be covered. 
 
 A need for the generalization described in the paper first 
 appeared when we were considering some problems of pattern recognition 
 and signal detection theory. 
 
 2. NOTATION AND DEFINITION OF AN INTERVAL COMPLEX 
 
 Elements of a discrete finite vector space E will be referred 
 to as events eJ = (x-, , . . . ,x n ) . Components x- take their values from 
 the sets {0,1,2 ,... ,h^-l} , i = l,2,...,n. We will assume that the 
 index j is given by 
 
 n-1 i-1 
 
 J = x + *T{x . ("I h J . (1) 
 
 n f-z n-i ' ' n-k » 
 
 i=l k=0 
 
 thus, as can be verified, j uniquely determines a vector e . The 
 value j will be called the number of the event eJ. When an event e£E 
 is given, its number will be denoted by y(e). For example, if 
 e = (2,2,2,1), assuming that h = 5 , h = 3 > h = 3 and hi = 2, 
 
 Y (e) = 1 + 2(2) + 2 (2-3) + 2 (2-3'3) = 53. 
 
 E kj E and E a E (or E E ) will denote the set-theoretic 
 
 union and intersection of E.. and E respectively, where E and E_ are 
 
 sets of events. E. is the complement of an event set E. , defined as 
 
 1 1 ' 
 
 C \ E. , where \ is the set-theoretic subtraction. The cardinality of 
 a set L will be denoted by c(L). 
 
 0; b 
 
 Definition 1 By X., called a literal , we will denote the set 
 
 of all events (x,,...,x ) from the space E such that a. < x. < b. i.e. 
 1' ' n ^ 1—1—1' 
 
 a; X b .' = {(x 14 ...,x ) a. < x. < b.} (2) 
 
 1 J'n'i— 1— 1 
 
 u b _j 
 
 If a. > b. then 'X. is the empty set 0. 
 11 1 * J 
 
 Literals ^'xf^ a. e {0 ,1,2 , . . . ,h. -1} , denoted briefly by 
 
 X., will be called elementary literal s. 
 
 -3- 
 
bion 2. '-. t of events L which c n presented 
 
 as a product of literals, i.- . 
 
 L = Pi X ' , I 4 {1,... ,n} (3) 
 
 i€l a 
 
 will be called an interval i- ... ; 2 x (or simply interva l) . 
 
 We can easily see that interval complexes constitute 
 
 in the space E, n-dimentional intervals, i.e. sets of all vectors 
 
 which lie between some two arbitrary vectors, say, e 1 and e\ 
 
 e l , e 2 € E. Namely we would have e 1 = (x 1 ,x* , . . . ,x* ) and e 2 = 
 
 v 1* 2 n 
 
 (x ? x ? . . . ,x ? ) , where x! = a. and x? = b. for i € I, and x 1 . = 
 V ^1» A ^» ' n 11 11 l 
 
 and x ? = h.-l for i £ T. 
 
 11 T 
 
 A product E- of n elementary literals, i.e. E. = X^ X^ "**^ n n 
 
 is an interval which consists of only one event, namely 
 
 ,,a n a ), thus E. = {e}. 
 
 12 n J 
 
 I = (a 
 
 3. SET-THEORETIC OPERATIONS ON INTERVAL COMPLEXES 
 
 In this section we state a series of theorems which 
 summarize the basic rules of set-theoretic operations on intervals. 
 First, recall that union and intersection of sets are idempotent , 
 commutative, associative and distributive; and the absorption laws 
 hold. Next we have: 
 
 L \j = L 
 
 L A E = L 
 
 L v E = E 
 L r\ = 
 
 L \J L = E 
 L A l = 
 
 L = L 
 
 (identities) 
 
 (h) 
 
 (null elements) 
 
 (5) 
 
 (complements ) 
 
 (6) 
 
 (involution) 
 
 (7) 
 
 L l v V L l A L 2 
 
 L L = L UL (de Morgan's Laws) (8) 
 
 L ± h 2 VLjL = L^ \j KjL V L g L (consensus) (9) 
 
 -h- 
 
Let Ll = p •**. l 2 - n c,x i' 
 
 id. 
 
 kl, 
 
 where L,I <; {l,2,...,n}. We will assume that for 
 
 i e {1,2,... ,n} \1,, a. = and b. = h.-l, and for i C {l,2,...,n} 
 
 c. = and d. = h.-l. 
 l li 
 
 V 
 
 Theorem 1; 
 
 1. L, C L_. iff Vi € L, a. > c. and b. < d. 
 
 12 2 l — l l—i 
 
 2. If L £ L then L (L \j L) = L and Lg v L L = L g 
 
 Theorem 2: 
 
 l l 
 
 , min(o,c) max(b,d) . / v . /, , v . 
 X. , if max(a,c) <^ mm(b,dj + 1 
 
 °X? w c X d , otherwise. 
 i l 
 
 The next theorem is a generalization of theorem 2. 
 Let a= (a ,a ,...,a ), b = (b ,b ,...,b ), c = (c,,c 2 ...,c ) and 
 d = (d lS d 2 ,... ,d n ). 
 
 Theorem 3: If vector a is comparable with c (i.e. a. >_c or a <_ c) 
 and vector b is comparable with d , and max(a,c) < min(b,d) +1, 
 where 1 = (1,1,... ,1) , then: 
 
 L 1 UL 2 
 
 P| minfa.jCjh^rnaxCbj.d,) 
 
 idyi 
 
 1~2 
 
 If for every i € I, v I , min(a. ,c. ) = and max(b. ,d. ) = h.-l 
 
 1 2 1 i'i i'i i 
 
 then 
 
 l 2 v L 2 = II. 
 
 Theorem h, 
 
 Proof: 
 
 hfl 
 
 U (L *X°) = L 
 
 a=0 
 
 h.-l 
 
 U (L r\ X a ) = L r\ (X° w X'. v...wX V ') = LaE = L. Q.E.D. 
 a=0 
 
 -5- 
 
More generally: 
 Theorem S ; 
 
 (L/\ 0,i Xj")=L 
 
 iff Vat{0.1,...,h.-1}, |j such that a. . < a < b. . 
 
 l — ij. — — ij 
 
 Theorem 6: 
 
 4« l 2 = 
 
 max(aj ,Cj) min(bj,dj ) 
 
 iCl^Ig 
 
 If for some i, max(a.,c.) > min(b.,d.) then L A L = 0. 
 
 Theorem 7 ' 
 
 ai Y bi_ Y a r l bi+l Y h L -l 
 
 Aj - Aj v_/ Aj 
 
 Theorem 8: 
 
 i. rK" = 
 
 1 
 
 3€I 
 
 i*I 
 
 2. 
 
 X.' = 
 --> i 
 
 n**? 
 
 iel 
 
 id 
 
 ( a special case of (8) ) 
 
 From theorems 2, 6 and 7 we see that the set L of all 
 possible intervals in the space E (0 £ L) is closed under A but 
 not under \j and - . Since the operation A is associative, 
 commutative and idempotent, the system with the carrier set L, 
 binary operation r\ , and nullary operations E and 0. i.e. 
 
 < L, A , K, > 
 
 (10) 
 
 is a meet semi-lattice. 
 
 Let us find the cardinality of L. Every element of L 
 can be represented as a product of n literals. The number of 
 
 «i v b; 
 
 different literals 'X.' for a fixed ie{l,2 ,. . . ,n} is equal to 
 
 the number of possible pairs (a.,b.), a..b. {0 .1 , . . . ,h . -1} and 
 
 ii'i'i l 
 
 a. < b. , i.eJ i 
 
 h.\ h.(h.+l) 
 11 li 
 
 Thus the number of intervals 
 
 in L is 
 
 1 n 
 C < L ) = lTn 1 h.Ch.+l). 
 
 (11) 
 
 i=l 
 
 -6- 
 
1. 
 
 E ± £ E 2 __ 
 
 2. 
 
 E. C L (Ej 
 
 3. 
 
 L (L(E.)) = L (E) 
 
 Let L(E.), E- C ''■> denote the minimal interval 
 1 *■ / 
 under inclusion which contains Ej (i.e. the interval included in 
 
 any other interval with such property). The unary operation 
 
 Ej * L(K) 
 
 of 2' into itself has the following obvious properties: 
 
 L (E 1 ) £ L (Eg) (isotone) (12) 
 
 (extensive) (13) 
 
 ( i dempot ent ) ( 1 M 
 
 The above meens that the operation E — >- L (E) ... is a 
 
 E b. 
 
 closure operation on the poset < 2, £ >, i.e. the set 2 with 
 
 inclusion c; as a partial ordering operation. 
 
 Let us define an operation u on event sets E., called 
 the normalized union s as: 
 
 U E i = L ( U E i ) («> 
 
 i i 
 
 The set L is closed under the normalized union. Thus the system 
 <L, A S U ,E,0> is a lattice. Furthermore, it is a complete lattice 
 as the intersection and normalized union of intervals from any subset 
 of L also belong to L. 
 
 k. COVERS OF A MAPPING f 
 
 1 
 
 Assume that we are given two disjoint sets F and F of 
 events from the space E. These sets define a mapping 
 
 f: E —{1,0,*} (16) 
 
 where * denotes some unspecified value, and such that 
 F°= { e | f(e) = } and F ] = { e | f(e) = 1 }. 
 
 Definition 3. A set of intervals D(f) = {L.} is a cover of the 
 
 1 i=l 
 mapping f if : 
 
 F 1 ^ 
 
 L. C F 1 ^ F* (i T ) 
 
 i=l 
 
 * ... 
 where F = { e f(e) = * }. 
 
 Thus the cover D(f ) distingui she s the set F from F (we also 
 say D(f) is a cover of F against F°). A first covering problem 
 is hov to provide a cover D(f) with a minimum number of intervals. 
 More generally, specifying a 'cost' functional for sets of intervals, 
 
 -7- 
 
we can ask how a cover of r: ; I cost can be found. If 
 the space E is a space of binary vectors, then the Hatter 
 problem is parallel to the well known problem of finding the 
 minimal disjunctive normal form for an incompletely specified 
 switching function (where 'cost' is the number of literals, 
 in this case unprimed or primed variables). For the purpose 
 of the present paper, by the cost of a cover we will mean the 
 number of intervals in it. 
 
 Assume now that two given sets of events in E are 
 not disjoint, i.e. there exists a non-empty set of 'mixed' 
 events F* representing their intersection. 
 
 Formally, we will consider the extended mapping 
 
 f: E— {[1,0],*} (18) 
 
 and define: 
 
 F 1 = {e e E / f(e) = 1} (19) 
 
 F° = {e e E / f(e) = 0} (20) 
 
 *. 
 
 {e e E / < f(e) < 1} (21) 
 
 F* = {e e E / f(e) = *} = E \ ( F^ F°u F^ ) . (22) 
 
 Thus mixed events are here those events for which the mapping f 
 takes values properly between and 1. A mixed event e can be 
 interpreted as having a nonzero conditional probability, rep- a 
 resented by the value of f(e), of belonging to F\ Events of F^ 
 can be, on this basis, linearly ordered from those most F-*- like 
 to those least F -like (i.e. most F^-like). 
 
 Assuming some threshold X, where Xe[0,l], we define: 
 
 F 1X = {e 6 E / f(e) > X} 
 
 F° A = {e e E / f(e) < X} 
 
 Sets F and F are disjoint sets, so we can now determine a cover 
 F ' against F . To formalize the above idea we state 
 
 Definition U. D(f|x) = {L^} is a cover of f under X if 
 
 F 1X C \J L. g F 1X „ F* . (23) 
 
 i 
 If X = 1 then D (f|x) reduces to D(f), defined in (17). 
 
 It may turn out that two covers D(f|x ) and D(f|x ) 
 can have considerably different cost, although values X, and X« 
 are close. 
 
In order to prevent this we partition the set 
 F^=(e 1 > f(e) > } into some arbitrary number p of classes 
 of events F^ 1 .F^ 2 , . . . .F^ (see fig. l), defined as: 
 
 F* 1 = { e | 1 > f(e) > 1 - 1/p } 
 
 F* 2 = { e | 1 - 1/p 1 f(e) > 1 - 2/p } 
 
 p*P = 
 
 = { e 1 - (p-l)/p > f(e) > 
 
 The values of f for events in the same class can differ by not more 
 than ]/p. Events in the same class will be considered as equivalent. 
 
 Definition 5. 13 ( f* | X ,p) = {L.} is a free cover of f under ( A ,p) if 
 
 F 1XP C Ul. C F 1Ap w F* (2U) 
 
 where F 1Ap = F 1X \ (f^ ( X) ) 
 
 0(F^ (A) ) - a subset of F* j(X) 
 
 J-l 
 
 j(X) - a value j such that 1 - >_ X > 1 - j/p. 
 
 The concept of the free cover D(f|X,p) allows us to cover 
 only those events from the class F^ which can be covered with 
 minimal increment of cost over the cost of covering the set 
 F \ F^ J . To distinguish the covers D(f|x) from D(f|x,p), we 
 will call the former exact covers . 
 
 The next concept to be introduced is that of a cover whose 
 individual intervals cover events from F in an ordered manner. 
 
 Definition 6 . D(f|x,p) = <L ,L , . . . ,L > is called an ordered free 
 cover of f under (X,p) if 
 
 \ = { L l9 L 2 ,...,L k }, k = 1,2, ...,d 
 
 are free covers of f under 
 
 X = min f(e) , where E = L ^L w . . . V L 
 
 K 
 
 To distinguish the covers D(f), D(f|x) or D(f|x,p) from 
 the ordered covers D(f| X,jj) , we will call the former u nordered covers, 
 We denote D(f|x,l) by ~lHf|X) which, if X = 1, reduces to D(f). 
 
 -9- 
 
Let { E-. ,E , . . . ,E } be a family of event sets and E an 
 
 event set. We adopt the following notation: 
 
 t 
 
 1 — . 
 i=l 
 
 {E 1 ,E 2 ,...,E t } i; = U E. (25) 
 
 (j) = <|) (26) 
 
 E^= W /F" = L E (27) 
 
 where L„ = {L ,L ,... lis the set of all intervals which are 
 E 1 2 x) 
 
 maximal under inclusion ' with regard to condition L C E, 
 k = 1,2,... (briefly, maximal intervals included in E). 
 
 Thus: TY ={ max L, L, € E} (28) 
 
 K K. 
 
 It can easily been seen that: 
 
 1^, = U^,!^,... } V = E (29) 
 
 and 
 
 r^. L E (30) 
 
 J ia 
 According to the introduced notation, the set V F 
 
 is a (usually redundant) cover of F against F - if F =0, 
 
 or, if F* 4 - a cover F L against E \ F ' = F \j F . 
 
 efinition 7 » L is called a maximal interval in f under X if 
 it is uiaxiinal under inclusion with reeard to the condition: 
 
 L C F 1A ^ F* 
 
 Maximal intervals in f under A are denoted by 
 L , k = 1,2,...,. It can easily be seen that if E is a space 
 
 (31) 
 
 l) A set S satisfying condition p is maximal ( minimal ) under 
 inclusion with regard to p, denoted by S = max S/p (S = min S/p) , 
 if there does not exist a superset (subset) of it also satisfying 
 condition p. In general, there can be many maximal (minimal) 
 under inclusion sets satisfying certain condition p. Family of 
 such sets, i.e.{ S./S. = max S./p } is denoted briefly by { max S . /p } 
 
 -10- 
 
of binary vectors and the set of 'mixed' events F^ is empty, 
 then the maximal intervals k , (i.e. the maximal intervals in 
 f under X = l) correspond to prime implicants of an incompletely 
 specified switching function f: E — -{1,0,*}. 
 
 'efinition 8. An exact cover D(f|X) or a free cover D(f|x,p) 
 is called an irredundant cover if it consists of maximal intervals 
 
 L. and if it is minimal under inclusion. 
 k 
 
 Definition 8 implies that deleting any interval from an 
 irredundant cover D(f|x) or D(f|X,p) will cause them to no longer 
 be covers. An irredundant cover D(f|x) or D(f|X,p) may be obtained 
 from the set 
 
 a/ 
 
 ¥ U F* 
 
 by removing from it a maximal under inclusion subset of intervals 
 
 such that the union of intervals in the remainder still covers set 
 
 1 X l Xc 
 F or F , respectively. 
 
 In general there can be very many different irredundant 
 covers D(f|X) or D(f|X,p). It is easy to see that if E is the 
 space of binary vectors and F*= 0, an irredundant cover D(f|l) 
 corresponds to an irredundant disjunctive normal expression of a 
 switching function f: E — -{1,0,*}. 
 
 Definition 9 . 
 
 The minimal exact cover M(f|X), minimal free cover M(f|X,p) and 
 minimal free ordered cover M(f[X,p) is a cover D(f | X) , D(f|x,p) and 
 D(f | X,p) - respectively, which has a minimum number of intervals. 
 
 r >. SYNTHESIS OF QUASI-MINIMAL COVERS 
 
 In this section we will briefly describe an application 
 of the disjoint stars method [1,2,3,1+] to the synthesis of the 
 quasi-minimal covers. 
 
 5.1 rhe extension operation \r 
 
 Definition 10. An extension operation \j- on event set E relative 
 to event set E is defined as: 
 
 E l^~ E 2 = '^ (32) 
 
 where A = {1^* ^E^ | 1^ r\ E ± 4 0) . 
 According to (26), if A = then E \T E = 0. 
 
 -11- 
 
\J/ 
 
 Since the union of intervals from any subset of V E 
 
 is also included in E we can state the following: 
 Theorem 9: 
 
 (1) E^y E c E 2 , if E ][ n E 2 4 
 
 (2) E v~ E = 0, otherwise 
 
 If E is an interval then we have the stronger 
 Theorem 10 : 
 
 { 9i otherwise 
 
 where E is an event set and L an interval. 
 
 The extension operation satisfies the following asymetrical 
 distributive properties: 
 
 Theorem 11; 
 
 (a) E ^ (O E i ) = [ |(Evr E. ) 
 
 i i 
 
 (b) (U B.)vr E = U(E.vr E) 
 
 i . i 
 
 i i 
 
 Proof : 
 
 Part (a). If E. are intervals then assertion follows 
 immediately from the fact that any product of intervals is also 
 an interval and from theorem 10. 
 
 We denote E vr (I, IE.) = A^ (i) 
 
 and 
 
 O (EVTB.) = fV. 
 
 1 _• i 
 
 (ii) 
 
 According to definition 10 and (27): 
 
 A ■ (L k £ V/°\ E. | 1^ f\ E y 0} = 
 i 
 
 = { max L ! L H E ^ and L, $ [ | E.} (iii 
 
 K K K . 1 
 
 1 
 
 -12- 
 
A . = { L, € / E. | L, A E^ } = 
 i k 1 ' k 
 
 { max L, L, f\ E j and L c E. } (iv) 
 
 k ' k k i 
 
 Consider (i) and (iii) : L £ H E. implies that 
 
 L ^ E. for every E.. So if L € A then also L 6 A . , for every E. . 
 a i l a / -. a i ' * i 
 
 Thus A C f~^\ k ■ > which implies that A Si I A w . 
 i i 
 
 Consider now (ii) and (iv). Set f \ A. can be unique ly 
 
 represented by set of intervals / /^A u . Let L £ V ( ] A 
 
 i 1 a i — 
 
 L a E 4 0. From (28), L € { max L, L. £ ( l A u . } and then 
 
 3, 8. K. K. • 1 
 
 L $ A . for every A . . From that and (iv) we have L S E. for 
 
 a i l >~ a i 
 
 every E.. It implies that L £ / ] E., and finally, according to (iii), 
 
 L £ A 
 a 
 
 Therefore 
 
 /f\ A . £ A , what implies I ) A . £ A . 
 i .i 
 
 l i 
 
 Proof of part (b) is similar. Q.E.D, 
 
 Theorem 12; 
 
 E v/- E = U ^ ' ( (e . } vr {~} ) 
 
 e.€ E. e.<= E J X 
 
 j 1 l 2 
 
 where the order of the union and intersection is irrelevant. 
 
 Proof: 
 
 E- = VJ {e.} and E = [J {e } = 
 
 e^E J e € E 
 
 j 1 l 2 
 
 = ( ]_ {e~}. Thus: 
 e.€E 2 
 
 E vrE = U {e } u- ' J {?.} 
 
 e.€E 1 e.£E_ 
 
 -13- 
 
Apply now the distribution rules of the theorem 11 
 (in any order) to derive the desired result. Q.E.D. 
 
 The theorem 12 gives a theoretical rule to compute the 
 set L of all maximal intervals in f under X. Namely: 
 
 A 
 
 L x = 7f : Vf W (33) 
 
 L^= F 1A u-F^ =U 1A O 0A (e.V{r>) (3U) 
 X e 4 € F 1X e.€ F° A J X 
 
 Find the irredundant expression of L as a union of 
 
 A 
 
 intervals. Set L is the set of intervals in this expression'. 
 
 A 
 
 A more detailed description of the above procedure - in the case 
 when F consists of the only one element - is given in the next 
 section. 
 
 5.2 Definition of a star (G(e|x) and an algorithm G for its 
 generation ., 
 
 A. fundamental concept in our approach to the synthesis 
 of interval covers is of an interval star G(e|x). 
 
 Definition 11 . The interval star G(e|x) under X of an event 
 e€F isasetof all maximal intervals under X which cover 
 the event e, i.e. 
 
 G(e|x) = {L A | e € l£} (35) 
 
 We denote G(e|l) by G(e). G (e|x) is - according 
 to (25) - the union of intervals in G(e|X). An interval star 
 can be expressed as: 
 
 G(elx) = 7 {e}KT F° X (36) 
 
 -lU- 
 
Theorem 13: 
 
 < U (e\X) = f""] ({e} v-{e ± }) 
 
 'OX 
 
 e.€F 
 
 Proof: G u (e|X) = {e} kj~ T vn = {e} v I I {~} 
 
 e.£F 
 
 1 
 
 Applying now theorem 9 part (a) we complete the proof. Q.E.D 
 
 The following algorithm for the generation of a star 
 G(e|A) follows from theorem 13 (algorithm G). 
 
 Given e e F and F = {e.} 
 
 i. , 
 
 .-.. orithm G : - 1 L 
 
 1. Determine for i = l,2,...,z 
 
 {e. } and then form D. = {e} ir {e. } 
 i .11 
 
 z 
 
 2. Set up the function \ \ D. and find its irredundant 
 
 i=l X 
 expression by multiplying each term D. by all the others 
 and applying the absorption laws. 
 
 3. G(e|A) is the set of terms of the expression thus 
 obtained. 
 
 5 . 3 S ynthesis of a cover M (f /a) by algorithm A ^ 
 
 The problem of synthesis of a minimal cover M(f ( \) is a 
 particular case of the general covering problem described in [2], As 
 indicated in this paper even in case of a relatively simple covering 
 problem the number of operations required for its exact solution may 
 not be feasible even with the fastest computers. Consequently, the 
 most desirable are good methods for an approximate solution whic?i 
 allow to drastically reduce the number of operations but also give 
 some measure of distance to the minimum. 
 
 The previously mentioned method of disjoined stars, when 
 realized by locally optimal decisions, generates — with a relatively 
 small number of operations and memory requirements — a so-called quasi- 
 minima] cover, which is either minimal or approximately minimal. 
 Furthermore, when we cannot state that obtained solution is minimal, 
 it provides an estimate of the maximal possible distance between the 
 obtained solution and the minimal one. 
 
 The fundamental theorem from which this method stems 
 (expressed in our terms) is following. 
 
 Let us assume that we are given (by realizing some algorithm) 
 a family of stars G = {G.(e|x)}, ee F , such that any two stars 
 chosen from it are disjoint sets (we say G r is a family of disjoint 
 stars) . 
 
 -15- 
 
Theorem 1*+ : The number of intervals in the minimal cover M(f|x) 
 satisfies the relation: 
 
 c (M(f|x)) > c(G r ) 
 
 The theorem implies that if ve have a cover D ( f | A ) and 
 know the number § = e(G r ), then the difference A = (D(f|x)) - § 
 caii be viewed as an estimate of the difference between the number 
 of elements in this cover, and in a minimal one If we can next 
 find another family of disjoint stars with a greater number of elements, 
 say o , then we can improve our estimate, namely: 
 
 A JL = (D(f|x)) - § 1 < A 
 
 If on the other hand we can find another cover with a 
 smaller number of intervals, then obviously our estimate will also 
 improve, and it may turn out then A will become 0. This will mean 
 that we found the minimal solution. 
 
 A possible algorithm for accomplishing the above ideas 
 with a view toward a solution of the generally stated covering problem , 
 was described in paper [2] (algorithm A ). In the formulation given 
 there it was assumed that a cover consists of some sets (not con- 
 cretely specified), called complexes. In our case the complexes are 
 specified as intervals (interval complexes). Thus in order to apply the 
 algorithm A for our purpose we only need to make use of the algorithm 
 for generating a star G(f|X), described in section [5.2], 
 
 The flow diagram of algorithm A in the form adapted to 
 our notation is shown in fig. 2. 
 
 The sign := denotes assignment statement (as in Algol 60 ). 
 F is an auxiliary variable. 0(fP; e-j_) denotes the operation of 
 choosing the event with the smallest number from the set specified 
 by the current value of F? and assigning the notation e-, to that event. 
 L denotes an interval inG(e-jlx), called a quasi-extremal , which 
 covers the maximum number of events in the set constituting the current 
 value of variable F^ . It can be noticed that this is a locally 
 optimal decision about the choice of an interval from the given star 
 (generally not unique). The last value of MQ- constitutes our solution, 
 the quasi-minimal cover M^(f|x). Value A is an estimate of the max- 
 imal possible difference between the cover M ( l(f|x) and a minimal 
 cover M(f|X), expressed in number intervals, i.e. 
 
 c(M q (f|x)) - c(M(f|X)) < A . (27) 
 
 If after the first execution of the algorithm, A is con- 
 sidered to be too large, the better estimate (and/or solution) may 
 be obtained realizing next iterations, e.g. in the way described in 
 [3]. 
 
 To synthesize a quasi-minimal free cover M (f|X,p) using 
 this algorithm, we substitute in the flow diagram (fig. 2) the set 
 
 F ' by F , which means that we have to cover all elements from 
 
 F \F^ and some elements (at least one) from the set F f\ F^ . 
 
 -16- 
 
A geometrical interpretation of interval covers using 
 a generalized logical diagram and examples of interval covers 
 synthesis are given in [5]. An extension of the algorithm for 
 the synthesis of quasi-minimal ordered covers and application of the 
 concepts described in the paper to pattern recognition are given in [6]. 
 
 6. REMARKS ON APPLICATIONS 
 
 A cover of a mapping, f consisting of malt i dimensional 
 intervals can be interpreted as a set of 'filters' for recognizing 
 events from a signal class (represented by F 1 *) for pattern recog- 
 nition and picture processing purposes, e.g. for discriminating 
 regions of different textures, striping of background, local feature 
 extraction, border detection, etc. 
 
 In case of an ordered cc . or the individual filters 
 correspond to the consecutive points on the optimum receiver- 
 operating-characteristic (ROC), defined as in statistical decision 
 theory. The above ROC curve can be obtained by optimal ordering 
 events from the two classes to be distinguished, with regard to 
 the likelihood ratio of their frequency occurence [6]. 
 
 7 . REFERENCES 
 
 1. R. S. Michalski, Synthesis of Minimal Forms and Recognition of 
 Symmetry of Switching Functions, PhD Thesis, The Thechnical 
 University of Silesia 1969 (extended version in Proceedings of 
 the Institute of Automatic Control, Polish Academy of Sciences, 
 No 91, Warsaw, 1971). (in Polish) 
 
 2. R. S. Michalski, On the Quasi-Minimal Solution of the General 
 Covering Problem, Proceedings of the 5th International Symposium 
 on Information Processing (FCIP 69), Vol. A3, pp. 125-128, 
 Yugoslavia, Bled, October 8-11, 1969. 
 
 3. R. S. Michalski, Automatic Synthesis of the Quasi-Minimal Multiple- 
 Output Switching Circuits, Proceedings of the 6th Yugoslav 
 International Symposium on Information Processing (FCIP 70) ,Vol.Dl-2 , 
 Bled, September 23-26, 1970. 
 
 h. R. S. Michalski, Z. Kulpa, A System of Programs for the Synthesis 
 of Combinational Switching Circuits Using the Method of Disjoint 
 Stars, a paper accepted for presentation at the IFIP Congress 71, 
 August 23-28, 1971, Ljubljana, Yugoslavia. 
 
 5. R. S. Michalski, A Geometrical Model for the Synthesis of 
 Interval Covers, to appear as a Report of the Department of 
 Computer Science, University of Illinois, Urbana, Illinois. 
 
 6. B. H. McCormick, R. S. Michalski, Synthesis of Interval Covers 
 for Pattern Recognition and Picture Filtering, to appear. 
 
 -17- 
 
f(e)A 
 
 1 
 x- 
 
 
 
 
 
 
 
 • » • • 
 
 ^**1 — — — — 1 
 
 F 1 
 
 pOl 
 
 p02| p03 
 1 
 
 • • • • 
 
 pop 
 
 J 
 
 F* 
 
 V 
 
 Pix 
 
 
 
 pOX 
 
 
 
 
 E 
 
 Fig. 1 
 
Input data 
 <f,X> 
 
 m 
 
 ^ 1X L 0X 
 
 ermine F , ? 
 
 X_ 
 
 ' : = F 1X , M q := ; A : = 
 
 1 
 
 0(K P ;eJ 
 
 I 
 
 Part T 
 
 Generate G(e X) 
 
 Part II 
 
 Generate G(e /A) 
 
 M q (f/X) := M q 
 
 L 
 
 Determine L 
 
 B 
 
 M q :■ 
 
 M q v (L q ), F U := F 1X 
 
 F := F \J L 1 , A := A + 1 
 
 Fig. 2 
 
[aEC-427 U.S. ATOMIC ENERGY COMMISSION 
 
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