The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books 
 are reasons for disciplinary action and may 
 result in dismissal from the University. 
 
 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
/Report No. U6l 
 
 r 
 
 coo-2118-0013 
 
 June 2U, 1971 
 
 A GEOMETRICAL MODEL FOR THE SYNTHESIS 
 OF INTERVAL COVERS 
 
 R. S. Michalski 
 
 JHE LIBRARY QBIHB 
 
 NOV 9 1972 
 
 .UNIVERSITY OF ILLINOIS 
 AT URBAWA-CHAMPAIGN 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMP 
 
 URBANA, ILLINOIS 
 
The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books 
 are reasons for disciplinary action and may 
 result in dismissal from the University 
 
 DIVERSITY OF IUINQ IS LIBRARY AT URBANA-CHAMPAIGN 
 
 L161 — O-1096 
 
p. 2- 
 
 coo-2118-0013 
 
 Report No. k6l 
 
 A GEOMETRICAL MODEL FOR THE SYNTHESIS 
 OF INTERVAL COVERS 
 
 by 
 R. S. Michalski 
 
 June 2k, 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. 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/geometricalmodel461mich 
 
ACKNOWLEDGMENT 
 
 The author would like to acknowledge his gratitude to 
 Professor B. H. McCormick for the valuable discussions pertaining 
 to this paper, his reading the manuscript and comments toward its 
 improvement . 
 
 The author wishes also to thank Mrs. Roberta Andre' for 
 the accurate typing and Mr. Stanley Zundo for the excellent drawings. 
 
A GEOMETRICAL MODEL FOR THE SYNTHESIS 
 OF INTERVAL COVERS 
 
 R. S. Michalski 
 Department of Computer Science 
 University of Illinois 
 
 ABSTRACT 
 
 The paper presents a planar geometrical model of a discrete 
 finite vector space E, called a generalized logical diagram (GLD), 
 and then uses the GLD to interpret and illustrate different concepts 
 and algorithms described in [l] (the concept of an interval complex, 
 of a mapping f: E — - { [0 9 l] ,*}, of an interval cover of f, of the 
 
 extension operation \j~ , algorithm G for star generation and algorithm 
 
 »Q \ 
 
 A for interval cover synthesis). 
 
 In simple cases , the GLD can also be used for the direct 
 
 (graphical) synthesis of interval covers using a simple rule for the 
 
 recognition of interval complexes, given in the paper. 
 
1. INTRODUCTION 
 
 The interval generalization of switching theory described 
 in [l] discussed a Boolean algebra of event sets in a discrete finite 
 vector space E and mappings f from E into {[0,1],*}, where * 
 represents an -unspecified value. The Boolean algebra and Boolean 
 functions considered in switching theory are a special case of the 
 above, in which E is a space of binary vectors and f maps E into the 
 endpoints of the interval [0,l] and *. The basic problem investigated 
 in [l] was how to find a minimal collection of multidimensional intervals, 
 whose union covers a set F , defined as {e€E|f(e) >_ A}, where X is an 
 assumed threshold value (0 _< A _< l), and does not cover any element of the 
 set F ={e<sE|f(e) < A}. A set satisfying the above conditions was 
 called a minimal exact unordered interval cover of the set F against F 
 (the names 'exact , unordered 1 distinguish this cover from free and ordered 
 covers, also introduced in [l]). 
 
 The purpose of the present paper is to present a planar 
 geometrical model of the space E which can serve as a useful visual aid 
 for representing interval covers of mappings f and geometrically interpreting 
 algorithms for their synthesis. In simple cases this model can also be used 
 for the direct graphical synthesis of interval covers. 
 
 To permit one to understand this paper without prior, though 
 desirable, knowledge of [l], those concepts from [l] which are relevant to 
 this paper have been included (in most cases in their equivalent geometrical 
 form) . 
 
2. THE FINITE DISCRETE VECTOR SPACE E 
 
 Assume that some objects or processes are described by a 
 set of n parameters (discrete or continuous) and each parameter is 
 mapped into a discrete variable x. , x. £ { 0, 1, 2, . .., h.-l }, 
 i = 1, 2, . .., n. Values h. represent the number of discrete units 
 into which the range of variability of each parameter is resolved to 
 achieve desired accuracy. 
 
 Variables x.. , x_, . .. s x are grouped together into vectors 
 e = (x.. , x , ...» x ), called events. The finite vector space con- 
 sisting of all possible events e is denoted by E(h , h , ..., h ) or 
 briefly by E: 
 
 H-l 
 E= { (x l9 x 2 , ..., x n ) } = { e J } . = Q (1) 
 
 where H = h h •••]! and j is the value of a function 
 1 2 n 
 
 1_ , J+l 
 
 y: E — ^ { 0, 1, 2, ..., H-l }: 
 
 J = Y (e) =x n+ ^ x Pl\ (2) 
 
 j=n-l k=n 
 
 y(e) is called the number of an event e. For example, if in the space 
 
 E(5, 6, k, 3) an event e = (3, U 9 1, 2) then y(e) = 2 + 1*3 + h- k- 3 + 3*6«H«3 
 
 269 and the event e is denoted by e . 
 
 3. A MAPPING f: E — »- { [0,l], * } AND A COVER OF THE MAPPING f 
 
 Let f be a mapping from the space E into { [0,l], *}, where * 
 denotes an unspecified value: 
 
 f: E -*{ [0,1], * } (3) 
 
The mapping f can "be interpreted, for example, in the following way. 
 Assume we are given two classes of objects, class 1 and 0. By measuring 
 some assumed parameters and specifying values for x.. , x_, ..., x , each 
 object can be characterized by an event e $ E. Assume that samples are 
 taken from class 1 and and the sets E and E are formed from distinguished 
 events characterizing the objects of class 1 and 0, respectively. The sets 
 E and E , in general, may not be disjoint. Thus, if an event e € E r\ E 
 is given, we cannot infer from this alone to which class the object 
 characterized by e belongs. 
 
 Assume that in the above case we are able to estimate the pro- 
 bability f(e) that the object belongs to class 1. Thus, the example can 
 be formally described by a mapping f defined by (3), if: 
 
 F 1 = { e € E | f(e) = 1 } = E 1 \ E° (li) 
 
 F^ = { e € E | < f(e) < 1 } = E 1 r\ E° (5) 
 
 F° =' { e €E I f(e) = } = E° \ E 1 (6) 
 
 F* = { e € E | f(e) = * } = E \ (E 1 \J E° ) (7) 
 
 Events from the set F correspond to no objects in the sample 
 sets; they are called unspecified events . Events from the set F^ are 
 called mixed events. By assuming some threshold A, _< A _< 1, we can 
 refer the objects characterized by mixed events to class 1 if f(e) >_ X , 
 and to class 0, otherwise. We define: 
 
 F 1A = { e I f(e) > X } (8) 
 
 F° X = {el f(e)< A } (9) 
 
An important concept defined in [l] is that of a literal 'XV. 
 A literal ° i X. i is the set of all events e J = (x, , x , . .., x ) <~ E, whose 
 x.th component takes values "between a. and b.: 
 
 V j = { (x l9 x 2 , ..., x n ) I a. _< x ± _<b. } (10) 
 
 When a. = "b. the literal is denoted briefly by X.' and called an 
 11 J 1 
 
 elementary literal . A set-theoretic product of literals 
 
 L = f ] 'X. 1 , I <s{ 1, 2, . . . ,. n } is called an interval . It constitutes 
 
 i€I X 
 in the space E an n-dimensional interval, i.e. a set of events lying 
 
 between two arbitrary vectors. For example, L is a set of events lying 
 
 between vectors a = (a n , a.., ..., a ) and b = (b n , b^ , ..., b ) where » 
 
 1' 2 ' n 1' 2 n 
 
 for i <£ I, a. = and b. = h.-l: 
 r l li 
 
 L={e€EE!a<e<b}={ 
 
 (x ,x ,...,x ) | V i?a. _<x. _<b. } (ll) 
 
 The basic concept of the paper [l] is that of an interval cover 
 of a mapping f. A set D(f|A) = {L. } of intervals is called an unordered 
 
 J 
 
 exact interval cover of f under A (or an interval cover of F against F ) 
 
 if the set-theoretic union of intervals L. covers every element of F 
 
 J 
 
 and does not cover any element of F : 
 
 F 1A C U L.C F 1X W F * (12) 
 
 J 
 
 An interval cover D(f|x) which has the minimum number of intervals is 
 called a minimal cover of f (there can be many equivalent minimal covers). 
 Free and ordered interval covers of a mapping f were also defined in [l]. 
 The present paper, however, will only consider unordered exact interval 
 covers. 
 
 In the case when a cost functional is specified for intervals, a 
 minimal cover means a cover of minimal cost. 
 
Sets F and F define a mapping f : 
 
 f A : E— *{0,1,*} (13) 
 
 where: {e|f (e) = 1,0,*} = F ,F ,F* respectively. 
 
 If F* = (j), the mapping F can he expressed in a notation similar 
 to that used to represent a Boolean function in switching theory: 
 
 f A = V l L -l ; !i 
 
 L.€D(f) J 
 J 
 
 where: 
 
 I L.l denotes the characteristic function of the interval L. in E, 
 i.e. 
 
 1, if e€ L. 
 
 ],,|(r-) - <| J 
 
 0, otherwise 
 D(f) is an interval cover of F against F , 
 
 denotes disjunction (logical sum). 
 
 Expression {lk) parallels the disjunctive normal form (sum-of- 
 products) of a Boolean function. It will therefore be called the interval 
 disjunctive normal form of f (alternatively, of f under A). 
 
 Similarly as in switching theory we can express f in another 
 form, corresponding to the conjunctive normal form of a Boolean function: 
 
 f X = I \ S(L.) 
 
 L.€ K(f) J 
 
 J 
 
 where S(L.) is the disjunction of the characteristic functions of 
 
 J 
 
 literals obtained by applying de Morgan's laws to the complement 
 
 of the interval L... For example, if L. = f| °>X b J , I ^ {l,2,...,n}, 
 
 then, S(L.) = |L.| = ll J °^< I = \/ {\°xV Iv^'x^' |) 
 J J is- I iel 
 
) 
 
 1 
 
 3 
 
 '2 
 
 i 
 
 
 h|= 
 
 4, h 2 = 
 
 4, 
 
 h 
 
 3 = 4 
 
 
 
 r. 
 
 •\ 
 
 
 y 
 
 /i 
 
 ^ 
 
 /" 
 
 /I 
 
 1 
 
 ) 
 
 n 
 
 ^2 
 
 
 ,J* 
 
 %T 
 
 /I 
 
 *Z 
 
 
 
 s 
 
 
 
 
 ''o.e 
 
 
 <?/ 
 
 
 
 r^ 
 
 ^0.9 
 
 
 
 
 
 
 
 y 
 
 A 
 
 > 
 
 
 
 
 
 f 
 
 
 
 « 
 
 1 
 
 
 
 
 ) 
 
 
 
 
 
 
 4 
 
 ^ 
 
 p 
 
 
 (. 
 
 < 
 
 
 
 
 
 
 
 
 
 
 A 
 
 p 
 
 
 
 f* 
 
 
 4 
 
 
 
 
 
 
 
 /^ 
 
 z 
 
 
 
 r 
 
 / 
 
 
 ^ 
 
 9 
 
 / 
 
 i 
 
 
 
 
 
 <s 
 
 V 
 
 
 r A 
 
 
 
 
 
 
 j 
 
 2 
 
 
 
 
 2 
 
 ^ 
 
 
 i 
 
 / 
 
 
 
 V 
 
 
 (\ 
 
 ^ 
 
 1 
 
 
 g 
 
 2 
 
 
 
 
 s 
 
 
 c 
 
 
 
 
 z 
 
 ''o.e 
 
 
 
 A 
 
 F 
 
 
 /I 
 
 *r 
 
 
 V 
 
 2 
 
 i 
 
 
 
 
 
 
 
 j/_ 
 
 
 
 
 2_ 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 v> 
 
 
 
 
 
 
 5, 
 
 
 ^ r, 
 
 
 
 
 f(e)=l 
 
 — 0<f(e)<l 
 
 O -f(e) = 
 
 • - f(e)=* 
 
 Fig. 1. Space E(U,4,U) and a mapping f: E(U,U,U) — >{[0, l],*} 
 
(for a detailed explanation of the rules of the transformation 
 
 see [l] ). 
 
 K(f) is a cover of F against F 
 The above expression, called a interval conjunctive normal expression 
 of f is formed by applying de Morgan's laws to the equation F = F 
 
 h. GENERALIZED LOGICAL DIAGRAM (GLD) AND A FUNCTION IMAGE T(f) 
 
 A discrete-euclidean geometrical representation of the space E 
 would be in the form of an n-dimensional 'grid' , spanned from the h ,h ,••• ,h 
 points on axes X-^x^. . . ,x n , respectively. A mapping f could then be 
 represented by assigning values f(e J ) to the nodes corresponding to vectors 
 e j (fig. l). 
 
 The above geometrical model of the space E is, however, not 
 easily visualized when n > 3. We therefore introduce another representation, 
 a planar one, which may be extended to any value of n. 
 
 Let us divide an arbitrary rectangle into h h •••h rows and 
 h v+l h v+2*" h n columns > where v is the maximal value for which h h • • «h _<H/2, 
 and assign vectors (x^,. . . ,x y ) , x. € { 0,1,. . . ,h . }, i _< v, to the rows, 
 
 and vectors U v+ l' X v+2" * * 'V > X i € { °>l,...,h. ) , v < i _< n to the columns, 
 according to the rules: 
 
 1. In the first step the rectangle is divided by horizontal lines 
 into h 1 rows which, in order from top to bottom, are assigned values 
 0, 1, ..., h.j-1, respectively (values of the component x of vectors e <= E). 
 In step i each row generated in step i-1 is divided into h. rows. The rows, 
 in order from top to bottom, within each row generated in step i-1, are 
 assigned values 0, 1, ..., h ± -l respectively (the values of the component x.). 
 
In toto, v steps are executed. 
 
 2. Steps v+1, v+2, . .., n are executed analogously to steps 
 1, 2, . .., v, "but now the rectangle is divided by vertical lines into 
 co-Lumns and the columns are assigned values in order from left to right. 
 
 The lines which divide the rectangle in step i, i€ {1,2,..., n} 
 are called axes of the component x. ( axes of different components are 
 graphically distinguished "by different thicknesses of. the lines ) . 
 
 The diagram so obtained, is called a generalized logical diagram 
 (GLD). Fig. 2. illustrates the GLD for n = k. 
 
 A unique vector (x ,x_,...,x ) corresponds to each row in the 
 GLD and a unique vector (x , ,x ,...,x ) corresponds to each column. 
 The intersection of any row with any column is called a cell of the 
 diagram. To be precise, we will assume that the cells do not include 
 points belonging to any axis nor to the perimeter of the rectange. The 
 obtained diagram comprises H = h h •••h cells (number of events in E). 
 Each cell of the diagram represents a unique event e of E, determined by 
 concatenating vectors (x^Xg,. . . .x^) and (x y+1 ,x ,. . . ,x ) which correspond 
 to row and column respectively, such that their intersection is the given 
 cell. 
 
 Thus the GLD is a geometrical model of the space E. 
 
 Cells of the GLD will also be denoted by e J and it will be clear 
 from the context if e J denotes an event or a cell. If e^ denotes a cell e, 
 then the index j = Y (e) will be called the number of the cell e. 
 
 It can easily be verified that the numbers of the cells are dis- 
 tributed in a GLD, in lexicographical order, i.e. from left to right and 
 from top to bottom, as is shown for E(i|,3,4,2) in fig. 3. If E is the 
 
X J X 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 o . 
 
 • 
 • 
 
 i 
 
 i 
 i 
 i 
 
 
 
 i 
 i 
 i 
 1 
 
 
 
 i 
 I 
 i 
 
 i 
 
 i 
 
 ! 
 
 1 
 
 i 
 
 1 
 1 
 
 1 
 1 
 
 h 2 -l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 1 • 
 
 • 
 
 ; 
 i 
 i 
 
 
 
 I 
 I 
 
 
 
 i 
 i 
 
 
 i 
 
 ! 
 
 1 
 
 ! 
 
 • 
 
 i 
 
 
 
 i 
 
 
 
 l 
 
 
 
 
 i 
 
 I 
 
 h,-l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 | 
 
 
 
 I 
 
 
 
 l | 
 1 1 
 
 ! 
 i 
 
 
 
 1 
 1 
 
 1 
 1 
 
 
 
 
 1 1 
 
 1 1 
 
 1 1 
 
 1 1 
 1 
 
 * 
 
 I 1 1 
 
 II 1 
 
 I i i 1 
 
 II II 
 
 : 
 i i i 
 i i i 
 i i i 
 ii i 
 
 1 
 
 1 
 
 1 
 i 
 
 1 
 1 
 1 
 
 l 
 1 
 
 ! 
 
 ! 
 
 1 
 
 : 
 
 ; ! 
 
 1 1 
 1 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ : 
 
 • 
 
 ! 
 
 i 
 i 
 i 
 
 
 
 i 
 
 : 
 
 i 
 
 
 
 i 
 i 
 I 
 
 i 
 
 I 
 1 
 
 1 
 1 
 
 1 
 1 
 1 
 1 
 
 1 
 
 1 
 1 
 
 h 2 -l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 l 
 
 • • » h 4 -l 
 
 
 
 i 
 
 • • • h 4 -l 
 
 
 
 1 
 
 • • • h 4 ~l 
 
 
 
 
 i 
 
 • • • h 4 -l 
 
 
 
 1 
 
 2 
 
 
 h 3 -l 
 
 i 
 
 I 
 
 n ig. 
 
 2. G 
 
 ene 
 
 •ral 
 
 ize 
 
 d Log 
 
 ica 
 
 1 D 
 
 iag 
 
 ram r 
 
 epr 
 
 esenting E(h ,] 
 
 V 1 
 
 .3.1 
 
 V 
 
 
 x 4 
 
 X, 
 
10 
 
 hi = 4 , h 2 ^3, h 3 = 4, h 4 = 2 
 
 Xj x 2 
 
 
 
 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 1 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 2 
 
 16 
 
 17 
 
 18 
 
 19 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 
 
 
 1 1 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 1 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 76 
 
 77 
 
 78 
 
 79 
 
 3 1 
 
 80 
 
 81 
 
 82 
 
 83 
 
 84 
 
 85 
 
 86 
 
 87 
 
 2 
 
 88 
 
 89 
 
 90 
 
 91 
 
 92 
 
 93 
 
 94 
 
 95 
 
 
 
 
 c 
 
 1 
 
 ) 
 
 
 
 1 
 
 l 
 
 L 
 
 
 
 1 
 > 
 
 
 
 l 
 5 
 
 x 4 
 x 3 
 
 Fig. 3. Distribution of cell numbers in the GLD for £(^,3,^,2) 
 
11 
 
 E(3,4,2,2,4) 
 
 X l X 2 
 
 
 
 ^- < 
 
 \ s 
 
 ^ 
 
 N> 
 
 N 
 
 \ 
 
 s 
 
 \; 
 
 s> 
 
 
 8: 
 
 k\ 
 
 \\\\\ s 
 
 
 x? 
 
 1 
 
 \ 
 
 
 A\ 
 
 \ V 
 
 s \ 
 
 N 
 
 \ 
 
 i 
 
 A 
 
 X 
 
 \ 
 
 \ 
 
 X 
 
 \\ 
 
 
 ^ X 
 
 
 U 
 2 
 
 
 
 
 
 'yv. 
 
 
 
 
 
 fX/j 
 
 yLS. 
 
 
 
 
 
 
 
 3 
 
 X 
 
 v 
 
 N 
 
 
 < 
 
 
 V 
 
 \ 
 
 
 \ 
 
 V 
 
 
 S5 
 
 V s ! 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S A 
 
 1 
 2 
 
 
 Y/y 
 
 /// 
 
 v/y 
 
 
 y/A 
 
 vY, 
 
 VA 
 
 y//< 
 
 77/ 
 
 y/A 
 ///< 
 
 i 
 
 y/A 
 //// 
 
 ^ 
 
 V7X 
 
 //^ 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 c. 
 
 2 
 
 '/// 
 
 
 
 % 
 
 % 
 
 
 
 A% 
 
 y//, 
 
 ^ 
 
 y/y 
 
 i 
 
 
 y/A 
 y/y 
 
 
 '/// 
 
 /// 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 l 
 ( 
 
 2 
 
 3 
 < 
 
 
 
 ) 
 
 i 
 
 2 
 
 I 
 
 3 
 
 
 
 1 
 ( 
 
 2 
 
 ) 
 
 3 
 
 J 
 
 
 
 L 
 
 l 
 
 1 
 
 2 
 
 t 
 
 3 
 
 X 5 
 
 x 4 
 
 X 3 
 
 
 2 
 Fig. k. The GLD representation of X and If. 
 
 Fig. 5. The GLD representation of X^ and xf 
 
12 
 
 space of binary vectors, i.e. E(2,2,. . . ,2) , the GLD "becomes the logical 
 diagram defined in [2]. The logical diagram of [2] is similar to a 
 diagram for the solution of logical problems first constructed by Alan 
 Marquand in l88l [3]*. The GLD can therefore be viewed as an extension of 
 Marquand' s idea. 
 
 It can easily be verified that an elementary literal X 
 k € { 0,l,...,h -1} is represented in the GLD by a set of cells contained 
 in the row generated in step 1 and assigned the value k. An elementary 
 literal XV, i€ { 2,3,...,v }, k€ { ,1,2 ,. . . ,h. -1 } is represented by the 
 union of cell sets comprising cells from the rows generated in step i and 
 assigned the value k (fig. ^ ) . The literals XV, i = v+l,v+2,. . . ,n, 
 k = 0,1, . . ,h.-l, are represented analogously (fig. 5). 
 
 Because 
 
 ■'<'- *£. (i 5 ) 
 
 k =a. 
 i 
 
 a non-elementary literal 'X.' is represented by the union of cell sets 
 
 corresponding to the elementary literals X a, 4 XT 1 * 1 ..., X. 1 . 
 
 l l ' l 
 
 Generally, the set-theoretic operations on any event sets (thus 
 also on literals and intervals, i.e. products of literals) are equivalent 
 
 l) 'Charts' of E. W. Veitch [k] , discovered or rediscovered in 1952, 
 are diagrams of the same kind, constructed for simplifying switching function 
 expressions. They have not become as popular as Karnaugh maps (in which rows 
 and columns are ordered according to Grey's code instead of the natural binary 
 code) because simple rules for detecting sets of cells corresponding to prime 
 implicants were lacking. However, if such rules are formulated [6, see also 
 section 5] and axes of different variables are distinguished by differing 
 their thicknesses (as in the diagrams of [6,2] and in the GLD), the diagrams 
 become highly useful for simplifying switching function expressions as well 
 as for detecting switching function symmetry [2] and for converting normal 
 Boolean expressions exclusive-or-polynomial expressions and vice versa [7]. 
 Important properties of these diagrams are that all the rules remain the same 
 for any number of variables and that the cell numbers have lexicographical 
 order (which Karnaugh's maps do not have). 
 
13 
 
 >- .3 
 
 "5 
 
 12 1 3 2 
 
 L 2 =X 1 X 2 X 5 
 
 denotes L l r\L 2 
 Fig. 6 . An intersection of interval complexes 
 
 13 10 2 
 
 Li=X 2 X, X, 
 
 1 1 3 
 
 L 2 =X, X, 
 
 denotes LjwL 2 
 
 Fig. 7 . A union of interval complexes 
 
Ik 
 
 12 1 12100 
 
 L = X t X 3 L=X,X 3 =X,vX J 
 
 denotes L 
 
 Fig. 8. Complement of an interval complex 
 
15 
 
 to the set-theoretic operations on the cell sets representing them. 
 For convenience, we will preserve the same notation for intervals and the 
 cell sets which represent them. However, to distinguish the intervals 
 from their geometrical equivalents we will call the latter interval 
 complexes . Fig. 6, 7 and 8 give examples of an intersection, a union and 
 a complement of interval complexes, respectively. 
 
 To represent a mapping f: E — -{ [ 0,1 ]»*}, for each event 
 e € E, the value f(e) is assigned to the cell representing e. The set of 
 cells of the GLD with their assigned values is called a function image of f 
 and denoted by T(f). The function image T(f) which corresponds to the 
 'grid' representation of f in fig. 1, is shown in fig. 9. A cover 
 D(f|X) of a mapping f under X can be represented by marking in T(f) 
 the interval complexes corresponding to the intervals of the cover. 
 
 Using T(f), we can in simple cases (when the GLD is not too large) 
 visually determine a cover D(f|x). First, the set F^ of cells with values 
 f(e), < f(e) < 1, is partitioned into the sets F^ = { e|A _< f(e) <l} 
 and F^° = { | < f(e) < X } and the sets F 1X = F 1 U F^ 1 and F° A = F° \J F^° 
 are determined (e.g., by changing the values of the cells of F^ to 1 and 
 those of F^ to 0). Then we visually find a set of interval complexes 
 
 whose union covers every cell of F and does not cover any cell of F 
 
 i * 
 
 lit may, however, cover cells of F ) . Section 5 gives a rule for the 
 
 easy recognition of cell sets which are interval complexes. The above pro- 
 cedure for determining a cover, based on the intuitive grouping of cells into 
 interval complexes , parallels the graphical way of finding a simplified 
 disjunctive normal form of a Boolean function in switching theory. It does 
 not guarantee that the obtained cover is minimal; in general it gives an 
 approximately minimal one, without, however, any estimation of the distance 
 
16 
 
 "1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 0.8 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 1 
 
 0.8 
 
 
 
 1 
 
 
 1 
 
 
 0.7 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 1 
 
 
 1 
 
 1 
 
 
 
 
 
 
 0.9 
 
 1 
 
 
 3 
 
 
 
 1 
 
 
 
 i 
 
 
 0.4 
 
 
 0.5 
 
 
 
 
 
 
 0.3 
 
 
 
 
 
 
 
 l 
 c 
 
 2 
 
 > 
 
 3 
 
 
 
 1 
 
 1 
 
 2 
 
 3 
 
 
 
 1 
 
 2 
 
 > 
 
 3 
 
 
 
 1 
 
 2 
 5 
 
 3 
 
 X 
 X 
 
 Fig. 9. T(f) of the mapping f from Fig. 1. The empty cells have value * 
 
17 
 
 (in terms of the number of intervals) to the minimal cover. The minimal, 
 
 or approximately minimal cover with an estimate of the maximal possible 
 
 Q 
 distance to the minimum can be found by applying the algorithm A 
 
 (see section T)» 
 
 5. RECOGNITION OF INTERVAL COMPLEXES IN THE GLD 
 
 The synthesis of a minimal interval cover is computationally 
 
 a very complex problem, particularly if the number of dimensions n and 
 
 the number of distinct values for each dimension h. , are not small numbers, 
 
 1' 
 
 e.g. n >_ 8 and h. >_ k. Therefore, the synthesis of interval covers 
 normally has to be performed by a computer. However, for checking the 
 results and understanding the synthesis algorithms, a geometrical 
 representation of a mapping f by a function image T(f) and of its interval 
 cover, provided by the GLD, is very useful. Also, in simple cases, the 
 image T(f) can be directly used for (graphical) synthesis of an interval 
 cover. This can be simplified by having rules for easy recognition of 
 interval complexes in the GLD. 
 
 In order to formulate such rules, we shall first define some 
 simple geometrical objects in the GLD. 
 
 A set of cells included in one row (column) or in two or more 
 adjacent rows (columns) generated in step i = l,2,...,v (i = v+l,v+2,. . . ,n) 
 and, if i ^ 1 (i 4 v+l) , contained in a single row (column) generated in 
 step i-1, is called a regular row ( regular column ) (fig.io). 
 
 The intersection of any regular row and any regular column is 
 called a regular rectangle (fig. ll) . 
 
18 
 
 Fig. 10, A ,A - regular rows, A ^»A^ - regular columns 
 
 B 1 - not a regular row, B 2 - not a regular column 
 
 Fig. 11. A ,A 2 ,...,A - regular rectangles 
 
 B, ,B ,B_,Bi - not regular rectangles 
 
19 
 
 Regular rectangles which can be made to cover each other by 
 translation are called identical . 
 
 Let E be a set of cells. The minimal-under- in elusion regular 
 rectangle which includes E (i.e. the regular rectangle contained in every 
 other regular rectangle which includes E), is called a covering rectangle 
 for E and is denoted by R(E) (fig. 12). 
 
 Let R and R 9 be identical regular rectangles containing event 
 sets E and E , respectively. E is said to have the same placement in 
 R, as E in R , if, when R and R are superimposed E and E cover each 
 other. E and E having the same placement in R and R respectively, can 
 be expressed by: 
 
 E = R 1 A L and E g = R 2 C\ L (l6) 
 
 where L is an interval complex, called an interval complex addressing 
 E in R (notation: L(E ,R ) ) , or E in R (L(E ,R )). For example, in 
 fig. 13 the set E has the same placement in R.. as E in R_ and we have: 
 
 R 1 L = L(E 1 ,R 1 ) 
 
 r^ e A v 
 
 1 l 1 ? 
 
 e i = x i x 2 x 3 x u x ; 
 
 j . 1 2 
 
 E = X Xp X„ Xi X 
 
 R 2 L = L(E 2 ,R 2 ) 
 
 Now we will formulate a theorem which gives a rule for recognizing 
 
 interval complexes in the GLD. Let E n ,E_,...,E. be some event sets. 
 
 12 k 
 
 Theorem 1: 
 
 The union E v E_ v . . . \j E, is an interval complex, if E-,E_,...,E. 
 12 K r 1 12k 
 
 are interval complexes and the covering rectangle R(E ^ E v . ,.^E ) can 
 
20 
 
 R(E 2 ) 
 
 Xj x 2 
 
 Fig. 12. Event sets E. and their covering 
 rectangles mE .) , i = 1,2,..,, 5 
 
 R(EiWE 2 ) 
 
 R(E,) 
 
 R(E 4 ) 
 
 Fig. 13. E^ E 2 ,E ,E, ,E are interval complexes 
 (Theorem l) 
 
21 
 
 be partitioned into k identical regular rectangles R,R ,...,! in 
 which E ,E ,...,E have the same placement, respectively. 
 
 J- c. K. 
 
 Proof 
 
 If E. ,E^,...,E, are interval complexes and have the same 
 1' 2' ' k 
 
 placement in R ,R ,,,,,R , respectively then we have: 
 
 E ± = R 1 AL, E 2 = R 2 AL, ..., E k = R^L 
 
 where : L = LfE^I^) = L(E 2 ,R 2 )= ... = L(E fc ,R k ) 
 
 k 
 i=l 
 
 \^J E = (R a L) yj (RgA L)u o..v; (Rj^aL) = L f\(R \JR \J... UR k ) 
 i=l 
 
 But (R v R g v ...ur ) is the covering rectangle R(E \J E v ...^E ), therefore 
 
 k 
 
 1 I E. = LAR(E n W E V ...WE ) 
 ^- / i 12 n 
 
 i=l 
 
 The intersection of interval complexes is also an interval complex, 
 
 thus v_y E. is an interval complex. Q.E.D. 
 
 i=l x 
 
 Theorem 1 is illustrated in fig. 13. 
 
 6. THE EXTENSION OPERATION ^r AND A STAR G(e|x) 
 
 An important concept introduced in [l] is that of the extension 
 operation w~ on event sets. Here we will give the GLD interpretation of 
 this operation. 
 
 Recall from [l] that {E ,E , . . . } W denotes the union of event 
 sets E iS i = 1,2,... and ViTthe set of all maximal (under inclusion) 
 intervals contained in E: 
 
22 
 
 {E ,E ,...) U ={J\ (IT) 
 
 i 
 
 U }/~T = {L Jl.^E andjl/ £E,L C i/.} (l8) 
 
 j J i J J J 
 
 Let E and E be event sets. The extension operation \T on E 
 relative to E (or the interval extension of E in E ) is defined: 
 
 E^ E 2 = {L \L.Q E 2 and L.A E 1 4 ty? (19) 
 
 Clearly, E vE C E (if E^ E g = <|) , E X ^E 2 = (|)). 
 
 If in (19) the set. {L.} includes only maximal (under inclusion) 
 
 J 
 
 intervals which are contained in E and have a non-empty intersection 
 with E , the union {L.} will he the same. Therefore, we have the 
 equivalent definition: 
 
 E 2 yE 2 =/Y (20) 
 
 where A = {L . I L . <S 7e~ and L.aE, 4 6} 
 
 Thus we have: 
 
 ^ EA/-E =A (2i; 
 
 To form EVE using the GLD, mark the intersection E a E 
 and then, by applying theorem 1, find all the maximal interval complexes 
 which cover at least one event e^ E a E and are contained in E . The 
 union of these interval complexes is E v/"E (fig. ik) . 
 
 If E 2 is an interval complex L, then clearly: 
 
 E v- L 
 
 fli., if E AL 4= ty 
 -< r 1 (22) 
 
 (j), otherwise 
 
23 
 
 X 1 X 2 
 
 Fig. lU, Extension operation v/~ 
 
 15 L, 
 
 Fig. 15. Examples of maximal interval complexes. 
 The empty cells have Value * 
 
2h 
 
 Interval complexes 
 
 L^7f 1 Vf° A (23) 
 
 J 
 
 are called maximal interval complexes in T(f ) under X (.or in T(f ) ) . 
 
 In other words, maximal interval complexes L. are maximal under inclusion 
 
 J 
 
 interval complexes which cover some events from F and do not cover any 
 
 r\\ ~\ % 
 
 event from F , i.e. they are contained in F \J F . The L. (i = 1,2,3) in 
 
 fig. 15 are examples of maximal interval complexes in a function image 
 T(f ). They were generated "by starting with some arbitrary cell e. € F 
 {where e = e ,e = e ,e = e ) and then adding more cells from F kj F , 
 as long as their union is still an interval complex according to theorem 1. 
 
 In general, there can he many different maximal interval complexes 
 in a T(f ) which cover the same event e€F . The set of all maximal interval 
 complexes in T(f ) covering a given event e€F is called the interval 
 star G(e|X) under X of that event. Clearly: 
 
 \J 
 
 G(e|x) = / {e}^F° A (2k) 
 
 The concept of an interval star is fundamental in our approach 
 to the synthesis of quasi-minimal interval covers described in Section 7. 
 We will here briefly explain the algorithm G of [l] for generating a star 
 G(e|x) and illustrate it by a simple example. 
 
 Denoting (G(elx)) by G (elx), we have from (2k): 
 
 G U (e|x) = {e}^F UA (25) 
 
 which is equivalent (see theorem 13 of [l]) to: 
 
 G v (e|x) = H nl ({s}v^{e v }) (26) 
 
 e^F 
 
25 
 
 Equation (26) is the theoretical basis for algorithm G. 
 
 r — i mO^ 
 
 First, the second de Morgan's law is applied to le ), e^€ F : 
 
 {e~} = X' X° 2 ---X an = X>X>...WX°" . (27) 
 
 k 1 2 n 1 2 n 
 
 0- 
 
 and then the X. are expressed as: 
 
 X.' = X- v ' X.' (28) 
 
 ill 
 
 By applying the distribution property of Vover ^([l]) and 
 the rule (22), each ({e}w~{e }) in (26) is transformed into a sum of 
 literals, denoted by S, . Then, by multiplying the S by each other and 
 
 K. K. 
 
 applying the absorption laws: 
 
 L(L wL v l \j •••) = L (29) 
 
 LU(LLL.---)*L (30) 
 
 (where L,L ,K , ••• are any literals or products of literals), the 
 G (e|A) is expressed as the irredundant sum of intervals: 
 
 G V (e|A) = L^yy (31) 
 
 The star G(elx) is then: 
 
 G(e|x) = {L.}, J = 1,2,... (32) 
 
 J 
 
 Example 1 
 
 Generate the star G(e|x) of the event e = e € E(3,3,3,3) if 
 F °* = {e )5 = {e 2\ e 3T ;e W 5e 67 je 75 K 
 
 k=l 
 
26 
 
 E(3, 3,3,3) 
 
 '1 A 2 / 1 2 3 4 5/6 7/8 
 
 U 
 
 denotes G (e X 
 
 i S9 
 
 Fig. 16. The star G(e X) of event e = e (Example l) 
 
27 
 
 From the GLD for E(3,3,3,3) the products of literals corres- 
 ponding to the events e and e , k = 1,2,. ..,5, are determined, and then 
 
 the sums S. are formed: 
 k 
 
 
 {e 39 } = X* X l Xl Xj. 
 
 u x ) 
 
 = x» 
 
 X 2 
 X 2 
 
 ^ 
 
 X U 
 
 9 
 
 S l 
 
 = {e}\y{e~} 
 
 = 
 
 ! x 2 v °x 2 
 
 {e 2 } 
 
 = xi 
 
 X 2 
 
 x° 
 
 X 1 
 
 x l+ 
 
 9 
 
 S 2 
 
 = ' fel^-fe^} 
 
 = 
 
 *X* V x° 
 
 {e 3 } 
 
 = 4 
 
 X 2 
 
 x l 
 
 X J 
 
 9 
 
 S 3 
 
 = ' {e}v/-{e^} 
 
 = 
 
 °xi v °xj 
 
 {e u } 
 
 = x i 
 
 X 1 
 
 x 2 
 
 Y 1 
 
 X 3 
 
 x i 
 
 9 
 
 S ^ 
 
 = {e}w~{e,} 
 
 = 
 
 °xi w xj 
 
 {e 5 } 
 
 = x i 
 
 Y 2 
 
 x 2 
 
 X 1 
 
 X 3 
 
 x £ 
 
 9 
 
 S 5 
 
 = ' {e}^-{e^} 
 
 = 
 
 °X1 v °X 2 
 
 According to 
 5 
 
 (26): 
 
 
 
 
 
 
 
 
 
 G V (e|A) =0 S k = ( 1 x2wO x l)(l x 2 v; X ) ( X 1 w Xj L ) ( X 1 w X 0)(0 X 1 W X 1 ) ^ 
 
 k=l 
 
 After multiplication with the use of the rule: 
 
 
 
 r °x b 
 
 i 
 
 V' V 2 
 \ X 2 
 
 =< 
 
 * 
 
 9 if ^ = I 2 
 
 , if i 1 = i g 
 
 i and a _< b 
 i and a > b 
 
 ^ 1 2 ' 
 
 (3U) 
 
 where 
 
 a = max(a ,a ) and b = min(b ,b ) , and the absorption laws 
 we have : 
 
 5 
 
 ■\) 
 
 (e|X) = U L. 
 
 (35: 
 
 j=l 
 
28 
 
 where 
 
 L x = X^ X l , L 2 = X 1 X 1 X l } ^ = X l 1 X 2 X 1 
 
 h = X 1 X 1 1 X 2 X 1 9 ^ = X l X f ^ = X 1 X 
 
 Thus the star G(e|x) - {L.} (see fig. 16). 
 
 Algorithm G was developed for the machine synthesis of interval 
 covers. The most tedious part of the algorithm, the transformation from 
 G w (e|A) = (S S •••) to (31) (in the example above from (33) to (35)), 
 can be simplified by using the special rules described in [8], When n and. 
 h. are small, a star G(e|x) usually consists of only a few elements and 
 therefore can be determined graphically without using the algorithm, simply 
 by visual inspection of the function image T(f). 
 
 7. GRAPHICAL SYNTHESIS OF QUASI-MINIMAL INTERVAL COVERS 
 
 Paper [l] described an adaptation of the algorithm A [9] (which 
 provides a solution for the general covering problem) to the synthesis of 
 quasi-minimal interval covers. From now on we will denote interval adaptations 
 
 of A q by A q -INTERCOVER or, briefly, by A Q . 
 
 Q 
 In this section we will geometrically describe a version of A , 
 
 using the GLD and will give examples of the graphical synthesis of interval 
 
 covers. The version described here differs slightly from the one in [l] in 
 
 that we assume the criterion of cover minimality be not only the number of 
 
 intervals (as in [l]), but also, with secondary priority, the number of literals, 
 
29 
 
 Q 
 The flow-diagram of our version of A is given in fig. IT. 
 
 Input to the algorithm is a function image T(f) and an assumed 
 
 value A. The output is a quasi-minimal interval cover M (f|A) (equa3 
 
 to the last value of the variable M ) , and parameters A and 6 which 
 
 estimate the maximal possible difference between the number of intervals 
 
 and literals, respectively, in the cover M (f|\) and in the minimal 
 
 cover M(f | A) : 
 
 c(M q (f|A)) - c(M(f|A)) _< A (36) 
 
 z(M q (f|A)) - z(M(f|A)) _<<5 (37) 
 
 By c(M), z(M) is denoted the number of intervals and number of literals 
 respectively^ in the cover M. 
 
 := denotes the assignment statement, 
 
 F is auxiliary variable, whose initial value is the 
 
 set F 1A defined by (8), 
 
 OCF^e..) denotes the operation of choosing the cell with the 
 
 smallest number from the set specified by the current 
 value of F and naming it e , 
 
 L is an interval complex in G(e n |AJ, called a quasi- 
 extremal, which covers the maximum number of events 
 
 from the current set F (in flow-diagram F designates 
 
 1A 
 a variable whose initial value is the set F given by 
 
 (8) and following values are the sets which remain to, 
 
 be covered after each step of the algorithm , similarly 
 
 F* is also a variable whose values are sets; its initial 
 
 value is given by (7)), 
 
30 
 
 Input Data 
 T(f),X 
 
 I 
 
 Determine T(f ) 
 
 I 
 
 F P : = F 1X , M q : = 0; A: = 6: = 
 
 v 
 
 0(F P ; ei ) 
 
 I 
 
 Generate G(e | A) 
 
 1 
 
 Determine L and 6 = z(L q ) - 
 
 o 
 
 - z(L m ) 
 
 I 
 
 M q : = M q U{L q }, F 1X : = F 1X \L q 
 F P : = F P \G U (e 1 |A), F : = F u L q 
 
 0(F 1A ; e x ) 
 
 I 
 
 M q (f A): = M q 
 
 Generate G(e 1 |A) and Determine L q 
 
 I 
 
 (END ) 
 
 M q : = M q u{ L q k F 1A : = F 1A \L q 
 
 * * 
 
 F : = F U L H , A: = A+l, 6: = 6 + z(l' 
 
 Fig. 17. Algorithm A^ (a graphical version) 
 
31 
 
 L is an interval complex in G(e |x) which has the minimum 
 
 number of literals , 
 ztL^) ,z(L m ) are numbers of literals in L q and L , respectively, 
 iVr- is the variable which stores the set of determined quasi- 
 extremals. Its final value is called a quasi -minimal cover 
 of f X and denoted by M q (f|x). 
 The algorithm consists of 2 parts. 
 
 Part I. 
 
 After T(f \ is determined, stars G(elx) are successively generated 
 
 for events which have the smallest number in the current set F , and L ,L 
 and 6 = z(L ) - z(L ) are found for each star. If a star contains more 
 than one L , the one with the smaller number of literals is preferable. 
 Since F .in each. step includes only those events from F which were not 
 covered by an interval complex from any star generated in previous steps 
 (due to the operation F P r = F \G^(e| X) ) , any two stars are disjoint, i.e. 
 they have no common interval complexes. 
 
 Part I is terminated when F P = and therefore the family of stars 
 which were generated (we will denote this family by G ) is a maximal under 
 inclusion family of disjoint stars (i.e. there does not exist an e € F 
 such that G(e|x) <jr G and is disjoint with all its members). 
 
 Part II. 
 
 This part (executed if F ^ and F P = 0) determines additional 
 interval complexes which cover the remaining cells of F . Their number 
 is the value A in (36). 
 
 To prove (36) let F denote the set of cells e, for which stars were 
 
 v 
 
 generated in Part I (note that F r £ {ej G(e^) <S G r } ) . Since the stars of any 
 
32 
 
 r r 
 
 two events of F have no common interval complexes, every cover of F 
 
 against F has to include at least c(G ) elements. But F $ F , and 
 
 therefore any cover of F against F , thus also a minimal cover 
 
 M(f|x) (which here means a cover with the minimum number of elements and 
 
 the minimum number of literals for that number of elements) cannot have 
 
 less elements than c(G ). Thus, the number A of interval complexes which 
 
 have to he used to cover the remaining set F in part II of the 
 
 algorithm can he viewed as an estimate of the maximal possible difference 
 
 in number of elements between the cover M (f|x) and the minimal cover 
 
 M(f|x). The estimate 6 in (37) is obtained by summing the differences 
 
 6 determined in part I and then adding the number of literals in intervals 
 
 determined in part II. If A = and 6 = 0, M q (f|x) = M(f|x). 
 
 If after the first execution of the algorithm, A and 6 are 
 
 considered to be too large, a better estimate (and/or solution) may be 
 
 obtained by performing other iterations. A simple and often successful 
 
 method for realizing further iterations is described in [8]. 
 
 Example 2 
 
 Determine a quasi-minimal interval cover M ((f|x) for the 
 mapping f A defined by T(f A ) in fig. 18. 
 
 -i o 
 
 In the first step, the star G(e | \ ) is determined. It consists 
 of interval complexes L,,L ,l' (fig. 19). Since L covers the maximum 
 number of cells of F X (8 cells) it is chosen to be the quasi-extremal. The 
 operation F P : = F P \G w (e |x) is realized by marking the cells of 
 
 -| Q -I 1 
 
 G (e |X) r\ F with a *. Part I terminates after determining quasi -extremal 
 
 -i Q(- 
 
 L in the star G(e |X). 
 
 at 
 
 c(K) denotes the cardinality of K. 
 
33 
 
 X l X 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 * 
 
 
 
 1 
 
 
 1 
 
 
 1 
 
 
 
 1 
 
 
 
 
 1 
 
 
 
 U 
 2 
 
 
 1 
 
 
 
 
 
 1 
 
 
 
 
 1 
 
 
 
 
 
 
 3 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 1 
 1 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 1 
 
 1 
 2 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 l 
 
 c 
 
 2 
 
 ) 
 
 3 
 
 c 
 
 
 
 ) 
 
 1 
 
 : 
 
 2 
 
 I 
 
 3 
 
 
 
 1 2 
 
 
 3 
 
 \ 
 
 
 
 L 
 
 1 
 
 2 
 
 1 
 
 3 
 
 5 
 
 x 4 
 
 *3 
 
 Fig. 18. T(f ) for example 2 
 
 A = 1 
 S = 3 
 
 M q (f|A) = {L lS L 2 ,...,L L^ } 
 
 Fig. 19. Interval cover M q (f|x) of T(f A ) ( fig. 18 ) 
 
3h 
 
 51 82 83 , _,1A 
 In part II, the remaining cells e ,e , and e of F are 
 
 covered by i) . Thus A = 1 and 6 = zd^ ) = 3 (fig. 19). 
 
 It can easily be verified that if the algorithm is started with 
 
 the cell e , the obtained cover will be the same but A = 6 = 0. Thus 
 
 the above determined quasi -minimal cover M^flx) is, in fact, the 
 
 minimal cover. 
 
 IX 
 
 Example 3 
 
 Assume that objects of class 1 are characterized by set F 
 of events e = (x ,x ,x ,x« \ € E(U,4,U,U) : 
 
 F 1A = {(0,3,3,0), (0,3,2,1), (1,2,3,0), (l, 3,3,1), 
 
 (0,2,3,1), (1,3,2,1), (1,2,2,0), (l,2,3,l), 
 
 (3,0,0,3), (2,1,0,3), (2,0,1,2), (3,1,1,2), 
 
 (2,1,1,2), (3,1,0,10, (2,0,0,2), (3,0,1,2)} 
 
 and objects of class by set F : 
 
 F° X = {(0,0,1,0), (1,2,1,2), (0,2,0,1), (1,1,1,0), 
 
 (3,3,3,2), (2,3,3,2), (2,3,3,0), (3,3,2,3), 
 
 (3,0,3,0), (2,1,3,1), (0,3,1,3), (0,2,0,3), 
 
 (3,3,3,0), (2,0,3,2), (0,1,1,1), (l, 1,0,1)} 
 
 The above events can be represented graphically as small 'pictures' 
 (fig. 20 a) by interpreting the variables x. , i = 1,2,3,1+ as elements 
 of a square: 
 
 x l 
 
 X 2 
 
 X 3 
 
 x h 
 
 and giving them different shades corresponding to the values of x. (x. € {0,1,2,3} ) 
 
a) F 
 
 IX 
 
 ,60 
 
 »57 
 
 H 
 
 .195 ~I47 
 
 s 
 
 -ox 
 
 .102 
 
 
 
 i 
 
 
 .204 a l57 
 
 E 
 
 e 33 e 84 
 
 
 i 
 
 pi 
 
 ^ 
 
 
 .55 
 
 e 55 e 252 e 142 
 
 35 
 
 e l08 e P5 g45 e .2l g.04 e l09 
 
 gl34 g2!4 gl50 g 2IO gISO g l98 
 
 g234 gl90 gl88 g25l 
 
 .21 
 
 
 w7 
 
 w% 
 
 e 81 
 
 
 V/A 
 
 Y//t 
 
 
 1 
 
 b) 
 
 M*(f): 
 
 c)F 
 
 *l 
 
 .105 ,.110 
 
 1 
 
 n 
 
 g63 g243 
 
 •*0 
 
 e 255 g,70 
 
 g206 g 232 
 
 Li 
 
 g247 g (2 
 
 L 2 
 
 gl26 g2,5 gl35 gl5, 
 
 gl92 g72 
 
 EH 
 
 e 85 e° 
 
 g 22 | g.19 
 
 I 
 
 I 
 
 m 
 
 7/A 
 
 gl.5 g7, 
 
 e" 
 
 
 H 
 
 m 
 
 
 
 .65 
 
 n-o 
 
 - 2 
 
 - 3 
 
 e 240 e^ 
 
 BE 
 
 J32 A I6I 
 
 gl36 g68 
 
 V77 
 
 
 i 
 
 
 . 2 ^203 
 
 H 
 
 
 
 H 
 
 
 1 
 
 1 
 
 
 Fig. 20 o a. 'Pictures' respresenting events from sets F ' and F 
 for example 3. 
 
 b. Cover M q (f|X) 
 
 *1 *0 
 
 c. Samples of 'pictures' representing sets F and F 
 
36 
 
 
 x i 
 
 
 
 x 2 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 3 
 
 6 
 
 7 
 
 
 
 9 1 
 
 L 
 
 n i 
 
 '12 
 
 13 
 
 14 
 
 IS 
 
 
 J± 
 
 
 *0 
 
 
 
 
 
 
 
 
 
 
 1 
 
 =-=r 
 
 
 *l 
 
 
 
 *r 
 
 13 
 31 
 47 
 
 
 
 l 
 
 
 
 
 
 *0 
 
 
 
 
 
 N 
 l| 
 
 1 
 
 1 
 1 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 li 
 
 +J. 
 
 
 
 
 1 
 
 
 
 e 41 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 '! 
 
 
 
 1 
 
 
 
 *l 
 
 63 
 79 
 95 
 
 III 
 
 127 
 
 143 
 
 159 
 
 175 
 
 191 
 
 207 
 
 223 
 
 239 
 
 255 
 
 
 
 1 
 
 
 
 
 *0 
 
 
 
 *0 
 
 
 
 *0 
 
 i*i 
 ii 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 *0 
 
 
 
 
 *0 
 
 
 
 I! 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 i 
 ii 
 
 1 
 
 *n 
 
 1 
 
 
 
 1 
 
 1 
 
 *l 
 
 
 
 
 3 
 
 
 
 
 *0 
 
 
 
 
 *0 
 
 i 
 
 1 1 
 
 
 
 
 1 
 
 *l 
 
 J 
 
 
 e l30 
 
 
 JL 
 
 /* 
 
 
 -+ 
 
 
 *l 
 
 
 1 
 
 «l 
 
 4-- 
 
 i*0 
 
 "1 
 
 1 
 
 — — - 
 
 4, 
 
 
 
 
 
 
 
 
 2 
 
 
 | 
 
 
 
 1 
 
 
 
 1 
 
 *l 
 
 ll 
 
 ■ I 
 
 I 
 1 
 
 
 
 
 
 
 
 
 
 . 1 
 
 2 
 
 
 *l 
 
 
 
 
 
 
 
 H 
 ■ 1 
 
 1 
 1 
 
 *0 
 
 
 
 
 
 
 
 L 2 
 
 3 
 
 \ 
 
 
 
 
 
 
 
 
 
 1 
 
 —A 
 
 5S.-S.' 
 
 ■**' 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 *l 
 
 
 
 1 
 
 
 
 1 
 
 
 #0 
 
 ! 
 I 
 
 
 *0 
 
 
 
 
 *0 
 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 1 
 
 *l 
 
 1 
 
 1 
 
 1 
 I 
 
 
 
 
 *0 
 
 
 
 
 L 2 
 
 _2_ 
 
 
 
 
 
 
 
 
 
 *-o 
 
 1 
 
 1 
 1 
 
 
 
 
 
 
 
 
 
 3 
 
 (*l 
 
 
 
 *l 
 
 
 
 
 »lj 
 
 1 
 
 1 
 
 1 
 
 I 
 
 
 
 
 
 
 
 
 
 *0 
 
 
 
 
 
 1 
 
 c 
 
 2 
 ) 
 
 3 
 
 
 
 1 
 
 i 
 
 2 
 
 L 
 
 3 
 
 
 
 1 
 
 2 
 
 > 
 
 3 
 
 
 
 i 
 
 2 
 
 3 
 
 x 4 
 
 X 3 
 
 
 M q (f|A) = {h 1% L 2 ) 
 
 A = 
 8 = 
 
 Fig. 21. Cover M q (f|x) for example 3 
 
37 
 
 Find the minimal set of features , in the form of intervals in 
 E(k 9 k t k 9 h) 9 which permit classification of any event from F \j F into 
 the set F or F 
 
 Fig. 21 shows the function image T(f ) defined by F and F 
 
 and the cover M (f|x) found by realizing the algorithm A . First the 
 
 Ul /• ■ i 
 
 star for e was generated (L ,L and L in fig. 21 are 3 of 5 elements 
 
 of G(e "|x)), a second star was generated for e (L ,L are 2 of 5 
 
 elements of (e 13 °|x). Since A = 6 = 0, M q (f|x) = M(f|x). 
 
 Figo 20 b shows the intervals L and L of M (f|x) as features 
 
 for distinguishing events from F and F . The cover M (f|x) partitions 
 
 all events of H(k 9 k 9 k 9 h) into class F of events which are covered by 
 
 L U L , and class F of events which are not covered by L U L . Clearly, 
 
 F £F and F $ F . Thus, the events previously unspecified (events 
 
 * *1 1 IX *0 * *1 
 
 of F ) are now included in either the set F =F\F or F =F\F . 
 
 *1 
 Assuming that events e6F are classified as events of class 1 and events 
 
 *0 10 
 
 e€F as events of class 0, the sets F and F can be viewed as generalized 
 
 sample sets F and F . To illustrate the above generalization, samples of 
 
 *1 *0 / 
 
 events from sets F and F were shown in fig. 20c. (In fig. 21 these 
 
 events are marked by *1 and *0, respectively.) 
 
38 
 
 ■8. CONCLUSION 
 
 The GLD model of the space E is a useful visual aid for 
 geometrically representing mappings f and their interval covers , and for 
 interpreting algorithms for interval covering synthesis. It has proved 
 to be particularly useful for developing and verifying computer 
 implementations of algorithm A [8]. 
 
 If the number n of dimensions of E and the values h. are 
 
 i 
 
 small, e.g. n _< 6, h. _<k, the GLD can easily be used for the direct 
 
 graphical synthesis of interval covers. 
 
 This geometric model can also be applied for representing 
 
 functions and algorithms of many-valued logic. 
 
39 
 
 9 . REFERENCES 
 
 1. R. S. Michalski, B. H. McCormick, "interval Generalization of Switching 
 Theory," Proceedings of the Third Annual Houston Conference on Computer 
 and System Science , Houston (Texas), April 26-27, 19T1» (an extended 
 version in Report No. kk2 9 Department of Computer .Science, University of 
 Illinois, Urbana, Illinois, May 3, 19Tl) • 
 
 2. R. S. Michalski, "Recognition of Total or Partial Symmetry in a Completely 
 or Incompletely Specified Switching Function," Proceedings of the IV 
 Congress of International Federation on Automatic Control (IFAC), 
 
 Vol7 27, pp. 109-129, Warsaw, June 16-21, 1969 . 
 
 3. A. Marquand, "On Logical Diagrams for n terms," Edinburgh and Dublin 
 Philosophical Magazine and Journal of Science, Vol. XII, 5th Series, 
 No. 75, pp. 266-270, 1881. 
 
 k. E. W. Veitch, "A Chart Method for Simplifying Truth Functions," Proceedings 
 of the Association for Computing Machinery , Pittsburgh, Pa., pp. 127-133, 
 May 2-3, 1952. 
 
 5. A. Svoboda, "Grafico - Mechanicke Pomucky Uzirane Pri Analise a Synthese 
 Kontaktowych Obvodu," Strode Zpracov. Inf., No. k, 1956 (in Czechoslovakia^ 
 
 6. R. S. Michalski, "Synthesis of Minimal Forms and Recognition of Symmetry 
 of Switching Functions',' Proceedings of the Institute of Automatic Control , 
 Polish Academy of Sciences, No. 91, Warsaw, 1971 (in Polish). 
 
 7. R. S. Michalski, "Conversion of Normal Forms of Switching Functions into 
 Exclusive-or-Polynomial Forms," to appear in Archivum Automatyki i 
 Telemechaniki , Polska Akademia Nauk, 1971 (in Polish). 
 
 8. R. S. Michalski, V. G. Tareski, "Interval Covers Synthesis: Computer 
 Implementation of and Experiments with Algorithm A ",to appear as a 
 report of the Department of Computer Science, University of Illinois. 
 
 9. R. S. Michalski, "On the Quasi-Minimal Solution of the General Covering 
 Problem," Proceedings of the 5th Internation Symposium on Information 
 Processing (FCIP 69), Vol. A3, pp. 125-128, Yugoslavia, Bled, 
 
 October 8-11, 1969. 
 

 rm AEC-427 
 
 (6/68) 
 VECM 3201 
 
 U.S. ATOMIC ENERGY COMMISSION 
 
 UNIVERSITY-TYPE CONTRACTOR'S RECOMMENDATION FOR 
 
 DISPOSITION OF SCIENTIFIC AND TECHNICAL DOCUMENT 
 
 i S*e Instructions on Revtrsa Side ) 
 
 AEC REPORT NO. k6l 
 
 COO-2118-0013 
 
 2. TITLE 
 
 A GEOMETRICAL MODEL FOR THE SYNTHESIS 
 OF INTERVAL COVERS 
 
 TYPE OF DOCUMENT (Check one): 
 
 Q a. Scientific and technical report 
 
 II b. Conference paper not to be published in a journal: 
 
 Title of conference 
 
 Date of conference 
 
 Exact location of conference. 
 
 Sponsoring organization 
 
 □ c. Other (Specify) 
 
 ! RECOMMENDED ANNOUNCEMENT AND DISTRIBUTION (Check one): 
 
 L_j a. AEC's normal announcement and distribution procedures may be followed. 
 
 (~~l b - Make available only within AEC and to AEC contractors and other U.S. Government agencies and their contractors. 
 
 I~1 c. Make no announcement or distrubution. 
 
 I REASON FOR RECOMMENDED RESTRICTIONS: 
 
 ^SUBMITTED BY: NAME AND POSITION (Please print or type) 
 
 R. S. Michalski 
 
 Visiting Asst. Professor of Computer Science 
 
 Organization 
 
 I University of Illinois 
 
 Department of Computer Science 
 Urbana, Illinois 
 
 Signature 
 
 Ut^*4*> 
 
 Date 
 
 June 2k 9 1971 
 
 FOR AEC USE ONLY 
 
 l\EC CONTRACT ADMINISTRATOR'S COMMENTS, IF ANY, ON ABOVE ANNOUNCEMENT AND DISTRIBUTION 
 IECOMMENDATION: 
 
 Latent clearance: 
 
 I 
 
 j □ a. AEC patent clearance has been granted by responsible AEC patent group. 
 I LJ b. Report has been sent to responsible AEC patent group for clearance. 
 i LJ c. Patent clearance not required. 
 
1 
 
'■; 
 
 *•>■ 
 
 <fc 
 
 •V