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R^orrTNo. UIUCDCS-R-78-897 
 
 yyuU^ 
 
 UILU-ENG 78 1707 
 
 A PLANAR GEOMETRICAL MODEL FOR REPRESENTING MULTIDIMENSIONAL 
 DISCRETE SPACES AND MULTIPLE-VALUED LOGIC FUNCTIONS 
 
 by 
 
 luary 1978 
 
 ■CO— i 
 
 
 Ryszard S. Michalski 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
 The Library of the 
 
Report No. UIUCDCS-R-78-897 
 
 A PLANAR GEOMETRICAL MODEL 
 FOR REPRESENTING MULTIDIMENSIONAL 
 DISCRETE SPACES AND MULTIPLE- VALUED 
 LOGIC FUNCTIONS 
 
 by 
 Ryszard S. Michalski 
 
 January 1978 
 
 Department of Computer Science 
 
 University of Illinois at Urbana-Champaign 
 
 Urbana, Illinois 61801 
 
 This work was supported in part by the National Science Founadation 
 under grant NSF MCS 76-22940. 
 
ABSTRACT 
 
 A general planar model of multidimensional discrete spaces (a 
 diagram) is described which can be used for geometrically representing binary, 
 multiple- valued, discrete and variable-valued logic functions. It is es- 
 sentially a multiple-valued extension of Marquand's binary diagram, with an 
 additional feature of varying thickness of lines representing axes of variables. 
 
 The diagram has been used extensively over the years by the author 
 and his collaborators, and has proved to be useful for such tasks as: logic 
 design and development of efficient algorithms for optimization of switching 
 circuits of many variables, detection of symmetry in binary or multiple-valued 
 Logic functions, fast conversion of normal forms of switching functions to 
 =xclusive-or polynomial forms, design of algorithms for inductive inference and 
 pattern recognition, optimization of decision tables and their conversion to 
 aptimal decision trees. 
 
 A method for recognizing in the diagram certain constructs (cartesian 
 complexes), which are important for various applications, is presented and 
 illustrated by two examples: one, involving a determination of a classification 
 rule, and another, involving a synthesis of a variable-valued logic expression. 
 
 tey words and phrases : Logic diagram, Marquand diagram, Karnaugh map, Veitch 
 liagram, discrete spaces, multiple-valued logic, variable- valued logic, logic 
 iesign. 
 
 :R categories: 3.61, 3.63, 5.20, 5.21, 6.1 
 
Motto: A picture is worth a thousand 
 words. Especially when it is 
 a right one. 
 
 1. INTRODUCTION 
 
 There exists a large number of problems in which there is a need to 
 
 deal with functions of the general form: 
 
 f : D., x D~ x . . . x D ■+ D (1) 
 
 12 n 
 
 where 
 
 and 
 
 D n , D_, ..., D are finite non-empty sets 
 1 I n 
 
 D is a finite or infinite non-empty set. 
 
 For example, a binary switching function is a special case of f when 
 
 I, = D - ... = D = {0,1} and D = {0,1,*}, where * denotes a 'DON'T CARE' 
 12 n 
 
 value. A k-valued switching function is a special case of f when D = D = 
 
 ... = D = {0,1, ..., k-1} and D = {0,1, ..., k-1,*}. Often used in operations 
 n 
 
 research are the so-called psuedo-Boolean functions, which are a special case 
 
 of f, when 
 
 D » D - ... - D = {0,1} and D = [0,1] (the closed interval). 
 12 n 
 
 When sets D. and D are finite sets of integers, {0,1,2,...}, then f represents 
 a discrete function (e.g., Deschamps and Thayse [2]). 
 
 In pattern recognition and decision theory many problems can be 
 phrased as a search for an efficient expression of a function f, where D. are 
 domains of values of certain features or descriptors which are used to char- 
 acterize objects or situations, and D is either a finite set of decision 
 classes to which the objects belong or a set of 'degrees of membership' of an 
 object in a class (usually the closed interval [0,1]). Considered mainly 
 from this point of view, functions f , in which D. and D are arbitrary finite 
 sets with or without any order, were studied by Michalski and his collaborators 
 (e.g., [12]- [14]) under the name of variable-valued logic functions. 
 
When analyzing a function f (which often is only partially defined), 
 
 or designing algorithms involving such a function, it can be very useful to 
 
 represent it in a form of a geometrical pattern. 
 
 The advantage of such representation, as compared to a symbolic re- 
 presentation, is that it is easier for humans to perceive in this form the 
 global properties of the function, i.e., to "see" the function as a whole. 
 Also, it is easier to detect various regularities in the function, if it is done 
 by comparing geometrical configurations rather than strings of symbols or numbers. 
 
 The above explains the popularity of Venn diagrams, Karnaugh maps , histograms or 
 other graphical aids for representing functions. 
 
 This paper describes a general planar geometrical model (a diagram) 
 for representing functions type (1) and gives a method for easily recognizing 
 constructs in the diagram which are important for applications. The 
 presented diagram has been extensively used over the years by the author and 
 his collaborators, and has proved to be very useful for various tasks such as: 
 
 • logic design and a development of efficient algorithms for optimization 
 of switching circuits of many variables (Michalski [8], Michalski and 
 Kulpa [17]) 
 
 • detection of symmetry in binary and multiple-valued logic functions 
 (Michalski [9], Jensen [3]) 
 
 « fast conversion of normal forms of switching functions to exclusive-or 
 polynomial forms (Michalski [10]) 
 
 • design of algorithms for inductive inference and pattern recognition 
 (Michalski [12]-[15], Larson and Michalski [6]) 
 
 • optimization of decision tables and their conversion to optimal 
 decision trees (Michalski [16]). 
 
 Let us define fi(d , d , ..., d ), written also as E, as the cartesian 
 1 I n 
 
 product of sets D , i - 1,2, ... n: 
 
 E(d , d , .... d )= E = D_ x D x ... x D (2) 
 
 1 ^ n 12 n 
 
 where c^ - the number of elements in D , and call it the universe of events 
 
 (or event space ). Let us assume, for simplicity, that the domains D are 
 
 sets of positive integers: 
 
 D. = (0, 1, .... d^, i = 1, 2, ..., n (3) 
 
where i ■ d -1. This assumption causes no loss in generality because any 
 
 finite set can be isomorphically mapped into D given by (3) Let us also 
 
 assume that D , D , ..., D are value sets (domains) of certain variables 
 
 Li x 2 , ..., x r , respectively (i.e., x. can take values only from D , x - 
 
 only from D«, and so on). 
 
 A discrete-Euclidean representation of the space E would be in the 
 
 form of an n-dimensional 'grid', spanned from the d. , d_, ..., d points on 
 
 12 n 
 
 axes associated with variables x , x , ..., x , respectively (Fig. 1). The 
 above geometrical model of E is, however, difficult to visualize when n > 3, 
 and therefore it is n©t practical for a larger number of variables. 
 
 In the past, for the case when x are binary variables, many dif- 
 ferent planar representations of the space E have been proposed. Among 
 well known early constructions are Euler's circles (1768) [1] and Venn 
 diagrams (1880) [20], The earliest known constructions for representing 
 logical operations were developed in XIII century by Raymond Lull [1]. 
 
 A diagram of rectangular shape, which is divided into cells corres- 
 ponding to single combinations of binary logic values was first pro- 
 posed by Marquand in 1881 [7]. The regularity and simplicity of such a 
 diagram made it useful for a larger number of variables than the Venn diagram. 
 It has not become popular, however, because there was no pressing need at that 
 time for representing complex logical functions. Such a need arose many years 
 later when Shannon discovered the applicability of Boolean algebra for des- 
 cribing switching systems. And then a Marquand-type diagram was independently 
 rediscovered by Veitch in 1952 [19] (a diagram of this type was also developed 
 by Svoboda [18]). Soon afterward in 1953, Karnaugh [4] reorganized the Veitch 
 diagram, assigning variable values to rows and columns according to the 
 Grey's code rather than to the regular binary code, used in Marquand and 
 Veitch diagrams. In such an arrangement, any two neighboring cells in the diagram 
 
 . 
 

 x 2 
 
 II 
 
 J -7- 
 
 , d 2 =4, d 3 =4 
 
 Jkf 
 
 : zzz^ z 
 
 2 n 
 
 f = X 
 
 
 A 
 
 ^ ^ 
 
 — -y^- -& 
 
 IT 
 
 &- 
 
 *n 
 
 
 
 y 
 
 -- ^- 
 
 — -*3^ -#! 
 
 *T 
 
 
 — ^ 
 
 ,,1 
 
 
 -■M — - 
 
 • - f(e)=l 
 O — f(e) = 
 
 CD- f(e) = ± 
 • - f(e)=* 
 
 Fig. 1. Discrete-Euclidean representation of the space E(U,U,M and 
 a mapping f: E(4,4,4) -► {0, h ,, 1, *}. 
 
correspond to adjacent conjunctions (i.e., conjunctions which differ only in one 
 literal (variable or its negation) and, therefore, can be reduced to one con- 
 junction). In this form, the diagram (called the Karnaugh's map) became very 
 popular as an aid in logical design of electronic circuitry. It is quite con- 
 venient for minimization of Boolean functions up to 4 variables. When there are 
 more than 4 variables, however, the rules of using the map change and quickly 
 become rather complicated. 
 
 This paper describes a diagram for representing discrete spaces spanned 
 over not only binary variables, but variables with any discrete values. The 
 rules for constructing and using the diagram are the same for any number of 
 variables and any number of values which variables can take. Consequently, 
 the diagram provides a general geometrical model of discrete spaces. 
 
 The diagram is essentially a multiple-valued extension of Marquand's 
 binary diagram (although the author was not aware of this when he developed it) • 
 In addition, it has a new feature which is a varying thickness of lines re- 
 presenting axes of variables. This feature greatly improves the clarity of 
 the diagram, especially when there are many variables. The diagram was orig- 
 inally described in a departmental report by Michalski [11] which is out of 
 print. The binary form of this diagram was described earlier (Michalski [8]). 
 The intention of this paper is to make available written information about 
 the diagram and describe various new results not yet published. 
 
 2. CONSTRUCTION OF THE DIAGRAM 
 
 Suppose a task is to represent the space E(d , d , ..., d^), that 
 
 is a space spanned over n variables, which take d , d», ..., d values, respectively 
 First, determine v which is the maximal number satisfying relation: 
 
 d • ' d„ ... d < d . ' d ., ... d ( 4 ) 
 
 12 v — v+1 v+2 n 
 
This is an arbitrary step, but it leads to a unique diagram for a 
 
 given E. Also, the diagram is visually more satisfying if the products on 
 
 both sides of (4) are approximately equal. (To achieve this a permutation 
 
 of the d. -s may be helpful.) Next, draw an arbitrary rectangle and divide 
 
 it into d * d„ . . . d rows and d , * d ,_ ... d columns according to the 
 12 v v+1 v+2 n 
 
 following rules: 
 
 (i) In the first step, divide the rectangle by horizontal 
 lines into d rows and assign to the rows values 0, 1, 
 .. ., &. (values of variable x ) in order from the top 
 to the bottom. In the step i, 1 < 1 < v, divide each 
 row generated in step i-1 into d rows, and assign to 
 them values 0, 1, ..., 4 . , in order from the top to the 
 bottom (&. = d. -1). 
 (ii) Do the steps v+1, v+2, ..., n } in the similar way as 
 above, but divide the rectangle by vertical lines into 
 columns . 
 (iii) The lines generated in step i, i = 1, 2, . . . , n, are 
 
 called axes of x . . Vary the thickness of axes of dif- 
 ferent variables: the thinnest should be axes of x 
 
 v 
 
 and x , next thinnest - the axes of x , and x ., , and 
 n v-1 n-1 
 
 so on, until exhausting all the axes. (Thus, if n 
 
 is even and v = n/2, the thickest will be the axes 
 
 x, and x , n ) . 
 1 v+1 
 
 Figure 2 presents a general form of the diagram for n = 4. A unique 
 
 vector (x , x , ..., x ) corresponds to each row and a unique vector 
 
 (x ... x ,„, .... x ) corresponds to each column of the diagram. The inter- 
 v+1 v+2 n 
 
 section of any row with any column is called a cell . We will assume that the 
 
x 1 x 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 I 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 o . 
 
 • 
 • 
 
 1 
 
 
 
 1 
 1 
 
 1 
 1 
 
 
 
 1 
 1 
 
 1 
 
 1 
 
 
 i 
 
 1 
 
 1 
 
 1 
 1 
 
 d 2 -i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 1 • 
 
 • 
 
 
 
 
 
 1 
 1 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 1 
 
 i 
 
 1 
 1 
 1 
 
 1 
 
 1 
 
 i 
 
 d 2 -l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 1 
 
 1 
 
 i 
 i 
 i 
 i 
 i 
 
 
 
 l i 
 
 ! ! 
 
 1 l 
 I i 
 i i 
 I i 
 • I 
 
 
 
 1 
 1 
 
 1 
 
 ! 
 
 i 
 
 1 
 
 1 
 1 
 
 
 • 
 
 1 
 
 1 1 
 1 
 1 1 
 
 i! 
 
 1 1 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 J r> : 
 
 • 
 
 1 
 1 
 1 
 
 
 
 1 
 
 : 
 
 i 
 
 
 
 1 
 1 
 1 
 1 
 
 i 
 i 
 
 i 
 
 1 
 1 
 1 
 1 
 
 I 
 l 
 l 
 
 i 
 
 d 2 -l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 l 
 
 • • • C 
 
 
 
 4-1 
 
 
 
 l 
 
 • • • c 
 
 1 
 
 4-1 
 
 
 
 1 
 
 • • • c 
 
 2 
 
 4-1 
 
 
 
 
 i 
 
 . . . ( 
 d 3 -l 
 
 u-i 
 
 Fig, 
 
 A diagram representing E(d ,d ,d ,d ), 
 
8 
 
 cells do not include points belonging to any axis nor to the perimeter of the 
 rectangle. Each cell represents an element of E ( event ) determined by con- 
 catenating vectors (x^ * r ..., x y ) and (x^, x^, ..., X fl ) , corresponding 
 to the row and the column, respectively, whose intersection produces the given 
 cell. The diagram comprises d=d 1 ' d r " d n cells (that is the number of events in 
 
 E). 
 
 Each event e = (x n , x_, ..., x ) from E can be assigned a unique 
 1 z n 
 
 number y(e) according to the formula: 
 
 1 
 
 y(e) =x n + y x ± p|d k (5) 
 
 i=n-l k=n 
 
 For example, in the space E(5,6,4,3) the value of the function y(e) 
 
 (y- number ) for the event e = (3,4,1,2) is 
 
 y(e) = 2 + 1-3 + 4-4-3 + 3-6-4-3 = 269 
 
 It is easy to see that from a given value y(e) and values 
 
 d, ,d_, ..., d , one can determine the corresponding event e=(x 1 ,x_, ..., x ). 
 J- z n 1 Z n 
 
 Namely, first divide y(e) by d ; the remainder is the value of x . Next, divide 
 
 n n 
 
 the result by d . ; the remainder is the value of x , , and so on, until the 
 n-1 n-1 
 
 value of x is obtained. It can be easily verified that if one assigns the 
 corresponding y-number to each cell, the order of the numbers in any diagram 
 will be lexicographical (i.e., from left to right and from top to bottom), no 
 matter what is the number of variables or the number of values the variables 
 take. 
 
 Figures 3 and 4 present diagrams for two different event spaces and 
 the distribution of y-numbers in them. 
 
a. 
 
 b. 
 
 X l X 2 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 
 1 
 
 4 
 
 5 
 
 6 
 
 7 
 
 
 
 
 1 
 
 8 
 
 9 
 
 10 
 
 11 
 
 
 i 
 1 
 
 12 
 
 13 
 
 14 
 
 15 
 
 
 
 
 
 c 
 
 1 
 ) 
 
 
 
 : 
 
 1 
 
 X 
 X 
 
 X 3 X 4 
 x x 00 01 II 10 
 
 00 
 01 
 
 II 
 
 10 
 
 
 
 1 
 
 3 
 
 2 
 
 4 
 
 5 
 
 7 
 
 6 
 
 12 
 
 13 
 
 15 
 
 14 
 
 8 
 
 9 
 
 11 
 
 10 
 
 A diagram representing 
 binary space E(2,2,2,2) 
 
 An equivalent 
 Karnaugh map 
 
 Fig. 3. Distribution of Y-numbers in a diagram and 
 aa equivalent Karnaugh map. 
 
10 
 
 Xj x 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 1 
 
 12 
 
 13 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 58 
 
 59 
 
 c 
 
 1 
 
 60 
 
 61 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 
 
 67 
 
 68 
 
 69 
 
 70 
 
 71 
 
 
 
 
 1 
 
 
 2 
 
 
 
 1 
 
 1 
 
 2 
 
 
 
 1 
 2 
 
 2 
 
 
 
 1 
 3 
 
 2 
 
 x 4 
 
 X, 
 
 Fig. 4. Diagram for space E(3, 2,4, 3). 
 
11 
 
 3. RECOGNITION OF INTERVAL AND CARTESIAN COMPLEXES IN A DIAGRAM 
 
 3.1 Definitions 
 
 We will introduce now certain concepts which play an important role 
 
 in various applications of the diagram. 
 
 Let {x. = a, }, i e l,2,...,n , where a is a subset of D., denote the 
 i 1 l i 
 
 set of all events from an E, whose x. component takes a value from ot . : 
 
 {x. = a. } = {e = (x. , x , ..., x ) |x. e a } (6) 
 
 11 12 n ' l i 
 
 Such an event set is called a cartesian literal . If the subset a . is a se- 
 quence of consecutive integers, a+1, ..., b, then the cartesian literal is 
 
 called an interval literal and denoted {x. = a..b}. If a . consists of only 
 l i 
 
 one element, then {x. = ol.} is called an elementary literal . 
 
 A set-theoretical product of cartesian (interval) literals is called 
 a cartesian complex ( interval complex ) : 
 
 n 
 
 L= II {x ± = a}, I c {1,2,. ..,n} (7) 
 
 i e I 
 
 A cartesian complex which is a product of elementary literals is 
 called an elementary complex . 
 
 Set-theoretical operations on cartesian complexes: complement, 
 product and sum are equivalent to the complement, intersection and union of 
 the corresponding sets of cells in a diagram* (Fig. 5, 6, 7). 
 
 In the binary case (i.e., when d = d = ... = d =2) cartesian 
 complexes reduce to sets of events corresponding to single conjunctions of 
 binary literals. For example, complex {x =0}{x =1}{x =0} corresponds to 
 conjunction x x.x,.. 
 3. 2 Supporting concepts 
 
 A function 
 
 f : I-* D (8) 
 
 
 * In the sequel, whenever it does not lead to confusion, a set of cells 
 corresponding to a cartesian complex will be called simply a cartesian 
 complex. 
 
12 
 
 L*{x x «l f 2}{x s «l} L : {x 1 =0}u{x 3 =0} 
 
 denotes L 
 
 Fig. 5. Complement of a cartesian complex. 
 
13 
 
 Lj»{vl}K-0.l} L 2 :{ Xl =l,2}{x 2 =1..3}{x 5 =O..E} 
 
 denotes L i r\L z 
 Fig. 6. An intersection of cartesian complexes. 
 
 L-r {x 2 =1..3}{x 3 =l}{x 5 =0..2} L 2 : {x 1 -l}{x a -1..3} 
 
 denotes L 1 \jL z 
 
 Fig. 7» A union of cartesian complexes. 
 
14 
 
 can be represented in a diagram by marking cells representing events e e E 
 by corresponding value f(e). For example, Fig. 8 presents a diagramatic 
 representation of the function from Fig. 1: 
 
 f: E(4, 4, 4, 4) ■* {0, % , 1, *} (9) 
 
 In solving various logical or combinatorial problems occurring. 7 
 e.g., in logic design, pattern classification, decision theory, etc., there 
 is often a need for expressing a function f (8) in terms of concepts which 
 are special cases of cartesian or interval complexes. For example, in 
 logical design such concepts are prime implicants; in pattern classification 
 and artificial intelligence - property lists or logical products of condi- 
 tions: 'does feature x have value a' or 'does feature x have value in the 
 set A' (Michalski [12]). In using diagrams for manual solving or illustrating 
 such problems, or as an aid in designing and testing algorithms for a com- 
 puter solution, a question arises of how to visually recognize in a diagram 
 the configurations of cells which correspond to cartesian or interval com- 
 plexes. In order to develop such a rule, we will first introduce some geo- 
 metrical concepts which are very easy to recognize in the diagram, and then 
 express the rule in terms of these concepts. 
 
 Definition 1 : A set of cells included in one row {column} or in two or more 
 adjacent rows {columns} generated in step i = 1, 2, ..., v {i= v+1, v+2, ..., n} 
 and, if i ^ 1 {i ^ v + 1}, contained in a single row {column} generated in step 
 i-1, is called a regular row {regular column } (Fig. 9). 
 
 Definition 2 : The intersection of any regular row and any regular column is 
 called a regular rectangle (Fig. 10). 
 
 Definitions 1 and 2 imply that any regular rectangle can be ex- 
 pressed as a single product of interval literals, that is as an interval 
 complex. A regular rectangle can be considered a diagram itself, representing 
 
15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 i 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 \ 
 
 1 
 
 
 1 
 
 
 1 
 
 
 
 v 2 
 
 
 
 
 2 
 
 
 
 
 
 
 
 1 
 
 1 
 
 % 
 
 
 1 
 
 1 
 
 
 
 
 
 v 2 
 
 
 
 
 3 
 
 
 
 
 
 
 
 v 2 
 
 Vz 
 
 
 
 v 2 
 
 v 2 
 
 
 
 
 v 2 
 
 
 
 
 
 
 
 l 
 C 
 
 2 
 
 ) 
 
 3 
 
 
 
 i 
 
 i 
 
 2 
 
 3 
 
 
 
 i 
 
 2 
 
 > 
 
 3 
 
 
 
 i 
 
 - 
 
 2 
 
 5 
 
 3 
 
 X 
 X 
 
 Fig. 8. Diagramatic representation of the mapping f from Fig. 1, 
 The empty cells have value *, 
 
16 
 
 Fig. 9. A t» a ? - regular rows, A ,A, - regular columns 
 
 B - not a regular row, B - not a regular column, 
 
 Fig. 10- A ,A , ...,A - regular rectangles 
 
 K 1»B_,B ,B< - not regular rectangleu 
 
17 
 
 a subspace of the total event space, spanned over the axes which cross the 
 rectangle. If we assume that the perimeter of a diagram is made of the 
 thickest line, then it is easy to observe that the thickest axis crossing 
 any regular rectangle is never thicker than the thickest, parallel to it, 
 borderline axis. 
 
 From now on, whenever we use the name rectangle, we will mean a 
 regular rectangle. 
 
 Definition 3 : A regular partition of a rectangle is a set of rectangles 
 obtained by splitting the original rectangle along the thickest horizontal 
 or vertical axes crossing the rectangle (Fig. 11). 
 
 It follows from the definition 3 that expressions of rectangles in 
 a partition differ only in one literal (Fig. 11). 
 
 Definition 4 : Let E be a set of cells. The minimal-under- inclusion regular 
 rectangle which includes E (i.e. the regular rectangle contained in every 
 other rectangle which includes E) , is called the covering rectangle of E_ 
 (Fig. 12). 
 
 Suppose a configuration of cells, E, corresponds to a simple car- 
 tesian complex L(E), i.e., a product of literals involving variables from the 
 
 set {x. , . . . ,x }. 
 1 n 
 
 Lemma 1 If E is a proper subset of rectangle R, then L(E) can be expressed 
 as 
 
 L(E) = L(R) n L (10) 
 
 where L(R) is a product of literals expressing rectangle R, and L a product 
 of literals called an expression of E ±n context of R. 
 
 Proof Since an intersection of sets of cells corresponds to a product of 
 expressions representing sets and E C R, therefore the expression L(R) must be a 
 part of the product L(E); the remaining part is L. M 
 
Xi x 2 
 
 
 
 \ir 
 
 "V 
 
 1 
 
 
 V= 
 
 l 
 2 
 
 X 4 
 X, 
 
 Fig. 11. R 2 : { X;L =0}{x 3=1,2} 
 R 3 . : { X;L =2}{x =1,2} 
 
 R 2 : {x 1 =l}{x 3 =l,2} 
 
 The set {R , R , R } is a regular partition of 
 rectangle R. 
 
 18 
 
19 
 
 R(Ej) 
 
 R(E 2 ) 
 
 x l 
 
 Xo 
 
 
 
 v 
 
 J 
 
 1 
 
 
 
 
 
 
 \ 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 r 
 
 1 
 
 
 
 
 
 
 
 L 
 
 / 
 
 
 
 
 
 
 
 l 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 i> 
 
 
 
 
 
 2 
 
 
 
 
 
 % 
 
 h 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 E 3 
 
 
 
 
 
 
 
 
 
 
 
 
 (^ 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 i 
 
 ^ 
 
 % 
 
 -»■ 
 
 
 
 2 
 
 
 
 
 
 4 
 
 % 
 
 
 
 V 
 
 
 /^/ 
 
 1 
 
 
 
 
 
 R(E 3 ) 
 
 
 3 
 
 
 
 
 
 V 
 
 *J 
 
 
 
 
 
 
 
 
 
 E 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 R(E 5 ) 
 
 2 
 
 1 
 
 p 
 
 
 
 b 
 
 ^ 
 
 
 
 -\ 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 E 5 
 
 R(E 4 )^ 
 
 3 
 
 
 
 
 ^ 
 
 
 
 
 ^) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 ( 
 
 2 
 ) 
 
 3 
 
 c 
 
 
 
 ) 
 
 i 
 
 2 
 
 L 
 
 3 
 
 
 
 1 
 
 C 
 
 2 
 ) 
 
 3 
 
 i 
 
 
 
 i 
 
 2 
 
 L 
 
 3 
 
 x 5 
 x 4 
 x. 
 
 
 Fig. 12 , Event sets E. and their covering 
 rectangles RtE,), i = 1,2,. ..,5. 
 
20 
 
 The product L is a cartesian complex in the subspace of event 
 space E, determined by rectangle R, i.e., spanned over the axes crossing the 
 rectangle. Lemma 1 implies that in order to determine whether a set E cor- 
 responds to a cartesian complex, it is sufficient to determine whether E is a 
 cartesian complex in the context of a rectangle which includes E (rather than 
 in the context of the whole diagram) . 
 
 Let E be a subset of cells of some regular rectangle R. E can be 
 represented in R by marking cells which belong to E and leaving the remaining 
 cells of R unmarked. 
 
 Definition 5 ; A regular rectangle R with marked cells belonging to a set of 
 cells E is called an image of E and denoted I(R,E). If E is a cartesian com- 
 plex in the context of R, then R is called an image of a_ cartesian complex . 
 For completeness, a rectangle with no marked cells is alternatively called 
 an empty image . 
 
 Definition 6 : Two or more images are congruent if they can be superimposed 
 by translation and, when superimposed, the corresponding cells in the images 
 are all marked or all empty (Fig. 13). 
 3.3 A cartesian complex recognition theorem and a recognition rule 
 
 Suppose E , E , ... are configurations of cells corresponding to 
 
 cartesian complexes. 
 
 Lemma 2 If and only if images I (R ,E ), I (R ,E ), ... are congruent then 
 
 expressions of E..,E ... in the context of R ,R , ... are identical. 
 
 Proof According to lemma 1, a cartesian complex L(E.) corresponding to E., 
 j = 1, 2, .. } can be expressed as 
 
 (i) L(E.) = L(R.) n L. 
 
 3 3 3 
 
 where L(F.) is the expression of the rectangle R • , and L^ is the expression of 
 E. in the context of R.. Since images I(R^,E^) are congruent, then each E. , 
 j 1, 2, .., has an identical location with regard to the borderlines of the 
 
21 
 
 Fig. 13. Images I , ..,1 are congruent. 
 
22 
 
 corresponding rectangle and, consequently, the differences between expres- 
 sions L(E.), j = 1, 2,..., can only be due to the differences in expressions of 
 rectangles, i.e., L(R.). Thus, all L. must be identical. 
 
 To prove the reverse implication, observe, that if all L. are 
 identical, then expressions L(E.), given by (i), can differ only in expres- 
 sions of rectangles, i.e., L(R.), and, consequently, images I(R.,E.) have 
 to be congruent. fl| 
 
 Let L.,j =1, 2,... be cartesian complexes which differ only in 
 one literal: 
 
 L. = T{x =a.}, j =1, 2, ... 
 where T is the common part. Using the distributive property of the set- 
 theoretical union over the intersection and the definition of literal, we 
 can write: 
 
 UL. - U T{x-a.} - T{x-U .a.} (11) 
 
 J J J 1 
 
 Consequently, the union of such cartesian complexes is also a cartesian 
 
 complex. The equation (11) is called the combining rule . If U a. = D , 
 
 . J k 
 
 where D is the domain of x, , then {x. =Ua . } is equivalent to the event space 
 k k' k j M v 
 
 E and rule (11) reduces to the simplification rule : 
 
 U T{x =a.} = T ( 12 ) 
 
 j R J 
 
 We can now formulate a theorem which leads to a recursive recogni- 
 tion rule for cartesian and interval complexes in an arbitrary diagram. 
 Theorem 1 A configuration E of cells in a diagram is a 
 cartesian complex if and only if: 
 
 (1) E is a regular rectangle (in this case E is also an 
 interval complex) , or 
 
 (2) A regular partition of the covering rectangle of E 
 consists of congruent images of cartesian complexes 
 
23 
 
 and possibly some empty images. If the partition has 
 no empty images, and consists of congruent images of 
 interval complexes, then E is also an interval complex. 
 Proof : Condition 1: The definition of a regular rectangle (definition 2) 
 implies that a rectangle can be expressed as an interval complex. 
 
 Condition 2: Definition 3 implies that a regular partition of a 
 rectangle consists of rectangles whose expressions differ only in one lit- 
 eral. Let R_, R , ... denote nonempty rectangles and E , E^ t ... event sets 
 contained in them, respectively. Because images I(R.,E.), j = 1, 2, ..., 
 are congruent, therefore, according to lemma 2, expressions of E in the 
 
 context of R. are identical, and, therefore, complete expressions of E , L(E ), 
 J J J 
 
 differ only in one literal. We have 
 
 E = E U E 2 U ... (i) 
 
 therefore, the expression of E, L(E), is the union of L(E.) and, according 
 
 to the combining rule (11), is a cartesian complex- 
 
 If a partition of the covering rectangle does not include empty 
 
 images, that means that in the expressions 
 
 L(E.) = T{x.=a.}, j = 1, 2, ... 
 j i J 
 
 a.-s form a set of consecutive integers. Consequently, the union UL(E.) = 
 
 T{x.=a.}, where a. = U a is an interval complex, if T is an interval com- 
 ii i j 
 3 
 
 plex, i.e., if rectangles of the partition are images of interval complexes. 
 Now we have to prove that if E corresponds to a cartesian complex 
 then it satisfies condition 2 (if it satisfies condition 1 it also satisfies 
 condition 2). Let L(E) denote a cartesian complex corresponding to E: 
 
 n 
 
 L(E) = n {x.=ci.}, a. C D. 
 lii—i 
 1=1 
 
24 
 
 Suppose sets a. are transformed to new sets a 3 a . , by filling 
 'gaps' in a. or substituting D. for a., such that 
 
 n 1 
 R(E) = {x.=a.}, 
 
 1-1 * X 
 defines the covering rectangle of E. If k is selected in such a way that x 
 
 k 
 
 are the thickest vertical or horizontal axes crossing rectangle R(E) , then 
 
 rectangles R defined by complexes 
 
 3. 
 
 L(R ) - L 1 {x =a}, a e 0.} 
 
 a rC K. 
 
 where 
 
 1 n 1 
 
 L = {x.=at> 
 
 ifk 
 i=l 
 
 constitute a regular partition of R(E). A rectangle defined by L(R ), contains 
 for a e ol , <l c ol , a set of cells, E c E, defined by the cartesian complex 
 
 L(E ) = L{x=a} 
 a k 
 
 where 
 
 n 
 L = fl {x.=a. } 
 1=1 X X 
 i^k 
 
 and for a e a, \a. .where \ denotes set subtraction, no cells of E. Because 
 k k 
 
 all L(E a ), a e a^^, differ only in one literal (involving x ), therefore the ex- 
 
 pressions of E^ in the context of R (which are parts of L) are identical, and, 
 
 according to lemma 2, images I(R ,E ), for a e a. \ a. are empty. If E is an 
 
 interval complex then a = a t there are no empty images I(R ,E ), and all 
 
 k k a a 
 
 images are images of interval complexes. ■ 
 
 Based on the above theorem, the following rule can be suggested for 
 determining whether a given set E of cells in a diagram represents a car- 
 tesian complex: 
 
25 
 Rule 1 ; 
 
 1. Find the covering rectangle, R(E), of E. If R(E) is 
 identical with E then go to YES. 
 
 2. Determine a vertical or horizontal partition of R(E) 
 which produces the most square-like images. 
 
 3. Check if the nonempty images of the partition are 
 congruent. If answer is no, go to NO . 
 
 4. Select any nonempty image and treat the subset of 
 cells from E contained in the image as a new set 
 
 E. Go to 1. 
 
 YES: E is a cartesian complex. 
 
 NO : E is not a cartesian complex. 
 
 If an application of the rule terminates with YES and there were no 
 empty images at any step of the rule, then E is an interval complex. 
 
 The rule can be used for 'building up' complexes or determining 
 maximal possible complexes, by starting with any set of cells which can be 
 directly recognized as representing a complex (in the worst case, an indi- 
 vidual cell), then adding to it more cells and applying the rule. 
 
 Fig. 14 illustrates the rule by a few examples of cartesian com- 
 plexes. 
 3.4 Recognition rule for elementary complexes 
 
 In many applications, for example, in optimizing decision tables 
 (Michalski [16]), one deals with a special case of cartesian complexes, which 
 are elementary complexes (i.e., complexes L. =M{x. = L.}, where L. is just 
 a single value (see sec. 3.1)). For such applications, rather than general rule 1, 
 one needs a rule for recognizing elementary complexes. Such a rule can be easily 
 obtained by appropriately modifying rule 1. 
 
 To do so, we will first introduce concepts of an elementary row, 
 elementary column and an elementary rectangle (which are special cases of the 
 
26 
 
 R(E!WE 2 ) 
 
 R(E,) 
 
 R(EJ 
 
 E \J E ,E ,E,,E ,E,, and E ? are cartesian complexes 
 
 (E.,, ..., E, are interval complexes) 
 3 o 
 
 Illustration of Theorem 1. 
 
 Fig. 14 
 
27 
 
 regular row, regular column and regular rectangle, respectively - see definition 
 1 and 2). 
 
 Definition la ; A set of cells included in a single row {column} created at any 
 step of constructing a given diagram, is called an elementary row { column }. 
 Definition 2a : The intersection of any elementary row with any elementary column 
 is called an elementary rectangle . 
 
 Next we define: 
 Definition 4a : The minimal-under-inclusion elementary rectangle which contains 
 a given set of cells, E, is called an elementary covering rectangle of the set E. 
 
 It follows from definitions la and 2a, that an elementary rectangle 
 is an elementary complex. Therefore it is easy to see, that if in theorem 1 
 every regular rectangle is restricted to be an elementary rectangle, then the 
 theorem describes conditions for recognizing elementary complexes. Thus, we have: 
 Theorem la : A configuration of cells, E, is an elementary cartesian complex, iff: 
 
 (1) E is an elementary rectangle, or 
 
 (2) a regular partition of the elementary covering rectangle of E 
 consists of congruent images of elementary cartesian complexes. 
 
 The proof is left to the reader. Using theorem la, it is easy now to 
 appropriately modify rule 1 to obtain a recognition rule for elementary complexes. 
 Figure 15 gives a few examples of elementary complexes. 
 
 4. REMARKS ON APPLICATIONS AND EXAMPLES 
 
 The diagram presented here is a general model of discrete spaces 
 and therefore can be useful as a geometrical representation in any situa- 
 tion involving such spaces. In particular, it can be used as an educational 
 a id in various areas of computer science, logic, discrete mathematics, etc. 
 
 For research purposes, it can be useful in design and testing com- 
 puter algorithms involving discrete spaces and functions determined on them 
 (e.g., binary, many-valued, discrete and variable-valued logic functions). 
 
28 
 
 L 2 : 
 
 {X^OHX^OHX^l} 
 {X 2 =0}{X 3 =2}{X 4 =0} 
 
 {x^iHx^i} 
 
 {X 1 =2}{X 3 =0} 
 {X 1 =2}{X 2 =1}{X 3 =3} 
 
 Fig. 15. Examples of elementary cartesian complexes. 
 
29 
 
 An example of such an application is the development of a series of AQVAL 
 programs for computer-aided inductive inference (Michalski [15]). The 
 diagram was extensively used there for developing algorithms, solving test 
 examples, illustrating obtained solutions, etc. 
 
 In binary logic design the diagram can be used similarly to the 
 Karnaugh ma p (it can b e especially useful in the case of more than 4 variables 
 and up to 8-10 variables). And, of course, the diagram can also be used in 
 many-valued logic design. 
 
 Another possible application is to designing decision tables and de- 
 cision trees, in particular for quick testing decision tables for redundancy, 
 consistency and completeness; for reduction of decision tables and for converting 
 them into decision trees. This application is described in detail in 
 Michalski [16]. 
 
 In conclusion, to give the reader an opportunity to get more practice 
 in using the diagram we will consider two examples. 
 Example 1 : Find a classification rule. 
 
 This, somewhat whimsical, example gives an illustration of a less- 
 traditional use of the diagram. Suppose we were given 4 different bottles of 
 wine produced by company A and 4 different bottles of wine produced by company 
 B (Fig. 16). Suppose, for the sake of the example, that the wines did not 
 have labels with company names but each bottle had a special geometrical pattern 
 on it, as shown in Fig. 16. The problem is to determine the 'simplest' rule(s) 
 which will permit one to distinguish the wines of each company based on the geo- 
 metrical pattern. A way to sblve the problem is to determine first a set of 
 possible relevant characteristics ( descriptors ) of patterns, and then to construct 
 the simplest (according to some criteria) 'discriminant' description of each 
 class of wines in terms of these descriptors (or their subset). Although 
 determining a set of possibly relevant descriptors is, in general, a difficult 
 problem itself, we will assume here that it can be done, and will consider only 
 the problem of determining the final rule. 
 
30 
 
 A: 
 
 B: 
 
 * * 
 
 A A 
 
 ® 
 
 ® 
 
 A A 
 
 x - # of squares 
 
 x - # of triangles 
 
 x - it of cirles 
 
 x, - H of asterisks 
 
 d( X;l ) " (0,1) 
 
 Fig. 16. 
 
 A: {x 3 = 1} W {x 2 = 1} 
 Bottles of wine produced by company A and B. 
 
 D(x 2 ) - (0,1,2,3) 
 D(x ) - {0,1,2} 
 D(x u ) - {0,1,2} 
 
31 
 
 Suppose the following descriptors were selected as a possible try: 
 
 x - number of squares in the pattern, 
 
 x„ - number of triangles, 
 
 x~ - number of circles, 
 
 x. - number of asterisks. 
 4 
 
 From Fig. 16, we can determine that the domains D(x.) of these de- 
 scriptors, sufficient for describing each bottle, are: D(x ) = {0, 1}, D(x~) = 
 {0, 1, 2, 3}, D(x 3 ) = {0, 1, 2} and D(x.) = {0, 1, 2}. 
 
 Figure 17 shows the diagram for the space D(x..)x...x D(x.) with 
 marks indicating the correspondence between the cells and individual bottles. 
 If the number of conjunctive statements is accepted as a measure of complexity 
 of a description, then the simplest discriminant description of bottles of 
 company, say A, can be obtained by grouping cells marked A into the fewest 
 elementary cartesian complexes, whose union includes every cell A but does not 
 include any of the cells B. Such a grouping is shown in Fig. 17. 
 
 The union of complexes L.. and L~ gives us a description of the bottles 
 
 of company A: 
 
 A: {x = 1} U fas = 1} 
 2 3 
 
 It can be interpreted: 'if a bottle has 1 triangle or 1 circle then the wine 
 
 was made by company A. ' In a similar way, we can obtain a description of wines 
 
 of company B: 
 
 {x. = lHx. = 1} 
 1 4 
 
 That is: 'if a bottle has 1 square and 1 asterisk then the wine was made by 
 company B. ' 
 
 By grouping cells differently, we can obtain alternative descriptions, 
 for example: 
 
 A: { X;L = 0} U {x 2 = 1} 
 
 or 
 
 A: {x. = 0} U { X/ = 2} 
 1 4 
 
32 
 
 L^L 2 
 
 A 
 
 where L 
 
 1 
 
 x„ = 1} , L„: {x = 1} 
 
 Fig. 17. Diagram representing a discriminant 
 description of bottles A, 
 
33 
 
 B: { X;L = lHx 2 + 1} 
 
 or 
 
 B: {x x = 1}{x 4 + 2} 
 
 The interpretation of these descriptions is left to the reader. 
 
 The above example gave an elementary and informal introduction to 
 
 the kind of problems which occur in the application of the variable-valued 
 
 logic system VI^ to problems of computer induction and pattern recognition 
 (Michalski[12-15] , Larson and Michalski[6] ). 
 
 It is obvious that when the number of descriptors is large, de- 
 scriptions of this kind should be synthesized using a computer. (Paper [14] 
 presents synthesis algorithms and papers [6], [15] describe various computer imple- 
 mentations in the logic system VL , and, also, much richer system VL 2 [15] ). 
 Example 2 : Determine a DVL expression of the function represented in Fig. 1 
 (and Fig. 8). 
 
 Any function f : E ■* D, where D is a linearly ordered set, can be 
 expressed as a disjunctive normal expression in the variable-valued logic 
 system VL.. , i.e., a DVL expression (Michalski [13]). 
 
 Assuming that D = {0,...,«O, a DVL expression is a disjunction 
 (maximum function) of terms , where term is a product (minimum function) of se- 
 lectors. A selector is either a constant from D or a function [x.=a.], from 
 
 E into endpoints of D, defined: 
 
 [4, if e e {x.=a.} 
 
 [x.=a.](e) = { n . . X 1 
 
 i i ^0, otherwise 
 
 To obtain a DVL expression of the function f from Fig. 8 one groups 
 the cells marked by 1 into cartesian complexes (including cells marked by *, 
 whenever it is convenient). 
 
 Next, all cells marked by 1 obtain mark *i and cells marked by 
 1/2 are grouped into cartesian complexes (Fig. 18). Expressing complexes as 
 terms with appropriate constants one obtains the following DVL.. formula: 
 
34 
 
 
 L 
 
 1 
 
 
 
 
 
 
 L 
 
 2 
 
 
 
 
 Lj 
 
 
 
 
 
 *1 
 
 / 
 
 / 
 
 
 
 f 
 
 \ 
 
 
 
 f 
 
 
 
 
 
 
 
 f 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 
 _ 
 
 
 
 
 
 J 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 7 s * 
 
 
 
 \ 
 
 r- 
 
 -< 
 
 
 r\ 
 
 <~~ 
 
 "\ 
 
 
 Pi 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 l 
 
 
 
 1 
 
 \ 
 
 i 
 
 
 1 
 
 
 i 
 
 
 
 \ 
 
 
 
 
 
 k 
 
 
 
 / 
 
 
 
 
 1 ' 
 
 ' 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 l 
 
 V 
 
 1 
 
 w 
 
 \J 
 
 1 
 
 1 
 
 
 \j 
 
 
 
 \ 
 
 
 
 
 3 
 
 
 
 
 
 
 
 '4 
 
 \ 
 
 
 
 Hs 
 
 \ 
 
 
 
 
 \ 
 
 
 
 
 
 
 \, 
 
 J 
 
 
 
 V 
 
 / 
 
 
 
 
 
 
 I 
 
 ) 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 
 
 1 
 
 2 
 
 3 
 
 
 
 1 
 
 2 
 
 3 
 
 o|i 
 
 2 
 
 3 
 
 X 
 
 
 
 c 
 
 ) 
 
 
 
 ] 
 
 I 
 
 
 
 4 
 
 > 
 
 
 
 
 5 
 
 
 X 
 
 Fig. 18. Cartesian complexes for determining a DVL. ; 
 expression of function from Fig. 8. 
 
35 
 
 f: l[x =l][x 2 =0]V l[x =l,2][x 2 =l,2][x +2] V Jg[x 3 -1,2] 
 
 V _ J V. j 
 
 L l L 2 L 3 
 
 An equivalent way of expressing f is by an ordered set of produc- 
 tion rules (i.e., rules which consist of a condition part and an action 
 part; the action part is evoked if the condition part is satisfied). Rules 
 in an ordered set are applied sequentially, the first rule satisfied produces 
 the action: 
 
 [ X;L =l][x 2 =0] => [f=l] 
 [x 1 =l,2][x 2 =l,2][x 3 +2] => [f=l] 
 [x 3 =l,2] => [f= h ] 
 
 Sometimes an unordered set of rules is preferable (i.e., rules can 
 be applied in any order). In this case, each set of cells with the same mark 
 is treated independently. For example, function f from Fig. 8 can be ex- 
 pressed by the following unordered set of rules: 
 
 [ Xl =l][x 2 =0] => [f=l] 
 [ Xl =l,2][x 2 =l,2][x 3 +2] => [f-1] 
 
 [x 3 =2] «> [f- h ] 
 [ Xl =3][x 3 =l,2] => [f= h ] 
 
 To see clearly the difference between the two sets of rules, the reader is 
 advised to represent them in the diagram. 
 5. SUMMARY 
 
 We have presented here a general geometrical model of multidimensional 
 discrete spaces, and introduced several concepts associated with the model, such 
 as regular row, column, rectangle, a regular partition, etc. These concepts were 
 subsequently used for determining a rule for recognizing constructs, specifically 
 cartesian and interval complexes, that are important in various applications of the 
 model. 
 
36 
 
 The model can serve as an aid in designing, testing, describing, and, 
 also, in many practical problems, in hand executing algorithms involving discrete, 
 many valued or variable-valued logic functions. It can also be useful in 
 education, for teaching concepts and algorithms involving such functions. 
 
 ACKNOWLEDGMENTS 
 
 This work was supported in part by the National Science Foundation 
 
 Grant NSF MCS- 76-22940. 
 
 The author owes many thanks to his collaborators and students who 
 shared with him the experiences of using the diagram over the years and con- 
 tributed to the final form of the recognition rule of cartesian complexes. 
 Thanks are also due to the former CACM Technical Department Editor Professor 
 Glenn Manacher for his comments and suggestions. 
 
37 
 
 REFERENCES 
 
 Bochenski, J. M. Formal Logic . Frelburg-Munchen, 1956 
 
 Deschamp, J. P. and Thayse A. Representation of Discrete Functions. 
 Proceedings of the 1975 International Symposium on Multiple-Valued Logic , 
 Indiana University, Bloomington, Indiana, May 13-16, 1975, pp. 99-111. 
 
 Jensen, G. M. The Determination of Symmetric VL^ Formulas. Master's 
 Thesis, Department of Computer Science, University of Illinois, Urbana, 1975. 
 
 Karnough, M. The Map Method for Synthesis of Combinational Logic Circuits, 
 Commun. Electron . November 1953, pp. 593-599. 
 
 Kuzichev, A. S. Diagramy Venna . Izd. Nauka, 1968 (in Russian). 
 
 Larson, J. and Michalski, R. S. AQVAL/1 (AQ7) User's Guide and Program 
 Description. Report No. 731 . Department of Computer Science, University 
 of Illinois, June 1975. 
 
 Marquand, A. On Logical Diagrams for n Terms. The London, Edinburgh 
 and Dublin Philosophical Magazine and Journal of Science . Vol. XII, 5th 
 Series, No. 75, 1881, 266-270. 
 
 Michalski, R. S. Graphical Minimization of Switching Functions in Class 
 of Disjunctive Normal Forms. Journal of the Institute of Automatic 
 Control . Polish Academy of Sciences. No. 52, Warsaw, 1967 (in Polish). 
 
 Michalski, R. S. Recognition of Total or Partial Symmetry in a Switching 
 Function Completely or Incompletely Specified. Proceedings of the IV 
 Congress of International Federation on Automatic Control (IFAC ), Vol. 27 
 (Finite Automata and Switching Systems), Warsaw, June 16-21, 1969, pp. 109-129. 
 
 Michalski, R. S. Conversion of Normal Forms of Switching Functions into 
 Exclusive-Or-Polynomial Forms. Archiwum Automatyki i Telemechaniki , 
 Polish Academy of Sciences, No. 3, 1971, 263-278 (in Polish). 
 
 Michalski, R. S. A Geometrical Model for the Synthesis of Interval 
 Covers. Report No. 461, Department of Computer Science, University of 
 Illinois, Urbana, Illinois, June 24, 1971. 
 
 Michalski, R. S. A Variable-Valued Logic System as Applied to Picture 
 Description and Recognition," Chapter in the book, GRAPHIC LANGUAGES , 
 eds. F. Nake and A. Rosenfeld, North-Holland Publishers, 1972. 
 
 Michalski, R. S. Variable-Valued Logic and Its Applications to Pattern 
 Recognition and Machine Learning. Chapter in the monograph: Multiple- 
 Valued Logic and Computer Science , edt. David Rine, North-Holland Publishers, 
 1975. 
 
 Michalski, R. S. Synthesis of Optimal and Quasi-Optimal Variable-Valued 
 Logic Formulas. Proceedings of the 5th International Symposium on Multiple- 
 Valued Logic , Bloomington, Indiana, May 13-16, 1975, pp. 76-87. 
 
38 
 
 15. Michalski, R. S. A System of Programs for Computer-aided Induction: A 
 Summary. Proceedings of the Fifth Intern. Joint Conf. on Artificial 
 Intelligence, MIT, 22-26 August, 1977. 
 
 16. Michalski, R. S. Designing Extended Entry Decision Tables and Optimal 
 Decision Trees Using Decision Diagrams. Report No. UIUCDCS-R-78-898, 
 Department of Computer Science, University of Illinois, Urbana, IL., 
 March 1978. 
 
 17. Michalski, R. S., Kulpa A. A system of programs for the synthesis of 
 switching circuits using the method of disjoint stars, Foundations of 
 Information Processing, IFIP Congress, Ljubliana, August 1971. 
 
 18. Svoboda, A. Grafico-mechanicke pomucky uzirane pri analise a synthese 
 kontaktowych obvodu. Stroje Zpracov . Inf. No. 4, 1956 (in Chech). 
 
 19. Veitch, E. W. A Chart Method for Simplifying Truth Functions. Proc. Assoc. 
 Comput . Machine Conf . , May 2-3, 1952, pp. 127-133. 
 
 20. Venn, J. On the Diagrammatic and Mechanical Representations of Propositions 
 and Reasoning. The London, Edinburgh and Dublin Philosophical Magazine and 
 Journal of Science, vol. 10, 1880. 
 
ILIOGRAPHIC DATA 
 EET 
 
 1. Report No. 
 
 UIUCDCS-R-78-897 
 
 ritle and Subtitle 
 
 [ PLANAR GEOMETRICAL MODEL FOR REPRESENTING MULTIDIMENSIONAL 
 DISCRETE SPACES AND MULTIPLE- VALUED LOGIC FUNCTIONS 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 January 1978 
 
 6. 
 
 Luthor(s) 
 
 tyszard S. Michalski 
 
 8- Performing Organization Rept. 
 
 No - UIUCDCS-R-78-897 
 
 'erforming Organization Name and Address 
 
 Iniversity of Illinois at Urbana-Champaign 
 )epartment of Computer Science 
 Jrbana, Illinois 61801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 Sponsoring Organization Name and Address 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 Supplementary Notes 
 
 tpstracts ^ general planar model of multidimensional discrete spaces (a diagram) is des- 
 ibed which can be used for geometrically representing binary, multiple-valued, dis- 
 ete and variable-valued logic functions. It is essentially a multiple-valued exten- 
 an of Marquand's binary diagram, with an additional feature of varying thickness of 
 ties representing axes of variables. 
 
 The diagram has been used extensively over the years by the author and his collabo- 
 tors, and has proved to be useful for such tasks as: logic design and development of 
 ficient algorithms for optimization of switching circuits of many variables, detection 
 
 symmetry in binary or multiple-valued logic functions, fast conversion of normal 
 rms of switching functions to exclusive-or polynomial forms, design of algorithms for 
 iuctive inference and pattern recognition, optimization of decision tables and their 
 iversion to optimal decision trees. 
 
 A method for recognizing in the diagram certain constructs (cartesian complexes) , 
 Lch are important for various applications, is presented and illustrated by two 
 amples: one, involving a determination of a classification rule, and another, involv- 
 g a synthesis of a variable-valued logic expression. 
 
 Key Words and Document Analysis. 17a. Descriptors 
 
 Logic diagram, Marquand diagram, Karnaugh map, Veitch diagram, discrete spaces, 
 multiple-valued logic, variable-valued logic, logic design. 
 
 CR categories: 3.61, 3.63, 5.20, 5.21, 6.1 
 
 Identifiers/Open-Ended Terms 
 
 COSATI Field/Group 
 
 Availability Statement 
 
 Please Unlimited 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 42 
 
 22. Price 
 
 M NTIS-39 ( 10-701 
 
 USCOMM-DC 40329-P7 1 
 
t 
 
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