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}U>'9^f UIUCDCS-R-75-729 
 
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 SYM-1 
 
 A Program That Detects 
 Symmetry of Variable-Valued Logic Functions 
 
 by 
 
 Gerald M. Jensen 
 
 May 1975 
 
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 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/sym1programthatd729jens 
 
UIUCDCS-R-75-729 
 
 SYM-1 
 
 A Program That Detects 
 Symmetry of Variable -Valued Logic Functions 
 
 Gerald M. Jensen 
 
 Department of Computer Science 
 
 University of Illinois at Urbana- Champaign 
 
 Urbana, Illinois 6l801 
 
 February 1975 
 
 This work was partially supported by NSF DCR 7^- O351U 
 
ACKNOWLEDGMENT 
 
 The author would like to thank Thomas Tan for his initial 
 development of the current SYM-1 program, and Professor R. S. Michalski 
 for his valuable assistance in writing this report. 
 
 This work was partially supported by NSF DCR 7^-0351^. 
 
SYM-1: 
 
 User's Guide and Program Description 
 
 ABSTRACT 
 
 This paper describes an algorithm, and a PL/l program ( SYM-1 ) 
 implementing it, that detects symmetry of a variable -valued logic 
 function with regard to the variables or their inverses. If symmetry 
 exists, symmetric variables are then used in symmetric selectors which 
 will describe the set of events in a more compact way than corresponding 
 non-symmetric selectors. 
 
I . Introduction 
 
 This paper contains a user's guide and brief description of an 
 algorithm that detects symmetry among variables in a VL, function. The 
 algorithm is implemented as a PL/l program, SYM-1. If symmetry exists, 
 the program creates a symmetric selector using the symmetric variables. 
 
 A definition of the concept of selectors is given in paper-'-, 
 however, to make this paper self-contained, a brief review of the 
 definition of a VL-, selector is necessary. 
 
 A selector is defined as the following form 
 
 [L # R] 
 
 The left part L or referee is a sequence of literals, i.e., variables, x., 
 
 or their compliments, x.' . If the sequence has more than one literal, then 
 
 they are separated by the symbol '+' which denotes normal arithmetic sum. 
 
 The value of xj is <z(.-x., where d. is the maximum domain element for the i"^ 
 ill 7 
 
 domain and x. is the value of x. . The selector relation (#) is one of the 
 
 1 1 v " ' 
 
 following relations =, ^, ^, g. The right part R or reference is a sequence 
 of one or more non-negative integers in increasing order. They are 
 separated by ', ' or ' : ' . The form c :c is a compressed way to represent 
 a sequence of consecutive integers from c to c , inclusive. A selector 
 is said to be satisfied if the left part L is in relation § with the right 
 part R. The following is an example of a selector: 
 
 [^1,3,5:7] 
 
 This selector is satisfied if the value of x does not equal 1,3,5*6, or 7- 
 
 A symmetric selector is a selector in which there is more 
 than one literal in the referee (L) . The following is an example of a 
 symmetric selector: 
 
Domains of Input Variables 
 [x 1+ x 2 +x 5 =0,5,7] \ = {0,1,2,3,4} 
 
 1=1,2,5 
 
 The above formula is satisfied if the variables x , x and 
 
 x,_ arithmetically sum to or 5 or 7- The function expressed by the 
 5 
 
 above formula is symmetric with regard to variables {x ,x ,x } since the 
 value of these variables can be exchanged one for another without changing 
 the value of the function, i.e., whether it is satisfied or not. 
 
 For a given function, total symmetry is defined as symmetry 
 with regard to all variables with the same domain. Partial symmetry is 
 defined as symmetry with regard to some, but not all, variables with the 
 same domain. A minimal symmetric selector is a symmetric selector which 
 covers the given event set but has a minimum number of references, and 
 therefore a minimal symmetric covering. 
 
 VL formulas are made up of selectors and are used to describe 
 groups of events. Constants outside the selectors are elements of the 
 output domain, D. The juxtaposition of the selectors and constants is 
 interpreted as a logical product, and is called a term. If all the 
 selectors in a term are satisfied, the term has the value of the constant 
 in front of it. This is also the value of the formula. The following is 
 an example of a VL formula and its evaluation. 
 
 Given that the domains of the variables x , x ,x ,x, , and x 
 are respectively: 
 
 D 1 = {0,1,2} 
 
 D 2 = {0,1,2,3,4} 
 
 D 5 = {0,1,2,3,4} 
 
 L^ = {0,1,2,3} 
 
 D 5 = {0,1,2,3,4,5,6,7,8} 
 
The domain of the values of formula (output domain) is D = [0,1,2,3,4], 
 and the formula can be interpreted as the following function: 
 
 f : D n XD xD z xD, XD C -+ D 
 -L ^ 3 ^ 3 
 
 (where x is the cartesian product) 
 
 4[x 3 =2][x 1 ^l][x^0:3^] j V3[x 2 +x 3 =4] v2[x ± =l][yLj>2] V l[x^0,2] 
 
 a term 
 
 This formula takes on values as follows 
 
 Assigned Value 
 
 k 
 3 
 
 Condition # » 
 
 Description of Condition 
 
 1 
 2 
 
 J 
 \ 
 
 x-,=2 and x.,^1 and x tr =0 or 1 or 2 or 3 or 5 
 3 1 3 
 
 if condition (l) doesn't hold (one or more 
 selectors are not satisfied) and 
 
 if condition (l) and (2) do not hold and 
 
 x n =1 and x_> 2 
 1 3 
 
 if condition (l) and (2) and (3) do not 
 hold and x, /o or 2. 
 
 If all h conditions do not hold, the function has value 0. 
 
 A"c the present time the program can generate only minimal symmetric 
 selectors. A planned extension of this is the generation of VL, formulas 
 containing non-symmetric selectors. This would lead to more efficient 
 coverings of specified events. For example, if no events in a given 
 set have x-.=l, and symmetry exists between x , x , and x , then the 
 formula [x -,^1] • [x +x +x=4, 5] would be a better representation than 
 
[x +x +x^,=K,5]. The former is better because it covers less unspecified 
 events. Work is continuing in this direction. 
 
 II. The Algorithm 
 
 The algorithm operates on user specified, disjoint, event 
 classes. 
 
 For each event class: 
 
 1. Partition variables into D-groups 
 (groups of variables with the same domain) 
 
 2. Check all variables in each D-group for 'total 
 symmetry' (see test below) 
 
 3» For each D-group . that step 2 fails, further 
 partition into CS-groups. 
 
 (A CS-group is defined as groups of variables 
 with the same column sum. The column sum of a 
 variable, Xj_, is the sum of the i^h component 
 of events in one class.) 
 
 k. Check all variables in each CS-group for 'total 
 symmetry', with respect to this group 
 
 5. For each CS-group that step h fails, find 'partial 
 symmetry' of variables in each CS-group, using 
 pairwise symmetry test. 
 
 The total symmetry condition is satisfied. 
 
 1. If all possible ways for the symmetric variables 
 to sum to a particular number are contained in the 
 event class. 
 
 OR 
 
 Ratio of number of events in this event class to 
 total number of events satisfying condition #1 
 is greater than a user specified compact ratio 
 (i.e., the selector is compact). 
 
 AND 
 
 Events in other event classes do not satisfy this 
 condition 
 
Example of symmetry condition 
 
 If one event in an event class has x, +x +x =3> then to specify 
 the symmetric selector (x +x +x =3) requires all other 
 possible ways to sum to 3j with these variables, must be 
 present in this event class. 
 
 OR 
 
 - event in this event class that sum to 3 user specified 
 # events satisfying above condition compact ratio 
 
 (typically 1.0 or 0.5) 
 AND 
 Events in all other event classes do not satisfy x^ +x +x =3 • 
 
 Program statistics 
 
 The current version of the program consists of about ^50 PL/l 
 statements and the object module produced by the PL/l (F) compiler requires 
 memory capacity of about 7°K bytes, with PL/l library routines. The 
 execution of the following example on an IBM 360/75 required l8.0 sec. 
 compile time and 2.2 sec. execution time. 
 
 ! : br - users, the program and necessary JCL (job control 
 language) are found in ^113,1376 under files 'SYMl' and 'JCL' respectively. 
 To run on I.B.M. 360 use a I JO I statement with the list 'JCL, SYML, DATA', where 
 data is the input data of the form given in the following section. 
 
 III. Users Manual 
 
 The use of the program requires the specification of program 
 parameters. This is best illustrated by an example. 
 
Example of input data 
 
 r 
 
 in PL/l 
 list format 
 
 in FL/l 
 data format 
 
 ( 
 
 r 
 
 in PL/l 
 list format 
 
 Variable 
 Value 
 
 4- 
 11 
 
 1 
 
 4 
 
 MAXPAT=500; 
 
 0.5 
 4 4 5 5 
 
 7 4 
 
 3 4 2 
 
 12 3 1 
 
 2 12 
 
 3 4 3 
 
 13 3 2 
 
 2 2 2 1 
 
 3 110 
 
 3 
 112 
 
 10 2 1 
 
 3 3 3Q 
 
 Variable 
 Name 
 
 > 
 
 <NV> 
 <NR0W> 
 
 <MAXCL> 
 
 <^AXD> 
 
 <MAXPAT> 
 
 <CRITERIA1> 
 <DLIST(1:NV)> 
 
 <NE(0:MAXCL)> 
 
 \<E(1:NR0W,1:NV)> 
 
 Comments 
 
 Number of variables 
 Total number of events in 
 
 all classes 
 Number of last class (i.e., 
 
 # classes=MAXCL+l) 
 Largest domain of all input 
 
 variables 
 Maximum number of events 
 
 satisfying a given selector 
 User specified compact ratio 
 Domain sizes of variables 
 
 (i.e., # elements of the 
 
 domain) 
 Number of events per class 
 Event array 
 
 Class has 7 events 
 Formula is 0((x +x =3, 4) & 
 
 (x 3 +x i ;=5,6)) 
 
 Class 1 has 4 events 
 Formula is l( (x +x =0, 1, 6) & 
 
 (x 5 +x^=3)) 
 
 12 
 
 end of input 
 
 The data in the bold-faced box is the actual input to the 
 program. In this example the variables are partitioned into two 
 D-groups, (4, 5) • 
 
 IV. Output Examples with G.L.D.s 
 
 The following pages are examples of actual program output with 
 corresponding G.L.D.s. 
 
^AX T MUM OA^-EFN S*ZE 
 CfMPACT_PAT T "l C'TFRIA = 
 Nt'MBErt °F VARIABLES = 
 NUMBER 3T EVEN'S 
 NliMBEP r F CLASSES 
 MAXIMUM LEVEL 
 
 NI.IMBFR 
 NUMBER 
 CI ASS 
 
 I 
 CI ASS 
 EVEN" 
 EVE*'~ 
 EVENT 
 EVENT 
 EVEN" 
 EVEN" 
 EVEN- 
 CLASS 
 EVFN T 
 EVEN T 
 EVEN- 
 EVENT 
 
 TT"E: 
 TOTAL 
 
 r F LEVELS FOR EACH 
 CF EVENT SPECIFIED 
 •EVENTS 
 
 500 
 
 0.50 
 
 4 
 11 
 
 2 
 
 4 
 VARIABLE 
 
 FOR EACH 
 
 : 4 
 CLASS 
 
 F( 0) 
 
 NO. 
 
 NO, 
 
 MS. 
 
 NC. 
 
 N ). 
 
 NC. 
 
 NT. 
 
 F( 1) 
 
 NC. 
 
 NC. 
 
 NC. 
 
 N?. 
 
 7 
 4 
 
 1 = 
 
 2= 
 
 3 = 
 
 4 = 
 
 5 = 
 
 6 = 
 
 7 = 
 
 1 = 
 
 -> - 
 
 3 
 2 
 
 1 
 
 3 
 ? 
 
 1 
 
 3 = 
 
 4 = 
 
 15175S78C 
 HHMM5S W S 
 
 NUMBER n F SY^ME T RIC 
 
 3 
 ? 
 4 
 3 
 2 
 I 
 
 
 I 
 ?. 
 3 
 
 2 
 
 1 
 
 
 3 
 
 2 
 
 L 
 
 
 3 
 2 
 
 I 
 
 
 VARIABLES IN CLASS = 
 
 F rt ?MULA U 0(X3+X4» =5,6) C (X1»X2=3,4) 
 
 CLASS 
 
 MTNIMIJM 
 
 TIME: 
 
 r TAL 
 
 CrwoACT f ATin = 
 COMPACT c A X IT = 
 15175«;81C 
 
 HFMMSS v r 
 NUMB E& OF ^Y^ME T RIC 
 
 0.14 
 I. 00 
 
 VA^IAdLES IN CLASS 1 = 
 
 FPQMULA TS 1(X3+X4=3) B ( XH- X2= , 1 , 6 ) 
 
 CLASS C p MPACT P A'TF = 
 MINIMUM C° M PACT RATIO = 
 -tme: 15l7Sq88C 
 HhMMSS MS 
 
 0.25 
 
 1.00 
 
 Output for User's Guide Example 
 

 4 3 z I o|c 
 
 -~m 
 
 4UI 
 
 ^>gjH^^7[us)H^r»^^uV|^ ^3+*/ 
 
 ^^^^0;::::;: 8 •::;;::: 
 
 e 
 
 o ; 
 
 1_3 
 
 2 
 
 5 
 
 x,+x, 
 
 I 2 1 w 
 
 3 
 
 / 
 
 3 1 IIS 
 
 rj- 
 
 D. 
 
 D 
 U 
 
 -SI 
 
 £1 
 
 D 
 15" 
 
 : i 
 
 Class O ■■::: 
 
 
 - — ft *•*./* 3} 
 
 t 
 
 |* ; +X,0.'>j 
 
 \ 
 
 \ 
 \ 
 
 wsu7 Y50 7 y| X^-rVy 
 
 X. y^ 
 
 CD 
 
 7 
 
 u 
 
 U 
 
 u 
 
 1 
 
 1 ' 
 
 
 
 1 ' 
 1 I 
 
 a 
 
 u 
 
 UJ 
 
 J ^ 
 
 — - 
 
 -i ' 
 
 3 
 
 
 
 
 - • 
 
 - 
 
 3 ; 
 
 n 
 
 n 
 
 n 
 
 £_ 
 
 - ■ - 
 
 
 O - 3 H 
 
 
 
 / 
 
 •2. 
 
 ^ / Z. 2 w 
 
 3; 
 
 oi z 3 V 
 
 V: 
 
 Cl4ss: j:::;::::::: : . 
 
 Xz 
 
 GLD's for User's Guide Example 
 
 Circled areas are events covered by the particular 
 selector. . . 
 
 
 , i , 
 

 
 
 
 
 
 
 
 9 
 
 MAXIMUM "AT T 
 
 EF N SIZE 
 n p'JTTE^Ift 
 
 500 
 = P. 25 
 
 
 
 NUMBFF np V«*T?BLES = 
 NUM«FP PF EVENTS = 
 
 2 
 31 
 
 
 
 
 NUMBER ~ c CL^SSf 
 v«e xtmuv i.F v c t. 
 
 ES 
 
 ~ 
 
 4 
 
 
 
 
 MIjmupr -p l PV El S F0° E***H VA D HRLE : 4 4 4 
 NUMREP <"<F EVEN' ^PEflFTEO F<~.F F.'CH CL'SS : 
 
 CLASS 
 
 
 ¥FVE 
 
 nITS 
 1 
 
 
 
 
 
 
 
 1 
 2 
 
 
 6 
 12 
 
 
 
 
 
 
 
 Cl*SS 
 
 F( 0) 
 
 12 
 
 
 
 
 
 
 
 EVE iv r 
 CI -vss 
 
 Mr. 
 F( i ) 
 
 1 = 
 
 
 -> 
 
 
 
 3 
 
 
 
 EVEN" 
 EVEN" 
 
 NC. 
 NO. 
 
 1 = 
 
 7 = 
 
 
 
 
 
 I 
 
 
 3 
 
 2 
 
 
 
 EVENT 
 EVFNli- 
 
 NO. 
 
 MP. 
 
 3 = 
 
 4- = 
 
 
 1 
 
 n 
 
 
 
 2 
 
 3 
 3 
 
 
 
 EVEN:* 
 EVENT 
 
 NQ . 
 
 5 = 
 
 6 = 
 
 
 
 
 
 
 1 
 
 
 2 
 
 1 
 
 
 
 CLASS 
 
 EVEN" 
 
 F( 2) 
 
 NC. 
 
 1 = 
 
 
 
 
 2 
 
 2 
 
 
 
 EVENT 
 EVENT 
 
 NC. 
 \ip. 
 
 2 = 
 
 •a = 
 
 
 
 
 2 
 
 1 
 
 
 
 
 FVFNT 
 E\' C NT 
 
 NC. 
 MP. 
 
 4= 
 5 = 
 
 
 I 
 1 
 
 2 
 2 
 
 _ Second 
 
 Example 
 
 
 EVEN" 
 EVENT 
 
 NO. 
 VD. 
 
 6 = 
 
 7 = 
 
 
 1 
 
 1 
 
 1 
 
 
 1 
 
 
 
 
 EVEN" 
 FVENT 
 
 NO. 
 
 MP. 
 
 8 = 
 
 9 = 
 
 
 2 
 2 
 
 2 
 i 
 
 3 
 7 
 
 
 
 FVENT 
 PVEN T 
 
 NC. 
 "C. 
 
 10= 
 
 11 = 
 
 
 2 
 3 
 
 
 1 
 
 1 
 3 
 
 
 
 EVEN T 
 CLASS 
 
 F( 2 ) 
 
 12 = 
 
 
 3 
 
 
 
 2 
 
 
 
 EVENT 
 cyPN" 1 " 
 
 NC. 
 MP. 
 
 1 = 
 
 2 = 
 
 
 G 
 
 
 3 
 2 
 
 1 
 
 
 
 
 
 EVFN" 
 EVEN^ 
 
 NO. 
 
 NO. 
 
 3 = 
 
 4 = 
 
 
 1 
 1 
 
 3 
 2 
 
 2 
 
 1 
 
 
 
 ^VEN^ 
 EVENT 
 
 NO. 
 
 N n . 
 
 5 = 
 
 6 = 
 
 
 i 
 2 
 
 i 
 
 3 
 
 
 3 
 
 
 
 EVEN" 
 pv/PNT 
 
 • in 
 ' v . 
 
 NO. 
 
 7 = 
 
 8 = 
 
 
 2 
 
 2 
 
 2 
 
 i 
 
 2 
 
 
 
 EV?N T 
 EVEN" 
 
 NC. 
 MP. 
 
 9 = 
 10 = 
 
 
 2 
 
 3 
 
 
 2 
 
 
 
 3 
 
 
 
 EVENT 
 EVENT 
 
 MP, . 
 NC. 
 
 11 = 
 
 12 = 
 
 
 3 
 3 
 
 1 
 
 c 
 
 2 
 1 
 
 
 
 T IME : 
 
 162 
 
 MM 1 ' 
 
 645930 
 M S S M !? 
 
 
 
 
 
 
 
 N l'-ip = ~ 
 
 " P 
 0< X. 
 
 . + X 
 
 : + x ? ' 
 
 -01 1 
 
 : T A«LES IN CLASS 
 
 = 3 
 
 
 r l a 3 s c r M p * r v 
 
 
 ' F = 
 
 " TO = 
 
 1.00 
 
 l .: r 
 
 
 
 T IM£: 
 
 162 
 
 6461 
 
 .30 
 u c 
 
 
 
 
 
 
 
 rr*~M 
 
 MIIMPep 
 
 i~F 
 
 *Y 
 
 y vPTr 
 
 -- VA 
 
 "TARLF5 IN CLfiSS 1 
 
 = 3 
 
 
 
 F^sv.iil 
 
 * T c 
 
 " ( X 
 
 .♦ < 
 
 :*x -• 
 
 = 1 ,'") 
 
 ! 
 
 MI NT M U M "<""*? 
 
 
 
 • ir = 
 
 - TP = 
 
 0.6 
 0.5 
 
 7 
 3 
 
 
 | 
 
 i 
 
 IMP 
 
 1 6?646' , 6n 
 HHMMS* MS 
 
-~T,n m).mppc ^f SY MM FTPTr V * CT tPt_FS T N CLASS 2 = 
 FTRMUL * T r 2( XHX7. + X3' =4) 
 
 10 
 
 1 ^ p 
 1,00 
 
 C L * S S r r m p ""~ ~ r h T 1 - 
 
 -TMF: i~6 2646 430 
 
 HH" M l fC M* 
 
 cr 5mi.il 4 ! c 3UH-X2+X3' = 5) 
 
 Z Lt S S ■ 
 i-tme: 
 
 pMP '. " " s ^ " jr = 
 
 r v p i* i'- - " * 7 to = 
 
 162647/60 
 HHMMSS MS 
 
 1 ,00 
 1.00 
 
 Note 
 
 GLD's for Second Example 
 
 The number in the cells of the GLD's represent the sum x-^x^+x' 
 Events circled by a solid line are the given events for 
 each class. Events circled by a dotted line are the events 
 covered by the particular selector. 
 
 X 
 
 tfi 
 
 1 
 
 5 
 
 1 
 
 ■2. 
 
 — 
 
 jo] 
 
 
 2 / 
 
 5 
 
 <V 
 
 5 
 
 1 
 
 to 
 
 s 
 
 H 
 
 3 
 
 / 
 
 H 3> 
 
 t 
 
 2 
 
 ( 
 
 < S 
 
 3 
 
 2 
 
 & 
 
 s 
 
 H 
 
 3. 
 
 7 
 
 c 
 
 5" 
 
 V 
 
 2 
 
 s k |3 
 
 2. 
 
 6is 
 
 </ 3 
 
 -7 
 
 to 
 
 s 
 
 W 
 
 fr 
 
 7 
 
 c 
 
 S 
 
 3 
 
 
 7 & 
 
 1 
 
 5 H 
 
 g* 
 
 7 
 
 c 
 
 5 
 
 T 
 
 <* 
 
 7 
 
 c 
 
 
 
 O / x 3 
 
 
 O 1 2. J 
 
 
 
 c 
 
 ) 
 
 
 i 
 
 / 
 
 
 
 *« 
 
 i 
 
 
 
 
 § 
 
 
 Class q 
 
 X? 
 
 X: 
 
 
 
 
 V, . 
 
 (2 
 
 
 
 
 V 
 
 3 
 
 
 /] 
 
 p 
 
 H |3 
 
 Ll! 
 
 G, j S 
 
 H 
 
 3 
 
 / 
 
 v 
 
 3 
 
 2, 
 
 • 
 
 
 5 
 
 V 
 
 l 
 
 3 : 
 
 (^ 
 
 k 
 
 | 
 
 ^ 
 
 -• 
 
 7 
 
 t 
 
 w 
 
 5" 
 
 H 
 
 2 
 
 5 
 
 H 
 
 I 
 
 3*. 
 
 -vs." 
 
 to- 
 
 5 
 
 H 
 
 ■3 
 
 7 
 
 
 <f 
 
 % 
 
 7 
 
 to 
 
 s 
 
 3 
 
 k? 
 
 £ 
 
 H 
 
 s 
 
 7 
 
 to 
 
 s* 
 
 4 
 
 V 
 
 4 
 
 £" 
 
 1 
 
 sr 
 
 7 
 
 u 
 
 
 11$ 
 
 O / 2 3 
 
 O / X 3 
 
 O / 2 3 
 
 fel CO (0 to) 
 
 (3) (2) CO ^) 
 
 (3^ ^ CO (<3 
 
 C3)C*) CO left 
 
 O 1 2 ' S 
 
 
 V3 
 
11 
 
 GLD's for Second Example (continued) 
 
 X, 
 
 UA 
 
 z 
 
 r^+^+x^j 
 
 cx 3 
 x*. 
 
 Class 3 
 
 £j| rt,+yx+yi»r] 
 
 
 
 X 
 
MAXIMUM PATTERN SIZE 
 :.nMP*CT WTin CRTTfqia 
 
 = 1 
 
 500" 
 .GO 
 
 12 
 
 MljMftPP 
 
 HF V^T^LES - 
 PF FVENTc 
 
 DF CLASSES 
 IM LEVEL 
 
 4 
 
 LEVELS FTP F* 
 C VFN T SPEflFI 
 
 CM 
 FP 
 
 VARIABL 
 
 FOP E& r 
 
 E : 
 
 H r 
 
 
 
 
 2 
 LASS 
 
 ] 
 
 1 
 
 
 
 n 
 
 IN CLASS = 
 
 IN CLASS 1 = 
 
 IN CLASS 2 = 
 
 >1ULA TS 2( Xl+X2+X3+X4=4) 
 
 CLAS 
 
 MTNTMU 
 
 rr 
 
 ~ r 
 T 
 
 H 
 
 MP.ACT 
 
 RA^ 
 
 P .<* T 
 
 Id = 
 IP = 
 
 .CO 
 .00 
 
 62644 
 HMMSS 
 
 710 
 
 Third Example 
 
13 
 
 ftAS5 O 
 
 C<L/bS / 
 
 Class* x 
 
 GLD for Third Example 
 
 Note: The numbers in the cells of the GLD 
 
 correspond to the sum x n +x_+x,+x, , not 
 
 l d $ 4 
 
 their respective class. 
 
v A X ! M 
 
 MJMBE 
 NU M RE 
 NU^BE 
 ''AX IN 
 NUMBE 
 NU W BE 
 CLASS 
 
 1 
 CLASS 
 r VENT 
 EVFNT 
 EVFNT 
 EVENT 
 EVENT 
 F V E NT 
 FVEN^ 
 CLASS 
 EVENT 
 E V E NT 
 EVENT 
 ▼ime: 
 
 C'_ 
 
 p i 
 Q n 
 
 R n 
 
 R n 
 
 P r 
 
 F( 
 
 Nn 
 
 NO 
 
 \ 
 
 NC 
 
 no 
 
 NC 
 
 NO 
 F( 
 NC 
 
 Ml 
 
 PA T 
 
 RAT 
 
 F V 
 F E 
 F C 
 LEV 
 F I 
 F E 
 *FV 
 
 0) 
 
 T EKN 
 
 : r ' c 
 
 AP I A 
 VENT 
 LASS 
 EL 
 
 EVEL 
 
 VENT 
 
 FN Tc: 
 
 7 
 
 3 
 
 2 = 
 
 3 = 
 
 4 = 
 
 5 = 
 
 6 = 
 
 7 = 
 
 SIZE 
 
 B T T E p T A = 
 
 BL ES = 
 S = 
 
 ES 
 
 S F°P FACH 
 SPECIFIED 
 
 500 
 1.00 
 
 3 
 10 
 
 2 
 
 2 
 
 VARIABLE 
 
 FOR EACH 
 
 11+ 
 
 1 ) 
 
 2 3 
 
 HH 
 
 1 = 
 
 2 = 
 
 3 = 
 
 0041910 
 
 2 
 
 1 
 
 2 
 1 
 
 1 
 
 2 
 
 1 
 2 
 
 : 3 
 
 CLASS 
 
 1 - 
 ? 
 
 I 
 ) 
 ? 
 L 
 
 
 2 
 2 
 
 1« 
 
 Same Event 
 
 V V 
 
 >b 
 
 V c 
 
 the PR3GFAV n ">ECATES ON DISJOINT EVEN T CLASSE C . 
 EVEN' NH. 1 IN CLAfc MO. IS THE SAME AS EVENT W 
 Y~ij9 EVE V T CLA^SE C AF E NC"" DISJOINT, 
 T "IS DATA VllL ^ nT BE PUN. 
 
 3 IN CLASS NC 1 
 
 Fourth Example 
 
 Note: This example shows the output message 
 
 when non-disjoint event-classes are run. 
 
15 
 
 V. Concluding Remarks 
 
 The most important attribute of symmetric selectors is the 
 ability to produce a compact representation of certain groups of events, as 
 compared with non-symmetric selectors. This attribute is most useful in large 
 pattern recognition problems or artificial intelligence generalization 
 problems, where numerous groups of events must be represented by 
 selectors. This would result in an obvious reduction of required 
 storage. 
 
 Further extensions of this program. 
 
 1. The partially symmetric case. 
 
 A. Generating selectors which include 
 unspecified events (don't cares) in 
 a consistent manner. 
 
 B. A better routine for determining if a 
 selector variable should be complimented. 
 
 2. The case of non-disjoint event classes. 
 
 3- Specifying VL formulas containing non-symmetric 
 selectors . 
 
 Reference 
 
 Michalski, R. S., "Variable Valued Logic: System VLq_, " Proceedings 
 of the 197*1 International Symposium on Multiple-Valued Logic , West 
 Virginia University, Morgantown, West Virginia, May 29-31, 197^- 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-75-729 
 
 3. Recipient's Accession N< 
 
 4. I 1; ie and Subi it le 
 
 SYM-1 
 A Program That Detects 
 Symmetry of Variable-Valued Logic Functions 
 
 5- Report Date 
 
 May 1975 
 
 7. Author's) 
 
 Gerald M. Jensen 
 
 8. Performing Organ iz.it ion Rept. 
 No. 
 
 9. Pi-H inning Organization Name and Address 
 
 Department of Computer Science 
 
 University of Illinois at Urbana-Champaign 
 
 Urbana, Illinois 6l801 
 
 10. Project/ Task/Work Unit N. 
 
 11. Contract /Grant No. 
 
 NSF DCR 7if-035lU- 
 
 I— 
 
 I ?. "n| iring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 1 pplrmentary Notes 
 
 16. ^hstrac t ; 
 
 This paper describes an algorithm, and a PL/l program (SYM-l) implementing 
 it, that detects symmetry of a variable-valued logic function with regard to the 
 variables or their inverses. If symmetry exists, symmetric variables are then 
 used in symmetric selectors which will describe the set of events in a more 
 compact way than corresponding non-symmetric selectors. 
 
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