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UIUCDCS-R-77-869 
 
 Inductive Inference in the Variable Valued Predicate Logic 
 
 UILU-ENG 77 1725 
 
 System VL ? . : Methodology and Computer Implementation 
 
 by 
 James B. Larson 
 May 1977 
 
 \4* 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
Inductive Inference in the Variable Valued Predicate Logic 
 System VL-. : Methodology and Computer Implementation 
 
 by 
 
 James B. Larson 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 
 
 May 1977 
 
 This work was submitted in partial fulfillment of the requirements for the 
 degree of Docotr of Philosophy in Computer Science in the Graduate College of 
 the University of Illinois and was supported in part by the National Science 
 Foundation, Grant No. NSF MCS 74-03514. 
 
no. ^^-813 
 
 INDUCTIVE INFERENCE IN THE VARIABLE VALUED PREDICATE LOGIC 
 
 SYSTEM VL : METHODOLOGY AND COMPUTER IMPLEMENTATION 
 21 
 
 James Burton Larson, Ph.D. 
 
 Department of Computer Science 
 
 University of Illinois at Urbana-Champaign, 1977 
 
 A formal methodology and computer program are 
 
 oresented for the transformation of a set of user supplied 
 
 logical decision rules into a new, generalized set of 
 
 decision rules which is near optimal according to a user 
 
 supplied criterion. The VL$2 logic system (a multi-valued 
 
 version of a first order predicate calculus) is used as the 
 
 framework for defining and expressing decision rules and 
 
 transformations on decision rules. The program INDUCE_1 
 
 which implements certain inductive inference rules using a 
 
 graphical representation of VL$2 expressions is described and 
 
 some examples of inductive problems solved by the program are 
 
 given. 
 
Ill 
 
 ACKNOWLEDGEMENTS 
 
 I would like to express my gratitude to the following 
 people for their help on this thesis: Pirst, Professor R. 
 S. Michalski for his inspiration, challenging problems, and 
 many significant suggestions especially regarding 
 meta-f unctions, my committee. Professor D. Plaisted and Dr. 
 Don Friesen foe encouragement and many helpful discussions, 
 I especially wish to recognize the valuable support of 
 Richard Chilausky during the early stages of this research, 
 Barr Segal for proofreading of the manuscript, and the many 
 predecessors in the area of inductive inference and variable 
 valued logic whose efforts have been much appreciated in the 
 development of this thesis, I am greateful for the financial 
 support of the National Science Foundation, the Department of 
 Computer Science, the Research Board of the University of 
 Illinois, and the Computing Services Offices, the latter for 
 supplying the computer time to implement the text formatter 
 used for this paper. 
 
 Finally, without the moral encouragement of my wife, 
 Rhonda, this work surely would have ended years ago from 
 utter frustration and discouragement. 
 
IV 
 
 PREFACE 
 
 This paper was prepared on a CYBEF 175 computer at 
 the University of Illinois using a text formatter written by 
 the author. Since several special characters were not 
 available on the print train of the printer, the following 
 character combinations have special meaning: 
 
 Character Meaning 
 
 v logical disjunction 
 
 6 logical conjunction 
 
 6_ set intersection 
 
 E the existential quantifier 
 
 1 the universal quantifier 
 
TABLE OP CONTENTS 
 
 CHAPTER PAGE 
 
 1. Introduction. •••••••••••••••••••••••••••••••• •• •• 1 
 
 1.1 The Problem 2 
 
 1.2 Previous Specific Applications. ••••••••••••••••••• • 4 
 
 1.3 Formal Systems for Inductive Inference. •••••••••• •• 7 
 
 1.4 Overview of the Following Chapters •••••••••• 12 
 
 2. Representing Decisions in the VL System. •••••••••••• • 14 
 
 2.1 VL System Structure. ••• • • ••• 16 
 
 2.2 Selector Formation and Interpretation Rules ••••••• 17 
 
 2.3 VL Formation Rule • • 21 
 
 2.4 Interpretation Rules. ••••••••••••••••••••••••••••• • 22 
 
 2.5 VL Decision Rules •• •••••••• 26 
 
 3. VL Transformation Rules. ••••••••••••• • 30 
 
 3.1 Eguivalence Transformation. • •••••••••••••• 31 
 
 3.2 Generalizing Transformations. •••••••••••••••••••• •• 33 
 
 3.3 Specializing Transformations. •••••••••••••••••••• •• 37 
 
 3.1 Transformation Rules Involving the Decision Part. • • 38 
 
 3.5 Example of Application of Transformation Rules..... 40 
 
 3.6 Efficient Application of Generalization Rules...... 45 
 
 4. Computer Representation of VL Decision Rules. •••••••• • 49 
 
 5. Algorithms and Computer Implementation................ 54 
 
 5.1 Input to the Program. •••••••••••••••••••••••••••• •• 54 
 
vi 
 
 5.2 Program Output, •••••••••••••••••••••••••••••••••• •• 61 
 
 5.3 Formation of a Complete Generalization. ••••••••••• • 64 
 
 5.4 Determine Cover and Intersection of 2 Formulas..... 6 5 
 5*5 Trimming a Set of c-f ormulas. • • • • •••••••••••• 70 
 
 5.6 Pormation of a Set of Consistent Generalizations. • • 73 
 
 5.7 Extending the References of a Consistent c2-formula 75 
 
 5.8 Adding Hew Functions and Predicates to c-f ormulas.. 79 
 
 6. Examples of Decision Rule Generation using INDUCE_1... 84 
 
 6.1 Figures Example (EX 1) • ••••••• 89 
 
 6.2 Arch Example (£1 2) 102 
 
 6.3 Trains (EX 3) 107 
 
 6.4 Textures (EX 4) 112 
 
 7. Current Limitations and Possible Extentions. ......... . 117 
 
 LIST OF REFERENCES 123 
 
 APPENDIX A 129 
 
 APPENDIX B. 138 
 
 VITA •••••• • 145 
 
1, Introduction 
 
 An important problem which is often presented to 
 computer systems is that of extracting relevant information 
 from complex data in order to gain a better understanding of 
 the meaning behind such data. Most current methods are 
 incapable of adeguately describing highly structured 
 situations and produce results which are difficult to 
 interpret, A selection of those systems which overcome these 
 difficulties is given in sections 1,2 and 1,3, 
 
 The following chapters deal with finding useful (as 
 defined by the user with an optimality criterion) , 
 generalized information about sets of situations represented 
 as logical VL decision rules, A decision rule is a form 
 
 CONDITION => DECISION 
 
 where CONDITION describes some set of situations and DECISION 
 describes some new situation or action which is indicated if 
 an existing situation satisfies the description in 
 CONDITION. If no situation satisfies the CONDITION in a 
 decision rule, the rule makes a NULL decision. The 
 descriptions in CONDITION and DECISION are represented in the 
 VL logic system. This system is a variable valued first 
 order predicate caluculus with a rich set of operators and 
 
the facility for allowing user defined domain sizes and 
 structures for variables and functions appropriate for the 
 problem at hand. The approach taken here is to apply 
 inductive inference rules to logical decision rules which 
 express some decisions made with sets of situations in order 
 to form new, near-optimal decision rules which retain the 
 decision making capabilities of the original rules. 
 
 1.1 The Problem 
 
 The specific induction problem being investigated is 
 as follows: 
 Given a set of decision rules: 
 
 C =>D ; C =>D ; •• • C =>D 
 
 1,1 1 1,2 1 1,t1 1 
 
 C =>D ; C =>D ; ... C ->D 
 
 2,1 2 2,2 2 2,t2 1 
 
 (1.1) 
 
 C =>D ; C =>D ; ... C =>D 
 n,1 n n,2 n n,tn n 
 
 where C and D are expressions in the VL system which 
 i,1 i 2 
 
 represent the CONDITION and DECISION parts of decision rules 
 respectively, then find, through an application of 
 generalization rules, a set of ?L decision rules: 
 
C =>D 
 
 1 1 
 
 C =>D 
 
 2 2 
 
 (1.2) 
 
 C =>D 
 n n 
 
 which are with regard to the rules (1.1): 
 
 1) consistent 
 
 2) complete 
 
 3) optimal with regard to a user supplied optimality 
 criteria. 
 
 The new rules are consistent if for any situation for which 
 the new rules assign a decision (a non-HULL decision) , the 
 initial rules assign the same decision or a NOLL decision. 
 They are complete if for any situation for which the initial 
 rules assign a decision, the new rules assign a decision. 
 From the initial rules, it is usually possible to derive many 
 sets of rules which are consistent and complete. Therefore, 
 a criterion of optiaality (defined by a user according to his 
 problem) is used to select a few alternatives which are most 
 
desirable according to the specific induction problem. The 
 attention is restricted to sets of rales which make only one 
 decision for a given situation* 
 
 1.2 Previous Specific Applications 
 
 Inductive inference is used here to describe a formal 
 method for rewriting or generalizing available data in order 
 to give new information about a problem and make new 
 decisions which could not be obtained before. Statistical 
 methods are probably the most widely used forms of inductive 
 inference. These methods reguire a great deal of a-priori 
 knowledge including the availability of a large set of data, 
 knowledge about the interpendence of variables on each other, 
 and an understanding of the type of underlying distribution 
 of the data [Croft-71 ]• In addition, statistical results may 
 be difficult to read and interpret [Larson 76] (e.g. a 
 conditional probability matrix). 
 
 The first approach to automated inductive inference 
 using logic was most likely developed by Hunt [Hunt 66]. He 
 described a number of different schemes for generating 
 decision trees which can be used to distinguish between sets 
 of letter sequences. Although a decision tree produces an 
 elegant procedure which can be easily executed on a computer, 
 it lacks the flexibility necessary to represent more general 
 concepts. 
 
The BEURISTIODENDRAL program [Buchanan et.al. 69] 
 provides a model appropriate for representing the structure 
 of chemical compounds and some transformations representing 
 possible chemical reactions which can be applied to the 
 compound representations under certain known physical 
 constraints. The program finds a set of possible structures 
 of a compound knowing its empirical formula and mass 
 spectrometer data by suggesting various structures for the 
 compound and applying transformations to the structures under 
 the guidence of a set of heuristics based on the mass 
 spectrometer data. The meta-DENDRAL program [Buchanan 
 et.al, 72] finds a general mechanism or theory which explains 
 the transformations which take place relying on the knowledge 
 of those transformations which are plausible and those which 
 are forbidden. 
 
 Computer aided medical diagnosis is another area in 
 which logical inductive inference methods have been 
 suggested. Pople [ Pople et.al, 72] have suggested a graph 
 structure representation of biomedical facts and an approach 
 to forming theories by finding common subgraphs using user 
 supplied suggestions. Of particular note is work in the area 
 of computer aided medical diagnosis and plant pathology 
 [Hichalski 73, 74, Chilausky et.al 76, Larson 76] use the VL 
 system (a variable valued logic system which is the precursor 
 
to the VL system used here) and the program AQTAL/1-AQ7 to 
 2 
 
 infer descriptions of clases of liver diseases and soybean 
 
 diseases. The latter work is a specific application of a the 
 
 VL logic system to several problems. 
 
 Winston [Winston 70] demonstrates specific procedures 
 
 vhich discover descriptions from examples in the toy blocks 
 
 world. A description which is descriminant (i.e. can be used 
 
 to distinguish one object or set of objects from other sets) 
 
 is formed by matching the similar parts of the object under 
 
 consideration with another object (near-miss) and then 
 
 isolating the structures which are different between the two 
 
 objects. This differs from the approach taken in the 
 
 following chapters in that matching is only done here in 
 
 order to adequately describe some specific feature which 
 
 distinguishes between two objects. (e.g. to specify the 
 
 second part from the top in an object, one may have to 
 
 include some predicates which define the second from the top 
 
 in terms of other descriptors. If the distinguishing feature 
 
 involves this part, then the definition of the part used in 
 
 the description must be common to both objects.) Winston also 
 
 uses modifiers such as •must 1 , 'may 1 , •must not 1 in 
 
 descriptions. A form of these modifiers is inherent in the 
 
 VL approach (e.g. discriminant descriptions involve the 
 2 
 
 •must 1 modifier, descriptive descriptions involving only set 
 
of objects yields descriptions involving a type of 'may 1 
 modifier) • 
 
 The program ARITHHETIC [Bongard 70] finds an 
 algebraic rule which explains sample relationships. The 
 program is given sets of tables, each table containing 
 3-tuples of a ternary relation. A set of 33 predicates are 
 used in the program (although not explicitly given in the 
 reference) where a predicate may be: e.g. if the guotient of 
 the first two elements of the 3-tuple is positive, then the 
 predicate is true. Por each row of a table, a set of 
 features is generated by finding Boolean combinations of the 
 predicates applied to the row. A feature describing the 
 table is generated for each Boolean combination by finding 
 the product of Boolean combinations for all rows. The set of 
 features which appear to be most useful in distinguishing one 
 table from the others is selected as the description of each 
 table. 
 
 1. 3 Formal Systems for Inductive Inference 
 
 A criticism of some of the above endeavors is that 
 they are problem specific. More general systems which have 
 many possible application areas have emphasis in two types of 
 approaches: 1) generation of descriptions of sets of objects 
 represented in a logic system of some kind , and 2) creation 
 
8 
 
 of nev concepts in a sequential manner by generating and 
 
 modifying hypotheses* k summary of several types of learning 
 
 systems can be found in [Banerji 75], Of particular interest 
 
 here is the work of Horgan [Horgan 72] in which a formal 
 
 system based on the first order predicate calculus with 
 
 falsehood preserving transformations is presented. Briefly, 
 
 the idea that inductive inference can be described as 
 
 backwards reasoning is apparently not sufficent for a 
 
 practical system. For example, if E was derived from the 
 
 2 
 
 assertions -»E and E v E , then backward reasoning would 
 1 1 2 
 
 somehow have to generate the two assertions above given only 
 E • There are far to many expressions E which can be applied 
 to a situation such as this to yield a practical system. 
 Instead, Horgan defines a falsehood preserving transformation 
 of a deductive inference rule into an inductive inference 
 rule (in Horgan^s notation) : 
 
 E |- E 
 1 P 2 
 
 where E is false for every interpretation in which E is 
 2 1 
 
 false. A set of transformations (P-rules) can be created to 
 
 convert a deductive logic systeoi into an inductive logic 
 
 system (e.g. if E and E are atomic expressions, then 
 
 1 2 
 
 E V E |- E E 
 1 2 P 1 2 
 
i.e.. from the disjunction E v E , one may infer the 
 
 1 2 
 
 con-junction E E while preserving falsehood)* With this 
 
 12 
 
 system, theorem proving technigues using falsehood preserving 
 rules may be applied to inductive problems, (In later 
 chapters, the symbol |< is used to denote such a 
 •generalizing 1 transformation.) 
 
 A number of authors have presented systems which use 
 a graph structure representation of expression in a type of 
 logic system. The 'parameterized structure representation 1 
 of Hayes-Roth [Hayes-Roth 76] is used in inductive tasks 
 which learn descriptions of sets cf objects and 
 transformations from one set of objects to another set of 
 objects from examples. The problems addressed are closely 
 akin to the work in the following chapters cf this paper, one 
 of the objectives being to find the common properties of all 
 examples of one class which are available to the system using 
 a graphical representation. The number of possible 
 alternatives in Hayes-Roth* s method is liirited by a fixed 
 utility function which evaluates intermediate results and 
 discards hypotheses of low utility. The work differs from 
 that presented in the following chapters in that only 
 independent descriptions are sought using a fixed utility 
 criterion (these are called descriptive descriptions in 
 
10 
 
 chapter 6.). The structure used for representation does not 
 take into account specific domain structures which are 
 inherent in the VL system and there is no facility for 
 generating new descriptors within the system. 
 
 A formal approach using the predicate logic system 
 [ Vere 75] produces the largest common set of descriptors of a 
 set of examples by representing examples using a graph 
 structure and finding the largest common subgraph of these 
 structures. Such methods suffer from the HP-complete nature 
 of graph isomorphism algorithms (this problem is addressed in 
 the following chapters by finding the smallest useful 
 subgraphs of graphs of examples instead of the largest 
 subgraph.) • Neither Vere nor Hayes-Roth use negative examples 
 heavily in their implementations. 
 
 Hedrick [Hedrick 74] uses a semantic net to represent 
 examples and to build and modify hypotheses as new examples 
 are given to the program. The semantic net supports only 
 binary relations but otherwise looks similar to the graph 
 structure of Vere and Hayes-Roth. 
 
 Kochen [ Kochen 74] presents a different type of 
 system with a set of initial events containing state 
 variables, actions and relations between the actions, a 
 learning program which applies certain transformations to 
 events at various time steps. At each tinre step, weighted 
 
11 
 
 hypotheses are formed which reduce the set of states stored 
 in memory (i.e those states which are explained by the 
 hypotheses ) • 
 
 Production systems provide a rich tool for the 
 introduction of many inference techniques (recall that VL 
 decision rules are similar to production rules) • Briefly, a 
 production system architecture contains a working memory, a 
 set of productions which modify working memory, and a 
 recognize-act cycle with conflict resolution to dictate the 
 order in which productions are applied to iremory and to add 
 new productions when necessary. 
 
 Waterman [ Waterman 70, 74, 75] uses these to solve 
 several problems. A program which plays poker has been 
 developed [Waterman 70] which designs betting strategies in 
 terms of production rules. More recently [Waterman 75], the 
 approach has been applied to recognizing letter seguences 
 with success. Rychener [ Rychener 76 ] has applied production 
 systems to chess end games and natural language input of a 
 toy blocks world. These two authors use distinct production 
 system architectures which differ in the ordering of working 
 memory, ordering of productions and in the way in which new 
 productions are added to an existing set of productions to 
 correct errors made by the system. 
 
 As a final note with regard to production systems. 
 
12 
 
 the MYCIN system [ Shortlieffe 74 ] has been shown to be a very 
 powerful tool in aiding physicians with regard to 
 antimicrobial therapy selection* In the HTCIM system, 
 deductive inference using a multi-valued truth model is 
 applied to expert supplied productions and a data base 
 consisting of a patient record* 
 
 1*4 Overview of the Following Chapters 
 
 The VL system (a subset of the VL system) is 
 21 2 
 
 described in chapter 2* The subset used here uses only the 
 
 truth values TRUE, FALSE, and UNKNOWN instead of the 
 
 mult i- valued truth domain of the VL systeir* Also, only a 
 
 2 
 
 very small subset of the operators available in VL are 
 
 2 
 
 used* The inductive rules used to transform initial decision 
 
 rules (1*1) into new decision rules (1*2) are given in 
 
 chapter 3* The rules involve selecting the most significant 
 
 features of the initial condition (C in 1.1), extending 
 
 i,1 
 
 the value set which each feature may assume under the domain 
 structure constraints, and adding new global functions which 
 describe certain characteristics of the condition (C )• 
 Chapter 4 contains a graphical representation of VL rules* 
 A subset of the graph structure is used in the computer 
 program INDDCE_1 described in chapter 5. The program accepts 
 as input: 
 
13 
 
 1) a set of decision rules representing certain examples 
 of sets of decisions, 
 
 2) a problem environment description in the form of VL 
 decision rules which describes certain 
 characteristics of functions and donrains which arise 
 from the particular application, and 
 
 3) a set of control parameters which supply the 
 optimality criterion and certain parameters which 
 limit the number of alternatives generated at various 
 points in the program. 
 
 The output from the program contains a set of complete, 
 consistent decision rules. Chapter 6 gives results of the 
 program as applied to some specific situations. Chapter 7 
 describes some limitations and possible extensions of the 
 program. Two appendices are given: appendix A, providing a 
 listing of the detailed output of the program applied to one 
 example of chapter 6, and appendix B, giving a brief review 
 of the precursor to this work (the program AQVAL/1-AQ7) which 
 is used as a procedure in IHDDCE_1. 
 
14 
 
 2. Representing Decisions in the VL System 
 
 2 
 
 Much of the information in this chapter is found in 
 
 [Larson et.al. 77, Nichalski 74b], It is included here to 
 
 give the reader a familiarity with the VL system. The 
 
 complete VL system contains a very rich set of operators and 
 
 domains. A subset of VL called VL which contains a basic 
 
 2 21 
 
 set of operators and domains is used here. Only the VL 
 
 system is described with notes indicating the extensions 
 
 which are possible in the full VL system. In later 
 
 2 
 
 chapters, the notation VL is used to refer to the system 
 
 VL . 
 21 
 
 The logic system VL is a language for describing 
 
 situations (e.g. objects, classes of objects) and expressing 
 
 decision and inference rules. The language provides for a 
 
 compact expression of descriptions which is both easily 
 
 readable and sufficiently precise to facilitate formal 
 
 manipulation (possibly by a computer) • 
 
 There are two major differences between VL and the 
 
 21 
 
 first order predicate calculus 
 
 1. Instead of predicates, selectors are used which can 
 be viewed as tests for membership of values of 
 predicates and functions in a certain set. 
 
15 
 
 2. Each variable, predicate and function symbol is 
 assigned a domain (or value set) together with a 
 characterization of the structure of the domain. 
 
 (This feature facilitates the process of rule 
 generalization and allows for the application of 
 different generalization transformations according to 
 the structure of the domain*) 
 
 There are three types of domains currently 
 distinguished: 
 
 T. Unordered or N om inal 
 
 Elements of the domain are considered to be 
 independent entities; no structure is assumed to 
 relate them. A variable or function symbol with this 
 domain is called nominal or cartesian (e.g. blood 
 type, names of objects, etc.) • 
 
 2» tiDS^rly. Ordered or Interval 
 
 The domain is a linearly ordered set. A 
 variable or function symbol with this domain is 
 called interval (e.g. military rank, temperature, 
 size) . 
 
 3. Tree Ordered 
 
16 
 
 Elements of the domain aire ordered into a 
 tree structure. 1 predecessor node in the tree 
 represents a concept which is more general than the 
 concepts represented by the descendent nodes (e.g., 
 the predecessor of the nodes •triangle 1 , •rectangle 1 , 
 •pentagon 1 may be a • polygon') • A variable or 
 function symbol with such a domain is called 
 structured. 
 
 2.1 VL System Structure 
 
 The »I 21 syste* used is a 5-tuple (V.B.S.B.I, .here: 
 
 V - is a set of variable symbols. Each variable symbol 
 
 is associated vith a domain D(i )• A £rou£ of 
 
 i 
 
 variables which have the same doirain are labelled 
 
 with the same variable symbol but a different 
 
 subscript (e.g. x ,x ,..,x , y ,y ,...,y are 
 
 specifications of variables in two variable groups 
 
 which assume values from two domains denoted D(x) and 
 
 D(y) or alternatively. Dfx ) and D(y )• 
 
 i i 
 
 P - is a set of n-ary functions and predicate symbols. 
 
 Each n-ary function symbol represents a mapping from 
 
 an argument space into a domain. For a function 
 
 f(z ,z ,...,x ), this is a mapping: 
 12 n 
 
17 
 
 D(x ) x D(x 2 ) X ... X D(X fc ) -> D(f) 
 
 where D (x ) , D (x ) , ... D(x ), D (f ) represent the 
 1 2 k 
 
 domains of the variables x , x, ... x and the 
 
 1 2 k 
 
 domain of the function f, respectively. A predicate 
 is a function whose domain is the set [ TRUE, FALSE ]• 
 Included in the domains of all function and variable 
 symbols is the value MA (not applicable), 
 s - is a set of symbols including: 
 
 ( ) [ 1 = < > -> <"> v => 6 v E E. A A. , . 
 
 R - is a set of formation rules described in section 2.3 
 I - is a set of interpretation rules described in section 
 2.4 
 
 2.2 Selector Formation and Interpretation Rules 
 
 A well formed VL formula (wff) is composed of 
 quantifier forms, selectors, and logical connective symbols. 
 
 A selector is a form: 
 
 fL # R] or [L» ] 
 
 where 
 
18 
 
 L, L* - each called the referee ace atomic forms. An atomic 
 form is a variable symbol or a function or predicate 
 symbol followed optionally by a list of atomic forms 
 enclosed in parentheses. In the above forms, the 
 atomic form L' must be a predicate symbol with 
 arguments following in parentheses. If L contains a 
 function symbol, then the related function is called 
 an atomic fanctioQ. 
 
 R - the reference is a set of values in the domain of the 
 atomic function of L. B may be in several forms: 
 
 Reference Example Description 
 
 a a constant in the domain of 
 
 the atomic function of L 
 
 a,b a list of values in the 
 
 domain of L separated by 
 commas 
 
 a..b a pair of values in the 
 
 domain of L separated by 
 
 * the symbol (*) representing 
 
19 
 
 all values in the domain of 
 L (except Nl) 
 
 NA the value NA (not 
 
 applicable) 
 
 # - is one of the symbol combinations 
 
 If R is a set of values, then L is related to 
 R by # if 
 
 when # is = or -»= L has a value (does not 
 
 have a value) in the set R 
 
 when # is <= >= < > L has a value related to 
 
 every value of R by #• 
 
 The selector is interpreted as a unit of 
 information about a situation with value or 
 truth-status TRUE if the relation R # L holds or 
 FALSE if the relation does not hold, or UNKNOWN in 
 which case the selector is interpreted as a question 
 about the situation which must be answered in order 
 to determine if the selector is satisfied. If some 
 
20 
 
 variables in the atomic form of the selector are 
 quantified, these quantifiers must he considered when 
 determininq the truth-status of a selector* 
 
 If R is *, then I is related to B for any 
 value of L except HA (in this case, # is always =) . 
 Below are some examples of a selector : 
 
 Selector Interpretation: truth-status TRUE 
 
 [ color (wall ) = white] The color of the wall represented 
 
 by wall is white. 
 1 
 
 [lenqth(box ) >1 ] The lenqth of the box represented 
 
 by box is qreater than or equal 
 to 1. 
 
 [box - 2 ..5 1 The variable box may have a value 
 
 1 1 
 
 between 2 and 5 inclusive. The 
 
 values of box may represent 
 
 various boxes in a situation. The 
 
 selector restricts the ranqe of 
 
 values of the variable box to the 
 
 1 
 
 values 2 throuqh 5. 
 
 [ ontop (x ,x ) ] The part represented by x is on 
 
 top of the part represented by 
 
 X i 
 
 2 
 
21 
 
 2. 3 VL Formation Rule 
 
 Formulas in the ?L logic system are used to describe 
 
 2 
 
 situations, and also to express decision rules and inference 
 rules. The VL formulas are defined by the following formation 
 rules: 
 
 1. A selector is a ?L formula (wf f ) • 
 
 2* If V, V and V are wff- then so are: 
 1 2 
 
 (V) a formula in parentheses 
 
 -•V inverse 
 
 V & V or V V con-junction (the symbol & is 
 
 12 12 
 
 used to represent conjunction) 
 
 V v V disjunction 
 
 1 2 
 
 V v V exclusive disjunction 
 
 1 _ 2 
 
 V — -^/V exception 
 
 12 
 
 V -> V V implies V 
 
 12 1 2 
 
 v <-> V V is eguivalent to V 
 
 12 1 2 
 
22 
 
 Ex ,x ,...,x (V) Existent ially quantified formula 
 12 k 
 
 (E is used to represent the 
 existential quantifier)* 
 
 E. x ,x ,...,x (v) Distinctly existential ly 
 
 quantified formula 
 
 Ax ,x ,***,x (v) Universally quantified formula 
 ~ 1 2 k 
 
 (A is used to represent the 
 universal quantifier). 
 
 Not all of these forms are considered in the 
 following chapters. In chapter 3, (VL inference 
 rules) only conjunction, disjunction, and quantifiers 
 are considered* Chapter 4 (Graph Representation) 
 presents a graph structure representation vhich 
 includes all of these forms but the types of formulas 
 actually included in the algorithm and the 
 implementation involve only conjunction and distinct 
 existential quantification* 
 
 2.4 Interpretation Rules 
 
 A VL formula may have truth-status TRUE, FALSE, or 
 
 unknown. In the full VL system, a truth- status domain with 
 
 2 
 
 interval structure may be defined, but here only the values 
 
23 
 
 above are considered* The connectives (-• v 6) are 
 interpreted in the normal manner: 
 
 VL formula Interpretation 
 
 -V PALSE if ? is TEUE, TRDE if V is 
 
 FALSE, UNKNOWN if V is UNKNOWN. 
 
 V v V TRUE IF EITHER V OR V IS TRUE, 
 
 12 12 
 
 UNKNOWN if both ? and V are 
 
 1 2 
 
 UNKNOWN or one is UNKNOWN and 
 the other FALSE, TRUE 
 otherwise. 
 
 V & V UNKNOWN if both V and V are 
 
 12 12 
 
 UNKNOWN or one is true and the 
 
 other is UNKNOfcN, TRUE if both 
 
 V and V are TRUE, FALSE 
 1 2 
 
 otherwise. 
 
 The remaining connectives may te rewritten in 
 equivalent forms: 
 
24 
 
 VL Formula Equivalent form 
 
 V -> V -V v ? 
 
 12 12 
 
 V <-> V (V -> ? ) S (V -> V ) 
 
 12 12 2 1 
 
 V v^ V ? ^.V 
 
 12 12 
 
 V v V V v ?^ , ? V 
 
 1 " 2 1 2^^ 1 2 
 
 A VL system is used to describe a set of situations* 
 
 In order to effectively apply a formula to a set of 
 
 situations, the VL system should contain variables, 
 
 2 
 
 functions, and predicates vhich adequately characterize the 
 situations. To determine the truth-status of a formula with 
 reqard to a specific situation, an event is created (an event 
 may be viewed as an interpretation of a situation in the VL 
 system) • An event is a sequence of assignments to variables, 
 functions and predicates in the system which characterize a 
 specific situation. Quantified variables nay be assigned a 
 set of values. One function assiqnment vay be made to a 
 given set of values of arguments if the value of the function 
 is known. If a function does not have an assignment for a 
 given set of values, then the value NA (not applicable) is 
 assumed. 
 
25 
 
 A selector [L # R] (or [ L • ]) is satisfied by an event 
 
 if there is a set of assignments to variables and functions 
 
 (or predicates) in L (or L*) such that L is related to R by # 
 
 (or L 1 has the value TRUE) • 
 
 A VL formula is satisfied by an event if it has 
 2 
 
 truth status TRUE when applied to the event. 
 
 The quantified formulas are interpreted: 
 The truth status of 
 
 Ex ,x ,«..,x (V) is TRUE (or FALSE) in a given 
 "12 n 
 
 situation if there exists (or 
 
 does not exist) values for 
 
 x ,x ,...,x in the event 
 12 n 
 
 assignments which makes the 
 truth-status of the formula V 
 egual to TRUE 
 
 ? - if it is not known whether 
 there exist values ••• 
 
 E.x ,x ,...,x (V) is TRUE (or FALSE) in a given 
 ""12 n 
 
 situation if there exists (or 
 
 does not exist) distinct 
 
 (different) values for 
 
 x ,x , ...,x in the event 
 12 n 
 
26 
 
 assignments which makes the 
 
 troth-status of the formula 
 
 equal to TRUE. This obviates 
 
 the need for extra predicates in 
 
 an expression like x -»=x , 
 
 1 2 
 
 x -»=x , x -»=x # etc. 
 2 3 13 
 
 ? - if it is not known whether 
 there exist values ••• 
 
 Ax ,x ,...,x (V) is TROE (or FALSE) in a given 
 12 n 
 
 situation if for all assignments 
 
 to the variales x ,x ,...,x , 
 
 1 2 n 
 
 the formula V has truth-status 
 equal to TRUE. 
 
 A VL formula is a description of a situation if 
 
 every event which can be derived from the situation satisfies 
 
 the VL formula and every event which satisfies the formula 
 2 
 
 is also an interpretation of the situation. 
 
 2*5 VL Decision Rules 
 
 If V and V are VL formulas, a general form of a VL 
 12 2 
 
 decision rule is 
 
 V => V (2.5.1) 
 
 1 2 
 
27 
 
 The formula V is called the cond it ion part and V is the 
 decision part. A restricted form of the VL decision rule 
 will be used in the following chapters. 
 
 (In the computer implementation, the formula V is 
 assumed to be a product of selectors which contain 0-ary 
 functions in the referee. The terminology is relaxed in this 
 case to allow the function symbol which appears in the 
 decision part to be called a decision variabl e. ) 
 
 A decision rule in the form 2.5.1 may be applied to a 
 
 set of situations as follows: If the condition part of the 
 
 decision rule (V ) is given truth-status TRUE, then the 
 
 decision part of the decision rule (V ) also assumes the 
 
 2 
 
 truth-status TRUE. For each event (assignment e:=L) which 
 
 satisfies V , a new set of assignments are made to the event 
 1 
 
 using the decision part of the rule to form the set of all 
 
 events which satifsfy the conjunction V 6 V • For example, 
 
 12 
 
 given the decision rule: 
 
 E. Xi ,X 2 [p(x ,x )] => [D=1] (2.5.2) 
 
 and the event 
 
 e: v =0,1; ,,-0.1 p,0.1),.«0« (2.5.3, 
 
 with variables functions and predicates: x, D, p, g with 
 
28 
 
 domains D(x)=[0,1], D(D)*[0,1], D (p) ,D (g) =[ IHIJE, FALSE ], anew 
 assignment is made to e to give: 
 
 e: x :=0,1; x :=0,1; p (0,1) :=TRUE; D:»1. (2,5. 4) 
 
 1 2 
 
 A decision rale: 
 
 E.x i# x 2 CPU #x ) ] => I»" 1 »* 2 [9(^»» 2 ) ] (2.5.5) 
 
 applied to the event 2.5.3 gives one of the two new events: 
 
 e : x :-0,1; x :=0,1; p (0,1) :=TR0E; g (0, 1) :=TRUE; 
 e 2 : ,,-0,1, x^O.1: p ,0.1, :=TRDE; , ,1.0, :=TE0E; 
 
 Note that q(1,1) and g(0,0) are not given status TBUE since 
 
 the quantifier E. insists that the tvo variables x and x 
 
 12 
 
 have different values. 
 
 Given a set of decision rules each in the form 2.5.1, 
 the set may be applied to a set of events. Initially, the 
 condition parts of all decision rules have value UNKNOWN. If 
 an event satisfies the condition part of a decision rule, nev 
 assignments are made to the event according to the decision 
 part of the satisfied rule and the condition part of the rule 
 returns to truth status UNKNOWN. 
 
 In the remaining chapters, events are only used as a 
 formal basis for defining certain concepts. Since the number 
 
29 
 
 of events necessary to completely describe a situation is 
 quite large, only the VL formulas themselves are manipulated 
 by the algorithms. 
 
30 
 
 3« VL Transformation Bales 
 
 From one set of decision roles (1. 1) , a new set of 
 decision roles (1*2) is obtained by applying certain 
 transformation roles (t-rules) • For now, we will restrict oar 
 attention to roles which transform the condition part of a 
 role. These t-rules may be grouped into three types of roles 
 based on the events which satisfy the condition part* 
 
 Given two roles: 
 
 H : V => D 
 
 1 1 
 
 H : V => D 
 
 2 2 
 
 V is more general than V if every event satisfying ¥ also 
 1 2 2 
 
 satisfies V • If the converse is also trae, then V is 
 1 1 
 
 egoivalent to V • Below, B and B are the input and ootpot 
 
 2 12 
 
 of a t-role. The three types of t-roles can be expressed as 
 follows: 
 
 h transformation T : B I # B (# being one of =, <, 
 >) is 
 
 1. An equivalence transformation (denoted B |= B ) if 
 
 1 2 
 
 the condition parts of both roles are egoivalent. 
 
 2. A generalizing transformation (denoted B^ |< B ) if 
 
 1 2 
 
31 
 
 the condition part V of R is more general than the 
 
 2 2 
 
 condition part V of R • 
 
 1 1 
 
 Rules 1, and 2. are called inductive inference 
 
 rules 
 
 3. A specializing transformation (deductive inference 
 
 rule denoted R |> R ) if the condition part V is 
 1 2 1 
 
 more general than the condition part V • 
 
 2 
 
 Here, we are most interested in the first two types 
 of t-rules (i.e., inductive inference rules). 
 
 3.1 Eguivalence Transformation 
 
 An eguivalence transformation rewrites a VL formula 
 
 2 
 
 into a different form either using equivalent VL operators 
 
 or introducing new functions which represent some information 
 
 already in the rule in a different manner. Below are some 
 
 examples of eguivalence transformations. The symbols L and 
 
 L represent atomic forms- V and V* represent VL formulas, D 
 2 2 
 
 represents a VL formula which has no variables in common 
 
 2 
 
 with V, V», L or L - and |= is used to represent an 
 
 1 2 
 
 eguivalence transformation. 
 
 E1. Eguivalent VL forms. 
 
 2 
 
 V(fL = 1] v [L = 2]) => D |= V[L = 1,2] => D 
 
32 
 
 ?[L -= 3,4] => D I* V[L ~ 0,1,2,5] *> D 
 
 (assaming that the domain of L is the interval [0..5] 
 
 and has nominal structure)* The dot operator (•) 
 
 between two atomic forms is called 'internal 
 
 conjunction* thus the expression on the right above 
 
 is read: 'If L and L both have the value 1 and V is 
 
 1 2 
 
 satisfied, then make decision D. ' • 
 
 V([L 1 = 1 ][L 2 = 1]) => D |= Y(£L .L = 1]) => D 
 
 (assuming that L and L have the same domain size 
 
 1 2 
 
 and structure) • 
 
 22. Internal Conjunction of Arguments 
 
 VV => D |= VV'[f'(l .X )=i] => D 
 
 12 
 
 V = [ff(x ) = i][f (x^ = i] 
 
 This rule introduces a new predicate f ' which 
 
 has the domain [TRUE, FALSE] and two arguments. The 
 
 (• ) operator instead of (,) indicates that the order 
 
 of arguments to f is irrelevant. The function f 
 
 assumes the value TfiOE if f has the value i for both 
 
 arguments x and x • 
 1 2 
 
 E3. Introducing New Predicates. 
 
 VV => D |= Y¥'[rel_f(x , x ) ] => D 
 
33 
 
 v = rf^) = i)m* 2 > = ji 
 
 where i is related to j by rel. Por example, 
 i<j, i>i# i> = j would result in new predicates LT_f, 
 GT_f r GE_f. 
 
 Ett. Splitting the Condition Part. 
 
 V v V 1 => D |= V => D, V 1 => D 
 
 This rule is used to form a set of decision 
 rules with product condition parts from decision 
 rules in disjunctive normal form. 
 
 3,2 Generalizing Transformations 
 
 This type of transformation usually produces not only 
 a more general decision rule from a set of decision rules but 
 also a 'simpler 1 one than the original. Some rules are 
 applied in context. That is, one rule is actually 
 transformed but the context consisting cf rules with a 
 related decision is used to obtain a more optimal result, 
 (These transformations may also be interpreted as 
 transformations from a set of rules into a new, more general 
 rule. The approach of focussing on one rule in the •context* 
 of others is taken here to more closely reflect the approach 
 taken in the implementation. ) In the following rules, the 
 symbol |< is used to indicate an inductive inference rule. 
 
3a 
 
 Gl Dropping a Selector* 
 
 VfL = H] => D |< ? => D 
 
 Although this rale is • interesting in a formal 
 sense, it should be applied with care since the 
 number of generalizations possible with successive 
 applications of this rule is very large. Hore is 
 said about this problem in section 3* 4* 
 
 G2, Extending the Reference 
 
 Hominal domain structure 
 
 VfL = a] => D |< V[L = a,b] => D 
 
 in context [ L = b 3 -> D 
 
 Interval domain structure 
 
 V(L = a] => D »< V[L = a..b] => D 
 
 in context [ L = b ] => D 
 
 Tree structured domain 
 
 VfL = a] => D |< ?[L = c] -> D 
 
 in context [L = b] => D 
 
 c is a predecessor of both a and b in the 
 generalization structure of the domain of L. 
 
 G3. Extension Against 
 
35 
 
 Nominal domain 
 
 V [L = R 1 -> D |< V [L -*=B ] => D 
 
 11 12 
 
 in context: V [L = R 1 => -»D 
 2 2 
 
 assuming R & R = null. (The symbol S 
 
 1 " 2 
 
 denotes set intersection.) 
 
 Interval domain structure 
 
 V [L = a..b] => D |< V [L = e. . f ] =>D 
 1 1 
 
 in context: V [ L = c. • d 1 => -»D 
 2 
 
 assuming fa..b] & [c.d] = null 
 
 if b < c then e = 0, f - c-1 
 
 if a > d then e = d+1, f = h (0 and h are 
 the minimum and maximum elements in the domain of 
 L) 
 
 Tree structured domain 
 
 V [L = a] => D l< V fL = c] => D 
 1 1 
 
 in context: V [L = b] => ~>D 
 2 
 
 assuming a & b = null 
 
 the constant (c) is the most distant 
 ancestor of (a) which is not an ancestor of (b) 
 (c may be egual to a) • 
 
 GU. Replace A with a constant 
 
36 
 
 (Ax V(x )) => D |< V(x ) [x = a] => D 
 
 i i i i 
 
 a - an element in the domain of x • (The 
 
 i 
 
 symbol A represents the universal quantifier*) For 
 
 * 
 
 example: 
 
 Ax [ffx^ = 1] => D |< [f<x ) = 1][x i = 2] => D 
 
 The left side of the expression requires that f have 
 
 the value 1 for all values of x before a decision be 
 
 1 
 
 made. The generalized expression (right side) only 
 requires examination of one value in the domain of 
 x , namely the value 2. ill other values in the 
 
 domain of x are irrelevant to the decision rule. 
 
 1 
 
 G5. Beplace a constant with E 
 
 V(x )fx = a] => D |< Ex V (x ) => D 
 
 i i i 1 
 
 a - an element in the domain of x • (The 
 
 i 
 
 symbol E represents the existential quantifier.) For 
 example: 
 
 [f(x ) = 1][x = 3] |< Ex [f(x^) = 1] => D 
 
 The left side of the expression makes a decision only 
 
 if f has the value 1 for x with value 3. The 
 
 1 
 
37 
 
 generalized expression makes a decision if f has the 
 
 value 3 or any value of x . 
 
 1 
 
 G6. Move E to the Right 
 
 Ex Ax ? (x ,x ) => D |< Ax Ex V (x ,x ) => D 
 
 ijij 1~iij 
 
 3. 3 Specializing Transformations 
 
 A specializing transformation may be used to apply a 
 decision rule to a new situation or to add certain 
 restrictions to decision rules* Below, the symbol |> 
 represents a deductive inference rule. 
 
 R1. Adding Restrictions 
 
 VV» => D |> VV» ¥••»>' D 
 
 in context V» => V« ■ 
 
 This rule is used to add restrictions to 
 descriptions and generalizations where the 
 restrictions represent some structure which is 
 imposed on a function or some relationship between 
 functions which always holds. For example. the 
 transitivity or symmetry of a function may be 
 introduced using a restriction. 
 
 rf (x ,x ) ][f (x ,x ) ] => [f (x ,x ) ] (transitivity) 
 
38 
 
 [f (x^x ) ] => [f (x^x ) ] (symmetry) 
 
 R2. Dropping Product Rule 
 
 V v V => D |>' ▼ -> D 
 
 1 2 1 
 
 This rule may be used to obtain a set of 
 decision rules with a product in the condition part 
 from a rule in disjunctive normal form. 
 
 3.4 Transformation Rules Involving the Decision Part 
 
 It is clear that reversing the roles of the input and 
 output of an inductive t-rule gives a deductive t-rule and 
 conversely. Therefore, each of the rules G1-G6 could be 
 inverted to get a deductive rule. The rules in sections 
 3, 1-3.3 were based on the events satisfying the condition 
 part of a decision rule* Similar rules may be applied to the 
 decision part of a rule to obtain equivalent, more general or 
 more restricted rules. 
 
 Given two rules: 
 
 R : V => D 
 
 1 1 
 
 R : V => D 
 2 2 
 
 R is more general than R and R is more restricted than R 
 2 11 2 
 
 if every assignment of event values made by R is also made 
 
39 
 
 by F . If the converse is also true, then R is equivalent 
 2 1 
 
 to P • To transform the decision part of a decision rule, 
 2 
 
 apply G1-G6 to the decision part of a rule to obtain a 
 deductive rule and R1-H2 to obtain an inductive rule 
 interchanging the roles of the condition and decision parts 
 in the transformation rules. A few examples are given 
 below. 
 
 DP1. Dropping Selector Rule Applied to the Decision Part 
 
 V => [L = R ][L = R ] |> V => [L = R ] 
 
 1 12 2 1 1 
 
 Though this is a generalization rule when 
 applied to the condition part, it is a deductive rule 
 when applied to the decision part of a decision 
 rule. 
 
 DP2. Splitting the Decision Part 
 
 V => [L = R ]D |= V => [L = R ], ? => D 
 
 2 2 11 
 
 This eguivalence preserving rule is used to 
 produce a set of decision rules which involve only 
 one decision variable. 
 
 DP3. Replace a Constant with A in the Decision Part 
 
 V => TL(x ) = R ][x = a] |> ? => Ax [ L (x ) =R ] 
 1 11 "111 
 
40 
 
 3*5 Example of Application of Transformation Bales 
 
 In this section, selected transformations will be 
 
 examined in more detail with respect to a specific example* 
 
 Consider the situation in Figure 3,1. • There are four objects 
 
 classified according to two decision variables DA and DB. 
 
 Each object is described in terms of the following VL 
 
 2 
 
 functions and predicates: 
 
 p ,p : variables each representing a part in an object 
 with domain : P = [0,1] with nominal structure 
 
 ontop: a predicate mapping P x P into [TBUB, FALSE] 
 
 The predicate is TRUE if the part represented 
 
 by the first argument is on top of the part 
 
 represented by the second argument. 
 
 shape: a function mapping P into [Triangle (T) , 
 Circle (C) , Rectangle (R) , Ellipse (E) , Polygon (P) , 
 Curved Figure (CF) ] with generalizations: 
 [shape=T,R ]=>[ shape=P ] and [shape=C,E ]=>[shape=CF ]• 
 
 The function specifies the shape of the part 
 represented by the argument. The domain is a tree 
 structured domain where Polygon is an ancestor (a 
 generalization) of Triangle and Rectangle and Curved 
 Figure is an ancestor of Circle and Ellipse. 
 
41 
 
 DA,DB: decision variables with domain [1,2] 
 
 DA = 1 DA = 2 
 
 1 Circle 2 Circle DB = 1 
 
 Triangle Circle 
 
 3 Ellipse H Rectangle DB = 2 
 
 Rectangle Rectangle 
 
 Figure 3. 1 Set of Objects 
 
 Each object in fig 3.1 may be described in the VL 
 system. The description of objects 1 and 4 arc: 
 
 [ontop(p ,p ) ][ shape (p ) =C ][ shape (p ) =T ] 
 
 => [DA=1 ][DB=1 ] (3. 5.1) 
 
 [ontop (P^Pj ]f shape (p)=R][ shape (p)=R] 
 
 => [DA=2 ][ CB=2] (3.5.2) 
 
 The variables p and p are assumed to be guantified with the 
 
 1 2 
 
 operator E. (distinct existential guantifier) • Using the 
 
42 
 
 rule DPI, the decision rale 3*5*1 can be transformed into two 
 new rules: 
 
 [ontoplp^p^ Hshape(p )=C][ shape (p)=T] => [DA=1 ] (3.5.3) 
 [ontopfp^p^ ][ shape (p ]=C][ shape (p)=T] => [DB=1] (3.5.4) 
 
 Concentrating now on 3.5.4, the dropping selector 
 rule may be applied twice in succession to obtain 
 
 [ shape (p )=C] => [DB=1]. (3.5.5) 
 
 This rule describes all objects in figure 3. 1 with the 
 decision DB=1. In addition, it does not describe any object 
 with DB=2; so it is a complete, consistent generalization of 
 the objects with DB=1. (Note that completeness and 
 consistency are in no way guaranteed by the application of 
 the dropping selector rule. These conditions can however be 
 checked after each application of the rule.) Applying the 
 extension against rule to 3.5.3 in the context 
 
 [onto(p ,p )][ shape (p ) =B ][ shape (p ) =B ] => [Dl=2] (3.5.6) 
 
 12 2 2 
 
 focusing on the selectors [ shape (p ) =C ] in 3.5.4 and 
 f shape (p )=B] in 3.5.6 one may obtain 
 
 [ shape (p =CP][ontop(p ,p )][ shape (p ) =T ] => [DA=1]. (3.5.7) 
 
43 
 
 Applying extension against now to 3.5.7 in the context 
 
 [ontopUyp , ]tshap e «p i ,.C] [ shape,P 2 ,»C] -> (Da-2] (3.5.8, 
 
 focusing on the selectors [ shape (p ) =T ] in 3, 5, 7 and 
 [ shape (p ) =C ] in 3.5,8, one obtains 
 
 [ s.ape,p i ,=CFMo„top ( p i ,p 2 ,Mshape,p 2 ,«P] [DA=1 ,. (3 . 5 . 9) 
 
 This rule is a consistent and complete generalization of all 
 rules in fugure 3.1 with DA=1 in the decision part. 
 
 Looking now at the description of object 4 (3.5.2), 
 two rules are obtained by applying t-rule R1: 
 
 [ontop( V P 2 , K shape,p i ,=HMshape«P 2 ,=R] => [D .-2] ,3.5.10, 
 [ontop(p if p 2 ) ][ shape ( Pi )=R][ shape (p 2 )=R] => [DB=2] (3.5.11) 
 
 An application of the dropping selector rule to 3.5.11 twice 
 in succession gives a complete, consistent rule: 
 
 [ shape (p )=R] => [ DB=2 ] (3.5.12) 
 
 An application of t-rule E3 to 3.5.10 with a relation egnals 
 produces: 
 
 [ontop(p ,p ) ]f shape (p)=R][ shape (p ) =R ]f EQ-shape (p . p ) ] 
 12 1 2 12 
 
44 
 
 =>[DA=2] (3.5.13) 
 
 The predicate EQ-shape specifies that the value of 
 the function shape is the same for arguments p and p • The 
 ,., separating the argents ^ and ^ of EQ-shape signif, 
 that the order of the arguments is irrelevant. (In the 
 implementation, a selector with this type of predicate is 
 written [ shape (p . p ) =same ). ) An application of the dropping 
 selector rule three times in succession to 3.5.13 gives: 
 
 tE0-.h.p.<p.p)l.>ID.-2] C3.5.1.) 
 
 which is a complete, consistent generalization of 3.5.10. 
 In summary, the new rules which were obtained are: 
 
 tontopOyp^Kshape^.-CPJtshape.p^P] => [0 »-1 ] 
 
 [ EQ-shape (p ,p ) ] => [DA = 2] (3.5.15) 
 
 1 2 
 
 [ shape (p )=C] => [DB=1] 
 [ shape (p )=R] => [ DB=2 ] 
 
 In the above discussion, the resulting simple 
 generalizations depended on knowing the proper rules to apply 
 and the proper portions of the decision rules to modify. Por 
 larger problems, this approach is infeasible because of the 
 large number of possible generalizations which could be 
 
45 
 
 made. Therefore, a more efficient approach is required. Such 
 an approach is given in the next section and described in 
 detail in the chapter on basic procedures (chapter 5) • 
 
 3.6 Efficient Application of Generalization Pules 
 
 There are two significant problems with using the 
 
 procedure described in section 3. 5 to apply t-rules to a set 
 
 of decision rules. The first is the large number of new 
 
 rules which can be generated from one rule. This problem 
 
 could be circumvented by trimming the intermediate lists of 
 
 new rules selecting only the most promising set of rules 
 
 according to a user specified criterion before further 
 
 application of t-rules (see section 5.3 for a description of 
 
 a trimming procedure). A second probleir which is not 
 
 surmounted as easily is the complexity involved in 
 
 determining whether a VL formula is consistent and in 
 
 2 
 
 evaluating the optimality cost functions during trimming. To 
 
 determine whether one VL formula is a generalization of 
 
 2 
 
 another or is consistent with respect to another formula, one 
 
 must determine whether all or any of the events which satisfy 
 
 one VL formula also satisfy another VL formula. 
 2 2 
 
 Using the current implementation, this involves 
 determining whether one graph structure representation is a 
 subgraph of another (see section 5.4 for a description of a 
 
46 
 
 subgraph isomorphism algorithm). This problem is exponential 
 in nature (i.e., the time to determine whether one graph is a 
 subgraph of another using a depth-first-search is 
 proportional to m raised to the power of n where m is the 
 number of nodes in the larger graph and n is the number of 
 nodes in the smaller graph). Actually, it is not guite so 
 time consuming since the graphs hare relatively few edges and 
 the edges and nodes are labelled. It is, however, important 
 to form simple generalizations which correspond to small 
 graph structures. 
 
 Since the cost criterion normally includes a cost 
 function which minimizes the number of selectors in a 
 product, the consistency and optimality of optimal rules 
 should be easily calculated. In the example of section 3.5, 
 the original rules were made smaller by dropping selectors. 
 An alternate approach is to grow the generalized rule 
 beginning with single selectors and adding new selectors 
 until a consistent rule or set of alternative rules is 
 created. A very general algorithm follows; chapter 5 gives a 
 complete description of the algorithm in the current 
 implementation. 
 
 1. Given a set of decision rules in disjunctive normal 
 form, create a new set of rules with a product in the 
 
47 
 
 condition part and one selector in the decision part 
 of each rule. 
 
 2. Select a rule which involves one value of a decision 
 variable. Add to the rule new selectors which 
 represent functional relations as specified by a 
 user. (i.e., multiply the condition part of the rule 
 by selectors containing new functional relations.) 
 
 3. Find a consistent generalization of the rule from 2 
 by locating the most promising selectors of the rule 
 from 2 and adding new selectors to each of these 
 selectors until a set of consistent generalizations 
 of the rule from 2 is obtained. 
 
 ft. Apply the extension against rule using an AQVAL/1 
 procedure to all consistent rules. 
 
 5. Select the best generalization from this set and 
 remove rules from the set produced in step 1 for 
 which this is a generalization. Repeat steps 2-5 
 until no more rules remain which involve the decision 
 variable and value of step 2. 
 
 6. Continue by selecting another value of the current 
 
48 
 
 decision triable or selecting another decision 
 variable until all decisions have been considered. 
 
49 
 
 4. Computer Representation of VL Decision Rules 
 
 Some of the information in this section appears in 
 [Larson et.al. 77] and is given here as a background for the 
 description of the computer implementation, A VL decision 
 rule can be represented as a graph with labelled nodes and 
 directed labelled edges. The labels on the nodes can be: a) 
 a selector containing k-ary descriptors without argument 
 lists, b) a k-ary descriptor without arguments, c) a 
 quantified variable with an optional subrange of values, d) a 
 logical operator, (From here on, a node is referred to by 
 its label, e.g., a selector node means a node with a selector 
 label.) The edges are labelled with integers from 0,1,... • 
 Edges not labelled refer to the position of an argument in 
 the label at the head of the edge. (Edges have non-zero 
 labels only if the position in the argument list of the head 
 node is important. Labels of may be dropped for 
 convenience.) 
 
 Several different types of relations may be 
 represented by edges. The type of relation is determined by 
 the label on the node at each end of the edge. The types of 
 relations are: 
 
 1. function Dependence - The label of the head node of 
 
50 
 
 the edge has a k-ary descriptor* The value 
 represented by the edge is the value of the atomic 
 form in the tail if the tail is a selector node, a 
 descriptor value if the tail is a descriptor node, or 
 one or all of a set of descriptor values if the tail 
 is a quantified variable* The edge label specifies 
 which argument of the head node assumes this value 
 (Figure 4.1). 
 
 Ex +* [ g = 1 . . 2 ] m» [ f = 1 ] - Ex 
 
 - 1 L * - 2 
 
 Functional Dependence: Ex^ ([g (x^ =1. . 2 ][f (g (x^ .^ = 1] 
 
 Figure 4.1 
 
 2» logical Dependence - The head node is a logical 
 operator (e.g. v, 6, =>) and the tail node is a 
 selector node, or a logical operator node. If the 
 tail node is a selector, then the value represented 
 by the edge is the truth value of the selector at the 
 tail (Figure 4.2) 
 
51 
 
 [f = 1] 
 
 [g = 2] 
 
 Logical Dependence : Ex (ff(x ) » 1 J v [ g(x ) =2] 
 
 Pigure 4,2 
 
 3« ISJBlicit Variable Dependence - The labels of the head 
 and tail nodes are quantified variables. This type 
 of dependence represents the implicit function (which 
 can be represented by a Skolem function) (Pigure 
 U. 3). 
 
 [P = 1] 
 
 Ax 
 
 -*• Ex 
 
 Implicit Variable Dependence: Ax Ex fp(x ,x ) =11 
 
 " r r r 2 
 
 Figure 4.3 
 
52 
 
 4. Scope of Var.±aJ>i£s - The head node is a logical 
 operator and the tail is a .quantified variable. This 
 type of dependence may be necessary for certain 
 binary loqical operators such as (->, <->) • For the 
 functions v and L this type of dependence is implicit 
 in the functional dependence of the arguments* 
 (Figure 4.4) 
 
 [P = 1] * *► =>- * [q - 1] 
 
 Ex Ex 
 
 ~ 1 2 
 
 Scope of Variables: Ex ,x ([ p (x ) = 1 ] => [g(x ) ■ 1 ]) 
 
 Figure 4.4 
 
 The graph of a more complex decision rule is given in 
 
 Figure 4.5. The value of x is dependent in an unspecified 
 
 way on the value of x (the edge labelled 1). The disjunction 
 
 2 
 
 (v) depends on the values of x and x , but this is clearly 
 
 2 3 
 
 specified by the functional dependence of £ and g on x and 
 x . Pinally, observe that the decision operator (=>) does not 
 
53 
 
 explicitly depend on the specific values of x , x , or x , 
 
 12 3 
 
 but instead depends on the truth value of the entire premise 
 
 using some set of value assignments for x ,x , and x • 
 
 12 3 
 
 Ex 
 
 rf = 1] 
 
 [f = 1] 
 
 Graph Structure Example: 
 
 Ex Ax Ex (([f(x ,x ) = 1 ] v [g(x ,x ) 
 12 3 2 3 2 3 
 
 [f(x ,x ) - 1J -> Id - 11 
 
 Figure 4,5 
 
 - 2,3]) 
 
5U 
 
 5, Algorithms and Computet Implementation 
 
 A computer program I!DUCE_1 has been written to find 
 a generalization of a set of decision rules. The algorithms 
 are described in the remaining sections of this chapter; 
 examples of generalizations produced by the program are given 
 in chapter 6 and a sample session with the program is given 
 in appendix A. The program does not perform all of the 
 transformations given in chapter 3 and does not accept the 
 full VL language given in chapter 2. The restrictions on 
 the form of the input and a description of the output are 
 given in the next two sections. 
 
 5.1 Input to the Program 
 
 The program accepts as input: 1) a set of decision 
 
 rules, 2) a problem environment description including a set 
 
 of restrictions, domain definitions, variable costs etc., and 
 
 3) a set of parameters which control certain aspects of the 
 
 program operation. Decision rules, restrictions, and domain 
 
 structures are entered a VL type formulas in the following 
 
 2 
 
 format : 
 
 Decision Rules 
 
 Decision rules must satisfy the following 
 grammar: 
 
55 
 
 11. <VIRULE> 
 
 12. <COHDITIOM> 
 
 13. <SELECTOR> 
 
 14. <ALIST> 
 
 15. <ELIST> 
 
 16. <REF> 
 
 17. <DECISION> 
 
 18, <FN-SYW> 
 
 19. <VARIABLE> 
 
 110. <NUMBER> 
 
 := <COMDITION> => <DECISIOH>. 
 := <COHDITION> <SELECTOR> | <SELECTOR> 
 := [ <FN-SYH> ( <ALIST> ) = <REP> ] | 
 [ <FN-SYM> ( <ALIST> ) ] | 
 [ <VARIABLE> = <REF> ] | 
 [ <PN-SYM> ( <ELIST> ) ] 
 := <ALIST> , <VARIABLE> l <VARIABLE> 
 := <ELIST> . <VARIABLE> | <VARIABLE> 
 := <REP> , <NUHBEB> | 
 
 <NOMBER> .. <NOHBEH> | 
 <NUMBER> | * 
 = [ <FN-SYM> = <REP> ] 
 = string of letter 
 = <PN-SYM> <NUBBER> 
 := string of digits. 
 
 Examples of decision rules: 
 
 TfU^) = 2,3][g<x ) = 1..4] => [D = 1], 
 
 In this expression, the structure of the 
 domain of the function g is set to interval by the 
 program (see note 4 below). The domain of f is 
 
 unchanged. All variables (e.g. 
 
 x and x ) are 
 1 2 
 
56 
 
 quantified by the operator f (distinct existential 
 quantifier - note 3) 
 
 [p(x ,x ) I* = 2,U] *> [D = 2], 
 
 In this example, the function p is assumed to 
 have two values (i.e., it is a predicate - note 6). 
 The selector containing p is satisfied if it has the 
 value 1. The second selector restricts the possible 
 values of x to the set of values [2,1] (note 3). 
 
 Several observations can be made about this grammar: 
 
 1. The condition part is a single product and the 
 decision part involves only one variable. It is 
 assumed that one decision variable has been selected 
 to be studied by the user. Also, the equivalence 
 rule has been applied which splits a condition part 
 of a formula in disjunctive normal form into a set of 
 decision rules with condition parts which are single 
 products (t-rules EU and DP2) • 
 
 2. Each atomic form is a function symbol with a list of 
 single variable arguments. It is assumed that the 
 user has converted forms such as 
 
 [g(f (x.)) ] 
 
 l 
 
57 
 
 into a form 
 
 [g(y ) ][p(y ri.) ] 
 1 j i 
 
 by introducing a new predicate p(y ,x ) f or each 
 
 J i 
 
 function symbol f which is in an argument list and a 
 
 variable y for each occurrence of the function f (x ) 
 J i 
 
 in an argument list. The predicate p is assumed to 
 
 have the value TRUE if y = f (x ) and FALSE 
 
 J i 
 
 otherwise. For example: 
 
 [shape (part (x ) ) = 1 ] 
 
 is assumed to be transformed into e.g. an expression 
 
 [ shape (p ) = 1 ][ contains (p ,x ) ] 
 
 3. ill variables (arguments) are assumed to be 
 existentially quantified. Variables with the same 
 function symbol part are assumed to have the same 
 domain. Furthermore, variables with values from the 
 same domain are assumed to take on distinct values 
 from the domain. Variables may be restricted to a 
 subrange of values by using the third option in the 
 
58 
 
 selector definition* Using this method, a constant 
 may be specified as an argument to a function. 
 
 ft. If a reference of the second form is specified at 
 least once in the input (production 16) (e.g. 
 [f(x )=2..2] or [f(x )=2..5]), a domain of type 
 interval is assumed; otherwise, the domain of a 
 variable or function is assumed to be of nominal type 
 or tree-structured if such a structure is specified. 
 
 5. The last form of the reference uses a value (*) . This 
 specifies the entire domain of the associated 
 function symbol. (E»g» [ P=* ] means that the 
 selector is satisfied for any value of p other than 
 the value NA.) If a selector is omitted from the 
 decision rule entirely, the program assumes that the 
 function has the value NA (not applicable). Such a 
 domain value has no generalization (other than NA 
 itself). Although NA is not specified in an input 
 rule, it can be a valid value of of a variable. 
 
 6. If the second form of the selector is used 
 (production 13, ,..,. [f !«,.*,» B. the program 
 assumes that the function symbol has the value 1. 
 This may be used to specify a TRUE value for a 
 
59 
 
 predicate (predicates are treated as functions with 
 
 domain [0,1]). In general, tc simplify the 
 
 expressions, only positive values cf predicates are 
 
 specified* The program then uses only positive 
 
 instances of the predicate in the generalizations. 
 
 If negative values of a predicate are desired in the 
 
 generalizations, these relations should be included 
 
 in the initial decision rule specification (see the 
 
 arch „„pl. in chapter 6). (e.g. [ontop ftyp^ ] 
 
 specifies that p is on top of p ; [ ontop (p ,p ) =0 ] 
 
 specifies that p is not on top of p •) 
 
 1 2 
 
 7. If the fourth form of the selector is used (e.g. 
 [p(x . x ) ], then the order of the arguments is 
 assumed to be irrelevant. 
 
 Restrictions: 
 
 Restrictions must satisfy the following 
 production: 
 
 <EEST> ::= <C0NDITI0N> - > <SELECT0R> 
 
 where the arguments of the selector part must all 
 appear in the condition part. CONDITION and SELECTOR 
 are the same as in the decision rule grammar. Por 
 example: 
 
60 
 
 [left(i ,x ) ][left(z ,z ) ] => [leftfz ,z ) ]. 
 
 Restrictions extend or modify decision rale 
 specifications. Por every occurrence of the CONDITION 
 of the restriction in the condition part of the 
 decision rule, the selector is added to the condition 
 part of the decision rule if it is not already 
 there. If the SELECTOR is already in the condition 
 part of a decision rule, then the values of the 
 SELECTOR replace the values in the occurrence of 
 SELECTOR in the decision rule. Using this feature, 
 the transitive closure of a transitive function may 
 be calculated. A restriction which adds equivalence 
 type predicates is included by default. Each 
 restriction is applied to each decision rule as it is 
 entered into the program. 
 
 Domain Generalization Structure Specification 
 
 This specification must satisfy the 
 production: 
 
 <T-STR0C> ::= [ <F«-SYH> = <REP> ]=>[ <PN-SYM> = <REP> ] 
 
 where the two function symbols are the same and 
 FN-SYM and REP are as in the decision rule grammar. 
 
61 
 Por example: 
 
 [s = 1,3,4] => [s = 6]. 
 
 [s = 0,2] => [s = 7]. 
 
 [s = 4] => [s = 8]. 
 
 [s = 6,8] => [s = 9], 
 
 Entering a T-STROC automatically sets the 
 type of the domain for the function symbol to the 
 structured type. Each element in the REF of the 
 condition part of this rule must either be a leaf of 
 the corresponding tree or have been previously 
 defined with another T-STROC specification. 
 
 Parameters: 
 
 Several parameters may be entered into the 
 program. In addition to the parameters mentioned in 
 the following sections (i.e., the MAXSTAR parameter 
 and optimality criterion parameters) , certain trace 
 reguests may be entered to follow the execution of 
 the program. 
 
 5.2 Program Output 
 
 Several trace features are available to obtain a list 
 of intermediate results in the program. An example of 
 
62 
 
 program output is given in appendix A. Although input 
 decision rules may be any formula which satisfies the grammar 
 given in the previous section, the program only searches for 
 generalizations which can be represented by a connected graph 
 structure with functional dependence edges. The user may 
 construct decision rules to satisfy this constraint by 
 including new predicates which link products which have no 
 arguments in common. (In general, it can be tested to see if 
 a product of selectors in a decision rule has a connected 
 graph structure by determining whether there is a 
 partitioning of selectors into two products which have no 
 argument in common. If there is such a partitioning, then 
 the graph structure is not connected.) 
 
 Different generalizations may be obtained by varying 
 the program parameters and reordering the decision rules. In 
 general, increasing any of the MAXSTAR, ALTER, or NCONSIST 
 parameters (described below) will cause the program to 
 require more memory and time but the resulting 
 generalizations may be more optimal. Rearranging the 
 optimality criteria may also produce a different result. A 
 higher tolerance associated with a particular cost function 
 will reduce the selection based on that function. The 
 default optimality criteria (or optimality cost functions) in 
 the order of application are the following: 
 
63 
 
 1. Minimize the inconsistencies of a rale with tolerance 
 0.30, i.e., the number of events covered by the rule 
 which are not supposed to be covered by the rule, 
 (this is cost function number 3 in section 5.5). This 
 
 allows the program to produce consistent 
 generalizations quickly. The high tolerance removes 
 highly inconsistent rules while leaving selection of 
 nearly consistent rules to the remaining cost 
 functions. 
 
 2. Minimize the number of products in the complete 
 generalization with tolerance O.CC (this is cost 
 function number 1 in section 5.5). 
 
 3. Minimize the number of selectors in each product 
 produced by the program (function 2 ir section 5.5). 
 
 *»• Minimize the cost of functions in each product 
 (function U in section 5.5). If costs are specified, 
 this criterion may be moved forward. If the user 
 wishes certain functions to appear in the resulting 
 products, the costs for these functions may be 
 specified (given negative cost). Similarly, functions 
 
64 
 
 which are very difficult or costly to measure may be 
 given appropriate positive costs. 
 
 5. Maximize the intersection of resulting rules (cost 
 
 function 5 in section 5.5). In real situations, the 
 
 separate products produced by the program may 
 
 represent a large number of common input decision 
 
 rules along with some peculiarities of specific 
 
 situations which arise. Use of this cost function 
 
 will favor the selection of a more representative 
 
 result as opposed to one which describes only a 
 
 particular set of situations. Appendix B contains a 
 
 more detailed discussion of a similar cost function 
 
 in the VL system. 
 1 
 
 The program contains about 40 procedures. 
 Five major tasks which are performed by some groups 
 of procedures are described below. 
 
 5.3 Formation of a Complete Generalization 
 
 A generalization is found of a set of decision rules 
 is found containing a specified value I in the decision 
 part. Two sets of products are generated: a set F1 which 
 contains all products in the CONDITION parts of rules with a 
 decision value of I, and a set F0 which contains all other 
 
65 
 
 products. Each product is called a c-formula 
 (conjunctive-formula) • 
 
 One c-formula E1 of F1 is selected at random and a 
 connected-con-junctive-f ormula (c2-formula) is generated which 
 is a generalization of E1, consistent with respect to the set 
 P0 r and near optimal with respect to a user defined 
 criterion. A c-formula is connected if its graph structure 
 representation is weakly connected by functional dependence 
 relations. A c-formula is consistent with respect to a set of 
 c-f ormulas FO if it does not intersect with any element of 
 the set FO (i.e., there is no event which satisfies both 
 c-formulas) • 
 
 Once a generalization of E1 is found, it is saved in 
 a set CQ and all elements of F1 which are covered by this 
 generalization are removed from F1. One c-formula E1 covers 
 another c-formula EO if E1 is a generalization of EO. 
 Another element of the new set F1 is selected and the 
 procedure repeated. When there are no more elements in F1, 
 the complete, consistent generalization of the set of 
 c-formulas F1 is the disjunction of all c2-formulas in CQ. 
 
 5.4 Determine Cover and Intersection of 2 Fornrulas 
 
 Two similar procedures are decribed here. The test 
 to determine whether a c2-f ormula E covers a c-formula E 1 is 
 
66 
 
 used when E* is an element of the set P1 . The test to 
 determine whether E intersects with E* is used when E' is an 
 element of FO (i.e., to determine if E and E' are 
 consistent) • The procedure uses the graph structure 
 representations of E and E* (6 and 6* with nodes and edges 
 V^E^V^E* respectively). The graph G is assumed to be weakly 
 connected. E cowers E 1 if there is a specia lizing isomorphism 
 (s- isomorphism) from G to a subgraph of G*» The reverse 
 mapping (from a subgraph of G' to G) is called a generaliz ing 
 isomorphism (g- isomorphism) • E intersects with E* if there is 
 an intersecting isomorphism (i- isomorphism) between G and a 
 subgraph of G'« Each isomorphism from G to a subgraph of G' 
 is a 1-to-1 correspondence between nodes and edges of G and a 
 subset of nodes and edges of G' where the correspondence (or 
 matching) of nodes and edges is defined as follows: 
 A node n of G matches a node n* of G' if: 
 
 1. They are both selector nodes or both quantified 
 variable nodes. 
 
 and 2. If they are selector nodes, then the function symbols 
 in both nodes are the same. If they are variable 
 nodes, they are of the same group of variables. 
 
67 
 
 and 3« With an s- isomorphism or g-isomorp hism, the set of 
 values associated with n is a generalization of the 
 set of values associated with n'« (lhe sets of values 
 may be equal.) With an i- isomorphism, the sets of 
 values intersect. In the case of selector nodes, 
 these values are the elements in the reference of the 
 selector. In the case of guantified variable nodes, 
 these values are the subranges of the variables. 
 
 An edge of G matches an edge of G* if : 
 
 1. They have the same label 
 
 and 2. The respective head nodes match and the tail nodes 
 match. 
 
 To speed rejection, a quick scan through the nodes is 
 made to see if there is a correspondence between nodes of G 
 and a subset of nodes of G* (ignoring links between nodes). 
 If there is a possible correspondence, a procedure is invoked 
 which locates a subgraph of G* which is isomorphic to G and 
 assigns each node of G to a corresponding node of G' • The 
 procedure is as follows: 
 
68 
 
 1* Select a starting node (n ) of 6 which contains the 
 
 
 
 most labelled incoming edges. (This is the selector 
 node with the largest number of arguments*) Selecting 
 a node of this type insures that there is a minimum 
 of backtracking through the starting node* 
 
 2* & rooted directed a-cyclic graph G* vith nodes and 
 
 edges V* and E* is constructed from G by copying all 
 
 nodes and edges of G to G* and assigning a direction 
 
 to each edge of G* so that G* has no cycles and for 
 
 each node x in f V*-n* 1, there is a path from n* to 
 
 
 
 x* (n* is the node of G* which corresponds to n in 
 
 
 G. ) A traversal of the graph G* is the list of edges 
 and nodes visited in a preorder traversal of G* vith 
 root n* • A preorder traversal of a subgraph with 
 root x visits the node x, visits each outgoing edge 
 of x and traverses the subgraph which has as the 
 root, the head node of the traversed edge* 
 
 3* The graph G is traversed in the order of the 
 traversal of corresponding nodes and edges of G** At 
 each step of the traversal of G, a node and new edge 
 of G* is found which match the node and edge of G* 
 If two nodes match, they are assigned to each other 
 
 
69 
 
 and a record of the matching nodes and edges is kept 
 for each assignment in a backtrack list. To 
 establish a 1-to-1 correspondence, nodes of one graph 
 which are previously assigned can only match 
 corresponding assigned nodes of the other graph. 
 
 4. If there is no node and edge of G ' vhich matches a 
 node and edge of G, the procedure backtracks to the 
 previous nodes and edges on the backtrack list, 
 erasing the last nodes and edges on the backtrack 
 list and the assignments associated with the nodes. 
 Another node and edge of G a are selected which match 
 the last node and edge of G on the backtrack list and 
 the traversal of G continues. If nc node and edge of 
 G* can be found at this point, then the procedure 
 again backtracks until a new match is found or the 
 backtrack list is exhausted. 
 
 5. If the traversal of G is complete, then G covers 
 (intersects) G 1 * If the backtrack list is exhausted, 
 then G does not cover (intersect) G'. 
 
 A feature is included which finds all subgraphs of G' 
 vhich are isomorphic to G. This feature is used in section 
 5.7 and in adding restrictions to c-formulas. 
 
70 
 
 6, If the traversal of G is complete, then the current 
 set of assignments is the desired napping. To find 
 the next isomorphism, the procedure returns to step 4 
 assuming that the last nodes and edges on the 
 backtrack list did not match* 
 
 5, 5 Trimming a Set of c-formulas 
 
 ItilDSJiDa is *& e process of selecting the BAXST1R best 
 
 elements of a set of c-formulas with regard to a user defined 
 
 criterion. The user specifies the cost functions which are to 
 be used, the order in which they should be applied, and the 
 
 tolerance associated with each cost function* Implemented 
 cost functions are: 
 
 1* The number of events of the current set P1 which are 
 covered by a c2-formula. (The negative of this 
 guantity is used to obtain a cost*) This function 
 minimizes the number of c-formulas in CQ* 
 
 2. The number of selectors in a c2-formula* The 
 function minimizes the number of selectors in the 
 
 c-formula. 
 
71 
 
 3. The number of events of FO which intersects with a 
 c2-formula. This function leads more rapidly to 
 consistent c-formulas. 
 
 4. The total cost of all functions contained in a 
 c2-f ormula. 
 
 5. The number of events of the original set PI which are 
 covered by a c2- formula. (The negative of this 
 guantity is used to obtain a cost.) This function 
 finds the most representative c-formulas. 
 
 A set of c-formulas is trimmed using n cost functions 
 
 (cf ,cf ,...cf ) and relative tolerance for each cost 
 12 n 
 
 function (tol ,tol ,...tol )• The costs are applied in the 
 12 n 
 
 order specified by the user (cf first, cf second, etc) • For 
 
 1 2 
 
 each cost function cf , the MAXSTAH best c2-formulas along 
 
 i 
 
 with all c2-formulas eguivalent in cost to the HAXSTAR best 
 
 c-formulas are passed to the evaluation using the next cost 
 
 function cf • Other c2-formulas in the set of c-formulas 
 i*1 
 
 are discarded. With the last specified cost function (cf ) , 
 
 n 
 
 only the MAXSTAH best c-formulas are retained. 
 
 For each cost function cf , i=1,2,...,n, eguivalence 
 
 i 
 
 of two c-formuas in cost is defined using an absolute 
 
72 
 
 tolerance (AT ) • Suppose the set of c-formulas P is composed 
 i 
 
 of a list p , p , ...p • After values for cost function cf 
 12m i 
 
 have been evaluated cf (p ) for each c-formula p , the 
 
 i J J 
 
 maximum and minimum cost functipn values are determined 
 
 cf (p ) and cf (p )• An absolute tolerance (AT ) is 
 i max i min i 
 
 calculated using the user specified tolerance tol as 
 
 i 
 
 follows: 
 
 AT. = tol. * (cf. (p ) - cf. (p )) 
 ixi max l mm 
 
 The MAXSTAR c-formulas of least cost are determined and the 
 
 list reordered (p ,p ,...,p ,...,p )• If i<n then 
 
 1 2 MAXSTAR m 
 
 c2-formulas which are not equivalent in cost to p are 
 
 MAXSTAR 
 
 discarded. 
 
 P = P - [p : cf (p ) - cf (p )>AT ] 
 j i j i MAXSTAR i 
 
 If i=n # (the last cost function) then only the MAXSTAR best 
 c2-forwulas are retained. 
 
 P = P - [ p : j >MAXSTAR], 
 
 The set of c2-formulas which remains is the desired trimmed 
 set of c2-formulas. 
 
73 
 
 5.6 Formation of a Set of Consistent Generalizations 
 
 A star (denoted by MQ) is formed which covers E1 • (A 
 star which covers E1 is a set of consistent c2-f ormulas which 
 cover E1.) The procedure begins by forming a partial star (P) 
 which contains a set of c-formulas each consisting of one 
 selector of E1 • (The £artial sta r may contain c2-formulas 
 which are not consistent with respect to FO. ) This partial 
 star is trimmed according to the user s applied optimality 
 criterion. The conjunction in each c-formula which remains 
 after trimming is multiplied by each selector of E1 which is 
 directly connected to it to form a new partial star. 
 Consistent c2-formulas are placed in WQ# The partial star is 
 again trimmed and new selectors added to each product until 
 the desired set MQ of c2-formulas is obtained. Several 
 paramenters control the sizes of sets in this procedure: 
 
 BAXSTAH - the number of c-formulas in a partial star after 
 
 trimming 
 NCOHSIST - the minimum number of consistent c2- formulas 
 
 which must be in HQ 
 ALTER - the maximum number of new alternatives which may be 
 
 formed by adding selectors to an element of a partial 
 
 star. 
 
7U 
 
 In the following discussion, equivalence type 
 
 selectors (i.e. , selectors of the form [f (z , x )=samel) are 
 
 1 2 
 
 treated differently from selectors involving a function 
 symbol and a set of values in the reference. 
 
 1. A partial star P is formed which contains all 
 selectors of E1 with unary functions. 
 
 2. P is trimmed to contain only the best MAXSTAK 
 c2- formulas. Consistent c-formulas are placed into 
 HQ. If fewer than NCOMSIST elements are in MQ, then 
 step 3 is executed. Otherwise, the AQ procedure is 
 applied to the elements of NQ (as described in 
 section 5.7) • 
 
 3. A new partial star P* is formed fronr the old one (P) • 
 
 Por each element p in P, a list of all variables 
 
 i 
 
 (i.e., arguments of selectors of p ) is formed. All 
 
 i 
 
 arguments of equivalence type selectors which occur 
 in the corresponding selector of E1 are also included 
 in the list. 
 
 U. A list of all selectors of E1 which are not already 
 
 in p and which have at least one argument in the 
 i 
 
75 
 
 variable list (found in step 3) is created* If there 
 are more than ALTER elements in this list, the best 
 ALTER selectors are retained (using as a criterion 
 the cost of functions in the selectors) • 
 
 5, For each of these selectors, a new c2-formula is 
 
 formed which contains the original selectors in p 
 
 i 
 
 and the new selector. If the new c-formula contains 
 an equivalence selector with only cne argument, then 
 the new c-formula is discarded; otherwise, it is 
 placed in P*. Steps 2 through 5 are repeated, setting 
 P - P 1 in step 2 until NCONSIST elements are in MQ or 
 until no new elements are in the new partial star P*. 
 
 5,7 Extending the References of a Consistent c2-formula 
 
 Each consistent c2-formula of MQ (obtained in section 
 5»6 step 2) contains an alternative, near optimal conjunction 
 of selectors of E1 which distinguishes E1 from any c-formula 
 of PO, Using some methods developed for the program AQ7, the 
 reference of each of these selectors may be generalized to 
 obtain a consistent c2-formula which will possibly cover 
 possibly more c-formulas of F1. 
 
 Given a graph G of a conistent c2-fcrmula mg in MQ, a 
 c^structure is created (G*) by replacing all references of 
 
76 
 
 nodes of G with * (the complete set of values for the 
 
 function in the selector) • The nodes of G* are enumerated n* 
 
 i 
 
 (i=1 ,2 ,. •• # m) and a TL system is created with each VL 
 
 variable x related to a node n* of G*. The domain (denoted 
 i i 
 
 D ) of variable x is the same as the domain of the function 
 i i 
 
 or variable in the node n* • 
 
 i 
 
 The VL events space may be defined: 
 
 E - D xD x • •• x D • 
 12 n 
 
 Two sets of VL complexes L and L* are formed from the events 
 
 1 
 
 of the current set F1 and the set FO respectively. 
 
 Individual complexes in these sets are denoted 1 and 1' • 
 
 i i 
 
 Each compelex covers a set of points in the space E. For 
 
 each element of F1 and FO all isomorphisms from the 
 
 c-structure G* to the graph representation G of a c-forraula 
 
 in F1 or FO are determined. For each isomorphism obtained 
 
 from F1 and FO, a VL complex is created (1 or 1' )• 
 
 1 i i 
 
 Denoting the value sets of the nodes in a subgraph of G which 
 
 is isomorphic to G* as R,R,...,B, the corresponding VL 
 
 12m 1 
 
 complex may be written: 
 
 [x =R ][x =R ] ... [x =R 1, 
 1 12 2 mm 
 
 This complex covers the VL events: 
 
 1 
 
 
77 
 
 B XB x . • . I B 
 1 2 m 
 
 in the event space E, 
 
 First, the complex 1 which results from extracting 
 the values from the nodes of the graph of iig is generated. 
 Then all other isomorphisms from G* to c-fcrmulas of F1 are 
 determined and a complex added to L for each isomorphism 
 which results in a new complex that is not already in L« The 
 set L* is created in a similar manner by generating all 
 distinct complexes resulting from isomorphisms from G* to 
 c-formulas of PO. 
 
 Since the c-formula mg is consistent with regard to 
 
 PO, the complex 1 is disjoint from all complexes in L 1 . 
 
 (That is, there is no point in the VL event space E which is 
 
 in both 1 and a complex of L'«) A near optimal extension of 
 1 
 
 1 against L' in E may be calculated using a version of the 
 AQ7 program. The best complex in this extension (1 ) is 
 
 g 
 
 calculated according to a user defined criterion. The 1 
 
 g 
 
 complex is converted to a c-formula by replacing the value 
 
 set of each node n* of G* by the reference of the selector 
 
 i 
 
 with variable x in 1 • This c-formula is consistent with 
 
 i g 
 
 respect to the set FO, (This is evident since, if there were 
 a VL event which satisfied both the c-formula from 1 and a 
 
 1 q 
 
 c-formula of FO, then one could find a VL complex — using 
 
78 
 
 an isomorphism between G* and the c- formula of FO — which 
 
 intersected with 1 and L*« But since 1 and L* are disjoint, 
 
 1 1 
 
 this can not happen.) 
 
 The cost function for the AQ7 procedure computes the 
 cost of a complex. Cost functions may be selected from the 
 following: 
 
 1. The number of elements in L which are covered by a 
 complex but not cowered by any previous 1 • This is 
 
 q 
 
 the AQ7 counterpart to the cost function 1 in section 
 5.5) (Use the negative of this value to get a cost.) 
 
 2. The number of selectors in a complex (the AQ7 
 counterpart to function 2 in section 5.5). 
 
 3. The number of elements of L covered by a complex 
 which are associated with different events of F1. 
 
 4. The total cost of variables which appear in a complex 
 (i.e., the cost of functions or variables in 
 associated nodes of 6*). This is the counterpart of 
 the function 4 in section 5.5. 
 
 5. The total number of events in L covered by a 
 
 complex. 
 
 
79 
 
 6. The number of events in L* covered by a complex (the 
 AQ7 counterpart to function 3 in 5.5) • 
 
 Trimming is done in the manner described above for c-formulas 
 (sec 5.5) • A HAXSTAR parameter is specified for this 
 procedure and the 1 is selected from the extension using a 
 
 q 
 
 MAXSTAR value of 1. Appendix B gives a more complete 
 description of the AQ7 procedure. 
 
 5.8 Adding New Punctions and Predicates to c-formulas 
 
 The program currently allows three types of new 
 functions to be added to existing c-fcrmulas: 1) global 
 descriptors (meta functions) which count the freguency of 
 occurrence of selectors with unary functions; 2) eguivalence 
 type predicates (of the form [f (x , x ) =same ] i.e., the value 
 of f is the same for x and x ); 3) extremity type predicates 
 ([lst-f (x ) ] or [mst-f (x ) ]) which indicate that the argument 
 x is at one end of a seguence of binary predicates. 
 Punctions are added at the user's discretion. 
 
 Beta Selectors - There are two types of meta functions 
 currently calculated and added to a c-formula as a 
 meta selector: One type (#PT(f=a) where f is an 
 atomic function and a is a value in D (f ) ) , counts the 
 
80 
 
 number of times a particular selector ([f (••)=*]) 
 appears in a c-formula. The second type 
 (FORALL (f =a) ) is a predicate which is true if a 
 function assumes only one ' value in a c-formula and 
 false otherwise. 
 
 For each atomic function-reference pair which 
 appears in any c-formula , a met a selector is added to 
 the c-formula which has a meta function in the 
 referee. For example, a c-formula 
 
 [ tx (x^ =1 ][ tx (x 2 ) =1 ][ sh (x^ =1 ][ sh (x 2 ) =0 ] 
 
 generates the four meta selectors: 
 
 [ #PT(tx=1)=2] - the number of parts with tx=1 is 2 
 
 [ #PT (sh=0) =1 ] - the number of parts with sh=0 is 1 
 
 [ #PT(sh=1) =1 ] - the number of parts with sh=1 is 1 
 [F0RALL(tx=1) ] - all parts have tx=1. 
 
 Since the number of such selectors may be 
 quite large, the list of meta functions is trimmed to 
 a small set. The size of the set is determined by a 
 parameter METATRIM supplied by the user using as 
 criteria the degree to which a value of the new 
 function will separate the sets F1 and F0. For each 
 
 
81 
 
 meta function- value combination (meta selector) which 
 is generated, the number of c- formulas of F1 and PO 
 which satisfy the selector is calculated. Associated 
 with the meta function (mf) are two numbers: P1C0V 
 and FOCOV. F1C0V is the maximum nunrber of c-formulas 
 of F1 cowered by a meta selector arising from the 
 meta function mf, FOCOV is the nurrber of c- formulas 
 of FO which are cowered by the meta selector which 
 gave the highest F1C0V value. 
 
 The list of possible meta functions is 
 trimmed to METATRIM remaining meta functions by 
 sorting in descending order, the list of meta 
 functions according to the primary field F1C0V and 
 the secondary field FOCOV and selecting the first 
 HETATRIH functions from this list. The meta 
 selectors which result from applying each of the 
 selected meta functions to each c-formula are 
 automatically appended to the c-forwulas and carried 
 with each c-formula during the generalization 
 process. Setting HETATRIH to bypasses the entire 
 meta selector generation process. 
 
 Eguivalence Predicates - These predicates may have 
 arbitrarily many arguments whose order is 
 
82 
 
 irrelevant. They are calculated by scanning all 
 selectors in each c- formula for sets of 2 or more 
 selectors with unary functions which have the same 
 atomic function and reference* Such a set of 
 selectors is said to be equivalent and a new 
 predicate is created which contains an argument from 
 each of the atomic forms of all equivalent 
 selectors* For example, a c-formula of the form: 
 
 [s(x )*1 ][s(x 2 )=1 ][s(x 3 ) = 1][s(x^)=2][s(x 5 )=2] 
 
 leads to the creation of predicates: 
 
 [s(x *x *x )=sameirs(x . x ) -same ] 
 12 3 4 5 
 
 Bxtremeties Predicates - These are unary predicates which 
 represent the ends of a sequence of selectors with 
 the same binary predicate: 
 
 [p(x »x ) ][p(x ,x ) ] . . . [p(> i _ 1 #x i ) ] 
 
 where, in the sequence, if i>2 then the first 
 
 argument of the j-th selector (1<=j<i) is the same as 
 
 the second argument of the j+1-st selector* Also, 
 
 the variables x (and x ) do not appear as the second 
 
 1 i 
 
 (or first) argument of another selector with atomic 
 
83 
 
 function p. The new predicates formed for the 
 arguments x and z of such a seguence are written: 
 
 [MST-p(x) ] and [LST-p («.)]• 
 
 1 x 
 
 Por example, the c-formula: 
 
 [ontopfx ,z ) tfontop(x ,x ¥ontcp(x ,x ) ] 
 
 would give rise to the new predicates: 
 
 [HST-ontop(x ) ][LST-ontop(x ) ]f LST-ontop (x ) ]. 
 1 3 4 
 
84 
 
 6. Examples of Decision Rale Generation using IND0CE_1 
 
 The program IMDUCE_1 is written in PASCAL for either 
 a CDC-CYBER 17 5 or a DEC-10 computer. (The two versions 
 differ in syntax and certain input output specifications.) 
 They contain approximately 3300 statements and occupy 34K 
 octal 60-bit words of CDC memory or 18K 36-bit words of 
 DEC-10 memory. The examples in this chapter were run on the 
 C7BER 175 using a field length of 130K octal words to include 
 both the program and data. Execution time estimates which 
 are given below are for the CYBER and should be accurate to 
 within 30%. One example was run on both the CYBER 175 and a 
 DEC-KI10 to obtain a time comparison between machines. 
 Execution time on the DEC-10 was 8 times larger than on the 
 CYBER. 
 
 Por each example, a sample of the input decision 
 rules is listed along with tables describing function domains 
 and program parameters. The resulting generalizations are 
 given as decision rules in disjunctive normal form. The set 
 of objects associated with a description below is given as 
 the value of the decision part of the rule (e.g. for a set 
 , the associated decision part is [d=1]). lith some 
 examples, several alternatives were produced by the program. 
 Each alternative which was produced is given as a separate 
 
85 
 
 decision rule* The various symbols contained in these tables 
 are defined belov: 
 
 Program Parameters: 
 
 Parameter 
 
 Descript ion 
 
 1QHAXSTAR the HAXSTAR parameter for the AQ7 
 
 procedure. This parameter specifies the 
 
 number of alternative complexes retained 
 
 in each partial star in the AQ7 procedure 
 
 (see appendix B) • 
 
 LQST 
 
 the LQST parameter in the AQ7 procedure* 
 If this parameter is set, then each 1 
 
 q 
 
 which is generated in the AQ7 procedure is 
 stripped so that the reference in the 
 selector for each variable in the 1 
 
 q 
 
 complex contains the minimum number of 
 elements under the constraints of the 
 domain structure vhile covering the same 
 number of events of L (see appendix B) • 
 
 AQCBIT 
 
 the list of cost functions used in the AQ7 
 procedure in the order of application (the 
 minus sign indicates that the quantity 
 
86 
 
 which is calculated by the function is 
 negated to obtain the cost of the complex) 
 (see section 5.7). 
 
 AQTOLERANCE the tolerance associated with each cost 
 function. If this figure is less than 1, 
 then it is a relative cost, otherwise it 
 is an absolute tolerance (see section 
 5.5). 
 
 AQNF the number of consistent generalizations 
 
 which are created in the procedure which 
 forms consistent generalizations (i.e., 
 the minimum number of elements in the star 
 NQ) (see section 5.6). 
 
 ALTER the number of alternatives which are 
 retained in a partial star (see section 
 5.6). 
 
 VLCRIT the list of cost functions used to trim a 
 partial star of c-formulas in the order of 
 application (a minus sign indicates that 
 the quantity which is caluculated by the 
 function is negated to obtain the cost of 
 the c-formula) (see section 5.5). 
 
 
87 
 
 VLTOLEB&HCE the tolerance associated with each cost 
 function. A value which is less than 1 is 
 assumed to be a relative tolerance; 
 otherwise, it is an absolute tolerance 
 (see section 5.5). 
 
 HBT1THIH the number of met a selectors of the type 
 FORALL and #PT (number of parts) which are 
 added to each formula. 
 
 EXTHTI the predicates of the type [HST-f (x ) ] and 
 
 i 
 
 [LST-f (x ) ] were added to input decision 
 i 
 
 rules. 
 
 EQOI? the predicates of the type [f (x *x )-same] 
 
 mm 
 
 were added to the input decision rules. 
 
 (These correspond to the predicates 
 
 [EQ f (x .x ) 1 used above) 
 12 
 
 Domain descriptions: 
 
 Column Description 
 
 HAHE the name of the function 
 
 NAFG the number of arguments of the function 
 
88 
 
 TYPE the domain struct ace: 1- nominal, 
 2-interval, 3- tree structured. 
 
 COST the cost of the function 
 
 HIH the minimum value in the domain of the 
 function 
 
 HAX the maximum value in the domain of the 
 function 
 
 STRUCTURE the allowable generalizations in tree 
 structure domains 
 
 Meta Selectors: 
 
 Column Description 
 
 HS the meta selector number associated with 
 this function 
 
 TYPE either #PT (number of parts) or FORALL 
 
 FUNCTION the associated function and value (i.e., 
 the associated selector ignoring arguments 
 e.g. PORALL TI = is a predicate which 
 
89 
 
 has the value 1 if all parts have a 
 texture of and otherwise) 
 
 F1C0V 
 
 the number of c- formulas in P1 covered by 
 one value of the meta function. 
 
 POCOV the number of c-formulas in PO which are 
 covered by the meta selector associated 
 with the value P1C07 above (see section 
 5.8). 
 
 6.1 Figures Example (EX 1) 
 
 The objects in figure 6.1 (from [Michalski 1977]) are 
 
 examples of three classes of situations: 0,0, and , each 
 
 12 3 
 
 class containing 3 examples. The figures are expressed as 
 VL formulas using 6 descriptors below. The values of 
 descriptors are translated to integers as follows: 
 
 Descriptor 
 
 Structure 
 
 Domain 
 
 ontop 
 
 nominal 
 
 0-false, 1-true 
 
 inside 
 
 nominal 
 
 0-false, 1-true 
 
 next 
 
 nominal 
 
 0-false, 1-true 
 
 size 
 
 interval 
 
 -small, 1-ffedium, 2- large 
 
90 
 
 tx (texture) nominal 0-clear, 1- shaded 
 
 shape tree-struc O-'triangle, 1-circle, 
 
 1,5 => 8 2-rectangle, 3-diamond 
 0,2 # 3,7=>9 4-open-top, 5-ellipse 
 
 6-open bottom, 7-square 
 
 The input decision rule for the first object in has the 
 
 1 
 
 following form: 
 
 [ontop(p f p ) ] [ontop(p ,p ) ] [size(p )=1 ] [size(p )=1 ] 
 
 [size(p 3 )=2] [tx(p )=0] [tx(p )=1] [tx(p 3 )=0] [shape (p^a ] 
 
 [ shape (p )=1] f shape (p) =4] [ inside (p , p ) ]=>[d=1 ]. 
 2 3 2 3 
 
 with domain structure: 
 
 [shape=1,5] => [shape=8], 
 
 [shape=0,2,3 # 7 ] => [shape=9]. 
 
 The resulting domain table is: 
 
 NAME HARG TYPE COST BIN HAX STRUCTURE 
 PORALL 10 1 
 #PT 2 
 
 
91 
 
 shape 
 
 1 
 
 3 
 
 
 
 
 
 9 1 5«> 8; 
 
 2 3 7*> 9; 
 
 ontop 
 
 2 
 
 
 
 
 
 1 
 
 
 P 
 
 
 
 
 
 
 
 4 
 
 
 p i 
 
 
 
 
 
 
 
 4 
 
 
 P 2 
 
 
 
 
 
 
 
 4 
 
 
 P 3 
 
 
 
 
 
 
 
 4 
 
 
 P » 
 
 
 
 
 
 
 
 4 
 
 
 size 
 
 1 
 
 
 
 
 
 
 2 
 
 
 tx 
 
 1 
 
 
 
 
 
 
 1 
 
 
 inside 
 
 2 
 
 
 
 
 1 
 
 1 
 
 
 d 
 
 
 
 
 
 
 1 
 
 3 
 
 
 next 
 
 2 
 
 
 
 
 1 
 
 1 
 
 
 Generalized decision roles using no new f anct ions (8 seconds) 
 
 The program was run vith the above decision rules 
 several times vith different parameters for each ran* The 
 first run through the program used modest values for the 
 parameters AQMAXSTAE, ALTER, VLHAXSTAR and NCONSIST and 
 default values and ordering of criteria vith no new functions 
 added* The parameter values and resulting generalizations are 
 given belov: 
 
 Parameters: 
 
 AQPARHS 
 
 VLPARMS 
 
92 
 
 AQHAXSTAR = 4 
 
 LQST HCONSIST = 4 ALTER * 4 VLHAISTAR = 4 
 
 AQCRIT 
 
 
 AQTOLERAHCE 
 
 
 VLCRIT 
 
 VLTOLERANCE 
 
 -1 
 
 
 
 
 
 3 
 
 0.30 
 
 2 
 
 
 
 
 
 -1 
 
 
 
 4 
 
 
 
 
 
 2 
 
 
 
 NBP OP 
 
 CRIT: 
 
 AQNF = 2 
 
 
 
 7LNF = 3 
 
 NEW FNCTNS: 
 
 METATRIH » 
 
 
 
 
 
 For the set , the program discovered one product 
 
 which covers all elements of but no element of O or • 
 
 1 2 3 
 
 The product represented as a decision rule is: 
 
 [ontop( Pi ,P 2 ) ir size ( Pi )=1][ size <P 2 )=1Xtx(P i )=0] => [d=1]. 
 (If there is a clear, medium size part en top of a medium 
 size part, then make decision d=1.) 
 
 Using a different set of parameters (MCOHSIST-2) , the 
 description (which is less optimal according to the criteria 
 specified) is obtained: 
 
 [ontop(p ,p ) ][shape(p )=9 ][shape(p )=9 ] v 
 
 [shape( Pi ) = 3J=>[d=1] 
 
 The description of the set produced several 
 
 2 
 
 alternative generalizations which describe one or two parts 
 
93 
 
 of bat only one generalization which covered ail 3 parts 
 
 in : 
 2 
 
 [ ontop (p^ , p 2 ) K size (pj =1 J[ shape (p ) *8 ]«>£d=2 ] 
 (there is an ellipsoid on top of a part with medium size) 
 
 there vas no single product which covered all objects in • 
 The program found one generalization which cowered two 
 examples of , namely: 
 
 [shape ( ?1 ) =5 ][ tx ( P<| ) =1 ]=>[ d«3 ] 
 (there is a shaded ellipse) 
 
 In covering the second object in , the program found a 
 number of combinations of shape, texture and size of parts of 
 
 the example of • The rule which was selected is: 
 
 3 
 
 [ ontop (P 1 . P 2 K shape (p ) =2 tf tx (p^ =1 ]=>[ d= 3 ] 
 (there is a rectangle on top of a shaded part) 
 
 giving a final description of the set : 
 
 [shape(p i )=5][tx(p i )=1] v [ ontop (p , p^ )[ shape (p^) =2 ] 
 
 [tx(p 2 )=1]=>[d=3) 
 
 Generalization with EITflST type predicate s (8 seconds) 
 
9a 
 
 In the second application of the program to the 
 example in figure 6.1 # the BZTHTT type predicates were added 
 to each input decision rule. The only difference in 
 parameters is that these new predicates are added to the 
 input decision rules. The results are the following: 
 
 For the set , there were two alternative generalizations 
 
 each of which cower all three parts in : 
 
 1 
 
 T shape (p^ ) =9 ][ tx (p^ ) =0 ][ HST-ontop (p^ ) ]=>[ d=1 ] 
 (the top part is a clear polygon) 
 
 [size( Pi )=1 ][tx( Pi )=0]CMST-ontop( Pi ) ]=>[d»1 ] 
 (the top part has medium size) 
 
 Only one product was found to cower the set : 
 
 [ shape (p ) =8 ]f HST-ontop (p ) ]=>[ d=2 ] 
 
 (the top part is an ellipsoid) 
 
 There is still no single product which cowers all 
 
 ob-jects in however, the description of the second ob-Ject 
 3 
 
 in was somewhat simplified: 
 
 3 
 
 fshape(p )=5 )[tx(p )=1 ] v [ shape(p <| ) =2 ][ M ST-ontop (p^ ] 
 
 = >£d=3] 
 
95 
 
 Gene raliz ations with meta selectors (9 seconds) 
 
 The next pass of the program included the three meta 
 selectors which were determined by the program to be the best 
 for describing each set of objects (using as a criteria the 
 values of F1C0V and FOCOV) • The results along with the 
 interpretation of the selected meta selectors are: 
 
 [ms1=2]=>[d«1 ] 
 (there are 2 clear parts) 
 
 The selected meta-selectors are: 
 
 HS TTPE PDNCTION P1C0? FOCOY 
 
 1 #PT tx =0 3, 
 
 2 #PT size =1 3, 3 
 
 3 #PT shape =6 3, 5 
 
 The three meta selectors count the number of parts with each 
 texture and the number of sguare parts* 
 
 The set O was cowered by the disjunction of tvo 
 selectors: 
 
 [ms3=1] ▼ [ms3=3]=>[d=2] 
 (there are either one or 3 clear parts) 
 
 The selected meta-selectors are: 
 
96 
 
 s 
 
 TYPE 
 
 PDMCTIOM 
 
 
 P1CO? 
 
 POCOV 
 
 1 
 
 #PT 
 
 shape 
 
 s 
 
 7 
 
 3, 
 
 4 
 
 2 
 
 #PT 
 
 shape 
 
 = 
 
 6 
 
 3, 
 
 5 
 
 3 
 
 #PT 
 
 tx 
 
 = 
 
 
 
 . 2, 
 
 
 
 Recall that tPT has an interval domain structure. Giving tPT 
 a nominal domain structure, the rale is: 
 
 [ms3=1,3]=>[d=3 ] 
 
 The most interesting simplification is with the set 
 
 • Tvo alternatives were discovered: 
 
 3 
 
 [ms1=0]=>[d=3 ] 
 [ms2=1]=>[d=3] 
 (all parts have shaded texture) 
 
 The selected meta-selectors are: 
 
 HS 
 
 TYPE 
 
 FUNCTION 
 
 
 F1COV 
 
 FOCOV 
 
 1 
 
 #PT 
 
 tx 
 
 = 
 
 
 
 3, 
 
 
 
 2 
 
 FOBALL 
 
 tx 
 
 = 
 
 1 
 
 3, 
 
 
 
 3 
 
 #PT 
 
 shape 
 
 = 
 
 5 
 
 2. 
 
 2 
 
 In summary, alternative descriptions for each set of 
 objects are: 
 
 f tPT (tx=0)=2 ]=>[d = 1 ] or 
 
 
97 
 
 [ shape (p )=9][size(p ) =1 ][ HST-ontop (p ) J=>[d=1 ] or 
 [shape ( Pi )-9][tx(p)=0][HST-ontop(p) ]=*>[ d=1 ] 
 (there are 2 clear parts or the top part is a clear 
 polygon or a polygon of medium size) 
 
 [ shape (p ) =8 ][ HST-ontop (p )]=>[d=2] or 
 
 [ ♦ PT (t x=0 ]=1 , 3 ]=>[ d=2 ] 
 
 (the top part is an ellipsoid or there are one or three 
 
 clear parts) 
 
 [F0HALL(tx=1) ]=>[d=3) or 
 [#PT(tx=0)=0]=>[d=3] 
 (all parts are shaded) 
 
 Descriptive generaliza tio ns (10 seconds each set) 
 
 If the program is given the description of only one 
 set of objects at a time, then a descriptive generalization 
 of sorts may be found by adjusting some parameters. The 
 significant modification of parameters is that the cost 
 function 2 (number of selectors) is replaced by -2 (i.e. , the 
 more selectors, the lower the cost). In addition, a 
 restriction was added to the predicate •on top* to implement 
 the transitive closure for this predicate: 
 
 [ ontop(p ,p ) ][ontop(p ,p ) ] => [ontop (p ,p ) ]. 
 
98 
 
 The resulting descriptions contain a product which covers all 
 objects in a set and contains as many selectors as possible* 
 Of course, since the descriptions • of the other two sets of 
 objects is not included, the descriptions are not 
 discriminant but they are complete. 
 
 AQPARHS VLPARHS 
 
 AQMAXSTAR = H LQST NCOHSIST = 70 ALTER = 3 VLHAXSTAR = 3 
 
 AQCRIT 
 
 
 AQTOLERA] 
 
 ICE 
 
 
 VLCBIT 
 
 VLTOLERAHCE 
 
 -1 
 
 
 
 
 
 
 3 
 
 0.30 
 
 -2 
 
 
 C 
 
 
 
 -1 
 
 
 
 H 
 
 
 
 
 
 
 -2 
 
 
 
 NBR OP 
 
 CRIT: 
 
 AQNF = 
 
 2 
 
 
 
 ?LNP = 3 
 
 NEW FHCTNS: 
 
 METATRIM ■ 
 
 3 
 
 EXTMTY 
 
 
 For each set, the products with the maximum number of 
 selectors which the program found to be com son to all objects 
 of the set are in the following decision rules; 
 
 rontop( Pi ,p 2 ) ]rsize( Pi )=1 tftxtp^l ][ shape ^ ) =9 ][ tx (p^ =0 ] 
 
 [HST-ontop(p )][size(p ) =1 , 2 1C shape (p ) = 2,4,5][tx(p )=0] 
 12 2 2 
 
 [LST-ontop(p ) ][ms1=0][ms2 = 2 Hms3=2]=>[d=1 ] 
 
 The selected meta-selectors are: 
 
99 
 
 s 
 
 TYPE 
 
 PUICTIOH 
 
 
 P1CO? 
 
 POC 
 
 1 
 
 #PT 
 
 shape 
 
 s 
 
 6 
 
 3, 
 
 
 
 2 
 
 #PT 
 
 size 
 
 - 
 
 1 
 
 3, 
 
 
 
 3 
 
 #PT 
 
 tx 
 
 = 
 
 
 
 3, 
 
 
 
 (The bottom part is clear and has size mediant or large* 
 The top part is a clear polygon, has medium size and 
 there are no parts with an open bottom, 2 parts of medium 
 size, and 2 clear parts*) 
 
 [ontop(p ,p ) ][ontop(p ,p ) ][ size (p ) =1 ,2 Xsize (p ) -1, 2 ] 
 
 r shape (p ) =8 )[ shape (p ) -0, 1 , 2, 4 ][ LST-ontop (p ) ] 
 12 2 
 
 [ ms1 =0 ][ ms2=0 ][ ms3=0 , 1 ]=>[ d= 2 ] 
 
 The selected meta-selectors are: 
 
 MS 
 
 TYPE 
 
 POICTIOH 
 
 
 P1COV 
 
 P0COV 
 
 1 
 
 #PT 
 
 shape 
 
 s 
 
 6 
 
 3, 
 
 
 
 2 
 
 #PT 
 
 shape 
 
 = 
 
 7 
 
 3, 
 
 
 
 3 
 
 #PT 
 
 shape 
 
 = 
 
 5 
 
 2, 
 
 c 
 
 (The top part is an ellipsoid vhich is not small and is 
 on top of two parts* The bottom part is of medium or 
 large size* Also, there are no sguare or open bottom 
 parts and no more than 1 ellipse*) 
 
 [ontop( Pi ,P 2 ) ][ size (p^) =0,1 K size (p^ =1 ,2 ]Ctx(p^)=1 ] 
 [tx(p )=1 ][HST-0HT0P(P ) J[ SHAPE (P ) =9 ] 
 [ LST-ontop (p ) ][ms1=0](ms2=1 ][ms3=0,1 ]=>[d=3 ] 
 
100 
 
 The selected Beta-selectors are: 
 HS TIPE PUHCTIOH P1CO? P0CO? 
 
 1 #PT tx 
 
 2 FORALL tx 
 
 3 tPT shape 
 
 =0.3, 
 = 1 3, C 
 = 6 2, C 
 
 (There are at least two parts both shaded vith the top 
 part being a polygon of small or medium size and the 
 bottom part being of medium or large size* ill parts are 
 shaded and there is not more than 1 open bottom part*) 
 
101 
 
 V, 
 
 21 
 
 Figures for Example EI 1 
 Figure 6, 1 
 
102 
 
 6.2 Arch Example (EX 2) 
 
 The objects in figure 6*2 are examples which describe 
 
 an arch much like the arch which Winston describes [Winston 
 
 70]* Some additional shapes and relations have been added to 
 
 give a more interesting example. The set 1 includes 
 
 examples of arches and the set k contains examples of 
 
 2 
 
 objects which are not arches. The description used in this 
 example are given below: 
 
 Descriptor 
 
 Structure 
 
 Domain 
 
 rel 
 
 nominal 
 
 0-on top, 1 -under 
 2-left, 3-right 
 d -other 
 
 touch 
 
 nominal 
 
 0-false # 1-true 
 
 or (ientation) nominal 
 
 0-horizontal, 1-vertical 
 
 shape 
 
 nominal 
 
 0-rectangle, 1-triangle 
 2-ellipse 
 
 The decision rule for the first arch in A as given to the 
 program is: 
 
 
103 
 
 [relfp^p )=0] [rel(p ,p )=0] [ rel (P 2 #P 3 > s 2 ] 
 [touch (p ,p)»1 ] [touch (p ,p ) = 1] [ touch( p ,p ) =0 ] 
 [or( Pi )=0] [or(p )=1] [or(p )=1 ] [shape (p )=0] 
 [ shape (p ) -0 ] [ shape (p ) =0 ] => [ d=1 ]. 
 
 mm -J 
 
 Restrictions describing the character of the 
 
 functions rel and touch are added to each rule: 
 
 [rel(p ,p )=4]=>[rel(p ,pj=4]. 
 12 2 
 
 [ rel (P.P J =0 ]=>[ rel (p ,p 
 12 2 
 
 [rel (p ,p )=1 }=>[rel(p ,p 
 
 [rel(p ,p )=2]=>[rel(p ,p 
 12 2 
 
 [ rel (p ,p ) =3 ]=>[ rel (p ,p 
 12 2 
 
 )-1]. 
 
 )=0]. 
 1-3]. 
 
 )=2]. 
 
 [ touch (p,p)=1 ]=>[touch(p ,p )=1 ]. 
 
 [ touch (p ,p )=0 ]=>[ touch (p , p ) =0 1. 
 **V 2 2 1 
 
 The program was run twice with these examples: once 
 to find an optimal discriminant generalization, and again to 
 find a descriptive generalization. The parameters and 
 results are as follows: 
 
 AQPARHS YLPARHS 
 
 AQMAISTAR = 2 LQST NCOHSIST = 12 ALTER = 1 YLMAXSTAR = 1 
 
 AQCRIT 
 -1 
 
 AQTOLERAHCE 
 
 
 VLCRIT 
 
 VLTOLERAHCE 
 
 0.30 
 
-2 
 
 
 
 4 
 
 
 
 NBR OP CRIT: 
 
 AQHF = 2 
 
 NEW FNCTNS: 
 
 METATRIH 
 
 104 
 
 -1 
 
 -2 
 
 YLMP = 3 
 
 DiscrinjinaQt description o£ sgt 4 ^ (12 seconds) 
 
 [rel(p ,p )=0] [touch (p,p)=1] [ touch (p ,p)=1 ] 
 
 [ touch (p,p) =0 ] [or(p ) = 1] [or(p .p )=same] 
 2 3 3 2 3 
 
 [ shape (p .p ) =same ]=>[d=1 ] 
 
 (There is a part which is touching two others and on top 
 of one. The other two parts are not touching but they 
 both have the same shape and a vertical orientation). 
 
 Descriptive decision rule for A i (2 seconds) 
 
 [rel(p ,p )=0] [rel(p ,p ) =0 ] [ rel (p ,p ) =2,3 ] 
 
 f touch (p ,p)=1 ] [touch (p ,p) = 1 ] [touch(p ,p )=0] 
 
 [or(p )»0] [or(p )=1] [or(p ) =1 ] [shape (p) =0,1 ] 
 12 3 1 
 
 [or(p .p ) =same ] [ shape (p . p )=same] =>[ d=1 ] 
 
 It so happens that the descriptive decision rules are 
 also discriminant in this instance but this is normally not 
 the case. In comparison to Winston's description of an arch, 
 the results relate very closely to his result, namely that an 
 
105 
 
 arch has two supporting members of a top piece which are not 
 touching. 
 
106 
 
 Arches for Example BZ 2 
 Figure 6.2 
 
107 
 
 6.3 Trains (EI 3) 
 
 The trains in figure 6.3 (from [Larson Michalski 
 1977]) represent two sets of trains, one set going vest and 
 the other set going east. There are 12 descriptors vhich 
 seem relevant to the situation in figure 6.3: 
 
 Descriptor 
 
 ncars 
 
 nwhl (twhl/car) 
 
 In (length) 
 
 cshape 
 
 Structure 
 
 interval 
 
 interval 
 tree-struc 
 0,1,2,5=>10 
 3,4,6,7,8,9= 
 
 npl (tpts/load) 
 lshape 
 
 if (in front) 
 
 ccont 
 
 lcont 
 
 t (train) 
 
 car 
 
 nominal 
 
 nominal 
 nominal 
 nominal 
 nominal 
 nominal 
 
 Domain 
 [3:5] 
 
 interval [2,3 ] 
 0-short, 1-long 
 0-open rctngl, 1-open trap. 
 2-0-shaped, 3-hexagon 
 =>11 4-ellipse, 5-dbl open rctngl 
 6-closed rctngl, 7-jagged top 
 8-sloping top, 9- locomotive 
 1 0-open top, 11-closed top 
 interval [0:3] 
 0-circle, 1 -hexagon 
 2-triangle, 3-rectangle 
 0-false, 1-true 
 0-false, 1-true 
 0-false, 1-true 
 
 [1] 
 [1:5] 
 
108 
 lod nominal [1:5] 
 
 The variable (t) and the predicate ccont (train 
 contains car) are introduced to form weakly connected graph 
 structure representations of input decision rules. The 
 description of each train contains about 60 selectors so only 
 the description of the first train is given below: 
 
 ncar(t ) =5 1 [ ccont (t r car )=11 [ ccont (t # car ) =1 1 
 1 1 1 12 
 
 ccont (t ,car ) =1 ] [ccont (t ,car )=1] [ccont (t ,car ) = 1] 
 1 3 1 4 1 5 
 
 if (car # car ) =1 ] [ if (car # car ) = 1 ] [if (car ,car ) = 1] 
 
 if (car ,car ) =1 1 [loc(car ) =1 ] [loc(car ) =2 ] 
 
 loc (car )=3] [ loc (car )=4] [loc(car )=5] [nwhl(car )=2] 
 3 4 5 1 
 
 nwhl(car ) =2 ] [nwhl(car ) =2 ] [ nwhl (car ) =3 ] [nwhl(car )=2] 
 
 [ In (car ) =1 ] [In (car ) =1 ] [In (car ) =0 ] [In (car ) *1 ] 
 12 3 4 
 
 In (car ) =0 ] [cshape(car ) =9 ] [cshape(car )-0] 
 
 cshape(car ) =8 ] reshape (car )=0] [cshape(car ) =0 ] 
 
 npl(car ) =0 ] fnpl(car )=3] [npl(car ) = 1 ] [npl (car^) =1 ] 
 
 npl (car ) =1 1 [lcont(car , lod ) =1 ] [ lcont (car ,lod )=1] 
 
 lcont (car f lod )=1] [ lcont (car , lod )=1] 
 2 3 3 4 
 
 lcont (car , lod )=1] [lcont (car , lod )=1] [lshape(lod )=3] 
 4 5 5 6 1 
 
 lshape(lod ) =3 ] [lshape(lod )=3) [lshape(lod )*2] 
 
 lshape(lod ) =1 ] [lshape(lod )=0] ->[d=1]. 
 5 6 
 
 The parameters and discriminant generalizations from this 
 example are now presented* 
 
 
109 
 
 AQPARHS 
 
 
 AQHAXSTAR = 
 
 4 
 
 LQST MCO 
 
 AQCRIT 
 
 
 AQTOLERANCE 
 
 -1 
 
 
 
 
 2 
 
 
 
 
 4 
 
 
 
 
 NBR OP CRIT: 
 
 
 AQHF = 2 
 
 NEW FHCTNS: 
 
 
 METATRIM = 
 
 VLPARHS 
 LQST MCOISIST = 8 ALTER ■ 8 VLHAXSTAR = 1 
 
 VLCRIT 
 
 3 
 -1 
 
 2 
 
 VLTOLERANCE 
 
 0.30 
 
 
 
 
 
 
 
 
 YLNP = 
 
 3 
 
 The discrimin an t generalizations o btai ned were the following: 
 (10 seconds) 
 
 [cshape(car )=11][ln(car )=0]=>[d=1] 
 
 (there is a short, closed top car) 
 
 [ncar(t ) =3 ] v [cshape(car ) =7 J=>[d=2 ] 
 (there are three cars or a car with a jagged top) 
 
 Some other interesting generalizations which were 
 obtained by using different parameters are: 
 
 [ncars(t ) =4, 5 ][lshape (lod )=2]=>[d=1] 
 
 (If there are at least 4 cars and one car has a load 
 which is a triangle then the train is going east*) 
 
110 
 
 [loc (car ) =4 ][cshape (car )=6] v 
 [loc (car )=5 ]f cshape (car (1) =0 ]=>[d=1 ] 
 
 (If the fourth car is a closed rectangle or the fifth is 
 an open rectangle then the train is going east*) 
 
 [ # PT (ln=1) =2 ][ POEALL (nwhl=2) ] v 
 
 [$PT (ln=1 ]=2 ][ loc (car ) =3 ][cshape (car ) =1 ]«>[d=2 ]. 
 (If there are two long cars and either all cars have 2 
 wheels or the third car has an open top, then the train 
 is goinq west.) 
 
 [ #PT(ln=1)=2 ][cshape(car )=1,2] w 
 [ loc (car ) =2 ][ cshape (car )=6]=>[d=2] 
 
 (If there are exactly 2 long cars, one of which has shape 
 open trapazoid or D-shape or car number 2 is a closed 
 rectangle, then the train is going west*) 
 
 
1. TRAINS GOING EAST 
 
 111 
 
 »■ l, P„ K, Q J - ^ - l D D D ^ -P° 
 
 
 JL 
 
 TJ 
 
 V 
 
 73 0" 
 
 k^W^Q 
 
 D 
 
 JGL 
 
 o 
 
 JL 
 
 77 
 
 .. lsJ-(^H^aa^ 
 
 dM 
 
 O 
 
 MLM-C 
 
 *-0 U 
 
 H 
 
 "73 — o — 0"" ' o o' ^0 — 0" 
 2. TRAINS GOING WEST 
 
 »■ lA. NoooU O^ 1 
 
 73 CT ^3 0"~^ O CT 
 
 ^S O- 1 7T 73 !T L=i 
 
 JL 
 
 CT 
 
 3. 
 
 ^AM 
 
 T3 W 
 
 ME 
 
 Dl 0- 
 
 TT 
 
 4 - \.Q.A^'='. M i ' 
 
 d — u 73 — tr ^3 a 
 
 AoAJ ^ 
 
 rr-u 
 
 ] C 
 
 TT 
 
 ^1 
 
 JL 
 
 TT 
 Trains for Example EX 3 
 
 Figure 6.3 
 
112 
 
 6.4 Textures (EX 4) 
 
 This example is included to give some idea of the 
 limitations of the program* The figure 6.4 (from [Hichalski 
 1972]) contains examples of figures with 9 boxes numbered 
 from 1 to 9 from the center to the right and around the 
 outside in a clockwise direction each with one of 4 
 textures. There are several ways to represent the figures. 
 The first representation takes advantage of the ordering of 
 arguments in a function to specify the location of each box 
 (i.e, the third box corresponds to the third argument of the 
 predicate p) • Since there is no apparant structure to each 
 object, the program relies heavily on the AQ7 procedure to 
 find a simple description of each set of objects. A 
 description of the first object in the set T the results 
 follow : 
 
 Cp( VVWVVWV ] ttx( V = ° 3 C tx < x 2 )=31 
 
 [tx(x )=1] [tx(x )=3] [tx(x )=1] [tx(x ) = 3] [tx(x )*0] 
 3 4 5 6/ 
 
 Ttx(x )=2] [tx(x )=0] =>[d=1] 
 8 9 
 
 Resulting decision rules ; (15 seconds) 
 
 [p(x^x ,x ,x^,z ,x .t^x^x^) ] [tx(x 3 ) =0,2,3] [ tx (x^) =1,2 ] 
 
 rtifs i-o,i] v 
 
113 
 
 tp {x^z ,t^,x^,t ,x .JiyX^zJ ] [tx(x 2 )=0,3] [ tx (x^ *1 ,2 ] 
 => [d=1] 
 
 [p(x i'VVVV x 6'VVV ][tx( V =1][tx( V =23v 
 
 £ p( VV W W WV ] C tx < x 2 > =1 ' 2 ] [tx(x 3 )=1 # 2] 
 [tx(x )=0,1,2] v 
 
 [p(x ,x r x f x ,x ,x ,x ,x ,x ) ] [tx(x )=0,3] [tx (x ) =0, 2, 3 ] 
 
 [tx(x 5 )=0,3] => [d=2] 
 
 If the program is now allowed to generalize over 
 
 location as well as texture, then the results are much more 
 
 difficult to attain. The above representation may be 
 
 modified by specifying that the order of arguments is 
 
 irrelevant to the predicate p (using a (•) between arguments 
 
 instead of a (,)) and specifying the location of a box in the 
 
 object by assigning a value to the variable representing the 
 
 box (assigning a subrange x -2, for example). The description 
 
 i 
 
 of an object then appears: 
 
 [p(x .x .x .x .x .x .x .x .x )][x =1 ] [x =2] [x =3 ] [ x =4] 
 
 1 2 3 4 5 6 7 8 91 2 3 4 
 
 [x =5] [x =6] [x =7] [x =8] [x =9] |tl(l }«fl] [tx(x )=3] 
 5 6 7 8 9 1 2 
 
 [tx(x )=1 ] [tx(x )=3] [tx(x )=1 ] [tx(x ) = 3] [tx(x ) =0 ] 
 
 O U D O / 
 
 [tx(x )=2] [tx(x )=0] ->[d=1]. 
 8 9 
 
 The program is allowed to generalize over values of both 
 
114 
 
 tz(z ) and z • The features of the graph Batching algorithm 
 i i 
 
 which before lead to efficient comparisons between objects 
 are no longer useful (i.e., the fact that all nodes are 
 labelled and that many edges .are labelled). Since the 
 c-structure of a formula of the form 
 
 1x^3) [tz(z i )=2] [p(x .x ) ] C* 2 =1] [tz(z 2 )=1] 
 
 is 
 
 [z^*] [tzfz^*] CPOy* ) ] [z 2 =*] [tz(z 2 )=*] 
 
 there are very many occurrences of such a c-structure in each 
 
 object resulting in a large number of VL events supplied to 
 
 the AQ7 procedure (200 events in each set with from 6 to 8 
 
 variables) • The program was run with the value 1 for each 
 
 parameter but unfotunately the program ran for over 1 minute 
 
 with no results* The input set of decision rules was then 
 
 cut to only the first 5 objects in set T and the last five 
 
 objects in set T and the program was run again with this 
 
 2 
 
 reduced set of objects. The results of the program ezecution 
 using this formulation are: (1 minute) 
 
 rp(z .z ) ][z =1 ][i =9][tz(z )=2][tz(z )=1 ] v 
 12 12 1 2 
 
 [p(X .X J ][z i =1][z 2 =5.9]Ctz(z i )=0.3][tz(Z 2 )=1.2.3] v 
 [x «3][tx(x )»3] => [d=1] 
 
115 
 
 [ p (x^ % 2 ) ][ x^ = 1 J[ x 2 =4 ][ tx (x^ =3 ][ tx (x 2 ) =3 ) * 
 
 [p(* .x ) Kl^l ][x 2 =9][tx(x i )=2][tx(x 2 )=0 ] ▼ 
 
 [x =1Jttx(x )=1] v 
 
 [i *9]ttx(x )=2J v 
 
 [p(x i .x 2 ) tfx^l ][x 2 =5]Ctx(x i )=0][tx(x 2 )=0 ] => [d=2J 
 
 (Presumably, if the parameters are increased, the results 
 would be much more interesting but this appears to be the 
 limit of the program's applicability with regard to the 
 exponential nature of the graph isomorphism algorithm*) 
 
 The time reguired to compute these results is 
 significantly greater than the time reguired when the 
 location of each box is known (15 seconds compared to one 
 minute) • 
 
116 
 
 
 
 r'/^SSi 
 
 
 uzzmk 
 
 1 
 
 n 
 
 ^ 
 
 _1 
 
 n 
 
 
 r— 
 
 ' 
 
 : 
 
 ^ 
 
 — ? 
 
 ^^ 
 
 10 
 
 
 llL 
 
 
 
 
 
 
 
 
 
 
 
 
 n-o 
 
 :A 
 
 - 1 
 
 - 2 
 
 - 3 
 
 Texture Figures for Example EX 4 
 Figure 6,4 
 
117 
 
 7« Current Limitations and Possible Extentions 
 
 In the preceeding chapters, a formal methodology has 
 been presented for solving a wide variety of inductive 
 tasks. It differs from previous work in it's formal basis, 
 flexible optimality criteria and domain definitions, ability 
 to add new descriptors to existing descriptions, and apparent 
 extensibility. Presented below are a few limitations of the 
 current implementation. Some of the limitations have 
 plausible solutions in light of some applications by other 
 authors in specific problem areas. Some extentions may be 
 added to the program with the current state of the art and 
 are omitted from the current implementation due to the higher 
 priority placed on evaluating the existing features. Some 
 features require algorithms which are not yet well understood 
 (especially those which involve the modification of 
 hypotheses in light of new information and the use of 
 multi- valued truth domains). Extensions are listed in the 
 author's estimate of increasing difficulty. 
 
 1. The restriction which reguires generalizations to 
 have weakly connected graph structure representations 
 has the advantage that the search space is limited 
 and resulting descriptions contain selectors which 
 
118 
 
 are related in some meaningful way. This constraint 
 should be applied at the option of the user* 
 
 2. The graph structure used in the current 
 implementation has not been fully analyzed. It 
 appears that unary functions represent a special type 
 of description of parts of objects and are well 
 suited to generalizations using a VL system* 
 Functions and predicates of higher order represent 
 relationships between parts and are more amenable to 
 generalization using a VL approach (i.e.,, 'growing' 
 
 mm 
 
 consistent generalizations) • These two roles of 
 functions and predicates have not been investigated* 
 
 3* The program has a potential for application to more 
 abstract objects (e.g. patient records used by the 
 HYCIN system). Op until now* only simple physical 
 objects have been used as examples* 
 
 U* There should be a facility to add new user defined 
 functions to a rule of the type: 
 
 V |= V[S0M_cost=3 ] 
 
 where V is [cost(x )=2][cost(x ) =1 ] 
 
 1' 2 
 
119 
 
 5. The tree structure domain may be extended to a 
 generalization structure as presented by Hichalski 
 [Hichalski 75]. 
 
 6. Only the operators (v and S) are currently understood 
 by the program. The TL system contains a very rich 
 set of operators, one of the most important being the 
 exception operator. In the current implementation, 
 there is no generalization of decision D=1 in the set 
 of rules: 
 
 [p(x ,x ) ] => [D=1] 
 
 [p(x ,xj )[s(x i )=1] => [D=2] 
 
 7. Decision rules which are used heavily in production 
 systems contain condition and decision parts which 
 share arguments. These should be handled by the 
 program in some way. e.g. 
 
 [p(* ,x ) ] => [q(x ,« ) ] 
 
 8. Descriptive generalizations only contain one 
 conjunctive expression. In some circumstances, it is 
 desirable to produce a disjunctive, descriptive 
 description (see Yuen for a program UNICLASS which 
 
120 
 
 generates descriptive descriptions for 7L 
 
 1 
 
 expressions) • 
 
 9* The order in which objects are selected from the set 
 P1 seems to be significant. The program ESEL whose 
 algorithm is described in Hichalski [Hichalski 75] 
 gives a method for selecting ?L type events. 
 
 10. Universal quantifiers have been considered only in a 
 global sense since a simple universally guantified 
 expression is less general than an existentially 
 guantified expression in the condition part of a 
 rule. A universal or distinct universal quantifier 
 could be added to the program. 
 
 11. Only the distinct existential quantifier is included 
 in the program. This decision vas made in order to 
 limit the search space as well as to simplify the 
 user input and the resulting expressions. Hayes-Roth 
 [Hayes-Roth 76] has pointed out the utility of a 
 •many to one* mapping instead of the current one to 
 one map. 
 
 12. No facility is included in the current implementation 
 to accomodate initial hypotheses in order to form a 
 
121 
 
 new consistent, complete generalization in light of 
 new information. Larson [Larson 76] and Hedrick 
 [Hedrick 74] give some approaches to this problem 
 using other systems* For example, given an 
 hypothesis: 
 
 CP(x i »x 2 ) Ifl|z )«1] => [D=1 ] 
 
 and the new information that 
 
 [p(x ,x ) ls(x 2 )=2] => [D=2], 
 
 the program should generate a nev rale which is 
 consistent (e. g. , 
 
 fp(x i ,x 2 ) ](s(X 2 )-=2] => [D=1] 
 
 13. Generalizations of subsets of parts of an object 
 should be usable by the program to form nev, higher 
 level generaliztions in terms of the these 
 generalized concepts (e.g. Winston [Winston 70] 
 generates the description of an arch and then uses 
 this description to describe a seguence of arches) • 
 
 14. Include a multi- valued truth domain to learn from 
 data of differing degrees of certainty. Very little 
 
122 
 
 is done in this area; one of the problems being that 
 a deductive system using multi-valued truth values is 
 not well understood* 
 
123 
 
 LIST OF REFERENCES 
 
 Janerji, R. 1975, "Learning with Structural Descriptions." 
 working paper Temple University, Philadelphia, Penn. 
 
 tongard, H. 1970, Pattern Recognition. . translated by 
 Theodore Chevon, Spartan books. New York, N. T. 
 
 Buchanan, B.G. , Sutherland, G.L., and Feigenbaum, E.I. 1969, 
 "Heuristic DENDRAL: a Program for Generating 
 Explanatory Hypotheses in Organic Chemistry", in 
 Machine Intelligence 4 t (B. Meltzer and D. Hichie 
 eds) , Edinburgh University Press. 
 
 3uchanan, B.G., Sutherland, G. L. , and Feigenbaum, E.A. 1972, 
 "Heuristic Theory Formation, Data Interpretation, and 
 Rule Formation." Machine I ntelligenc e It. (B. Heltzer 
 and D. Hichie eds) , John Hiley & Sons. 
 
 :hilausky R. , Jacobsen B. , Hichalski R.S. 1976, "An 
 application of Variable Valued Logic to Inductive 
 Learning of Plant Disease Diagnostic Rules", 
 Proceedings of the S_ixth Annual Symposium on flultjple 
 Valued Logic. Logan, Utah. 
 
124 
 
 Croft, J. A. 1971, "A comparative Study of Mathematical 
 lethods for Diagnosing Disease." Fa. D dissertation, 
 ■orth western University Evans ton, Illinois* 
 
 * 
 
 Hayes-Roth, F. and HcDeraott, J. 1976, "Knowledge 
 Acquisition from Structural Descriptions." Department 
 of Computer Science, Carnegie Mellon university. 
 
 Hedrick C.L. 1974, "A Computer Program to Learn Production 
 Systems Using a Semantic Net. " Department of Computer 
 Science, Carnegie Mellon University, Pittsburgh, 
 Penn. 
 
 Hunt, E. B. 1966, Experiments in Induct ion. Academic Press, 
 
 Kochen, M. 1974, "Representation and Algorithms for 
 Cognitive Learning. " Artificial I ntel ligence 5, 
 
 Larson J., Hichalski R.S., "AQVAL/1-AQ7 user's Guide and 
 Program Description." Department of Computer Science 
 report number 731, University of Illinois. 
 
 Larson J. 1976, "A Multi-step Pormation of Variable Valued 
 
 Logic Hypotheses." Proceedings of tjie Sixth 
 
 International Symposium on Multiple Valued Logic. 
 Logan, Utah. 
 
125 
 
 Larson J., Hichalski R.S. 1977, "Inductive Inference of VL 
 Decision Bales*** Workshop on Pattern Directed 
 Inference Systems, Hawaii* 
 
 Hichalski R.S. 1972, "1 Variable- Valued Logic System as 
 Applied to Picture Description and Recognition," 
 ££a£hj.c Lan^uages^ p roceedings of the %?%!? Wo rking 
 Conference on Graphic Languag es* Vancouver Canada* , 
 
 Hichalski R.S. 1973, "AQVAL/1 AQ7 Computer Implementation of 
 
 a Variable Valued Logic System and the Application to 
 
 Pattern Recognition." P roceedi ngs of the First 
 
 International J oin t Confere nce on Pattern 
 
 Recognition. Washington, D. C. 
 
 Hichalski R.S. 1974a, "Variable-Valued Logic: System VL •" 
 P roce edings gf Fourth International Symposium on 
 Hutifile^Valued Logic* West Virginia University, 
 Horgantovn, Virginia* 
 
 Hichalski R.S. 197Ub, "Learning by Inductive Inference." 
 HATO Advanced Study Institute on Computer Oriented 
 Learning Process. " Prance. 
 
 Hichalski R.S. 1974c, "Problems of Designing an Inferential 
 
126 
 
 Hedical Consulting System." Pirst Illinois Conference 
 on Medical Information Systems University of Illinois 
 Urbana, Illinois* 
 
 Hichalski R.S. 1975, "On the Selection of Representative 
 Samples From Large Relational Tables for Inductive 
 Inference." Department of Information Engineering 
 University of Illinois at Chicago Circle, Chicago, 
 Illinois. 
 
 Hichalski R.S. 1977, "Toward Computer-aided Induction: a 
 brief review of currently implemented AQVAL 
 programs" • 5th International Conference on Artificial 
 Intelligence, Cambridge, Mass. 
 
 Morgan C.G. 1972, "Inductive Resolution." Master^s thesis. 
 Department of Computer Science, Edmonton Alberta. 
 
 Pople, H., Werner G. 1972, "An Information Processing 
 Approach to Theory Formation in Biomedical Research." 
 AlflPS Conference Proceedings. Vol 4 0. 
 
 Rychener, M.D. 1976, "Production Systems as a Programming 
 Language for Artificial Intelligence Applications." 
 Ph.d dissertation. Department of Computer Science, 
 Carnegie Mellon University, Pittsburgh, Penn. 
 
127 
 
 Schubert K.L. 1976, "Extending the Expressive Power of 
 Semantic Networks." Artificial Intelligence 7. 
 
 Shortliffe, E.H. 1974, "A Rale Based Computer Program for 
 Advising Physicians Regarding Antimicrobial Therapy 
 Selection*" Ph.D dissertation." Stanford Artificial 
 Intelligence Laboratory, Memo AIH-251 • 
 
 Tar-jan R. 1972, "Depth-First Search and Linear Graph 
 Algorthms." SI AH Journal on Computers, Vol 1, No 2* 
 
 Vere S.A. 1975, "Induction of Concepts in the Predicate 
 Calculus," P roc eedings of the Fou rth International 
 Joint Conference on Art ificial Intelligence * 
 
 Waterman D.H. 1970, "Generalization of Learning Technigues 
 for Automating the Learning of Heurisics," Artificial 
 Intel ligence 1 • 
 
 Waterman D.H. 1974, "Adaptive Production Systems." working 
 paper #385, Department of Psychology, Carnegie Mellon 
 University, Pittsburgh, Penn. 
 
 Waterman D.A. 1975, "Serial Pattern Acguisition: A 
 Production System Approach. " working paper #286, 
 
128 
 
 Department of Psychology, Carnegie Hellon University, 
 
 Pittsburgh, Penn. 
 
 Winston, D.B. 197C, "Learning Structural Descriptions from 
 Examples, M AI-TR-76, Cambridge; MIT, Irtificial 
 Intelligence Laboratory* 
 
 Yuen H* , "Oniclass Synthesis: User's Guide" internal report* 
 Department of Coomputer Science, University of 
 Illinois* Urbana, Illinois* 
 
129 
 
 APPENDIX A 
 This appendix contains a trace of the program 
 execution through the solution of EX1 (section 6.1) covering 
 the set with c- formulas with the EXTHTT type predicates 
 added to each input rule. Host of the output is self 
 explanatory (vith reference to chapter 6 for the definition 
 of some terms) , with the possible exception of the output of 
 rules. Rules are printed with a rule number (a unique number 
 assigned by the program to each new graph data structure 
 which is needed). The event set which the rule is intended to 
 cover (i.e., the decision value in the decision D=i) is given 
 after the rule number the cost function indices used in 
 trimming (in parentheses after the word COSTS) is given. 
 Following this is a list of the actual costs of the 
 c-formula) • Comments are interspersed with the output to 
 clarify specific points. (Computer output is given in 
 capital letters). 
 
 Parameters: 
 
 AQPABMSS VLPAHMS 
 
 AQHAXSTAR = 7 LQST NCONSIST - 10 ALTER = • VLMAXSTAR = a 
 
 RQCRIT 
 
 
 AQTOLERAHCE 
 
 VLCRIT 
 
 ▼LTOLERANCE 
 
 -1 
 
 
 
 
 3 
 
 0.30 
 
 2 
 
 
 
 
 -1 
 
 
 
 4 
 
 
 
 
 2 
 
 
 
 NBR OP 
 
 CRIT: 
 
 AQNF = 2 
 
 
 VLHP = 3 
 
130 
 
 NEW PNCTNS: 
 
 METATRIH = 
 
 NOW COVEBIHG EVENT 
 
 ROLE 5 EVENT SETS: 1 COSTS ( 3-1 2) 
 
 fONTOP (P1,P2) ][ONTOP(P2,P3) X SIZE(P1)= 1][SIZE(P2)= 1] 
 
 [SIZE(P3) = 2][SHAPE(P1) = ][ SHAPE (P2) - 2 ][SHAPE (P3) = 5] 
 [TX(P1) = 0KTX(P2)= 1][TX(P3)= ][ HST-ONTOP (P1) ] 
 [LST-ONTOP(P3) ] 
 
 A c-formala has been selected from the set which 
 
 1 
 
 is represented by rale number 5, 
 
 THE FOLLOWING FORNOLAS ARE 
 
 ROLE 22 EVENT SETS: 
 
 TLST-ONTOP(PI) ] 
 
 ROLE 21 EVENT SETS: 
 
 [ HST-ONTOP (P1) ] 
 
 ROLE 20 EVENT SETS: 
 
 TTX(P1)= 0] 
 
 ROLE 19 EVENT SETS: 
 
 TTX(P1)= 1 ] 
 
 ROLE 18 EVENT SETS: 
 
 [TX (P1)= 01 
 
 ROLE 17 EVENT SETS: 
 
 [SHAPE (P1)= 5] 
 
 ROLE 16 EVENT SETS: 
 
 [ SHAPE (P1)= 2] 
 
 ROLE 15 EVENT SETS: 
 
 TSHAPE (P1)= ] 
 
 ROLE 14 EVENT SETS: 
 
 fSIZE(Pl)= 2] 
 
 ROLE 13 EVENT SETS: 
 
 rSIZE(Pl)= 1 ] 
 
 ROLE 12 EVENT SETS: 
 
 fSIZE(Pl)= 1] 
 
 IN THE ONTRIMMED STAR 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 1 COSTS 
 
 3-1 2) 
 
 3-1 2) 
 
 -3 
 -3 
 
 3-1 2) 3 -3 
 
 3-1 2) 6 -3 
 
 3-1 2) 3 -3 
 
 3-1 2) 3 -1 
 
 3-1 2) 2 -2 
 
 3-1 2) 2 -1 
 
 3-1 2) 4 -2 
 
 3-1 2) 5 -3 
 
 3-1 2) 5 -3 
 
131 
 
 All selectors of the event (condition of rale 5) are 
 place in the intial partial star. Since VLHAXSTAR has the 
 value 4, the above list is trimmed to a smaller list of the 4 
 best c-formulas in the partial star. 
 
 THE FOLLOWING FORMULAS BEBAIN AFTER TBIHMING 
 
 BOLE 16 EVENT SETS: 1 COSTS ( 3-1 2) 2 
 
 [SHAPE (P1) = 2 1 
 
 BOLE 14 EVENT SETS: 1 COSTS ( 3-1 2) 4 
 
 [SIZE(P1)= 2] 
 
 ROLE 12 EVENT SETS: 1 COSTS ( 3-1 2) 5 
 
 rsizE(pi)= 1 ] 
 
 BOLE 18 EVENT SETS: 1 COSTS ( 3-1 2) 3 
 [TX(P1)= 0] 
 
 The next partial star is formed by adding one 
 selector to all c-formalas in the previous partial star 
 forming a list of c-formalas which have two selectors. 
 
 -2 1 
 
 -2 1 
 
 -3 1 
 
 -3 1 
 
 THE FOLLOWING FORMULAS ARE IN THE ONTRIHMED STAR 
 
 BOLE 38 EVENT SETS: 
 
 [TX(P1) = 0][HST-OITOP(P1) ] 
 
 BOLE 37 EVENT SETS: 
 
 [SHAPE (P1)= 0][TX(P1)= 0] 
 
 BOLE 36 EVENT SETS: 
 
 [SIZE(P1)= 1 ]f TI(P1)= ] 
 
 BOLE 35 EVENT SETS: 
 
 [0NT0P(P1,P2) ][TX(P1) = 0] 
 
 BOLE 34 EVENT SETS: 
 
 [SIZE(P1)= 1 ][ HST-OITOP(PI) ] 
 
 BOLE 33 EVENT SETS: 1 COSTS ( 3-1 2) 
 
 1 COSTS ( 3-1 2) 
 
 1 COSTS ( 3-1 2) 
 
 1 COSTS ( 3-1 2) 
 
 1 COSTS ( 3-1 2) 
 
 1 COSTS ( 3-1 2) 
 
 -3 
 -1 
 -3 
 -3 
 -3 
 -3 
 
 2 
 2 
 
132 
 
 r SIZE(P1)= 
 
 RULE 
 
 [SIZE(P1) = 
 
 RULE 
 
 CONTOP (P1, 
 
 ROLE 
 
 [SIZE(P1)= 
 
 ROLE 
 
 [SIZE(P1)= 
 
 ROLE 
 
 rsizE(Pi)= 
 
 ROLE 
 
 [ONTOP (P1, 
 ROLE 
 
 [SHAPE (P1) 
 ROLE 
 
 [SIZE(P1) = 
 ROLE 
 
 [ONTOP (P1, 
 ROLE 
 [ONTOP (P1, 
 
 1][TX(P1) = 0] 
 
 32 ETENT SETS: 1 COSTS 
 
 1 ][ SHAPE (P1)= ] 
 
 31 EVENT SETS: 1 COSTS 
 P2) ][SIZE(P1)= 1 ] 
 
 30 EVENT SETS: 1 COSTS 
 
 2][LST-0HT0P(P1) ] 
 
 29 EVENT SETS: 1 COSTS 
 
 2][TX(P1) = 0] 
 
 28 EVENT SETS: 1 COSTS 
 
 2][SHAPE(P1)= 5] 
 
 27 EVENT SETS: 1 COSTS 
 P2) ][SIZE(P2)= 2] 
 
 26 EVENT SETS: 1 COSTS 
 = 21[TX (P1)= 1] 
 
 25 EVENT SETS: 1 COSTS 
 
 1 ][SHAPE(P1)= 2] 
 
 24 EVENT SETS: 1 COSTS 
 P2) ][ SHAPE (P1)= 2] 
 
 23 EVENT SETS: 1 COSTS 
 P2) ][ SHAPE (P2)= 2] 
 
 ( 3-1 2) 
 
 2 
 
 -1 
 
 2 
 
 ( 3-1 2) 
 
 4 
 
 -3 
 
 2 
 
 ( 3-1 2) 
 
 3 
 
 -2 
 
 2 
 
 ( 3-1 2) 
 
 1 
 
 -2 
 
 2 
 
 ( 3-1 2) 
 
 3 
 
 -1 
 
 2 
 
 ( 3-1 2) 
 
 3 
 
 -2 
 
 2 
 
 ( 3-1 2) 
 
 1 
 
 -1 
 
 2 
 
 ( 3-1 2) 
 
 2 
 
 -2 
 
 2 
 
 ( 3-1 2) 
 
 2 
 
 -1 
 
 2 
 
 ( 3-1 2) 
 
 1 
 
 -2 
 
 2 
 
 At most 4 new selectors were added to each c-formula 
 of the previous partial star (ALTER-4) • This partial star is 
 now trimmed to the 4 best elements according to the 
 optima lity criterion. 
 
 THE FOLLOWING PORMOLAS REMAIN AFTER TRIMMING 
 
 ROLE 25 EVENT SETS: 1 COSTS ( 3-1 2) 
 
 TSIZE(P1)= 1 ][SHAPE(P1)= 2] 
 
 ROLE 23 EVENT SETS: 1 COSTS ( 3-1 2) 
 
 [ONTOP (P1,P2) ][SHAPE(P2)= 2] 
 
 ROLE 29 EVENT SETS: 1 COSTS ( 3-1 2) 
 
 [ SIZF(P1)= 2 ][TX (P1)= ] 
 
 -2 2 
 
 -2 2 
 
 -2 2 
 
133 
 
 ROLE 38 EVENT SETS: 1 COSTS ( 3-1 2) 1 -3 
 
 [TX(P1) = 0][HST-OHTOP(P1) ] 
 
 THE FOLLOWING FOBMOLAS ABE IH THE OHTBIMHED STAB 
 
 RULE 51 EVENT SETS: 1 COSTS ( 3-1 2) -1 
 
 [SHAPE (P1)= 0][TI(P1)= 0][HST-ONTOP(P1) ] 
 
 RULE 50 EVENT SETS: 1 COSTS ( 3-1 2) -3 
 
 [SIZE(P1)= 1][TX(P1) = 0][HST-ONTOP(P1) ] 
 
 RULE 49 EVENT SETS: 1 COSTS ( 3-1 2) 1 -3 
 
 [ONTOP (P1,P2) ][TX(P1) = 0][MST-ONTOP(P1) ] 
 
 RULE 48 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 
 [SIZE(P1)= 2]fTI(Pl)= 0][LST-ONTOP(P1) ] 
 
 RULE 47 EVENT SETS: 1 COSTS ( 3-1 2) 1 -1 
 
 [SIZE(P1)= 2][SHAPE(P1) = 5][TX(P1) = 0] 
 
 RULE 46 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 
 [ONTOP (P1,P2) ][ SIZE (P2)= 2][TX(P2)= 0] 
 
 RULE 45 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 
 [ONTOP (P1,P2) ][SHAPE(P2) = 2][TX(P1) = 0] 
 
 RULE 44 EVENT SETS: 1 COSTS ( 3-1 2) -1 
 
 [ONTOP (P1,P2) ][SHAPE(P1) = ][ SHAPE (P2) = 2] 
 
 RULE 43 EVENT SETS: 1 COSTS ( 3-1 2) 1 -2 
 
 [ONTOP(P1,P2) ][SIZE(P2) = 1 ][SHAPB (P2) = 2] 
 
 RULE 42 EVENT SETS: 1 COSTS ( 3-1 2) 1 -2 
 
 [ONTOP (P1,P2) ][ SIZE (P1)= 1 ][ SHAPE (P2)= 2] 
 
 ROLE 41 EVENT SETS: 1 COSTS ( 3-1 2) 1 -1 
 
 [SIZE(P1)= 1 ][SHAPE(P1)= 2][TX(P1)= 1] 
 
 RULE 40 EVENT SETS: 1 COSTS ( 3-1 2) 2 -1 
 
 [ONTOP (P1,P2) ][SIZE(P1)= 1 ][SHAPE(P1)= 2] 
 
 RULE 39 EVENT SETS: 1 COSTS ( 3-1 2) 1 -2 
 
 [ONTOP (P1,P2) 1[ SIZE (P2)= 1 ][ SHAPE (P2) = 2] 
 
 THE FOLLOWING FOBHOLAS BEHAIN AFTEB TBIHMIHG 
 
 RULE 56 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 
 [ONTOP (P1,P2) 1[ SHAPE (P2) = 2][TX(P1)= 0] 
 
 RULE 55 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 
 [ONTOP (P1,P2) ][ SIZE (P2)= 2][TX(P2)= 0] 
 
 RULE 54 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 
 [SIZE(P1)= 2][TX(P1)= OXLST-ONTOP(PI) ] 
 
134 
 
 RULE 53 EVEBT SETS: 1 COSTS ( 3-1 2) 0-3 3 
 
 [SIZE(P1)= 1][TX(P1) = 0][HST-ONTOP(P1) ] 
 
 Since each selected c-formula is consistent, all 
 c-formulas are placed in the list HQ. The program now looks 
 for more descriptive generalizations to supply a larger set 
 of variables to the AQ7 procedure* 
 
 THE FOLLOWING POBHULAS ARE IN THE ONTRIHHED STAR 
 ROLE 69 EVENT SETS: 1 COSTS ( 3-1 2) 0-1 4 
 
 TSIZE(P1)= 1 ][SHAPE(P1) = 0][TX(P1) = X HST-OHTOP (P1) ] 
 RULE 68 E?ENT SETS: 1 COSTS ( 3-1 2) 0-3 4 
 
 [ONTOP (P1,P2) 1[SIZE(P1)= 1][TX(P1) = ][ HST-ONTOP (P1) ] 
 RULE 67 EVENT SETS: 1 COSTS ( 3-1 2) 0-14 
 
 TSIZE(P1)= 2][ SHAPE(P1) = 5][TX(P1) = ][ LST-ONIOP (P1) ] 
 RULE 66 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 4 
 
 [ONTOP (P1,P2) ][ SIZE (P2)= 2][TX(P2) = ][ LST-ONTOP (P2) ] 
 RULE 65 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 4 
 
 [ONTOP (P1.P2) ][SIZE(P2)= 2][TX(P1) = 1][TX(P2)= 0] 
 RULE 64 EVENT SETS: 1 COSTS ( 3-1 2) 0-1 4 
 
 [ONTOP (Pl r P2) ][ SIZE (P2)= 2 ][ SHAPE (P2) = 5][TX(E2)= 0] 
 RULE 63 EVENT SETS: 1 COSTS ( 3-1 2) 0-14 
 
 [ONTOP (P1,P2) ][ SIZE (P2)= 2 ][ SHAPE (P1) = 2][TX(P2)= 0] 
 RULE 62 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 4 
 
 [ONTOP (P1,P2) ][SIZE(P1)= 1][SIZE(P2)= 2][TX(P2)= 0] 
 RULE 61 EVENT SETS: 1 COSTS ( 3-1 2) 0-14 
 
 [ONTOP (P1,P2) ][ SHAPE (P2) = 2 K TX (P1) = ][ TX (P2)= 1] 
 RULE 60 EVENT SBTS: 1 COSTS ( 3-1 2) 0-1 4 
 
 [ONTOP (P1,P2) ][SHAPE(P1)= ][ SHAPE (P2) = 2][TX(P1)= 0] 
 RULE 59 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 4 
 
 [ONTOP (PI. P2) ][ SIZE (P2)= 1 ][ SHAPE (P2) = 2][TX(P1)= 0] 
 RULE 58 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 4 
 
 [ONTOP (P1,P2) ][ SIZE (P1)= 1 ][SHAPE (P2) = 2][TX(E1)= 0] 
 
135 
 
 At least 10 consistent formulas have be generated and 
 placed in HQ* The AQ procedure is then applied to each one 
 of the consistent formulas (NC0NSIST=10) • Only the trace of 
 one such application is given below. 
 
 THE CONSISTENT FORMULAS: 
 BEPOFE AQ: 
 
 ROLE 58 EVENT SETS: 1 COSTS ( 3-1 2) -2 4 
 
 [0NT0P (P1,P2) ][SIZE(P1) = 1 ][SHAPE(P2)= 2][TX(P1) = 0] 
 THE C-FORMOLA STRUCTURE IS: 
 
 ROLE 58 EVENT SETS: 1 COSTS ( 3-1 2) -2 4 
 
 [ONTOP (P1,P2) ][SIZE(P1) = * ][SHAPE(P2)= * ][TX(P1) = * ] 
 THERE ARE 6 VL1 TYPE VARIABLES X1,X2,...,X 6 
 
 VARIABLES ARE ASSOCIATED WITH NODES IN THE C-FORMOLA AS FOLLOWS: 
 NODE VARIABLE 
 
 ONTOP X1 
 
 P1 12 
 
 P2 X3 
 
 SIZE XU 
 
 SHAPE X5 
 
 TX X6 
 
 AQ IS APPLIED TO THE POLLOWING IHPOT CPIS/EVEiTS 
 
 SET 1 
 
 0] 
 
 1] 
 
 0] 
 
 0] 
 
 [X1 = 
 [11 = 
 [X1 = 
 [X1 = 
 
 [X1 = 
 [X1 = 
 [X1 = 
 
 rxi= 
 
 [X1 = 
 
 rxi= 
 
 [X4= 
 [X4= 
 [X4= 
 [X4= 
 
 1][X5= 2][I6 = 
 
 1)fX5= 4][X6 = 
 
 1][X5= 1]£X6= 
 
 1][X5= 2][X6 = 
 
 SET 2 
 
 [X4= 
 [X4 = 
 [X4= 
 [X4 = 
 [X4= 
 [X4= 
 
 2][X5= 
 1][X5= 
 2][X5 = 
 1 ][ X5= 
 1][I5= 
 1][X5= 
 
 1 ][ X6= 
 4 ][ X6= 
 0][X6= 
 1)[X6= 
 2]fX6 = 
 5KX6 = 
 
 0] 
 0] 
 
 1] 
 
 0) 
 
 1] 
 1] 
 
136 
 
 [X1 = 1][X4= 1][X5= 1]£X6 = 1] 
 
 r X1= 1][X4= 2][X5= 6]£X6= 1] 
 
 [X1= 1][X4 = 0][X5= 5J[X6= 1] 
 
 THE RESULTING COMPLEX FROM THIS PASS IS: 
 
 [X4 = 1][X5= 2 3 7 9](X6= 0] 
 
 AFTER AQ: 
 
 RULE 58 EVENT SETS: 1 COSTS f 3-1 2) 
 
 rONTOP(Pl,P2) ][SIZE(P1)= 1 ][SHAPE(P2)= 9]fTX(Pl) = 
 
 -2 
 
 0] 
 
 The results from the application of the AQ7 procedure 
 to each consistent formula are given below. These are the 
 consistent alternative solutions (note that only those with a 
 cost of -3 for cost function -1 are complete) • 
 
 THE FOLLOWING 
 RULE 58 
 r ONTOP (P1,P2) 
 RULE 59 
 f ONTOP (P1,P2) 
 RULE 60 
 ("ONTOP (P1,P2) 
 RULE 61 
 [ONTOP (P1,P2) 
 RULE 62 
 [ONTOP (P1,P2) 
 RULE 63 
 [ONTOP (P1,P2) 
 RULE 64 
 [ONTOP (P1,P2) 
 RULE 65 
 [ONTOP (P1,P2) 
 RULE 66 
 [ ONTOP (P1,P2) 
 RULE 67 
 
 ARE ALTERNATIVECONSISTENT GENERALIZATIONS 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 ][SIZE(P1)= 1 ][ SHAPE (P2)= 9][TX(E1)= 0] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 ][SIZE(P2)= * ][ SHAPE (P2)= 2][TX(P1)= 0] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 ][SHAPE(P1)= 9][SHAPE(P2)= 9][TX(P1)= * ] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 ][SHAPE(P2)= 2][TX(P1)= 0][TX(P2)= * ] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 1[SIZE(P1)= 1][SIZB(P2)= 2][TX(P2)= 0] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -1 
 1[SIZE(P2)= 2][SHAPE(P1)= 2 ][ TX ( B2) = * ] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -1 
 ][SIZE(P2)= 2 ][ SHAPE (P2)= 8][TX(P2)= 0] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 ][SIZE(P2)= 2][TX(P1)= 1][TX(P2)= 0] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -2 
 1[SIZE(P2)= 2][TX(P2)= ][ LST-ONTOP (P2) ] 
 
 EVENT SETS: 1 COSTS ( 3-1 2) -1 
 
137 
 
 [SIZE(P1)= 2][SHAPE(P1) = 8](TX(P1) = )[ LST-OMTOP (P1) ] 
 
 ROLB 68 E?EHT SETS: 1 COSTS ( 3-1 2) 0-3 3 
 
 [ ONTOP (P1,P2) ][SIZE(P1)= 1]£TX(P1)= ][ HST-ONTOP (P1) ] 
 
 ROLE 69 EVENT SETS: 1 COSTS ( 3-1 2) 0-3 3 
 
 [SIZE(P1)=. 1 ]f SHAPE (P1)= 9][TI(P1)= ][ HST-ONTOP (P1) ] 
 
 POLE 44 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 3 
 
 rONTOP (P1,P2) ][SHAPE(P1) = 9 ][ SHAPE (P2) = 9] 
 
 ROLE 45 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 3 
 
 [ONTOP (P1,P2) ][SHAPE(P2)= 2KTI(P1) = 0] 
 
 ROLE 46 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 3 
 
 [ONTOP (P1,P2) ][ SIZE (P2) = 2][TX(P2) = 0] 
 
 ROLE 48 EVENT SETS: 1 COSTS ( 3-1 2) 0-2 2 
 
 [SIZE(P1)= 2][TX(P1) = 0]£LST-ONTOP(P1) ] 
 
 ROLE 50 EVENT SETS: 1 COSTS ( 3-1 2) 0-3 2 
 
 [SIZE(P1)= 1 ][TI(P1)= ][BST-0MT0P(P1) ] 
 
 ROLE 51 EVENT SETS: 1 COSTS ( 3-1 2) 0-3 2 
 
 [SHAPE (P1)= 9][TX(P1)= ][ HST-ONTOP (P1) ] 
 
 THE SELECTED HQ PORHOLA IS: 
 
 ROLE 50 EVENT SETS: 1 COSTS ( 3-1 2) 0-3 2 
 
 [SIZE(P1)= 1]rTX(Pl)= 0][MST-ONTOP(P1) ] 
 
 The solution to the problem is the following rule. 
 Note however that there were four distinct solutions 
 generated (rules 68, 69, 50, and 51) • 
 
 THE FOLLOWING POBMOLAS COVER SET 1 
 
 THIS ROLE COVERS 3 NEW ROLES 
 
 ROLE 50 EVENT SETS: 1 COSTS { 3-1 2) 0-3 2 
 
 [SIZE(P1)= 1][TX(P1)= 0)f HST-ONTOP (P1) ] 
 
138 
 
 APPENDIX B 
 
 This appendix contains a very brief description of 
 
 the AQ7 procedure implemented in the program IHD0CE_1. The 
 
 reader is referred to [Larson et.al. 75] for farther 
 
 details. A set of variables (x ,x ,...,x ) and a domain 
 
 12 n 
 
 definition including size and structure (noodnal, interval or 
 tree-structured) is given. The space (even t space) defined 
 
 by 
 
 E = D(x ) x D(x ) x ... x D(x ) 
 12 n 
 
 contains a set of points (events) where each point may be 
 
 specified by supplying a list of values for the variables 
 
 x ,x ,..,x • A complex is the set of points in the event 
 12 n 
 
 space which satisfy a particular VL expression of the form 
 
 [x = E ][x = R ] ... [x = S ] 
 112 2 n n 
 
 where R is a representation for a subset of the domain D (x ) 
 i i 
 
 or the symbol (*) (if R =*, then the selector may be removed 
 
 i 
 
 from the expression). The procedure accepts two sets of ?L 
 expressions (L and L 1 ) representing disjoint complexes and 
 produces a near optimal generalization of the set of 
 expressions in L with regard to a set of optimality 
 criteria. Certain other parameters control the generalization 
 process. 
 
139 
 
 Let L=[l ,1 ,..,1 ] and L'=[l' ,1« ,...,1» ] (these 
 12m 12 m' 
 
 two sets are denoted P1 and PO respectively in other 
 
 references; the above notation vas selected to avoid 
 
 confusion with sets of c-formulas) . The program selects one 
 
 element (e ) from L and forms a star about this element (a 
 
 star about e is a set of complexes each of which contains e 
 
 
 but does not intersect with any element of L' and is nearly 
 maximal under inclusion). One element (1 ) is selected from 
 
 q 
 
 the star using the optimality criterion and placed in a set 
 of output complexes; all other elements of L which are 
 covered by this 1 are removed from L and the process is 
 
 q 
 
 repeated with a new e until the set L is exhausted* 
 
 
 
 I star is generated by forming a sequence of 
 
 elementary stars and partial stars, one for each element of 
 
 L'« An elementary star (BS (e ,1* ) or ES ) is a set of 
 
 i i 
 
 complexes which covers e does not intersect with 1' , and is 
 
 i 
 
 maximal under inclusion and under domain structure 
 
 constraints. A partial star (P (e ) or F ) is a set of 
 
 i i 
 
 complexes which contain e but do not intersect with any 1' , 
 
 j 
 
 j<=i (note that P is in fact a star) • To generate an 
 
 m* 
 
 elementary star ES , the extension against rule (see chapter 
 
 i 
 
 3) is applied to each selector of e in the context of 1* • 
 
 i 
 
 The result is a set of selectors, each one corresponding to 
 
 one selector of e • To form a partial star P from a 
 
 i*1 
 
mo 
 
 partial star P , each eleaent of P is multiplied by each 
 
 i i 
 
 element of ES (i.e., the set of complexes in P and ES 
 i*1 i i 
 
 may be viewed as a logical sum of products of selectors; the 
 
 multiplication is then the normal result of expanding a 
 
 product of sums in the VL system) • 
 
 The partial star P is initialized to be the entire 
 
 
 
 event space E. As each partial star F is generated, 
 
 i 
 
 absorption laws are applied to discard any complex of P 
 
 i 
 
 which is contained in another complex of P • The partial 
 
 i 
 
 star is then trimmed to AQHAXSTAR number of elements using a 
 
 procedure identical to that in section 5.4. The final 
 
 partial star (P ) is trimmed with an AQHAXSTAR value of 1 to 
 
 m 1 
 
 produce 1 • If a parameter LQST is set (has the value TRUE) , 
 
 q 
 
 then 1 is stripped down to an expression vith the following 
 
 q 
 
 properties: 
 
 1) the stripped 1 contains the same variables as the 
 
 q 
 
 original expression, 
 
 2) the stripped 1 covers the same eletrents of L as the 
 
 q 
 
 original, 
 
 3) the reference of each selector of 1 contains the 
 
 q 
 
 fewest elements of all complexes satisfying 1 and 2 
 above under domain structure constraints (i.e., 
 interval variables must have a range of values in the 
 reference) . 
 
141 
 
 If the set L' is null, then (1) above is replaced with: 
 
 1) the stripped 1 contains all variables x , x ,***,x • 
 
 g 1 2 n 
 
 The latter condition occurs when generating descriptive 
 descriptions covering only one set of complexes (see sections 
 6.1,6.2). Stripping is done by finding the disjunction of all 
 complexes in L covered by 1 and then adjusting the reference 
 
 q 
 
 of each selector in the sun to conforn to the domain 
 
 structure (i.e., for interval domains, this involves filling 
 the gaps to while form an interval, while for tree structured 
 
 domains, this involves finding the lowest level 
 generalization of all elements in the reference)* 
 
 A simple example may clarify the procedure* Given 3 
 variables with domains: 
 
 Variable 
 
 x 
 1 
 
 X 
 
 2 
 
 x 
 
 Structure 
 
 Values 
 
 nominal 
 
 [0:2] 
 
 interval 
 
 [0:3] 
 
 tree-struc 
 
 [0:6] 
 
 0,1,2=>5 
 
 
 3,4=>6 
 
 
 and input complexes and parameters: 
 
 L: 1 [x =0] [x =1 ] [x =2] 
 11 2 3 
 
1U2 
 
 1 [x =1 ] [x =1 ] [x =0] 
 
 2 1 2 3 
 
 1 [x =1 ] [x =2] £x =0] 
 
 3 1 2 3 
 
 L«: 1^ r« t -H C« 2 =2] [x 3 =3] 
 1' 2 [x^O] [x 2 =3] [x 3 =1] 
 1« 3 [x i= 3] [x 2 =2] [x 3 =4] 
 
 AQMAISTAR = 2 LQST = TBOE 
 
 cost functions: -1 (maximize number of complexes covered) 
 
 2 (minimize number of selectors) 
 tolerance = C for both functions* 
 
 Let e =1 , the elementary star ES and the partial 
 1 1 
 
 star P contain: 
 
 1 
 
 ES^^ [x -0,2], [x 2 =0..1], [x 3 =5] 
 
 since multiplication by the entire space E (P ) leaves each 
 
 element unchanged. The trimmed P is 
 
 1 
 
 P <i [x 2 =0..1], [x 3 =5] 
 
 since each of these selectors cover at least 2 elements of L 
 
 while the selector [x =0,2] covers only 1 element. The 
 
 elementary star ES is 
 
 2 
 
 ES [r =0..2 ], [x =2] 
 2 2 3 
 
 
143 
 
 since the references of the variable x intersect in 1 and 
 
 1 1 
 
 1' • The resulting partial star P is 
 2 2 
 
 P [x =0..1], [l»0..1Ix»2], [x «0.2)£x»5], [x =2] 
 2 2 2 3 2 3 3 
 
 which may be reduced by absorption to 
 
 P [i =0..1], [i«0..2Ii-5], [x =2]. 
 2 2 2 3 3 
 
 Since the third element of this partial star covers only one 
 element of L, the trined partial star leaves 
 
 P [x =0..1], [x =0..2J[x =5] 
 2 2 2 3 
 
 Nov considering 1* , ES is 
 
 3 3 
 
 ES 3 [x =0,1), [X =0..1], [x 3 =5] 
 
 and P after absorption becomes 
 3 
 
 P [x =0..1 ], [i=0..2Xi-5] 
 
 3 2 2 3 
 
 The 1 selected from the star P is 
 g 3 
 
 1 [x =0..2][x =5] 
 g 2 3 
 
 since this complex covers all three complexes in L# 
 Stripping reduces the complex to 
 
14U 
 
 1 [x *1..2l[x =5] 
 q 2 3 
 
 If the set L' is ignored, a descriptive generalization of L 
 
 can be formed 
 
 1 [x =0,1][x =1..2][x -5]. 
 q 1 2 3 
 
145 
 
 VITA 
 
 James B. Larson was born April 3, 1947 in Seattle 
 Washington. He received a B.S. from the University of 
 Washington in Chemistry and Mathematics in 1969 and an M. 
 A. from the University of Washington in Mathematics in 1971. 
 While at the University of Illinois, he was a teaching 
 assistant in the Department of Computer Science for 4 years 
 and a reasearch assistant under professor R.S. Michalski for 
 1 year. 
 
 The titles of his publications include: "A Multi-Step 
 formation of Variable-Valued Logic Hypotheses", "AQVAL/1-AQ7: 
 User*s Guide and Program Description" (with R. S. 
 Michalski) , "The DCS Information Storage and Retrieval System 
 for Graduate Applicants and Students" (with W.J. Kubitz) , 
 and "Inductive Inference of VL Decision Rules" (with R. S. 
 Michalski) • 
 
 James Larson is a member of the Association for 
 Computing Machinery and Sigma Xi. 
 
TlBLlOGRAPHIC DATA 
 SHEET 
 
 4. Title and Subtitle 
 
 Inductive Inference in the Variable Valued Predicate Logic 
 System VL>2± : Methodology and Computer Implementation 
 
 1. Report No. 
 
 U1UCDCS-R-77-869 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 May 1977 
 
 6. 
 
 7. Author(s) 
 
 James B. Larson 
 
 8- Performing Organization Rept. 
 No. 
 
 9. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 61801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 NSF MCS 74-03514 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, DC 
 
 13. Type of Report 81 Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 A formal methodology and computer program are presented for the transformation 
 of a set of user supplied logical decision rules into a new, generalized set of 
 decision rules which is near optimal according to a user supplied criterion. The 
 VL$2 logic system (a multi-valued version of a first order predicate calculus) is 
 used as the framework for defining and expressing decision rules and transformations 
 on decision rules. The program INDUCE-1 which implements certain inductive inference 
 rules using a graphical representation of VL$2 expressions is described and some ex- 
 amples of inductive problems solved by the program are given. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 inductive learning 
 variable valued logic 
 inductive inference 
 decision rules 
 production systems 
 
 17b. Identifiers/Open-Ended Terms 
 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 *ORM NTIS-35 (10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 curity Class (Thi 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 22. Price 
 
 USCOMM-DC 40329-P7 1 
 
SFP 
 
 1 S 1971 
 
AUG f ram