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 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAICN 
 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/graphtransformat368schw 
 
pi 
 
 COP 
 
 HC^Ul \J 11W • ^)W 
 
 GRAPH TRANSFORMATIONS 
 FOR COMPOSITE FORMATION 
 
 by 
 John C . Schwebel 
 
 December 12, 1969 
 
 n »\ s *» 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN • URBANA, ILLINOIS 
 
COO-1018-1195 
 
 Report No. 368 
 
 GRAPH TRANSFORMATIONS 
 FOR COMPOSITE FORMATION 
 
 by 
 John C . Schwebel 
 
 December 12, 1969 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 *This work was supported by the U.S. Atomic Energy Commission under 
 Contract No. USAEC AT(ll-l)-10l8. 
 
1 . INTRODUCTION 
 
 This paper defines and investigates transformations of labelled 
 directed graphs. The properties of composite formation under a binary 
 relation, which were defined in a previous report [l], are shown to be 
 sufficient conditions for some graph transformations. Implications 
 between the defined transformations are also determined. A method of 
 classifying relations involved in a transformation is demonstrated. 
 A partial classification is performed on the basis of the defined 
 transformations and on the abstract properties of the constituent 
 relations. 
 
 The motivation for the work in this area was the concept of 
 "precedence" of graph transformations, introduced by Professor Bruce H. 
 McCormick. These ideas were put forth at a computer graphics conference 
 [2] and in further frequent discussions since that time with this author. 
 
 Within the framework of these ideas, an abstract graph is 
 "parsed" by a series of composite forming transformations. In forming 
 composites some links, representing relations between objects, must be 
 assumed to have a higher binding strength, i.e. precedence, than others. 
 Each transformation simplifies the available information by deleting or 
 "cutting" links of lower precedence to objects on one level and extending 
 them in turn to a composite element introduced at a higher level in the 
 parse. Information supplied by relations within the composite element, of 
 course, remains available at the lower level in the parse. 
 
 -1- 
 
2. LABELLED GRAPHS AND TRANSFORMATIONS 
 
 We will consider finite directed graphs with labelled edges 
 corresponding to binary relations . 
 
 A labelled grap h, L = (X, H ) consists of an arbitrary set, X, 
 of n elements and a set , H , of m binary relations on X: 
 
 H A = {H x ! H. CX x X, HA 0, 6 e A} 
 A oo — 
 
 Here A = {6} is an index set of m elements. L is represented by a diagram 
 with n points corresponding to elements of X, and r labelled directed edges. 
 There is a directed edge labelled by 6 from point X to point X if and only 
 
 if (X. , X.) e H . Let r r = IhJ. Then r = I r_. 
 k 1 6 6 • J 6eA 6 
 
 The values of n, m and r will be used to enumerate some simple 
 
 graphs which will be considered in graph transformations. 
 
 A transformation , Q, is a graph replacement rule, denoted 
 
 L -|_ = :: Lg* where L j_ and L 2 axe labelled graphs involving -variables on the 
 same set. 
 
 For example we define a transformation Q2 by the rule: 
 
 L = : : L where 
 
 L ± = ({a,b,c}, H 1 = {(b,a)}, H 2 = {(b,c)}) 
 
 L 2 = ({a U b, c}, H 2 = {(a U b, c)}) 
 
 Q2 is represented by the diagram: 
 
 a U b 
 
 Q2 : a <g — 1 — , b _ . . 
 
 2 
 c 
 
 ]f. y 
 
 -2- 
 

 A labelled graph, L, or a replacement rule, Q, is a logical 
 expression about a system of relations on a set which may contain relations, 
 variables, and operations between variables. Analogous to [l], we assume 
 that the sets are lattices and that inverse and dual operators are defined 
 on the relational system and on statements about the system. 
 
 In this case, an inverse operator is defined for each 6 in A, 
 and, since each relation is defined on the same set, there is only one dual 
 operator, i.e. D T = D_ = D. The group G of operators is generated by 
 composition of inverse and dual operators defined in Table 1; where T is 
 an operator in G and A is a logical expression such as a graph or trans- 
 formation. 
 
 T 
 
 T (A) 
 
 E 
 
 A 
 
 D 
 
 A with lattice operations replaced by 
 
 
 their dual operations 
 
 h 
 
 A with H« replaced by H. 
 
 
 
 Table 1 
 
 The group G is a commutative group with 2 elements 
 obtained from m different options of identity and inverse operators and the 
 dual operator • The following notation will sometimes be used for composition 
 of inverse operators: 
 
 I denotes II I 
 
 6eA 6 
 
 I. . denotes I. I. 
 6 1 6 2 6 1 6 2 
 
 In this paper, we will restrict our attention to the case when 
 m = 2 and A = {1, 2}, where the eight operators in G are: 
 
 E, I r I 2 , I, D, LjD, I 2 D, ID. 
 Here I = II. 
 
 -3- 
 
ENUMERATION OF SIMPLE LABELLED GRAPHS 
 
 In order to choose some representative labelled graphs as 
 operands in transformations we will enumerate graphs with up to four 
 points, two relations, and four edges, i.e. n » h, mi2, r^ h: 
 
 (i) Graphs which are isomorphic, i.e. identical except for 
 a renaming of labels and points, are considered equivalent in 
 the enumeration. 
 
 (ii) Graphs with reflexive loops will not be considered. 
 Thus, edges will be exhibited between distinct points only 
 and n i 2. 
 
 (iii) Multiple relations between two points will be treated 
 as defining a new relation. Accordingly, graphs with multiple 
 edges will not be enumerated. For example, the graph: 
 
 with n, m, r = 3, 2, 3 will be considered equivalent in the 
 
 enumeration to the graph: 
 
 l' 2 
 « <: # ^~« 
 
 with n, m, r = 3, 2, 2 since H /may be defined by H /= H f\ H 
 
 (iv) Disconnected graphs are treated by considering their 
 connected components only and are omitted in the enumeration. 
 
 Figure 1 displays the enumerated graphs. Each row contains 
 a graph, L, and all the inverse graphs I (L), I (L), l(L). Isomorphisms 
 may exist between graphs in the same row only. Isomorphic graphs in a 
 row are listed on the same line to the right of the row. 
 
 Except for the 2,1,1 case, graphs with m = 1 are not listed 
 since they are special cases of graphs with m = 2 which are listed. For 
 n = 2 or 3 and m = 2, a graph with any possible value of r is equivalent 
 to one of the listed graphs. For n = k and m = 2, the U,2,l and U,2,2 
 graphs are disconnected. All U,2,3 graphs are listed. 
 
 -1+- 
 
Additional h,2,h graphs may be obtained by adding an edge to 
 any corner of the 3,2,3 graphs. These graphs are not listed since trans- 
 formations will be applied to their 3,2,3 subgraphs. Similarly, other 
 more complex graphs will be handled by a series of transformations on 
 simpler subgraphs. 
 
 -5- 
 
i m r 
 
 E 
 
 FIGURE 1 
 SHEET 1 
 
 I, I 2 
 
 I 
 
 
 1 1 1 
 
 1 ' -« ' ' ► 
 
 5 2 2 
 
 M l 
 \ 
 
 2 2 2 2 
 
 5 2 3 
 
 *1 
 
 I 2 ^ 
 
 
 
 ^2 
 l 2 
 
 * 2 3 
 
 y\ 
 
 ^ i 
 
 A 
 
 i 
 
 i 
 
 ? A 
 
 1 u 
 2 
 
 1 
 
 [• A 
 !• /I 
 
 2 
 
 1 
 
 -6- 
 
 '■7 
 
 2 
 
 1 
 
 2 
 2 
 
 2 
 
 » I ' 2 » 
 
 i|.i 2 
 
 E.Ij 
 I.I 2 
 
 E, I 
 I„I 2 
 
 E,I, 
 I2.I 
 
 E,I 2 
 I,. I 
 
MbUKL 
 SHEET 2 
 
 4 2 3 
 
 I 
 2 
 
 2 
 
 I 
 
 E, I 
 I,,I 2 
 
 I 
 
 4 2 4 
 
 I 
 
 2 
 
 2 2' 
 
 2 2 
 
 E,I,,Io,I 
 
 2 I 
 
 2 I 
 
 ^ 1 I 
 
 E.I 
 
 2 2 
 
 2 2 
 
 2 2 
 
 E,I„I?,I 
 
 2 
 
 I 
 
 2 2 
 
 2 2 
 
 2 2 
 
 E, I 2 
 I,. I 
 
 E,I 
 I..L 
 
 I 2 
 
 i' 
 
 i,.i. 
 
 2 2 - 2 
 
 -T- 
 
k. DEFINITION OF REPRESENTATIVE TRANSFORMATIONS 
 
 We choose three representative composite forming transformations 
 which use graphs in Figure 1 as operands. One graph is selected for each 
 r =2,3,^- Figure 2 displays the three transformations, Q2 , Q3 , OM and 
 also T(Q2), T(Q3), T(qU) for the eight T in G. There are a total of 
 fourteen non- isomorphic transformations generated by Q2 , Q3 , and QU . 
 The transformations form composites by taking the meet or the join of 
 elements on the lefthand side of the transformation. 
 
 The basic criteria for the choice of transformations were 
 simplicity and plausibility of composite formation. It is assumed that 
 simple transformations, such as Q2 , will occur most frequently and will be 
 basic to any application procedure. More complex transformations are 
 selected if one of the ways in which composites could be chosen appears 
 intuitively most reasonable. For example, of the two basic 3,2,3 graphs 
 in Figure 1, the one selected for Q3 gives an obvious choice of composite 
 formation assuming that a relation can be extended to a composite if it 
 holds with all the parts of the composite. 
 
T 
 
 D 
 
 I.D 
 
 Q2 
 
 FIGURE 2 T(Q) 
 Q3 
 
 ib 
 
 Qub 
 
 t 
 
 c c 
 
 au b 
 
 2 =:: 
 
 Q4 
 
 a 
 
 b aub 
 
 2 =:: 
 
 i 
 
 a-<- 
 
 aub 
 
 Q3 
 
 nb 
 
 aub 
 
 2 =:: 
 c 
 
 due 
 
 Q4 
 
 2 =:: 2 
 
 c 
 
 I . aub 
 2=:: 2 
 
 KQ3) 
 
 Q4 
 
 a 
 
 aub 
 
 2 = * • 
 
 Q^ ,b Qnb 
 
 2 =:: 
 
 c c 
 
 Q4 
 
 b 
 
 2 : 
 
 anb 
 2 
 
 IoD 
 
 anb 
 
 2 =:: 
 
 a 
 
 anb 
 
 b 
 2 =:: 
 
 c 
 
 D(Q3) 
 
 2 
 
 D(Q4) 
 
 ID 
 
 2 =:: 
 
 ID(Q3) 
 
 D(Q4) 
 
 anb 
 
 b a^b 
 
 2 
 c 
 
 '^■Vf 
 
 -9- 
 
 D(Q4) 
 
Reverse Transformations 
 
 We will also consider the reverse transformations of the 
 defined transformations. The reverse of a transformation Q will "be 
 
 A. A. 
 
 denoted by Q. Q is the transformation which restores all edges cut by 
 the composite forming transformation Q, while assuming that the relations 
 within the composite still exist. 
 
 Q2, Q3 and Q4 are shown below. 
 
 Q2 
 
 a.\Jh 
 
 a ^ 
 
 b = 
 
 Q3: 
 
 a Ub 
 
 b = 
 
 oh 
 
 3. w b a.^, b 
 
 dl/ 
 
 d<- 
 
 a 
 
 
 1 
 
 
 
 
 
 
 2 
 
 
 
 
 \ 
 
 I 
 
 
 V 
 
 
 *<^ 
 
 
 
 1 c 
 
 -10- 
 
Figure 3 represents the transformations and reverse trans- 
 formations by diagrams in which dotted straight lines indicate the lattice 
 ordering between a composite element and its parts. Relations which are 
 assumed by the transformation, i.e. those on the lefthand side, are 
 indicated by unbroken curved edges and relations added are indicated by 
 dashed curved edges. 
 
 -11- 
 
FIGURE 3 
 
 aub 
 
 aub 
 
 aub 
 
 -12- 
 
5. SUFFICIENT CONDITIONS FOR TRANSFORMATIONS 
 
 In this section we will determine sufficient conditions such 
 that the transformations defined earlier are always valid. A trans- 
 formation, Q = (L =:;L ), is valid if and only if all relations 
 contained in L and not in L are true. 
 
 Three different types of conditions are formulated: 
 
 1. Boolean expressions involving the composite formation 
 
 properties of a single relation H, which were defined in [l]i 
 
 -1 -1 -1 
 These properties are denoted by PI, PI , P2 , P2 , P3, P3 , 
 
 P4, PIT 1 , P5, P6, P7, P6" 1 . 
 
 2. The conditions that a single relation H is a subset of 
 the defining relations of the lattice or their union or inter- 
 section. We denote the lattice containment relations by GTE, 
 "greater than or equal," and LTE, "less than or equal." 
 
 Then the notation for the four conditions is given below. 
 
 Notation Condition 
 
 GLTE H £ GTE U LTE 
 
 LTE H ^ LTE 
 
 GTE H ^- GTE 
 
 E H ^ GTE (\ LTE 
 
 3. Boolean expressions involving other transformations Q. 
 These conditions express implications between transformations 
 and thus, unlike the previous two types of conditions, can 
 depend on the properties of more than one relation. 
 
 The group of operators G is used to simplify the establishment 
 of sufficient conditions for the transformations. As in[l], it is 
 easily seen that any statement, A, involving relations, lattice operations, 
 variables, and logical expressions is true if and only if T(A) is true, 
 where T is an operator in G. That is, A^^TlA). In particular, we 
 will use the fact: 
 
 -13- 
 
(A=>Q)<^>(T(A) =^.T(Q)) 
 where A is a condition and Q is a transformation. 
 
 Table 2 lists alternate sufficient conditions for the fourteen 
 non-isomorphic transformations T(Q2), T(Q3), T(qU) and the fourteen 
 reverse transformations T(t$2) , T(Q3) , T(Q^) for all T in G. All the 
 conditions T(A) for T(Q) are determined from the conditions A for Q in the 
 first row of Table 2. Each alternate condition for T(Q) .is 
 contained completely on one line within the row and column entry for T 
 and Q unless the condition is enclosed in parenthesis. In Table 2 the 
 conditions involving properties, P, (type 1 above) always apply to the 
 relation H of the transformation and the containment conditions (type 
 2 above) always apply to the relation H of the transformation. For 
 example, the meaning of the entry in row "E" and column "QU" of Table 2 
 is expanded below: 
 
 (H 2 satisfies P5) or (H is: GTE) or (H £L LTE) or ( ( Q2 is valid) 
 and (I (Q2) is valid)) =^ Q> is valid. 
 
 The purpose in establishing these conditions is to be able to 
 determine in as many cases as possible which transformations are valid 
 by considering only abstract properties of the relations involved in the 
 transformations. Then by characterizing all relations under consideration 
 in terms of an adequate set of properties s a name-independent or context-free 
 type of parse may be facilitated. 
 
 -Ik- 
 
Table 2 - Sufficient Conditions for Transformations 
 
 t\^ 
 
 Q2 
 
 Q2 
 
 Q3 
 
 A. 
 
 03 
 
 * 
 
 QU 
 
 E 
 
 PI" 1 
 GTE 
 
 P2 -l 
 GTE 
 
 Q3 
 
 I ± (03) 
 
 P3 _1 
 
 GLTE 
 
 Q2 
 
 i 1 (Q2) 
 
 P2" 1 
 
 E 
 
 (02 & 
 I 1 (&)) 
 
 P5 
 GTE 
 LTE 
 (02 & 
 I 2 (Q2)) 
 
 P2 & P2 1 
 E^_ ^_ 
 (Q2 & I (Q2)& 
 I 2 (Q2)& X 
 
 1(02)) 
 
 h 
 
 pi- 1 
 
 LTE 
 
 P2 -l 
 LTE 
 
 3^(03) 
 
 03 
 
 h 
 
 PI 
 GTE 
 
 P2 
 
 GTE^ 
 
 I 2 (Q3) 
 
 1(03) 
 
 P3 
 
 GLTE 
 1(02) 
 I 2 (Q2) 
 
 P2 
 
 (I(Q2)& 
 I 2 (Q2)) 
 
 I 
 
 PI 
 LTE 
 
 P2 
 
 LTE^ 
 I (Q3) 
 I 2 (&) 
 
 D 
 
 P2- 1 
 
 LTE 
 
 PI" 1 
 LTE 
 
 D(Q3) 
 1^(03) 
 
 PIT 1 
 GLTE 
 D(Q2) 
 1^(02) 
 
 PI" 1 
 
 E ^ 
 (D(Q2)& 
 
 I D(Q2)) 
 
 P7 
 LTE 
 GTE 
 
 (D(Q2) & 
 I 2 D(Q2)) 
 
 r 
 
 PI & Pi" 1 
 
 E ^ 
 (D(Q2) & 
 
 1^(02)8= 
 I D(^2)& 
 ID(Q2)) 
 
 I i D 
 
 P2- 1 
 
 GTE 
 
 PI" 1 
 GTE ^ 
 1^(03) 
 
 D(Q3) 
 
 I 2 D 
 
 P2 
 LTE 
 
 PI 
 
 LTE 
 
 I 2 D(Q3) 
 
 ID(Q3) 
 
 GLTE 
 
 ID(Q2) 
 
 I 2 D(Q2) 
 
 PI 
 
 E ^ 
 (ID(Q2)& 
 
 I 2 D(Q2)) 
 
 ID 
 
 P2 
 GTE 
 
 PI 
 
 GTE 
 
 H>($3) 
 
 I D(Q3) 
 
 -15- 
 
6. TRANSFORMATION INDUCED CLASSIFICATION OF RELATIONS 
 
 Assume we have a set of transformations and a set of relations 
 which may occur in the transformations. That is, the set H of relations 
 in a transformation represents m variables which may be chosen from the 
 class of relations under consideration. If we know the behavior (e.g. the 
 validity) of the transformations for all possible combinations of relations, 
 we can partition the relations into classes, such that all relations in the 
 same class behave the same in a position in a given transformation with 
 respect to all possible relations in the other positions in the trans- 
 formation. Then by intersecting the partitions induced from each position 
 in each transformation we will obtain a classification which is as fine- 
 grained as necessary for the given relations and transformations. Thus, 
 the only property of a relation now necessary for applying the transfor- 
 mations, as for example, in a name- independent parsing procedure, is the class 
 to which the relation belongs. 
 
 The classification described above is an exhaustive technique 
 which assumes complete information about all relations in all transfor- 
 mations and would be impractical for most situations. The approach 
 taken in this section is to determine as much as possible about the trans- 
 formation induced classification of relations on the basis of the pro- 
 perties defined previously. Although the exhaustive technique may be the 
 only way to verify that the finest level classification has been obtained, 
 a consideration of a priori properties will partially, and may completely, 
 determine the classification in some cases. 
 
 Induced Partitions 
 
 First, we define and show the uniqueness of the transformation 
 induced partitions. Given a set of relations, H, we consider a trans- 
 formation, Q, involving m relations, H r , 6 e A, where each H\ is a 
 
 6 o 
 
 variable over the set H. Assume there is a function f_: n -> N which 
 associates some element of a set, N, with the transformation Q for any 
 m relations in H. 
 
 -16- 
 
Elements of N may i idicate the validity or other properties 
 of the transformation. 
 
 The function f can be represented by an m-dimensional array, 
 
 A, where the size of each dimension is the order of H. Assume A = {1,2,..., 
 
 m} and g and h are fixed elements of H. Then we can define m equivalence 
 
 relations, E-. , E_ , . . . , E , on H by: 
 12 m 
 
 gE.h <*=» f(H r H 2 ...,H H , g, H. +1 ,...,H m ) 
 
 = f(H l' H 2"-' H j-l' h > Vl V 
 
 The m equivalence relations uniquely determine m partitions, II, II , . . . , n , 
 on H. In terms of the array, A, the classes of II. correspond to equal 
 hyperplanes perpendicular to the axis of the ith dimension. By combining 
 equal hyperplanes , the array A can be reduced to an array where the size 
 
 of the ith dimension is the number of classes in II . . 
 
 1 
 
 Thus, the partition n . uniquely classifies relations which are 
 indistinguishable from each other in the ith position of the transformation 
 for all relations in H. 
 
 Partial Classification 
 
 Now we consider the transformations Q2 , Q3, Ql* -and Q2 Q3* Q^T 
 Here m = 2, A = {1, 2}. Let N = {0, 1} and define f by: 
 
 f (g,h) =J 1 If Q is always valid with H = g and H = h, 
 
 ^ Otherwise 
 
 Then f n is represented by a matrix of binary values denoted by M . Rows 
 of M correspond to relations H and columns of M correspond to 
 relations H . The two partitions IL and II group equal rows and columns 
 respectively. 
 
 -17- 
 
We partially determ ne the matrix M by considering the con- 
 ditions of Table 2. That is, we determine the number of different classes 
 in n and II on the basis of all possible values of the conditions which 
 imply H and H p . These results are summarized by Table 3 which shows the 
 reduced matrix, M, for the six transformations Q2, Q2, Q3, Q3, Q^- , and Q^+. 
 This matrix contains all the information of the individual matrices 
 M , . . . , M-^ and the partitions, II and II of M are the intersections of 
 the partitions of the individual matrices. The five possible values of 
 
 the containment properties for H give five different classes in II (rows 
 
 —1 
 
 of M) and the twenty possible values of the properties PI , P3 , P5, 
 
 P2, P2 give fifteen different classes in II (columns of M). Since 
 the matrix is only partially filled out, these are lower bounds on the 
 total number of classes in II and n . 
 
 We know also that the seven partially filled out matrices 
 for the transformations T(Q2), T(Q2) , . . . ,T(qU) for the other seven T 
 in G, i.e. T ^ E, will have values identical to M. In these matrices 
 the conditions A are replaced by T(A) and the application of the dual 
 or inverse operators to all conditions A does not change the possible 
 values of the conditions. 
 
 -18- 
 
Table 3 
 
 M 
 
 
 Pl" 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ H 2 
 
 P3 -l 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 H l \ 
 
 P5 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 P2 
 
 1 
 
 
 
 X 
 
 1 
 
 
 
 X 
 
 1 
 
 
 
 X 
 
 1 
 
 
 
 X 
 
 1 
 
 
 
 X 
 
 
 P2" 1 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 
 Q2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 £ 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 
 Q3 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 Q3 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 
 Qh 
 
 1 
 
 1 
 
 1 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 qH 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 Q2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 GTE 
 
 € 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 U 
 
 Q3 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 LTE 
 
 03 
 
 oh 
 
 1 
 1 
 1 
 
 1 
 1 
 
 1 
 
 1 
 1 
 
 1 
 
 
 1 
 1 
 1 
 
 1 
 1 
 
 1 
 
 1 
 1 
 
 1 
 
 
 1 
 1 
 
 1 
 
 
 
 Q2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 >*- 
 
 02 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 LTE 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 ts 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 
 QU 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 0^ 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 GTE 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 ^3 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 
 oh 
 
 1 
 1 
 
 1 
 
 1 
 
 1 
 1 
 
 1 
 
 1 
 
 1 
 1 
 
 1 
 
 1 
 
 1 
 1 
 
 1 
 
 1 
 
 1 
 1 
 
 1 
 
 1 
 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 tt2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 E 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 0^ 
 ^> 
 0^ 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
 1 
 1 
 
7. EXAMPLE CLASSIFICATION 
 
 As an application of the ideas of the previous section, 
 
 we will consider an example in which a labelled directed graph 
 
 represents a simplified "house cartoon" of the type used in the 
 paper by Ledley and Wilson (3). 
 
 The nodes of the initial graph correspond to the primitive 
 elements, window, door, chimney, wall, roof and gable, of the cartoon. 
 The set of relations, H, between nodes is given below. 
 
 Relations 
 
 H directly left of 
 ij 
 
 H directly on top of 
 
 H contains 
 
 *a 
 
 near 
 
 Figure h shows a house cartoon with labelled primitives, and 
 the associated graph. Relations which are implied by other relations, 
 such as "near" implied by "directly on top of" , are not shown on the 
 graph. 
 
 The goal now is to parse the graph by forming composite elements 
 to obtain a final composite, "house", which can be represented by a 
 linear string of relations and primitives. 
 
 The graph should be parsed so that no information from the 
 original graph is lost. In considering the application of a composite 
 forming transformation, Q, to the graph, we know that Q will be 
 information lossless if both Q and the reverse of Q, Q, are always 
 valid. In order to parse the graph, we would like to determine the 
 validity of the transformations, Q2, Q2, Q3 , 03, QU , Q 1 * , for all 
 combinations of the relations, H. Thus, we want to obtain the validity 
 matrix, M, and a classification of the relations as described in 
 Section 6. 
 
 We first partially determine the matrix on the basis of the 
 abstract properties of the four relations. The properties used in M 
 are shown in Table h for each relation H„. 
 
 -20- 
 
/ 
 
 
 
 a 
 
 
 <\ 
 
 c\ -\ 
 
 
 e 
 f 
 
 
 h g 
 
 L±J < * 
 
 HOUSE 
 
 T 
 
 <b 
 
 L 
 
 L 
 
 PRIMITIVE GRAPH 
 
 Figure h 
 
 -21- 
 
6 
 
 Prop. 
 
 pi- 1 
 
 P3" 1 
 
 P5 
 
 P2 
 
 P2" 1 
 
 LTE 
 
 GTE 
 
 C 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 T 
 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 
 
 L 
 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 
 
 N 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 Table h 
 
 From these properties ve can use the entries in Table 3 
 of Section 6 directly, in order to partially fill in the matrix, M. 
 
 M is shown in Table 5. 
 
 By examining the matrix we can determine which transformations 
 are always information lossless and thus can be carried out without any 
 additional contextual or grammatical information. 
 
 Thus, Figure k is transformed into Figure 5 by applying only 
 information lossless transformations Q2, l(Q2), and Q3 with H = H , 
 to form the composite elements (euf), (gUh), (jUk) and then 
 (iuj uk). 
 
 Depending on the exact definitions of the relations used, more 
 information lossless transformations may be possible on the graph. 
 These could be determined by examining specific transformations to 
 complete the matrix M. All of these transformations depend only upon 
 the type of relations involved in the transformations and not on the 
 names of the objects or composites being formed. 
 
 After no more information lossless transformations can be 
 applied, the next level of parsing may have to be name dependent. For 
 example, a model directed parse based on Ledley's house grammar, 
 could now restrict the possible transformations so that the vertical 
 composites are formed first. Under such a model, a series of Q2 and QU 
 transformations would produce the graph shown in Figure 6, which is 
 expressed in linear form below: 
 ( bT( eCf ) )L( ( aTc )T( gCh) )L( dT(iC( jNk ) ) ) 
 
 -22 
 
\h 2 
 
 
 
 
 
 
 H l \^ 
 
 Q 
 
 c 
 
 T 
 
 L 
 
 N 
 
 
 Q2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 Q2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 C 
 
 Q3 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 03 
 
 
 1 
 
 1 
 
 
 
 Q^ 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 1 
 
 1 
 
 
 
 Q2 
 
 1 
 
 
 
 1 
 
 
 Q2 
 
 
 1 
 
 1 
 
 
 T 
 
 Q3 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 03 
 
 
 1 
 
 1 
 
 
 
 QU 
 
 1 
 
 
 
 1 
 
 
 ot 
 
 
 1 
 
 1 
 
 
 
 Q2 
 
 1 
 
 
 
 1 
 
 
 >*- 
 
 02 
 
 
 1 
 
 1 
 
 
 L 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 "© 
 
 
 1 
 
 1 
 
 
 
 0^ 
 
 1 
 
 
 
 1 
 
 
 & 
 
 
 1 
 
 1 
 
 
 
 02 
 
 1 
 
 
 
 1 
 
 
 02 
 
 
 1 
 
 1 
 
 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 N 
 
 03 
 
 
 1 
 
 1 
 
 
 
 0^ 
 
 1 
 
 
 
 1 
 
 
 tk 
 
 
 1 
 
 1 
 
 
 Table 5 
 Validity Matrix M For Transformations Q 
 
 -23- 
 
© 
 
 L ,A_J^@ 
 
 PRIMITIVE GRAPH AFTER INFORMATION LOSSLESS TRANSFORMATIONS 
 
 Figure 5 
 
 -2k- 
 
© 
 
 T 
 
 © 
 
 L T 
 
 GRAPH AFTER MODEL DIRECTED PARSING 
 
 Figure 6 
 
 -25- 
 
References 
 
 1. Schwebel, J. C. and McCormick, B. H. , Consistent Properties of 
 Composite Formation Under a Binary Relation, University of 
 Illinois, Department of Computer Science, Report Wo. 348, 
 
 August 12, 1969. 
 
 2. McCormick, B. H. , Syntax-Directed Recognition of Pictures: A 
 
 New Theoretical Model, Talk presented at the conference: Emerging 
 Concepts in Computer Graphics, University of Illinois, November T> 
 1967. 
 
 3. Ledley, R. S. and Wilson, J. B. , Concept Analysis by Syntax Processing. 
 Proceedings ADI , pp. 1-8, 1964. 
 
 -26- 
 
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