radial grammars and radial l-systems

COMPUTER ORAPHICS AND IMAGE PROCESSING (1975)4, (361-374)

Radial Grammars and Radial L-Systems

G I F T SIROMONEY* AND RANI SIROMONEY~"

Madras Christian College, Tambaram, Madras, India 600059

Communicated by A. Rosenfeld

Received February 10, 1975

Models to generate two-dimensional circular patterns are proposed. The first one generates circular patterns with a finite number of radii and growth takes place in parallel along the radii. Examples are given which reflect growth patterns on biological objects. The second system is an extension which takes care of growth along the radii, the number of radii being unlimited. Comparison is made with cycle languages. Catenation and star operation are extended to radial and transversal operations and closure is examined. The third system includes developmental models which reflect radial and transversal growth. Radial and transversal generations take place in parallel, controlled by a control set, and describe circular developmental patterns.

1. INTRODUCTION Lindenmayer [3] has introduced developmental models that generate

sequences of cellular descriptions of developing organisms. The L-systems describe developing organisms at given instants of time as arrays of symbols in one, two, or three dimensions, each symbol corresponding to a cell.

We have developed two-dimensional generative models for the description of digital pictures and kolam patterns. Complicated patterns of kolam, a South Indian folk art, appear similar to patterns found in nature. The two models we have defined [10,11] are powerful enough to describe a large class of patterns which basically have a reactangular format. We introduce in this paper a third model which can describe a class of pictures with a circular format, starting from one center and growing at several points on the circumference. We give a brief description of our models. The first model is referred to as the matrix model, the second as the array model and the third, the radial.

The matrix models [10] can be considered as a class of parallel sequential machines [1] on the one hand and as generalizations of equal matrix languages (EML) [13] on the other. A two-dimensional matrix language is generated as follows. First, a horizontal line of intermediate cells is generated by a phrase- structure, context-sensitive, context-free, or regular grammar [2]. Then starting with each intermediate cell a regular language is generated downward in a parallel manner from each cell in the horizontal line. Organisms with bilateral symmetry can be described by these models, which are closed under operations such

* The support of the Homi Bhabha Fellowships Council, India, is gratefully acknowledged. The authors wish to thank Professor Azriel Rosenfeld for providing a stimulating atmosphere and Mrs. Shelly Rowe for her invaluable help in the preparation of the manuscript.

t Supported by the Information Systems Branch, Office of Naval Research, under Contract N-0014-67A-0239-0012 with the University of Maryland.

361

Copyright ~ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved,

362 SIROMONEY AND SIROMONEY

as translation, reflection, and half-turn. These matrix models can be used to represent organisms which have unrestricted growth in two or three independent directions.

The matrix models do not adequately describe two-dimensional patterns of growth that maintain a fixed proportion between length and breadth. Such patterns can be described by our more satisfactory second set of models called the array models [11]. The array models are found useful in describing expanding kolam designs of intricate patterns that maintain the same basic shape of a square, rhombus, or hexagon [12]. Some of these kolam patterns are being generated by a computer, using the array models [ 11 ], by Narasimhan [4]. The three-dimensional models [15] generate the cube, the tetrahedron, and the octahedron, and these models can be interpreted to describe radiolarians whose skeletons exhibit such symmetrical forms.

The two models described so far generate arrays of cells that are in the form of a rectangle or a parallelogram. These models, though powerful, are not ade- quate for describing organisms with radial symmetry. The main purpose of this paper is to define a new class of models, called radial systems, for describing patterns that are basically circular. Such systems can be extended to describe organisms with cylindrical symmetry and those that have spiral forms.

In the first section, we introduce a model to generate two-dimensional patterns having radial symmetry. The model deals with the case where there are only a finite number of rays and growth takes place in parallel, outwardly along the rays. When the length of the radius increases, the area increases, but when growth (expansion) is along the rays only, there are gaps between the rays when the area is large. We therefore propose two more systems of models designed to remove these restrictions of finiteness and growth only along the rays.

In the second section, we introduce a model which is an extension of the cycle languages proposed recently by Rosenfeld [6]. We first generate the innermost circle of cycle languages by regular, context-free or context-sensitive rules. Then, generation along the radii takes place in parallel. Thus in this model, we are able to remove the restriction that there can be only a fixed finite number of rays.

In the third section, we introduce a system which provides for growth radially as well as tangentially. L-systems were introduced in [3] to describe development of filamentous organisms. In [ 16], we incorporated the developmental type of generation into two-dimensional rectangular arrays. There, controlled growth takes place in parallel only along the edges. Here, we introduce it into two- dimensional radial languages. The models proposed in the third section describe the generation of circular developmental patterns. Growth takes place in parallel, restricted by a table, and the radial and tangential generations are controlled by a control set.

2. FINITE-RAYED RADIAL MODELS

We now describe informally the radial grammars and the corresponding radial languages. We take S (the start symbol) at the center and imagine k finite state automata (FSA) facing k rays emanating radially from the center. Each FSA acts

RADIAL L-SYSTEMS 363

along a particular ray and generates a regular language the symbols of which are situated equidistant from each other. The actions of the k FSA are parallel and may be constrained by a matrix as in EML [13] or not. The angles between the different rays may be taken to be equal or different. If they are equal, symmetric figures such as the starfish as well as the growth of rings in wood can be described. By allowing the angle to vary, a much wider class, including seashells such as scallops, can be described. The same kind of interpretation can be extended to three dimensions, where the models can be modified to describe organisms having cylindrical symmetry (such as the earthworm or the leech) or conical symmetry.

We present examples to show that several variations of the models can be interpreted to describe the symmetry or growth of many living organisms. First, binary pictures are considered. Then examples with several symbols where each symbol stands for a different primitive are considered, and this yields many interesting pictures. Also, the symbols may be treated as labeled dots and in addition a finite set of instructions [ 12] on how to join the different kinds of labeled dots can be given. By this method we can generate a variety of ko lam designs which could not be generated by our earlier models. Finally, a new interpretation is given to the terminals in that they represent distances. The new interpretation makes it possible to describe the growth of shells such as the Nautilus in two dimensions and to describe the symmetry of conical spirals such as that of the shell of Tur i te l la dup l i ca ta in three dimensions.

Certain operations in formal language theory such as substitution are interpreted meaningfully to generate k o l a m patterns of different sizes that corre- spond to geometric designs found on the shell of the Starred Tortoise (Tes tudo

e l e g a n s [17]) commonly found in South India. The ko lam patterns which corre- spond to the growth of these designs are called the Tortoise Shell.

2.1. We now illustrate with examples. The grammar G is taken to be an equal matrix grammar [13]. The only difference is that the language generated is such that the terminals are situated in a two-dimensional plane. We assume S to be at the origin and each nonterminal along each ray.

E X A M P L E 2.1.1. Five-rayed stars of X's. Let G = ( N , T , P , S ) , where N = {S,<A1, . . . 415>}, T = {X} and P = {[S ---) A1, . . • ,As] , (Ax . . . . 4t5) ~ <X . . . . ,X)<A1, . . . ,As) ,

< i l , • • • 415) "-') <X . . . . , X ) } .

Then the string language generated is { X ' X " X " X n X n / n >1 l} and the radial language {Mn/n ~> 1} is shown in Fig. 1. We have made use of the vector notation for the EMG to make it compact.

The next example is chosen to illustrate the fact that simple "primitives" may be chosen in the place of terminals. The grammar is of the same type as in Ex- ample 1.1 with primitives in the place of terminals.

E X A M P L E 2.1.2. South Indian Y a n t r a design. Let G = ( N , T , P , S ) , where N = { S , ( A 1 , . . . ,As)}, T = {~0}, P = {[S--)AI . . . . ~ , ] . < A , , . . . 41~)--, < ~ . . . . ,q,)<A1 . . . . 41~>, <A1 . . . . ,As> --~ <~ . . . . . O)}. The radial language generated is { M , / n ~ 1 } shown in Fig. 2.


M x

M 2 x X

M 3 ~ '~)< x

x X

\ / M

4 ~(%'/"/" ~ Jf~'~r x x x

Fla. 1. Five-rayed stars of X's.

M 1 ®

M 2 ~ C

M 3 - ~

F~a. 2. South Indian Yantra design.

R A D I A L L - S Y S T E M S 3 6 5

M 1

M 2

FIG. 3. H e x a g o n s of different sizes.

The next two examples are chosen to show how labeled dots can be generated by radial grammars and additional instructions given to join them suitably to form hexagons or six-pointed stars of different sizes.

E X A M P L E 2.1.3. Hexagons of different sizes. Let G = ( N , T , P , S ) , where N = {S, (A1 . . . . . A6)}, T = {O}, and P = [S--*A~ . . . . . . A6], (A1 . . . . ,A,) ~ ( 0 , • • • ,O)(A~ . . . . ,A6), ( A ~ , . . . ,A0) .-,, (O . . . . . O) . The radial language generated is {Mn/n >~ 1} given in Fig. 3.

E X A M P L E 2.1.4. Six-pointed stars of different sizes. Let G = ( N , T , P , S ) where N = {S, (A~ . . . . ,A12)}, T = {O,X}, P = {[S ~ Ax, • • • rA12], (A1,A2 . . . . ~ll~A12) ( O , X , . . . , O , X ) ( A ~,A~, . . . , A I ~ , A ~ ) , (A1,4~ . . . . ,4~1,41~) ( o , x . . . . . O , X ) } . In addition, we give instructions on how to join the X in each ray to the X in the next ray by suitably labeling the X's. We then get the six-pointed stars drawn in Fig. 4.

2.2. So far we have taken the angle between the rays to be fixed. In fact the angle will be determined by the order k of the E M G generating the radial language and is equal to 27r/k. On the other hand we can allow the angle to vary and this can be incorporated into the initial rule. For example, the rule IS ~ Alc~A2q52 • . . ~bk-iAk] will imply that the angle between the first and second ray is ~b~ . . . . . between the (k-l) th and the kth ray is 6k-~. This additional information will enable us to describe the growth of scallops which follow this pattern.

2.3. Earlier we took the terminal symbols in a radial language to be either


M 1

M 2 . • , •

Fro. 4. Six-pointed stars of different sizes,

binary or labeled dots or to stand for simple primitives. We now show that if a new interpretation is given to the terminal symbols in that they represent distances, then a wider variety of patterns can be described. This will include the generation of an equiangular spiral (logarithmic spiral) which will help in the description of the growth of the snail or the Nautilus shell or other molluscan shells and when extended to three dimensions will help in the description of the shell of TuriteUa duplicata [18].

2.4. Last, we are interested in the pattern of the tortoise (Testudo elegans) found in South India, which is reflected in the kolam called the Tortoise Shell

Fxo. 5. The Tortoise Shell kolam.


(Fig. 5). In this figure we note that there is growth in the interior pattern (in the form of a star) as well as in the exterior (which is in the form of a rhombus). Here we make use of the idea of substitution, which is well known in formal language theory. We adapt it to two-dimensional pictures.

EXAMPLE 2.4.1. The Tortoise Shell kolam. Let L(G) be the finite radial language

A A A C A

A A.

r(C) = Lt, where L~ is the set of all six-pointed stars. ~-(A) = L~, where L2 is the set of all rhombuses. In order that the operation of substitution may be mean- ingful in the two-dimensional picture there must be enough space for the substi- tuted picture. In this particular example it is clear that this will be the case provided we assume that the rhombus of the smallest size fits into the star of the smallest size, the rhombus of the second size into the star of the second, and so on. This can be seen in Fig. 5.

3. THE RADIAL MATRIX MODELS

3.1. We now define the radial matrix models and define operations which are extensions of catenation and Kleene closure. We name these radial matrix models since we can show that this class is equivalent to the class obtained by taking the rectangular matrix models [10] and bending them round to form circular patterns.

DEFINITION 3.1. A Context-Sensitive Radial Matrix Grammar G (CSRMG) is a 2-tuple (G1,G~), where Gt is a context-sensitive grammar (CSG) and G2 = (_jk~=lG2~, with each G2t an E-free right-linear grammar corresponding to each terminal in G~ (called an intermediate as in [10]). We similarly define context-free radial matrix grammar (CFRMG) and regular radial matrix grammar (RRMG).

NOTATION. The rules of G1 are written a ~ / 3 and derivations are denoted B

by © and ~ . The rules of G2 are written A 1' a or A I' a and derivations are denoted by 41' and *,ft. We introduce this notation since in the next section, we need ~ to stand for transversal rules, ~ for the application of transversal rules in the clockwise direction, 1' for radial rules, and ~, for the application of radial rules.

We now describe informally how a generation proceeds. First, we begin with the generation of a cycle of intermediates as the innermost circle, making use of G1. This cycle can either be regular, CF, or CS. Once the generation of intermediates is over, radial generation takes place in parallel, starting with each intermediate and proceeding outward. Each intermediate generates radially a regular set.

DEFINITION 3.2. A circular language L is called a context-sensitive radial matrix language (CSRML) iff there exists a CSRMG G such that L = L(G) = {®/S ~ ® *'ff ®}, i.e.,


s ~* s. s ,~ .~'n 13 amn" "a212 aln al3a2 3 " "am3

S. ~ ' . Sir'' 1 4 . ,~4., %~

t

We can similarly define context-free radial matrix languages (CFRML) and regular radial matrix languages (RRML).

3.2. Rosenfeld [6] has shown that the class of CS (CF) cycle languages generated by taking CS (CF) string languages and bending them into cycles is equivalent to the class of cycle languages generated by considering cycles ("necklaces") as sentential forms and applying CS (CF) rules. More precisely, let L(G) be the string language generated by G and let L (G) be the language generated when sentential forms are regarded as cycles. Also, for any string language L, let L be the set of cycles that results from bending the strings of L. Then, if G is context-free, we have L(G) = L ~ ) . On the other hand, if G is context-sensitive, he has proved that there exist context-sensitive grammars G~ and G2 such that L(G) =/~(G~) and L~(-G) = L(G2). Making use of this result, we obtain the following theorem immediately.

NOTATION. Let M(G) be the set of matrices generated by G = (G~,G2), ]Q (G) the radial language generated as in Definition 2.1.2, i.e., when the sentential forms are annular in shape. Also, for any matrix language M, let 2~t be the radial language that results from bending the matrices of M.

T H E O R E M 3.1. If G = (G~,G2), where G~ is context-free, then ~ ( G ) = M'('G). On the other hand, if G = (G~,G2), where G~ is context-sensitive, then there exist context-sensitive grammars G~ and G~' such that IQI(G) = M(G') and M"-~) A~f(G"), where O' = (G1,G2), G" = (G~ ,G~).

Note. Rosenfeld has taken a specific grammar G (CF or CS) and compared the effect of generating the string language and bending it into a cycle in contrast to considering cycles as sentential forms and applying the rules of G. On the other hand, if we start with a specific language (CF or CS) and bend it, in the two-dimensional case we find that the results are different. In other words, if we start with a rectangular pattern (say T or I) generable by a context-free matrix grammar [10] and bend it round, then that pattern is generable by a regular radial matrix grammar. Also, when we start with a pattern (say 7r) generable by a context-sensitive matrix grammar [10] and bend it round, then that pattern is generable by a context-free radial matrix grammar. In other words, there are examples in which, if M = M (G) where G is context-free, then M = 37/(G') where G ' is regular; and if M = M(G) where G context-sensitive, then M = M(G ' ) where G ' is context-free.

The two-dimensional radial models can be extended to three-dimensional cylindrical models. First, a cycle language or a radial language (consisting of intermediates) can be generated in two dimensions and then, corresponding to

R A D I A L L - S Y S T E M S 369

each intermediate, a regular language can be generated perpendicular to the plane of the cycle language, as in [14,15]. The latter generation is parallel and the resultant is a three-dimensional cylindrical language•

3.3. The equivalence established in Section 3.2 will help us to transform some of the rectangular patterns into interesting circular patterns• In addition, it helps us to extend some of the operations like cantenation and Kleene closure to radial languages• For example, row and column catenation, row and column star defined for rectangular arrays can be reinterpreted as radial and transverse catenation and radial and transverse star as follows.

DEFINITION 3 .3 .1 • I f L =

a n d L ' =

*e. ~P

e, "? amn" 'a2naln a13a23" "am3

~ . . - "%,.

% .,4"

bm, n , . - b 2 n , b l n , b 1 3 b 2 3 - -bm, 3

-9 ' ~ , , . # .

radial catenation L ® L' is defined only when n =' n' and is equal to

& ~, .04 ~ v •"

"d .0.,¢ ~

~ . ~ •°

'~e ~ bm,n-.blnamn..aln al3''am3bl3.•bm, 3

' ' " %,5

2, "~

370 S I R O M O N E Y A N D S I R O M O N E Y

and transversal catenation is M ® M'.

defined only when rn = rn' and is given by

*; o,. b~,''b2n,bln, alna2n..amn

Similarly, we define radial and transversal star. It follows immediately (as in [10] ) that the families of regular, CF, and CS radial matrix languages are closed under transversal catenation and star but not under radial catenation and star.

4. EXTENDED CONTROLLED TABLE RADIAL L-SYSTEMS

4.I. In this section, we define the extended controlled table radial languages and show how as in the rectangular case [16], a hierarchy can be established, depending on the different types of control and table. In addition, an added ad- vantage is that there is tangential growth also. This helps to fill up the gaps between the radii. In a radial language, this is necessary and possible since the area of coverage increases as the radius increases whereas in a rectangular array, it remains constant.

We use the word "extended" to denote cases where the distinction between terminals and nonterminals is maintained, whereas in L-systems there is no distinction between sets of terminals and nonterminals. Table 0-L languages were introduced in [8] and the effect of control studied in [1].

D E F I N I T I O N 4.1.1. An extended controlled table radial grammar G is a 5-tuple (V , I ,~ ,C ,S ) , where

V is a finite alphabet; I C V is a finite set of terminals;

is a finite set of tables, each table consisting of either a finite set of radial rules or a finite set of transversal rules;

C is a control language over 9~; S @ V-I is the start symbol.

As in [16], (i) if V = I, and S is an axiom (a cycle with or without a center), we get the

L-systems; (ii) if ~ = {P}, there is a single table; and

(iii) if C is empty, there is no control. The rules in each table may be regular, CF (or 0-L) or context-dependent on

its three neighbors (or l-L), and the control language C may be regular, CF, or CS. In the L-systems, in every table there should be at least one rule corresponding to each symbol satisfying the "completeness" condition. Thus here again, several variations give rise to several radial models.


In analogy with the Chomsky and Kuroda normal forms, we assume that the right side in the radial rules consists of only one or two symbols. In the transversal rules, there may be several, but only one symbol is replaced by a rule, and the right side of that rule fits into the wedge filled by that symbol.

The radial rules act only along the radii and on the symbols on the outermost cycle. Thus in the extended system, the radial rules can only be of two types,

viz right linear (A 1' aB, A I' a) or context-dependent of the form

D B A C ~ B ~ C ,

where A, B, C, D are nonterminals and u a string of terminals which are the neighbors of A in the previous cycle. In the L-systems, the radial rules can be

c e either 0-L, i.e., a 1' b, a ]' b, or l-L; i.e., b a c 1' b d c. The transversal rules

can be either regular, CF (or 0-L), or CS (or l-L) and act in parallel, each symbol acting within the wedge available.

Here again, the L-systems are interesting and we illustrate them with examples and establish the hierarchy for L-systems.

4.2. The first set of examples consists of 0-L rules both radially and tangentially and we impose regular, CF, or CS control.

EXAMPLE 4.2.1. We shall take G = (V,~,C,Mo), where

V = {X}, Mo =

X n = { x ? X},

XX X X X X X ' and ~ = (R,T},

T = { x ~ x x} . Then, by taking C to be

(i) a regular set {(RT)n/n >- 1} or (ii) a C F L {RnTn/n >~ 1} or

(iii) a CSL {(RT)n'~/n >~ 1 }, we get circular patterns which reflect development of cells (by division into two)

(i) radially and tangentially alternately in equal proportions, (ii) radially and tangentially only along the outer edge, the number of cells

in the two directions being equal in number, (iii) radially and tangentially alternately but the growth of cells is of the

order of n 2. By changing X to V and C to {(RTR)n/n ~ 0}, we get Fig. 6.

These three control languages are so chosen that they reflect proper containment of the corresponding families of radial languages.

For the second set of illustrations, we can choose the terminals to stand for colors (for the sake of simplicity we have chosen only binary pictures) or for cells. Here we take the radial rules to be context-dependent but the tangential rules to be 0-L.

EXAMPLE 4.2.2. Let G = (V,~,C,Mo), where V = {B,W} (B stands for black, W for white).

372 SIROMONE'Y AND SIROMONEY

V

0

A

• ,.I V

v 3 . ~ ~ Z .z . I

0

,~ r,

',,.I x/ v p.

..~ ,4 -,I k ~ /-

~ ' ~ ~ V "~ z.

0

7 7 A ~ ~ . . ? --/"I ^ ~, ~%, .

FIG. 6. Illustration of Example 3.2.1.

W B B B and ~ = {R1,T,R~},

M ° = W B W B W

81 = {O ~ B, W ~ W}, T ~- {B ~ WB, W ~ WO},

B B W W R , = { W B ? WB, WB? WB, WB, WB? WB}.

B B B B W W W W

By choosing the control C to be (i) {(R1TR~)n/n >~ 0} a regular set (Fig. 7), or

(ii) {R'~T~R~/n >~ 0} a CFL, or (iii) {(RITR2)"2/n >I 0} a CSL,

we can show proper containment of the corresponding families of languages. We can include decreasing rules among the set of transversal rules, to take

care of cell death, thus avoiding e-rules. The strictly circular and radial lines can be deformed to denote cells, while maintaining neighborhood properties.

4.3. Comparison with the cycle languages and the radial matrix models. The family of context-free cycle languages [6] is properly contained in the family of radial 0-L systems with a single table and no control. This follows immediately from the result proved in [9] that if a 0-L system H contains the rule a ~ a for every symbol in H, then it is context-free.

Similarly, it can be proved that the family of context-free radial matrix languages is properly contained in the family of extended table radial languages with regular control, the radial rules being right linear and the tangential rules

RADIAL L-SYSTEMS

® 373

FIG. 7. Illustration of Example 3.2.2.

0-L. In fact, an extended table radial grammar with two tables, one a transversal table R consisting of all the CF rules in Gt, together with the rule a ~ a for every symbol a in G1 and the other a radial table T consisting of all the right linear rules in G.z, with control R * T*, generates the class of context-free radial matrix languages.

In the context-sensitive case, for cycle languages and the radial matrix models, the generation is sequential, whereas in the table radial models, it is parallel. Hence, comparison is. more difficult.

5. CONCLUSION

Extensions of string languages to array languages have posed several problems. Isotonic array rules are considered in [5], to overcome the problem of shearing. Row and column operators are introduced in [10] as extensions of the catenation operator. Table arrays with controlled growth only along the edges are studied in [16] to extend one-dimensional developmental languages to arrays. In this paper, we have extended string languages to circular patterns including developmental languages. The problem of shearing is overcome by controlled growth radially and tangentially. We have also extended the catenation operator to tangential and radial catenation operators. This system of circular languages describing certain types of development should prove useful in describing biological systems. Though we have given only very simple illustrations, the models we have proposed are powerful enough to include several variations and several circular developmental patterns or patterns with radial symmetry.


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radial grammars and radial l-systems

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