A syntactic basis of classification

A SYNTACTIC BASIS OF CLASSIFICATION

M.B. DALE

CSIRO, Division of Tropical Crops and Pastures, Cunningham Laboratory, St. Lucia, 4067, Australia

Keywords: Classification, Ecological grammar, Semantics, Syntax

Introduction

Numerical classification has been treated somewhat superficially by ecologists: they practice but they do not believe, they plant figures and reap ink. Since, as Shakes- pear says, 'There is occasions and causes why and where- fore in all things', some explanation of this is needed. The confusion existing over the plethora of similarity measures and the profusion of competing methodologies suggests that a stronger theoretical foundation is required, linked more closely to the objectives of the users, and providing sufficient explication to allow rational choices.

The earliest vegetation classifications were not based on formula or method. Instead you learnt to classify by classifying, by paying attention and doing what one thereby discovers has to be done an apposite description but originally applied by Huxley to love. Such an approach is, I feel, no longer tolerable at least for classification. Yet the present alternative which is offered, the statistical decision theoretic method which I too em- braced in my salad days when I was green in judgement, is also ineffectual. The difficulty is not that it does not suffice, but that it is bereft of any relationship to the aims of the ecologist. The pattern is no longer for the the agent and the search an endless exercise in artifice.

Now as Freud, almost, said something which leaves unsatisfied and drives to rebelliousness so large a number of

practitioners has neither the prospect of continual existence nor deserves it. Certainly the statistic~tl path is attractive and if we do not know (or care) where we are going we shall certainly arrive somewhere. I must add of course, that I would not wish to reject all stochastic elements. That would be rather like the architect in Morgenstern's poem, whose mansion was built of the holes between fence slats:

Es war einmal ein Lattenzaum mit Zwischenraum, hindurchzuschauen. Ein Architekt der dieses sah stand eines Abend pl6tzlich da und nahm den Zwischenraum hinaus und baute draus ein grosses Haus.

C. Morgenstern

Of objectives and grammar

If we start by examinig the objectives of ecologists it seems to me and my mind may of course be late maturing or rotting early - that phytosociology is concerned with the processes which generate the variety of vegetation. If classification is to be useful it must aid in elucidating these processes without dumosity, which means covered in bushes and briars, and will itself receive support and explication from an appropriate theoretical basis. One of the most convenient ways of describing processes, and specifically the kinds of discrete processes apparent in vegetation, is to develop an appropriate grammar. A grammar, talking as a linguist not a logician, is a formal tool for defining infinite or finite varation with finite, albeit recursive, rules. An interesting example is given by Lidov & Gabura (1973) in musical composition. As an aside I find myself ill attuned to deterministic differential equations which smear plants like butter across a land- scape as the filling in a biomass cake, or energy sandwich.

Since it is foolish to make a long prologue and be short in the story I shall for the rest of the paper glance briefly over the why and hows of grammars.

While Fu (1977) provides a simple introduction to the kinds of grammars, to demonstrate here I shall use a very simple example due to Knuth (1968). This is a context- free grammar for binary numbers - (Fig. la).

Vegetatio vol. 42: 93-98, 1980 93

P = N ~ E . E N ---, E E ---, EB E--*B B ~ I B ~ O

V N -= {N, L, B} V T --- {0, 1} S ---N

Structure of 1101.01

N -

/ \ E

E

0 o

E

I B

I 1

Fig. 1. a. A context free grammar for binary numbers. b. The structure of a binary number.

E B

The grammar is defined as a 4-tuple G = (V N, V r,

S, P) where V N is the set of nonterminals V r the set of terminals or primitives S the start symbol, a single nonterminal P the set of rules or productions.

I want to emphasise 8 things about this grammar, without going into rigorous definitions. 1. It is a sequential grammar. A rule is selected non-

deterministically and applied to one instance of the

string. 2. The grammar is context-free. There are no rules of

the form A X L ~ A Y L where X can be written as Y if surrounded by A and L, and only in that context.

3. The grammar is deterministic in the sense that there

94

are no probabilities attached to the rules to control their selection and use.

4. The only relation between primitives is 'next to'.

5. The grammar assigns a particular tree structure to every number - (Fig. lb).

6. This grammar shows only how the number is con- structed, but does not show what the constructions mean. There are no semantics.

7. The primitives are provided from outside the grammar.

8. The grammar is given. We do not have any means. given to infer such a grammar from examples of binary numbers.

These remarks are not intended to make you treat me like a cuckoo in June, heard but not regarded. They simply point up obvious problems, which I shall now attempt to resolve.

Grammars for ecology

The earliest biological examples of grammars are those of Lindenmayer (1968), whose L-systems were introduced to prescribe morphological development. Their most important feature is that L-systems are parallel grammars, that is a rule applied simultaneously to all the parts of the string which permit it. L-systems have now been generalised to three-dimensional problems (Reusch 1976,

Mayoh 1974), and Herman (1970) has discussed the role of environmental variations, though regrettably not from an ecological viewpoint. Perhaps most significant in our present context, Hogeweg & Hesper (1974) showed that numerical classification techniques could retrieve the

grammar underlying a set of examples, and that such techniques could be used to aid in identifying processes.

The first ecological grammars are apparently those of Haeffner (1975) who attempted to formalise a niche approach to ecosystems. He required an essentially encyclopaedic description of every species and of its interactions with abiotic and biotic environments; at the risk of rubbing the poor itch of my opinion I feel that this simply demonstrates the irrelevence of the niche concept. Partly Haeffner's problems arise from using a deterministic grammar whose complexity will approach the real system as its adequacy increases, whereas a stochastic gramroar need not be so complex (Wharton 1974). Partly it is because he adopts a sequential grammar when a parallel grammar is mQre appropriate. Significantly he later (1978) attempted to show that no class of grammars less than transformational grammars could be adequate for ecosystem description. Now it is true that every

Given ~ XXXX

rules

XXX ~ XYX ~ XX ~ ~ cX XXY ~ XaY X X ~ ~ Xc XYY ~ XbY cXc --+ ccc aX ~ aa bX ~ bb Xa ~ aa bY ~ bb aY ~ aa Xb ~ bb Sequential application ~ aaaa Parallel generation ¢¢ bbbb Parsed parallel ~ cccc

(i.e. xeverse the rules)

g~ is an end of string marker.

Fig. 2. A grammar applied in sequence and in parallel.

parallel grammar can be rewritten as a Sequential grammar

but Rosenfeld (1971) showed that parallel context free

grammars in general are equivalent to context sensitive

sequential grammars. Certainly the same grammar applied

sequentially and in parallel will give divergent results -

(Fig. 2). Parallel grammars are almost always simpler

than equivalent sequential grammars, as Gips (1974)

demonstrates so delightfully with his snowflake grammar.

Gips (loc. cit) also illustrates that grammars can be

applied to more complex relationships, in his case to

geometric shapes. Indeed there has been a considerable ef-

fort given in the last decade to developing grammars for

picture recognition, and in this case the rules of the

grammar must not only specify changes in the symbols

but also specify how the new symbol will be connected

to other components of the scene. For complex relation-

ships this is not a trivial task. However, Moayer & Fu

(.1977) in looking at grammars for fingerprints introduced

special rules simply to subdivide the picture. Such sub-

division was used to reduce the computat ion by concen-

trating attention on interesting regions. Kelly (1971)

had earlier made use of this approach with somewhat

greater finesse. So classification, in the sense of subdivi-

sion of an area, is here used to emphasise interesting

areas at the expense of less interesting areas, which

surely has some ecological analogue.

Grammars impose structure on examples, and it is

the existence of this structure which really distinguishes

syntactic from decision theoretic methods. The lattel

ignore any such structure, regarding it as unfortunate

logical dependency between primitives, a nuisance to be

avoided or at best tolerated plunge not the finger of

enquiry into the pie of imprudence. The former regard the

structure as being of particular importance, as it represents

a presumed process of generation to which the idea of

classes can be bound. Small changes at early stages can

produce massive changes at the terminal level. Fu and Lu

(1977) have suggested that similarity can be reasonably

defined by using this structure. For two entities A and B

they suggest that the rules for generating A, augmented by

specific error rules, be used for producing B. The similarity

of A and B is then a function of the number and kind of

errors which are necessarily introduced. It is interesting

to note that Sussman's (1975) program for skill acquisi-

tion also emphasises the importance of errors. Further,

the use of Manhat tan metrics in cladistic studies can be

seen as representing the single step nature of mutations,

which in a sense are errors.

S e m a n t i c s

Although grammars may well formally describe patterns,

this is a somewhat barren occupation unless interpreta-

tions can be given. Knuth (1968) presents one approach

for the binary numbers - (Fig. 3) which is to assign to

each symbol a finite set of attributes. Each attribute may

be synthetic or inherited: the start symbol has no in-

herited attributes, the primitives no synthetic attributes.

To each rule in the grammar further semantic rules

Each B has a 'value" v(B) which is a rational number Each B has a 'scale' s(B) which is an integer Each L has a 'value' v(L) which is a rational number Each L has a 'length' I(L) which is an integer Each L has a 'scale' s(L) which is an integer Each N has a 'value' v(L) which is a rational number.

Syntaetie Rules Semantic Rules B ~ 0 v(B): = 0 r B --, I v(B): ~s(B) L ~ B v(L): ~-- v(B)

s(B): = s(L) I(L): = 1

L 1 ~ L2B v(L1): = v(L2) -1- v(B) s(B): = s(L 0 s(Lz): = s(L1) + 1 I(L~): = I(Le) + 1

N ~ L v(N): = v(L) s(L): = 0

N ~ L1.L 2 v(N): = v(L:) + v(Lz) s(Ll): = 0 s(L2): = -l(Lz)

Synthetic attributes are v(B), v(L), I(L), v(N) Inherited attributes are s(B), s(L)

Fig. 3. Attribute grammar for se/nantics of binary numbers.

95

are attached to show how the attributes are related func- tionally.

Formal semantics is presently an area of great activity in computer science. Knuth ' s proposals are related to coordinate grammars, and, I suspect affix grammars. What is important here is not the precise formulation but that the semantic meanings can be coupled to the syntactic rules, leading to the possibility of automatic interpretation of patterns within some other domain.

For vegetation the most obvious domain might be the abiotic environment. The separation of synthetic and inherited attributes is, incidentally, for convenience rather than of necessity, and one of the interesting corollaries is that monothetic methods, or something close to them, might make interpretation easier since they relate class formation to specific primitives.

Inference and primitives

We are left with two trouble spots; can we infer grammars from examples, and what are the ecological primitives. It can be shown (e.g. Fu & Booth 1975, Chou & Fu 1976, Berger & Pair 1978) that inference is possible for some

classes of grammars and procedures have been developed to do this. The difficulty as Herman & Walker (1972) point out, is that we have first to infer the hypothesis space, and then the particular system within it, which is not a standard inference problem. In fact, most of the available procedures are computationally intractable, even for quite simple problems, although Cook & Rosenfeld (1976) have made some progress with heuristic methods. Indeed Cook and Rosenfeld's method is remarkably like agglomerative classification. The key difference is that there are 2 functions being examined. One relates to the complexity of the grammar, while the other relates to the adequacy with which it explains the examples. Haeffner (1978) has already indicated that adequacy also involves the existence of rules in the grammar which are not

falsifiable at the time, but this has semantic overtones. Klein (1973) has examined inference of semantic rules and and there is much work on this aspect in robot studies.

The selection of primitives forms a distinct problem, the usual solution to which is to adopt the species. I have previously (Dale & Clifford 1976, Dale 1978) suggested the use of other idiotaxonomic categories and other alternatives are of course available. As an experiment I have taken the data of Williams et al. (1969) on r~in forest in north Queensland. Each tree is firstdescribed by a list of its neighbours, using Delaunay triangulation

96

TREE NO. 81 TREE NO. 89 TREE NO. 97



Fig. 4. Initial labelling of 9 trees based on their Delaunay triangulation neighbours.


\ ! / i<!/i

TREE NO. 99 TREE NO. 107 TREE NO. 121 8


L/I\! Fig. 5. Relabelling of neighbours after 8 reclassifications.

(Rhynsburger 1973) to find the specific adjoining trees

(Fig. 4). After classifying, each tree is in a specific class,

and the new class labels can be used to provide a new des-

cription. Of course, this process can be iterated until

hopefully there is some convergence. The biggest problem

is fixing the number of primitives, that is the number of groups. A number of methods do exist but for this exercise

I have subjectively chosen 16 groups. This is less than the

original 25 species, and was based on an initial mono-

thetic analysis. The actual classification method used was

Indicator Species analysis (Hill, Bunce & Shaw 1975).

The results after 8 iterations (Fig. 5) show some convergence. In this particular example, the 9 trees were all of

the same species, and the final relabelling suggests that it exists in 5 different environmental neighbourhoods. At the

end I shall attempt to obtain a grammar for each species

of tree in terms of the class affinities of the neighbours.

Knowing the species gives some semantic handles to

attach to the rules of the grammar, but this is difficult

with the lack of knowledge of rainforest species.

Of course almost any classification of vegetation

might b e regarded as an attempt at grammatical inference, and hence as an attempt to identify the processes opera-

ting in the vegetation. A choice of an hierarchical classifi-

cation is an attempt to order the processes in terms of

their significance and clearly parallels the structure in-

duced by rules of grammars. As an example the work of Noble & Slatyer (1978) is a conscious attempt to identify

primitives related to processes of reproduction and

regeneration. The usual taxonomic primitives are more

strongly related to environmental interpretations, setting

spatial rather than temporal boundaries. These two approaches clearly contrast two different views of vegeta-

tion, and the resulting organisations of vegetation data

will be useful in different contexts. For still other contexts, still other primitives will be required. As examples histori-

cal processes may require the use o f t axa at other than the

species level, while relationship to climate may require the

use of structural descriptions.

Conclusion

I hope then that I have clothed grammars in just enough for

modesty and no more. Maybe I have committed all the

oldest sins in the newest kinds of ways, but I think gram-

mars do provide a useful way of looking at phytosociologi- cal processes, and relume the techniques of numerical classification. At least I hope you will no longer treat

them as an

egregious professor of Podunk

who once when overly drunk,

remarked I think

I can decline pink

Let me see it goes pink, pank and punk

and I will finish before I rhyme you to death.

References

Berger, J. & C. Pair. 1978. Inference for regular bilanguages. J. Comput. System Sci. 16: 100-122.

Chou, S.M. & K.S. Fu 1976. Inference for transition network grammars. Proc. 3rd Intern. J. Cong. Pattern Recogni- tion pp. 79 84.

Cook, C.M. & A, Rosenfeld. 1976. Some experiments in grammar inference. In: Computer Oriented Learning Programs, J.C. Simon (ed.) Noordhof-Leyden pp. 157-174.

Dale, M.B. 1978. Planning an adaptive numerical classificatien. Vegetatio 35:131 136.

Dale, M.B. & M.T. Clifford. 1976. On the effectiveness of higher taxonomic ranks for vegetation analysis. Austral. J. Ecol. 1 : 37 62.

Fu, K-S. 1977. Introduction to syntactic pattern recognition. In: Syntactic Pattern Recognition, Applications. K-S. Fu (ed.) Springer N.Y., p. 1-29.

Fu, K-S., & T.L. Booth. 1975. Grammatical Inference: Intro- duction and Survey. Pt. 1 & II. I.E.E.E. Trans. System, Man and Cybernetics SMC-5:95-111,406 423.

Fu, K-S., & S.Y. Lu. 1977. A clustering procedure for syntactic patterns. I.E.E.E. Trans. Systems, Man and Cybernetics SMC-7 : 734-742.

Gips, J. 1974. Shape grammars and their uses. Ph.D. thesis Dept. Computer Science. Stanford University 233 pp.

Haeffner, J.W. 1975. Generative grammars that simulate ecological systems. Simulation Council Proceedings Series 5: 189-211.

Haeffner, J.W. 1978. Ecosystem ~tssembly grammars: generative capacity and empirical adequacy. J. Theoret. Biol. 73: 293- 318.

Herman, G.T. 1970. The role of environment in development models. J. Theoret. Biol. 29: 329-341.

Herman, G.T. & A.D. Walker. 1972. The syntactic inference prob|em applied to biological system. In : Machine Intelligence 7. B. Meltzer & D. Michie (eds) Edinburgh University Press. pp. 341-356.

Hill, M.O., R.G.M. Bunce & M.W. Shaw. 1975. Indicator species analysis: a divisive polythetic method of classification and its application to a survey of native pinewoods in Scotland. J. Ecol. 63: 597-613.

Hogeweg, P. & B. Hesper. 1974. A model study of biomorpho- logical description. Pattern Recognition 6: 165-179.

Kelly, M.D. 1971. Edge detection in pictures using planning. In: Machine Intelligence 6. B. Meltzer & D. Michie (eds) Edin- burgh University Press pp. 397~,10.

Klein, S. 1973. Automatic inference of semantic deep structure rules in generativesemantic grammars. Proc..International

97

Conf. on Computer Linguistics PISA 27/VIII-I/IX No. 2 pp. 557-574.

Knuth, D.E. 1978. Semantics of context free languages. Mathem. System Theory 2: 127-145.

Lidov, D. & J. Gabura. 1973. A melody writing algorithm using a normal language model. Comput. Studies in the Humanities and Verbal Behaviour. 4:138 148.

Lindenmayer, A. 1968. Mathematical models for cellular interaction in development 1. Filaments with one-sided inputs. J. Theoret. Biol. 18:280-289.

Mayoh, B.H. 1974. Multidimensional Lindenmayer organisms. In : Lecture Notes in Computer Science No. 15. B. Groose & J. Hartman (eds) pp. 302-326.

Moayer, B. & K-S Fu. 1977. Fingerprint classification. In: Syntactic Pattern Recognition, Applications. K-S, Fu. (ed.) Springer N.Y. pp. 179-214.

Noble, 1.R. & R.O. Slatyer. 1978. The effect of disturbance on plant succession. Proc. Ecol. Soc. Australia 10: in press.

Reusch, P.J.A. 1976. A common approach to retrieval concepts and multidimensional developmental systems based on lattice- like structures. Gesellschaft ftir Mathematik und Data verarbeitung mBH, Bonn. Ber. Nr. 90 pp. 1 70.

Rhynsburger, D. 1973. Analytical delineation of Thiessen poly- gons. Geogr. Analysis 5: 133-144.

Rosenfeld, A. t97t. Isotonic grammars, parallel grammars and picture grammars. In : Machine Intelligence 6. B. Meltzer & D. Michie (eds) Edinburgh University Press. pp. 281 296.

Sussman, G. 1975. A computer model of skill acquisition. Else- vier 133 pp.

Wharton, R.M. 1974. Approximate language identification. Information and Control 26:369 374.

Williams, W.J., G.N. Lance, L.G.J. Webb, J.G. Tracy & J.H. Connell 1969. Studies in the numerical classification of complex rain forest communities IV A method for the elucidation of small scale pattern. J. Ecol. 57: 635-654,

Accepted 31 October 1979

98

Documents

A syntactic basis of classification