J. theor. Biol. (1976) 63, 19-31
Structural Stability, Alternate Descriptions and Information ROBERT ROSEN Center for Theoretical Biology, State University of New York at Buffalo, N. Y. 12210, U.S.A. (Received4 March 1975, and in revisedform
4 July 1975)
Biology, and indeed any science involving a study of complexity, deals with systems which can be described in many distinct ways. The basic question then is: how do such distinct descriptions of the same system relate to each other? More specifically, we may ask when two apparently distinct descriptions of a system are equivalent: i.e. convey the same “information”. To investigate these questions, we take as the basic ingredient of system description a single system observable, which we regard as a real-valued mapping defined on the states of the system. We show how different descriptions place different topologies on the set of states, and in particular, provide us with different notions of when two states are close to each other. Given two such descriptions, we can consider the set of all states such that a perturbation which is “small” in one of the descriptions is not small in the other. The bifurcation sets considered in the theory of structural stability are shown to be simple examples of this general phenomenon arising from alternate descriptions; in our frarnework, then, a bifurcation point is one at which two descriptions are conveying essentially different information. We then proceed to regard a statistical mechanical description of a system as an alternate to a purely
microscopic description, and argue that there must, in this sense, be bifurcation points of each description with respect to the other. In a previous paper (Rosen, 1972) we considered at length some of the problems involved in determining a state description for an empirically specified system, particularly those systems arising in biology and physics. Our point of departure was the observation that such state descriptions must invariably be built out of the results of measurements, or observations, carried out on the system in question. Each such measurement process, or set of such processes, therefore provides a kind of state description of the system, and all of our information about the system comes from such descriptions. Many of the basic problems of system theory arise from the need to compare or reconcile such descriptions, obtained through different sets of observational techniques. 19
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It should be said at the outset that our emphasis on measurement requires that we shall understand the term “system” in a wide sense. A measurement process can be applied meaningfully to many different kinds of objects, which we may ordinarily regard as being states of different systems. For instance, the discipline of taxonomy is concerned with making comparisons between organisms of highly disparate characters, exactly through the fact that the same types of numerical measurements can be carried out on all of them. Thus, when we shall speak of the “set of states” of a system, we shall mean a set of elements to which a given family of measuring procedures can be applied. This usage comprises the ordinary kinds of systems found in physics, but it also includes the kind of usage found in taxonomy and comparative anatomy; in particular, it comprises the basic ingredients necessary for a discussion of biological form, whether the forms to be compared are those of a single system (such as a single developing organism at different times) or those belonging to a family of different systems (such as an evolutionary line). Following the usage prevailing in physics, we shall regard a measurement as a procedure for associating numbers with the states of a system (understood in this wide sense). Thus, an observable defined by a measuring process is simply a real-valued mapping defined on the set of states. Since these observables are the basic ingredients in all system descriptions, we shall briefly review some of their properties. Let then Z be the set of states of our system, and f: Z --t 9 be an observable. We can introduce an equivalence relation RI into X, by writing rr1Rf12 if and only iff(a,) = f(oZ). Let us denote the equivalence class of a state c under this relation by [cl. The observablef introduces a metric into the set of equivalence classes X/R,, by writing II[ar], [oZ]II = If(or)-f(oJ1. Under this metric, there is a homeomorphism between X/Rr and the set of values off, considered as a subset of the real numbers under the ordinary topology; i.e. the values off parameterize E/R/. If f is the only observable at our disposal, then we will not in fact “see” the set Z, but only the set X/Rf; it is this space, or rather its parameterization through the values off; that we will call “the” state space of the system. We will then call the elements of Z/R, by the name “states”; through the metric induced on X/R/, we will then say that two states are close if and only if their correspondingf-values are close. Exactly the same kind of argument holds if we have a family $ = {fi, fi, . . . ,f,} at our disposal; we generate an equivalence relation R, by writing aIRso if and only if fi(cr,) = fi(az) for each i = 1, . . . , n. We then parameter& the quotient space E/RI through sets of n-tuples of numbers, and impose a metric on this space which specifies that two classes Co,], [c2] are close if and only iffi(al),fi(o,) are close for each i = 1, . . . , n.
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In general, we may say that any set of observables / defined on E gives rise to (a) a parameterization of the quotient set Ii/R,, and (b) a metric imposed on C/R,, which is the metric inherent in the space of parameters. It is this space of parameters which is termed “the” state space for the system; since we generally cannot inspect the set Z itself through other observables than those available in the set f under consideration, we generally, by the familiar ubus de langage, regard the elements of Z/R, as the “states” of our system, and in particular speak of the metric in Z]Rf as if it pertained to the set C itself. What is important to recognize is that any set of observables / gives rise to a description of our system, and indeed captures some aspect of reality of the system. The problem with which we shall be concerned in the present note, and which indeed is one of the basic questions of science, is the way in which different such descriptions can be compared with each other. As we shall see, many important features of the study of form arise as necessary features of alternate descriptions, and are thus not intrinsic to the system itself. Thorn (1972) uses such ideas as a point of departure for his discussions of organic form. He considers, in the simplest case, a set d of “geometric objects” parameterized by a manifold S through a l-1 mapping q: S + 6. Here, Thorn’s set d of “geometric objects” corresponds to our set ISfRy, and the parameter set corresponds to the space of values of the observables f a f. Thorn then goes on to say that a point a E S is generic if, for all a’ sufficiently close to a, ~(a’) has “the same form” as q(a). It follows immediately that the set of all generic points is an open subset of S; the complement of this open set is the closed set of bifurcation points. Much of Thorn’s study of structural stability involves the study of the bifurcation set in particular situations. In Thorn’s geometric picture, it is clear that we are precisely in the situation of comparing two different descriptions of the elements of the set 8’ of “geometric objects”. On the one hand, we have a description arising from the parameterization q. On the other hand, we have a tacit description, summed up in the use of the words “the same form”. The determination of the “form” of a geometric object in d can only arise from an alternate description of those objects, independent of the description arising from the parameterization. Indeed, as we shall argue below, the basic ingredients of structural stability arise precisely from the comparisons of alternate descriptions of the same set of states. As a result, the concept of structural stability is a property of pairs of descriptions, and is contingent on those descriptions. Related ideas were also developed by Pattee (1974). Returning to Thorn’s picture of geometric objects and their “forms”, we see that it involves the comparison of two different metrics on 6, arising
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from the two descriptions available for the elements of 8. Under these two metrics, we have available two different and independent criteria for asserting that two objects E, E’ are close. To understand more clearly what is happening, we must put these two descriptions on the same footing. We can do this by retreating to the empirical basis for a parameterization like u, which we have outlined above. Underlying the set d of “geometric objects” is some set of states X, on which a particular family 2 of observables has been defined. The set d is then just the set of equivalence classes CfRf induced by those observables, and q is the l-l map from d onto the corresponding set S of parametric values. We thus have a diagram Z4-h j
The second description, objects in 4’ have “the observables f’ defined above argument, obtain of the form
l-l
which is tacitly embodied in the assertion that two same form”, must proceed from a second family of on b, and hence also on E. We thus, by repeating the a corresponding diagram for the second description, d-d/R,. 3’
where S’ is the corresponding Now by definition, under homeomorphic, and q is a diagrams together, to obtain
: S’ l-l
set of parameter the description homeomorphism. a diagram of the
values. induced by f, S and d are Thus, we can put the two form
S : SIR/s : S’ l-l
The mapping 0 in the above diagram is just a set-theoretic mapping, and need in general have no relation to the topologies in S and S’. It is clear that a generic point a E S is precisely one at which the composite map 4’0 is continuous; i.e. such that, whenever a’ is close to Q in the metric on S, $0(a) and q’O(u’) are close in the metric on S’. But the metric on S’ arises from those observables entering into the description of the “form” of the underlying geometric objects; thus this is the precise meaning of the assertion of genericity of an element a e S. Let us note that the mapping $8 cannot be continuous everywhere unless S and S’ are already homeomorphic. Since this will not in general be the case, there will generally be points of discontinuity of the mapping q’fX These points of discontinuity are then the bifurcation points; i.e. elements a E S such that, in any neighborhood of a, there exist points u’ for which q’O(u) and $O(u’) are not closed in S’, and so, by definition, are of different “form”.
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Let us note further that we can in effect turn the entire argument around; i.e. instead of constructing a map from S to S’, we could similarly construct a map from s’ to S. This amounts to taking the description in terms of “form” as the primary description, and measuring the description arising from the observables j against it. This would define a decomposition of S’ to that obtained into generic and bifurcation points, complementary previously for S. With these remarks as background, we shall now turn to some explicit examples of alternate descriptions of the same set of objects (which may be variously called “states” or “forms” or “patterns”), arising from particular sets of observables (or “descriptors”, or “features”), with a particular view towards seeing how the metrics imposed by these choices give rise to questions of structural stability. Let us consider some explicit examples, which should make the concepts clear. (1) Let I = W’, the Euclidean plane, considered just as an abstract set of “geometric objects”. Let f, g be the respective projections on the x and y axes. Then under f, two points (x, y), (x’, y’) are close if f(x, y) = x and fW> Y’) = x’ are close, under g, two points (x, y), (x’, y’) are close if g(x, y) = y and g(x’, y’) = y’ are close. It is clear that in this case there are no generic points. For let r E W. Then f -l(r) = the set of all points of the form (r, y) for arbitrary y. No matter how close t ’ is to P, we can find elements in f-l(r), f-l(t)) such that, for these elements gf -l(r); d-l@‘) are as far apart in 99 as we please. Hence every point in 4E is a bifurcation point under these descriptions. (2) Let 8 be the first quadrant of W. Let f(x, y) = r, g(x, y) = r be two one-parameter families of curves. Then underf, two points (x, y), (x’, y’) are close if they lie on nearby curves of the familyf; and likewise for g. Consider the two families shown in Fig. 1.
FIG.
1.
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In this case, it is easy to see that the set of generic points (for either description relative to the other) are open intervals around the integers 0, 1,2,3, . . . (3) Let us take 8, f as in (2) above, but let g(x, u) be defined as follows: dx, Y) = 1 if x < 1; g(x, y) = 0 if x 2 1. Then under f, two points are close if their x-co-ordinates are close; under g, two points are close if their values under g are the same (this is a degenerate topology). In this case there are no generic points when we use the mapping g to define the intrinsic topology on 8, whereas if we use the mapping f to define the topology, every point is generic except the point r = 1. This example is a prototype of a discrete versus a continuous classification, which is at the root of much of the difficulties in theory of pattern recognition. (4) As an extension of the preceding example, we can consider one of Thorn’s important examples, the cusp catastrophe. Here we may take the elements of d as the set of curves representing cubic equations in one variable. In appropriate co-ordinates, these curves can all be described algebraically by expressions of the form x3 +ax-t b = 0, and hence by the pair of numbers (a, b). That is, these curves can be regarded as being parameter&d by the Euclidean plane W2 via a mapping f from the set of cubic curves to W’. On the other hand these curves may also be described in terms of their intersections with the X-axis; they may have either three distinct real roots, one distinct and two repeated real roots, or one real root. Let us define a mapping x from the set of all cubic curves C into the discrete set (0, 1, 2) as follows : (a) x(C) = 0 if C has one real root; (b) x(C) =i 1 if C has a pair of repeated real roots; (c) x(C) = 2 if C has three distinct real roots. Then according to the description imposed by x, two cubic curves Ci, C, are “close” if x(C,) = x(C,). If we use the mapping x to impose the intrinsic “topology” on 8, and look at the generic points of 9?’ induced by f, we find that these comprise all points (a, b) for which 4u3+27b2 # 0; the bifurcation points lie on the cusp which is the complement of the generic set. On the other hand, if we use f to define the intrinsic topology, we find that each of the three points of (0, 1,2) is a bifurcation point. This is again typical of the situation involving a discrete versus a continuous description. (5) As a final illustration, let us consider the class 8 of all dynamical systems on some differentiable manifold M. This was the traditional setting for the concept of structural stability. Let us suppose that a dynamics on M is specified locally by the usual system of first-order differential equations ii-i
=
fAxI
9 -
*
*
9 41)s
i=
l,...,n
(1)
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in some appropriate co-ordinate system. There are two independent ways in which such a dynamical system can be described : (a) in terms of the functions (fi, * - . , f.) which specify, at each point of M, how fast each of the coordinate vectors is changing at that point; and (b) in terms of the asymptotic properties of the trajectories determined by the dynamics (1). In the first description, we can say that two dynamical systems D, = (fi, . . . ,A,), Dz = h, . . . , g,) are close when eachfi is close to the corresponding gr in some appropriate norm in function space. Traditionally, it is usual to require not only that the functions be close, but that their corresponding partial derivatives with respect to the state variables up to some order r be also close. For r = 1, we have the Cl-norm considered by Thorn: the distance between the dynamical systems D, and D,, in this description, is given by
On the other hand, in the second description, two dynamical systems D,, D, are regarded as close if corresponding trajectories in S under the dynamics D,, D2 are close for all values of t (for exact definition, see e.g. Peixoto, 1959). Thus we again have the situation we have been considering: a set of objects 8 with two different measures of closeness imposed upon it, which may be compared with each other. In the usual investigations of structural stability, the “intrinsic” topology on 8 is chosen to be that involving the closeness of corresponding trajectories, and the topology imposed by the C’-norm is compared with it, in the manner we have described. The resulting bifurcation set is what Thorn refers to as the bifurcation set. However, using the above ideas, we would find another bifurcation set by choosing the C-topology as the intrinsic one, and comparing the topology in terms of trajectories against it. In the first case, the bifurcation set lies in a space of parameters determined by the functions fi which map M into the real line; in the second case, the bifurcation set lies in a set of parameters determined by mappings of the real line into A4 (representing the corresponding trajectories). It should be noted explicitly that, by taking the C’ topology for larger and larger values of r, the topology imposed on 6’ changes; dynamical systems that are close in the Co-topology need not be close in the C’-topology for r > ro. Thus, the character of both bifurcation sets depends on t; we shall return to this point below, in our discussion of “information” in partial descriptions. Let us now return to the situation in which 6’ is described by observables. If f: I + W is any observable, then any other observable which is a
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continuous function off(e.g. the observable g = f”) induces the same metric on I thatfdoes; that is, if E, E’ e 8’ are close underf, they are close under g. In other words, every point of W is a generic point for either description with respect to the topology induced by the other. In this case, the description of 8 provided by the observable g is redundant to that supplied by f; or in other words, we obtain no new information about d by supplementing the f-description by the g-description. Now the whole point of using alternate descriptions, or the supplementing of one description by another, is to gain new information about the system being described. We thus can formulate another criterion for the equivalence of two descriptionsf: d + S, g: 4’ --t S, where S, S are arbitrary parameter sets; namely, that the bifurcation sets in S, S induced by the descriptions pairs (g, f) and (f, g) respectively shall be empty. Conversely, two such descriptions intuitively convey maximal information if their respective bifurcation sets are all of S, S respectively; i.e. if the generic sets are empty. For instance, in example (1) above, the descriptions of the elements of d = W2 imposed by the two projections f and g on the co-ordinate axes are such that every point of 92 is a bifurcation point; here it is clear that the two descriptions separately convey entirely different information regarding the elements of 8. We shall call such descriptions orthogonal. In general, given a pair of descriptions f: d + S, g: d + S, the generic points in S and S represent elements of 8’ for which the two descriptions coincide; i.e. elements at which we gain no information by using both descriptions. On the other hand, on the bifurcation points, the two descriptions become orthogonal, and we gain information about the corresponding elements of d by employing both descriptions. On the generic points of either description with respect to the other, the properties of the second description are hidden; they become manifest only on the bifurcation points. How do we go about incorporating both descriptions into a new description which refines them both? The obvious way is to take their Cartesian product, as we do with individual orthogonal observables. More precisely, given two descriptionsf: d + S, g: B + S, let us form the Cartesian product S x $ and hence describe a given element E E8 by the corresponding pair cf(E), g(E)]. The product S x Swill be given the product topology, so that two elements E, E’ are close if and only if f(E) is close to f(E’) in S and simultaneously g(E) is close to g(E) in S. The new topology imposed on Q by the product Sx S is now simultaneously a refinement of both of the original topologies (i.e. introduces many more open sets into 8). Therefore, since the original topologies on 8 imposed by S and S separately are redundant with respect to their product, it follows that every point of Sx S
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is a generic point, when referred to the intrinsic topology imposed on 8 by either S or S. (On the other hand, the imposition of the topology of S x S as the intrinsic topology of 6 leads to the same set of bifurcation points in S as if S had imposed the intrinsic topology, and conversely.) From these considerations, we obtain, as a general result, the following abstract characterization of when a descriptionfimproves a description g: Theorem: A description fi 8 + S is an improvement of a description g : 8 + T if and only if every point of S is generic with respect to the topology imposed on d by g, while the bifurcation set in T arising from the topology imposed on 8 by f is not empty.
Such improvements of description are the analogs of adding more state variables, or observables, to the description of the states of a given dynamical system. As we saw in Rosen (lot. cit.), the introduction of new observing procedures may require radical revision of our modes of system description, generating a new representation of the state space (now represented by S x S) and a new topology on that space and on the original space S. Let us briefly discuss the meaning of these results for the examples (4) and (5) given above. In the example (4), it will be recalled, the set d was the family of cubic curves in a single variable, described in two ways: (a) via the coefficients (a, b) occurring in the algebraic representation of the curve in an appropriate set of co-ordinates, and (b) by assigning the number 0, 1, or 2 to a curve, according as whether it had one real root, a pair of repeated roots, or three real roots. Thus, the combination of these two descriptions yields the Cartesian product of the Euclidean plane W2 [the space of coefficients (a, b)] with the discrete set (0, 1,2}. In this description, we associate with every cubic curve cpthe triple (a, b, z), where (a, b) are the corresponding coefficients and z represents the character of the real roots of cp (note that not every such triple corresponds to a cubic curve). Two cubic curves cp, cp’, represented by the triples (a, b, z), (a’, b’, z’), are close in d if and only if (a, b) and (a’, b’) are close in W2, and simultaneously z = z’ (the only way two elements can be close in a discrete set is for them to coincide). Thus in this new description, the cubic curves corresponding to coefficients lying on the bifurcation set (i.e. on the cusp 4u3 + 27b2 = 0) can be close only to each other, and not to any cubits determined by coefficients off the bifurcation set. In the example (5), which was that of dynamical systems, we can apply similar reasoning, by applying both the description in terms of the functions locally governing the rates of change of the state variables, and in terms of the temporal behavior of corresponding trajectories.
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We shall conclude this analysis of the nature of alternate descriptions and the problems of structural stability associated with them with some remarks regarding partial descriptions and the concept of entropy. We have already used the word information in relation to the deviation of alternate descriptions of the same object from each other. A bifurcation point of a description is precisely a point at which a description fails, in some sense, with respect to some other description; hence we feel that there must be some intuitive relation between the concept of information (or entropy) and the bifurcation set of one description with respect to another. In the remainder of the present note, we shall consider some aspects of that relation. Let us briefly review the manner in which entropy is defined in statistical mechanics. Suppose we are given a large mechanical system, with Hamiltonian H, and phase space P. One mode of description of the states of the system would involve the specification of initial values of position and momentum for each constituent particle of the system. Knowing that in practice we cannot make such a complete specification, we are led to consider instead the probability that an initial state will lie in some arbitrary subset E of P. The attaching of such a probability to each (measmable) subset E is another way of describing the initial state; thus in statistical mechanics we consider our states to be specified by such probability measures a defined on P. We usually further stipulate that such probability measures are defined in terms of an observable p. which represents a density, so that on each set E c P we can write NJ9 = j P= dp E
where p is some appropriate (usually Lebesque) measure on P. In these terms, we can define the expected value of any observable f of the system on any subset E as
E=lf& and conversely, the specification of an expected value determines a subset of P for which the above relation is satisfied. The question generally asked in statistical mechanics is the following: what is the state tip which represents the maximal “uncertainty” when we know the expected value of the energy? This state is defined by two properties: it must maximize the integral S(a) = j P,, log U/P,) dp P
and simultaneously
satisfy the auxiliary condition
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It is shown in conventional required state is defined by
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29 mechanics that the
,-H/B pa,
=
j kH,B
,j/,
b where the constant B is related to the absolute temperature; B = kT. We could, of course, introduce further auxiliary conditions to be satisfied in addition to the expected value of the energy, and for each one maximize the “uncertainty” integral with respect to it. The resultant state maximizing the “uncertainty” integral with respect to all of the constraints would be derived from a density function which is the product of the density functions obtained from each auxiliary condition considered separately. The main point to notice is that the “most probable state” obtained in this fashion is specified in terms of one or a few observables, and that on this “most probable state”, or Gibbs canonical state, we can evaluate any observable of the system. We can thus employ this “most probable state” to make predictions about the value of any observable f of the system, on the basis of the (expected) values of the observables which were used in constructing that “most probable state”. The validity of such predictions then depends explicitly on an assumption that the state description arising from f, and that used in constructing CC,,,are not orthogonal. Thus, we once again find ourselves in a situation in which we have two ways of specifying when states are “close”. They may be close in the expected values of the observables entering into the initial auxiliary conditions, or they may be close in the value of some other observable, different from those used in constructing the “most probable state”. Let then f be such an observable. If indeed it is the case that CI close to a, implies that.f(a) is close to f(cQ, then we ordinarily say that we have gained no information about the system by using the observablef. On the other hand, if f(a) is not close to f(a&, then we say that the system is “ordered” with respect to f, and that we have gained information about it, always in the conventional sense of information theory (in which “information” is equated with a deviation of an observed behavior from one posited to be “most probable”). Thus we see that the definition of entropy imposes one kind of description, or measure of “closeness” on a set of states, which may be compared with a description of the same set of states obtained by evaluating any other observable which was not used in defining the entropy. Where these two descriptions coincide, i.e. when states close in one description are close in the other, we say that there is no information. Where the two descriptions differ, we say that information has been gained; but where the two descriptions
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differ is precisely on the set of bifurcation points of the description in terms of the observable f with respect to the entropic description. It is at these points that the descriptions in fact become orthogonal. At such points in fact the description in terms of probability densities on P breaks down; it is fundamentally incomplete, and needs to be supplemented by new state variables (which are “hidden” in the states corresponding to generic points of the description). We can then redefine our states as probability distributions on the bigger state space P’ which incorporates the new state variables along with the old ones and repeat the same kind of calculation, but the meaning of the concept of entropy will be quite different in the larger space than it was in the smaller. Thus we see in another way something which we asserted elsewhere from quite a different point of view: that entropy (or information) is a contingent concept, depending on a specific set of observables chosen to characterize the states of a physical system (Rosen, 1964). From the present treatment, we also see quite clearly how the physical concept of entropy (or information) arises from the interplay of two separate descriptions of the states of a system, and how this concept is, in fact, a special corollary of the general notion of bifurcation and structural stability. [It might be added parenthetically that the discussion of entropy provided above is quite different in character from that given by Thorn (lot. cit. p. 139) for the “entropy of a form”; our present concern is not with computing values of an “entropy function” in a given situation, but in seeing how the concept itself arises from pairs of descriptions.] It should be noted in conclusion that our above treatment of entropy can be applied, mutatis mutandis, to any other situation based on incomplete descriptions. For example, Thorn (lot. cit.) discusses the description of functions in terms of their coefficients in a Taylor expansion (p. 48). Here we have, in fact, three kinds of descriptions which may be compared, and to each of which an entropy may be assigned: (a) a description in terms of an initial segment of Taylor expansion coefficients; (b) a description in terms of a larger initial segment of Taylor expansion coefficients, which may be “predicted” from the first segment just as any observable in a mechanical system can be predicted from a specification of the expected values of one or more other observables; (c) a description in terms of the values of the functions specified by Taylor expansions whose initial segments are close, for arbitrary values of the arguments. Another example involves the employment of different topologies C’, C”, r’ > r on the set of all dynamical systems, as described earlier. We shall not pursue the details of these analyses here, but simply point out here the deep analogies between all these situations.
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paper was prepared with the support of NIH Grant no. 2ROl HDO513604, no. lPO1 HD7328-01, and NASA Grant no. NGR33015002, while the was in residence at the Salk Institute for Biological Studies. The hospitality Institute is gratefully acknowledged. REFERENCES (1974). Preprint for CSDI Conference on Structuralisms.
PATTEE, H. PEIXOTO, M. M. (1959). Ann. Math. 69, 199. ROSEN, R. (1964). Philosophy of Science 31, 232. ROSEN, R. (1972). Znt. J. Syst. Sci. 4, 65. THOM, R. (1972). Stabilite Structurelle et Morphogenese.
Inc.
Reading,Mass.: W. A. Benjamin,
Structural Stability, Alternate Descriptions and Information ROBERT ROSEN Center for Theoretical Biology, State University of New York at Buffalo, N. Y. 12210, U.S.A. (Received4 March 1975, and in revisedform
4 July 1975)
Biology, and indeed any science involving a study of complexity, deals with systems which can be described in many distinct ways. The basic question then is: how do such distinct descriptions of the same system relate to each other? More specifically, we may ask when two apparently distinct descriptions of a system are equivalent: i.e. convey the same “information”. To investigate these questions, we take as the basic ingredient of system description a single system observable, which we regard as a real-valued mapping defined on the states of the system. We show how different descriptions place different topologies on the set of states, and in particular, provide us with different notions of when two states are close to each other. Given two such descriptions, we can consider the set of all states such that a perturbation which is “small” in one of the descriptions is not small in the other. The bifurcation sets considered in the theory of structural stability are shown to be simple examples of this general phenomenon arising from alternate descriptions; in our frarnework, then, a bifurcation point is one at which two descriptions are conveying essentially different information. We then proceed to regard a statistical mechanical description of a system as an alternate to a purely
microscopic description, and argue that there must, in this sense, be bifurcation points of each description with respect to the other. In a previous paper (Rosen, 1972) we considered at length some of the problems involved in determining a state description for an empirically specified system, particularly those systems arising in biology and physics. Our point of departure was the observation that such state descriptions must invariably be built out of the results of measurements, or observations, carried out on the system in question. Each such measurement process, or set of such processes, therefore provides a kind of state description of the system, and all of our information about the system comes from such descriptions. Many of the basic problems of system theory arise from the need to compare or reconcile such descriptions, obtained through different sets of observational techniques. 19
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It should be said at the outset that our emphasis on measurement requires that we shall understand the term “system” in a wide sense. A measurement process can be applied meaningfully to many different kinds of objects, which we may ordinarily regard as being states of different systems. For instance, the discipline of taxonomy is concerned with making comparisons between organisms of highly disparate characters, exactly through the fact that the same types of numerical measurements can be carried out on all of them. Thus, when we shall speak of the “set of states” of a system, we shall mean a set of elements to which a given family of measuring procedures can be applied. This usage comprises the ordinary kinds of systems found in physics, but it also includes the kind of usage found in taxonomy and comparative anatomy; in particular, it comprises the basic ingredients necessary for a discussion of biological form, whether the forms to be compared are those of a single system (such as a single developing organism at different times) or those belonging to a family of different systems (such as an evolutionary line). Following the usage prevailing in physics, we shall regard a measurement as a procedure for associating numbers with the states of a system (understood in this wide sense). Thus, an observable defined by a measuring process is simply a real-valued mapping defined on the set of states. Since these observables are the basic ingredients in all system descriptions, we shall briefly review some of their properties. Let then Z be the set of states of our system, and f: Z --t 9 be an observable. We can introduce an equivalence relation RI into X, by writing rr1Rf12 if and only iff(a,) = f(oZ). Let us denote the equivalence class of a state c under this relation by [cl. The observablef introduces a metric into the set of equivalence classes X/R,, by writing II[ar], [oZ]II = If(or)-f(oJ1. Under this metric, there is a homeomorphism between X/Rr and the set of values off, considered as a subset of the real numbers under the ordinary topology; i.e. the values off parameterize E/R/. If f is the only observable at our disposal, then we will not in fact “see” the set Z, but only the set X/Rf; it is this space, or rather its parameterization through the values off; that we will call “the” state space of the system. We will then call the elements of Z/R, by the name “states”; through the metric induced on X/R/, we will then say that two states are close if and only if their correspondingf-values are close. Exactly the same kind of argument holds if we have a family $ = {fi, fi, . . . ,f,} at our disposal; we generate an equivalence relation R, by writing aIRso if and only if fi(cr,) = fi(az) for each i = 1, . . . , n. We then parameter& the quotient space E/RI through sets of n-tuples of numbers, and impose a metric on this space which specifies that two classes Co,], [c2] are close if and only iffi(al),fi(o,) are close for each i = 1, . . . , n.
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In general, we may say that any set of observables / defined on E gives rise to (a) a parameterization of the quotient set Ii/R,, and (b) a metric imposed on C/R,, which is the metric inherent in the space of parameters. It is this space of parameters which is termed “the” state space for the system; since we generally cannot inspect the set Z itself through other observables than those available in the set f under consideration, we generally, by the familiar ubus de langage, regard the elements of Z/R, as the “states” of our system, and in particular speak of the metric in Z]Rf as if it pertained to the set C itself. What is important to recognize is that any set of observables / gives rise to a description of our system, and indeed captures some aspect of reality of the system. The problem with which we shall be concerned in the present note, and which indeed is one of the basic questions of science, is the way in which different such descriptions can be compared with each other. As we shall see, many important features of the study of form arise as necessary features of alternate descriptions, and are thus not intrinsic to the system itself. Thorn (1972) uses such ideas as a point of departure for his discussions of organic form. He considers, in the simplest case, a set d of “geometric objects” parameterized by a manifold S through a l-1 mapping q: S + 6. Here, Thorn’s set d of “geometric objects” corresponds to our set ISfRy, and the parameter set corresponds to the space of values of the observables f a f. Thorn then goes on to say that a point a E S is generic if, for all a’ sufficiently close to a, ~(a’) has “the same form” as q(a). It follows immediately that the set of all generic points is an open subset of S; the complement of this open set is the closed set of bifurcation points. Much of Thorn’s study of structural stability involves the study of the bifurcation set in particular situations. In Thorn’s geometric picture, it is clear that we are precisely in the situation of comparing two different descriptions of the elements of the set 8’ of “geometric objects”. On the one hand, we have a description arising from the parameterization q. On the other hand, we have a tacit description, summed up in the use of the words “the same form”. The determination of the “form” of a geometric object in d can only arise from an alternate description of those objects, independent of the description arising from the parameterization. Indeed, as we shall argue below, the basic ingredients of structural stability arise precisely from the comparisons of alternate descriptions of the same set of states. As a result, the concept of structural stability is a property of pairs of descriptions, and is contingent on those descriptions. Related ideas were also developed by Pattee (1974). Returning to Thorn’s picture of geometric objects and their “forms”, we see that it involves the comparison of two different metrics on 6, arising
22
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from the two descriptions available for the elements of 8. Under these two metrics, we have available two different and independent criteria for asserting that two objects E, E’ are close. To understand more clearly what is happening, we must put these two descriptions on the same footing. We can do this by retreating to the empirical basis for a parameterization like u, which we have outlined above. Underlying the set d of “geometric objects” is some set of states X, on which a particular family 2 of observables has been defined. The set d is then just the set of equivalence classes CfRf induced by those observables, and q is the l-l map from d onto the corresponding set S of parametric values. We thus have a diagram Z4-h j
The second description, objects in 4’ have “the observables f’ defined above argument, obtain of the form
l-l
which is tacitly embodied in the assertion that two same form”, must proceed from a second family of on b, and hence also on E. We thus, by repeating the a corresponding diagram for the second description, d-d/R,. 3’
where S’ is the corresponding Now by definition, under homeomorphic, and q is a diagrams together, to obtain
: S’ l-l
set of parameter the description homeomorphism. a diagram of the
values. induced by f, S and d are Thus, we can put the two form
S : SIR/s : S’ l-l
The mapping 0 in the above diagram is just a set-theoretic mapping, and need in general have no relation to the topologies in S and S’. It is clear that a generic point a E S is precisely one at which the composite map 4’0 is continuous; i.e. such that, whenever a’ is close to Q in the metric on S, $0(a) and q’O(u’) are close in the metric on S’. But the metric on S’ arises from those observables entering into the description of the “form” of the underlying geometric objects; thus this is the precise meaning of the assertion of genericity of an element a e S. Let us note that the mapping $8 cannot be continuous everywhere unless S and S’ are already homeomorphic. Since this will not in general be the case, there will generally be points of discontinuity of the mapping q’fX These points of discontinuity are then the bifurcation points; i.e. elements a E S such that, in any neighborhood of a, there exist points u’ for which q’O(u) and $O(u’) are not closed in S’, and so, by definition, are of different “form”.
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Let us note further that we can in effect turn the entire argument around; i.e. instead of constructing a map from S to S’, we could similarly construct a map from s’ to S. This amounts to taking the description in terms of “form” as the primary description, and measuring the description arising from the observables j against it. This would define a decomposition of S’ to that obtained into generic and bifurcation points, complementary previously for S. With these remarks as background, we shall now turn to some explicit examples of alternate descriptions of the same set of objects (which may be variously called “states” or “forms” or “patterns”), arising from particular sets of observables (or “descriptors”, or “features”), with a particular view towards seeing how the metrics imposed by these choices give rise to questions of structural stability. Let us consider some explicit examples, which should make the concepts clear. (1) Let I = W’, the Euclidean plane, considered just as an abstract set of “geometric objects”. Let f, g be the respective projections on the x and y axes. Then under f, two points (x, y), (x’, y’) are close if f(x, y) = x and fW> Y’) = x’ are close, under g, two points (x, y), (x’, y’) are close if g(x, y) = y and g(x’, y’) = y’ are close. It is clear that in this case there are no generic points. For let r E W. Then f -l(r) = the set of all points of the form (r, y) for arbitrary y. No matter how close t ’ is to P, we can find elements in f-l(r), f-l(t)) such that, for these elements gf -l(r); d-l@‘) are as far apart in 99 as we please. Hence every point in 4E is a bifurcation point under these descriptions. (2) Let 8 be the first quadrant of W. Let f(x, y) = r, g(x, y) = r be two one-parameter families of curves. Then underf, two points (x, y), (x’, y’) are close if they lie on nearby curves of the familyf; and likewise for g. Consider the two families shown in Fig. 1.
FIG.
1.
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In this case, it is easy to see that the set of generic points (for either description relative to the other) are open intervals around the integers 0, 1,2,3, . . . (3) Let us take 8, f as in (2) above, but let g(x, u) be defined as follows: dx, Y) = 1 if x < 1; g(x, y) = 0 if x 2 1. Then under f, two points are close if their x-co-ordinates are close; under g, two points are close if their values under g are the same (this is a degenerate topology). In this case there are no generic points when we use the mapping g to define the intrinsic topology on 8, whereas if we use the mapping f to define the topology, every point is generic except the point r = 1. This example is a prototype of a discrete versus a continuous classification, which is at the root of much of the difficulties in theory of pattern recognition. (4) As an extension of the preceding example, we can consider one of Thorn’s important examples, the cusp catastrophe. Here we may take the elements of d as the set of curves representing cubic equations in one variable. In appropriate co-ordinates, these curves can all be described algebraically by expressions of the form x3 +ax-t b = 0, and hence by the pair of numbers (a, b). That is, these curves can be regarded as being parameter&d by the Euclidean plane W2 via a mapping f from the set of cubic curves to W’. On the other hand these curves may also be described in terms of their intersections with the X-axis; they may have either three distinct real roots, one distinct and two repeated real roots, or one real root. Let us define a mapping x from the set of all cubic curves C into the discrete set (0, 1, 2) as follows : (a) x(C) = 0 if C has one real root; (b) x(C) =i 1 if C has a pair of repeated real roots; (c) x(C) = 2 if C has three distinct real roots. Then according to the description imposed by x, two cubic curves Ci, C, are “close” if x(C,) = x(C,). If we use the mapping x to impose the intrinsic “topology” on 8, and look at the generic points of 9?’ induced by f, we find that these comprise all points (a, b) for which 4u3+27b2 # 0; the bifurcation points lie on the cusp which is the complement of the generic set. On the other hand, if we use f to define the intrinsic topology, we find that each of the three points of (0, 1,2) is a bifurcation point. This is again typical of the situation involving a discrete versus a continuous description. (5) As a final illustration, let us consider the class 8 of all dynamical systems on some differentiable manifold M. This was the traditional setting for the concept of structural stability. Let us suppose that a dynamics on M is specified locally by the usual system of first-order differential equations ii-i
=
fAxI
9 -
*
*
9 41)s
i=
l,...,n
(1)
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in some appropriate co-ordinate system. There are two independent ways in which such a dynamical system can be described : (a) in terms of the functions (fi, * - . , f.) which specify, at each point of M, how fast each of the coordinate vectors is changing at that point; and (b) in terms of the asymptotic properties of the trajectories determined by the dynamics (1). In the first description, we can say that two dynamical systems D, = (fi, . . . ,A,), Dz = h, . . . , g,) are close when eachfi is close to the corresponding gr in some appropriate norm in function space. Traditionally, it is usual to require not only that the functions be close, but that their corresponding partial derivatives with respect to the state variables up to some order r be also close. For r = 1, we have the Cl-norm considered by Thorn: the distance between the dynamical systems D, and D,, in this description, is given by
On the other hand, in the second description, two dynamical systems D,, D, are regarded as close if corresponding trajectories in S under the dynamics D,, D2 are close for all values of t (for exact definition, see e.g. Peixoto, 1959). Thus we again have the situation we have been considering: a set of objects 8 with two different measures of closeness imposed upon it, which may be compared with each other. In the usual investigations of structural stability, the “intrinsic” topology on 8 is chosen to be that involving the closeness of corresponding trajectories, and the topology imposed by the C’-norm is compared with it, in the manner we have described. The resulting bifurcation set is what Thorn refers to as the bifurcation set. However, using the above ideas, we would find another bifurcation set by choosing the C-topology as the intrinsic one, and comparing the topology in terms of trajectories against it. In the first case, the bifurcation set lies in a space of parameters determined by the functions fi which map M into the real line; in the second case, the bifurcation set lies in a set of parameters determined by mappings of the real line into A4 (representing the corresponding trajectories). It should be noted explicitly that, by taking the C’ topology for larger and larger values of r, the topology imposed on 6’ changes; dynamical systems that are close in the Co-topology need not be close in the C’-topology for r > ro. Thus, the character of both bifurcation sets depends on t; we shall return to this point below, in our discussion of “information” in partial descriptions. Let us now return to the situation in which 6’ is described by observables. If f: I + W is any observable, then any other observable which is a
26
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continuous function off(e.g. the observable g = f”) induces the same metric on I thatfdoes; that is, if E, E’ e 8’ are close underf, they are close under g. In other words, every point of W is a generic point for either description with respect to the topology induced by the other. In this case, the description of 8 provided by the observable g is redundant to that supplied by f; or in other words, we obtain no new information about d by supplementing the f-description by the g-description. Now the whole point of using alternate descriptions, or the supplementing of one description by another, is to gain new information about the system being described. We thus can formulate another criterion for the equivalence of two descriptionsf: d + S, g: 4’ --t S, where S, S are arbitrary parameter sets; namely, that the bifurcation sets in S, S induced by the descriptions pairs (g, f) and (f, g) respectively shall be empty. Conversely, two such descriptions intuitively convey maximal information if their respective bifurcation sets are all of S, S respectively; i.e. if the generic sets are empty. For instance, in example (1) above, the descriptions of the elements of d = W2 imposed by the two projections f and g on the co-ordinate axes are such that every point of 92 is a bifurcation point; here it is clear that the two descriptions separately convey entirely different information regarding the elements of 8. We shall call such descriptions orthogonal. In general, given a pair of descriptions f: d + S, g: d + S, the generic points in S and S represent elements of 8’ for which the two descriptions coincide; i.e. elements at which we gain no information by using both descriptions. On the other hand, on the bifurcation points, the two descriptions become orthogonal, and we gain information about the corresponding elements of d by employing both descriptions. On the generic points of either description with respect to the other, the properties of the second description are hidden; they become manifest only on the bifurcation points. How do we go about incorporating both descriptions into a new description which refines them both? The obvious way is to take their Cartesian product, as we do with individual orthogonal observables. More precisely, given two descriptionsf: d + S, g: B + S, let us form the Cartesian product S x $ and hence describe a given element E E8 by the corresponding pair cf(E), g(E)]. The product S x Swill be given the product topology, so that two elements E, E’ are close if and only if f(E) is close to f(E’) in S and simultaneously g(E) is close to g(E) in S. The new topology imposed on Q by the product Sx S is now simultaneously a refinement of both of the original topologies (i.e. introduces many more open sets into 8). Therefore, since the original topologies on 8 imposed by S and S separately are redundant with respect to their product, it follows that every point of Sx S
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is a generic point, when referred to the intrinsic topology imposed on 8 by either S or S. (On the other hand, the imposition of the topology of S x S as the intrinsic topology of 6 leads to the same set of bifurcation points in S as if S had imposed the intrinsic topology, and conversely.) From these considerations, we obtain, as a general result, the following abstract characterization of when a descriptionfimproves a description g: Theorem: A description fi 8 + S is an improvement of a description g : 8 + T if and only if every point of S is generic with respect to the topology imposed on d by g, while the bifurcation set in T arising from the topology imposed on 8 by f is not empty.
Such improvements of description are the analogs of adding more state variables, or observables, to the description of the states of a given dynamical system. As we saw in Rosen (lot. cit.), the introduction of new observing procedures may require radical revision of our modes of system description, generating a new representation of the state space (now represented by S x S) and a new topology on that space and on the original space S. Let us briefly discuss the meaning of these results for the examples (4) and (5) given above. In the example (4), it will be recalled, the set d was the family of cubic curves in a single variable, described in two ways: (a) via the coefficients (a, b) occurring in the algebraic representation of the curve in an appropriate set of co-ordinates, and (b) by assigning the number 0, 1, or 2 to a curve, according as whether it had one real root, a pair of repeated roots, or three real roots. Thus, the combination of these two descriptions yields the Cartesian product of the Euclidean plane W2 [the space of coefficients (a, b)] with the discrete set (0, 1,2}. In this description, we associate with every cubic curve cpthe triple (a, b, z), where (a, b) are the corresponding coefficients and z represents the character of the real roots of cp (note that not every such triple corresponds to a cubic curve). Two cubic curves cp, cp’, represented by the triples (a, b, z), (a’, b’, z’), are close in d if and only if (a, b) and (a’, b’) are close in W2, and simultaneously z = z’ (the only way two elements can be close in a discrete set is for them to coincide). Thus in this new description, the cubic curves corresponding to coefficients lying on the bifurcation set (i.e. on the cusp 4u3 + 27b2 = 0) can be close only to each other, and not to any cubits determined by coefficients off the bifurcation set. In the example (5), which was that of dynamical systems, we can apply similar reasoning, by applying both the description in terms of the functions locally governing the rates of change of the state variables, and in terms of the temporal behavior of corresponding trajectories.
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We shall conclude this analysis of the nature of alternate descriptions and the problems of structural stability associated with them with some remarks regarding partial descriptions and the concept of entropy. We have already used the word information in relation to the deviation of alternate descriptions of the same object from each other. A bifurcation point of a description is precisely a point at which a description fails, in some sense, with respect to some other description; hence we feel that there must be some intuitive relation between the concept of information (or entropy) and the bifurcation set of one description with respect to another. In the remainder of the present note, we shall consider some aspects of that relation. Let us briefly review the manner in which entropy is defined in statistical mechanics. Suppose we are given a large mechanical system, with Hamiltonian H, and phase space P. One mode of description of the states of the system would involve the specification of initial values of position and momentum for each constituent particle of the system. Knowing that in practice we cannot make such a complete specification, we are led to consider instead the probability that an initial state will lie in some arbitrary subset E of P. The attaching of such a probability to each (measmable) subset E is another way of describing the initial state; thus in statistical mechanics we consider our states to be specified by such probability measures a defined on P. We usually further stipulate that such probability measures are defined in terms of an observable p. which represents a density, so that on each set E c P we can write NJ9 = j P= dp E
where p is some appropriate (usually Lebesque) measure on P. In these terms, we can define the expected value of any observable f of the system on any subset E as
E=lf& and conversely, the specification of an expected value determines a subset of P for which the above relation is satisfied. The question generally asked in statistical mechanics is the following: what is the state tip which represents the maximal “uncertainty” when we know the expected value of the energy? This state is defined by two properties: it must maximize the integral S(a) = j P,, log U/P,) dp P
and simultaneously
satisfy the auxiliary condition
STRUCTURE,
It is shown in conventional required state is defined by
STABILITY
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treatments
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29 mechanics that the
,-H/B pa,
=
j kH,B
,j/,
b where the constant B is related to the absolute temperature; B = kT. We could, of course, introduce further auxiliary conditions to be satisfied in addition to the expected value of the energy, and for each one maximize the “uncertainty” integral with respect to it. The resultant state maximizing the “uncertainty” integral with respect to all of the constraints would be derived from a density function which is the product of the density functions obtained from each auxiliary condition considered separately. The main point to notice is that the “most probable state” obtained in this fashion is specified in terms of one or a few observables, and that on this “most probable state”, or Gibbs canonical state, we can evaluate any observable of the system. We can thus employ this “most probable state” to make predictions about the value of any observable f of the system, on the basis of the (expected) values of the observables which were used in constructing that “most probable state”. The validity of such predictions then depends explicitly on an assumption that the state description arising from f, and that used in constructing CC,,,are not orthogonal. Thus, we once again find ourselves in a situation in which we have two ways of specifying when states are “close”. They may be close in the expected values of the observables entering into the initial auxiliary conditions, or they may be close in the value of some other observable, different from those used in constructing the “most probable state”. Let then f be such an observable. If indeed it is the case that CI close to a, implies that.f(a) is close to f(cQ, then we ordinarily say that we have gained no information about the system by using the observablef. On the other hand, if f(a) is not close to f(a&, then we say that the system is “ordered” with respect to f, and that we have gained information about it, always in the conventional sense of information theory (in which “information” is equated with a deviation of an observed behavior from one posited to be “most probable”). Thus we see that the definition of entropy imposes one kind of description, or measure of “closeness” on a set of states, which may be compared with a description of the same set of states obtained by evaluating any other observable which was not used in defining the entropy. Where these two descriptions coincide, i.e. when states close in one description are close in the other, we say that there is no information. Where the two descriptions differ, we say that information has been gained; but where the two descriptions
30
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differ is precisely on the set of bifurcation points of the description in terms of the observable f with respect to the entropic description. It is at these points that the descriptions in fact become orthogonal. At such points in fact the description in terms of probability densities on P breaks down; it is fundamentally incomplete, and needs to be supplemented by new state variables (which are “hidden” in the states corresponding to generic points of the description). We can then redefine our states as probability distributions on the bigger state space P’ which incorporates the new state variables along with the old ones and repeat the same kind of calculation, but the meaning of the concept of entropy will be quite different in the larger space than it was in the smaller. Thus we see in another way something which we asserted elsewhere from quite a different point of view: that entropy (or information) is a contingent concept, depending on a specific set of observables chosen to characterize the states of a physical system (Rosen, 1964). From the present treatment, we also see quite clearly how the physical concept of entropy (or information) arises from the interplay of two separate descriptions of the states of a system, and how this concept is, in fact, a special corollary of the general notion of bifurcation and structural stability. [It might be added parenthetically that the discussion of entropy provided above is quite different in character from that given by Thorn (lot. cit. p. 139) for the “entropy of a form”; our present concern is not with computing values of an “entropy function” in a given situation, but in seeing how the concept itself arises from pairs of descriptions.] It should be noted in conclusion that our above treatment of entropy can be applied, mutatis mutandis, to any other situation based on incomplete descriptions. For example, Thorn (lot. cit.) discusses the description of functions in terms of their coefficients in a Taylor expansion (p. 48). Here we have, in fact, three kinds of descriptions which may be compared, and to each of which an entropy may be assigned: (a) a description in terms of an initial segment of Taylor expansion coefficients; (b) a description in terms of a larger initial segment of Taylor expansion coefficients, which may be “predicted” from the first segment just as any observable in a mechanical system can be predicted from a specification of the expected values of one or more other observables; (c) a description in terms of the values of the functions specified by Taylor expansions whose initial segments are close, for arbitrary values of the arguments. Another example involves the employment of different topologies C’, C”, r’ > r on the set of all dynamical systems, as described earlier. We shall not pursue the details of these analyses here, but simply point out here the deep analogies between all these situations.
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paper was prepared with the support of NIH Grant no. 2ROl HDO513604, no. lPO1 HD7328-01, and NASA Grant no. NGR33015002, while the was in residence at the Salk Institute for Biological Studies. The hospitality Institute is gratefully acknowledged. REFERENCES (1974). Preprint for CSDI Conference on Structuralisms.
PATTEE, H. PEIXOTO, M. M. (1959). Ann. Math. 69, 199. ROSEN, R. (1964). Philosophy of Science 31, 232. ROSEN, R. (1972). Znt. J. Syst. Sci. 4, 65. THOM, R. (1972). Stabilite Structurelle et Morphogenese.
Inc.
Reading,Mass.: W. A. Benjamin,
