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Biological adaptabilities and quantum entropies
BioSystems 64 (2002) 33 – 41 www.elsevier.com/locate/biosystems
Biological adaptabilities and quantum entropies Kevin G. Kirby Department of Mathematics and Computer Science, Northern Kentucky Uni6ersity, Highland Heights, KY 41099, USA Received 7 May 2001; received in revised form 25 July 2001; accepted 14 August 2001
Abstract The entropy-based theory of adaptability set forth by Michael Conrad in the early 1970s continued to appear in his work for over two decades, and was the subject of the only book he published in his lifetime. He applied this theory to a host of subjects ranging from enzyme dynamics to sociology. This paper reviews the formalism of adaptability theory, clarifying some of its mathematical and interpretive difficulties. The theory frames the computational tradeoff principle, a thesis that was the most frequently recurring claim in his work. The formulation of adaptability theory presented here allows the introduction of quantum entropy functions into the theory, revealing an interesting relationship between adaptability and another one of Conrad’s deep preoccupations, the role of quantum processes in life. © 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Adaptability; Entropy; Quantum information; Supercorrelation; Bioinformational science
1. Introduction The development and application of an entropybased theory of adaptability was a recurring element in the work of Michael Conrad for at least 23 years, and was the subject of Adaptability, the only book he published in his lifetime (Conrad, 1983). The formalism became the lingua franca connecting his various interests, and he used it to analyze everything from mutation buffering in enzymes (Conrad, 1979) to a society’s response to computer technology (Conrad, 1993). In presentations of various and sundry topics in papers from 1972 through 1995, the ubiquitous ‘H( *)’ — the marginal entropy that became the signature of his theory —was sure to make an appearance. E-mail address: [email protected] (K.G. Kirby).
While many of Conrad’s themes and ideas had widespread influence, the adaptability formalism itself was apparently not contagious. This may be due to the complexities of the formalism, the bewildering variety of reformulations of it even within the confines of his book, and the difficulties in what he himself would call the ‘epistemological status’ of the equations. Nevertheless, he stood by the formalism for over two decades. It is the thesis of this paper that the unusual use to which he put Shannon’s entropy-based information theory can indeed be made approachable, and, moreover, it reveals links to contemporary quantum information theory. Conrad (1972, 1983) derived his scheme from Khinchin’s presentation of Shannon’s theory, an approach based on the entropy of transitions in a
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Markov process (Shannon, 1948; Khinchin, 1957; modern treatment is Cover and Thomas, 1991). In the next section, I will explain Conrad’s formalism, modernizing the notation a bit and streamlining the definitions. In Section 3, I will review the ways the formalism has been interpreted and applied, showing in particular its connection to the computational tradeoff principle (Conrad, 1988). In Section 4, I will move to a new use of adaptability theory, relating it to quantum information theory and tying it in to one of the other great themes of his work, the role of quantum processes in living systems. I conclude by discussing the relationship between adaptability theory and the fluctuon theory.
used here is chosen to anticipate the theory’s extension into quantum density matrices later in this paper). If the system is in state qj at time t, the uncertainty about what state it will be in at time t+ 1 is given by hj (M) = − %mij log mij
This is the Shannon entropy of the jth column of M, viewed as a probability density vector. If at time t we do not know for sure which state qj the system is in, but only know the state density p(t), then we can only speak of the expected uncertainty as to what happens next: H(M)(p) = %pj hj (M) h(M)Tp
2. The ecosystem entropy formalism Conrad (1983) framed adaptability theory within a formal model of ecosystems based on Markov processes. The scene for the theory is a very large but finite state space Q = {q1, q2, …, qn }. At a loss to describe the unknowably complex dynamics on Q, we turn to the use of a state density 6ector p(t), each component pj (t) representing the probability that the system is in state qj at time t. Everything we know about the ecosystem’s dynamics is encapsulated in a matrix M whose columns sum to 1 and whose entries mij represent conditional probabilities: mij =Pr[system is in state qi at time t +1 system was in state qj at time t] The state density vector evolves in discrete time by a linear transformation: p(t +1)=Mp(t) Since the columns of M sum to 1, trajectories are confined to the probability simplex
!
Dn − 1 = pDn pi ]0 and %i pi =1
"
Adopting a helpful abuse of language, we can say p(t) itself is the state of the system. If p(t) is at one of the vertices of the simplex (i.e. one component is 1 and the others are 0), it is said to be a pure state; otherwise, it is mixed (the language
(1)
i
(2)
j
Thus, associated with each Markov process M we have a linear functional H(M) = h(M)T defined on the probability simplex, which tells us the ‘uncertainty about what happens next’ for each state density vector p. In the context of information theory, p lists the probabilities of emission of n different symbols by a source. The application of one step in the dynamics (multiplication by M) would represent a ‘data processing step.’ Importantly, for the purposes of information theory it is assumed that this Markov process is stationary, meaning there is a state density vector p* that is a fixed point of M, i.e. Mp* = p*. We define the number H(M) = H(M)(p*)
(3)
This is the ‘entropy rate’ of the Markov process M. Taking this as a starting point, Conrad decomposed Q into a Cartesian product of biota and en6ironment states: Q=B× E. The term biota is more general than ‘organism’ (we could be talking about anything from an organelle to a society), yet not so all-encompassing as ‘biological system’ (which might include an environment). As the environment itself may be partly living, the term biota ultimately denotes ‘the living system of interest.’ In this scheme, each ecosystem state qi becomes a pair (i, m)B×E. Following Conrad’s lead, we
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will represent the Markov process matrix here by V. Rather than keeping track of matrix subscripts, it is easier to write V as a map V: (B × E)×(B× E)[0, 1]. It brings out the sense of this expression as a conditional probability to write V(i%, m%, i, m) as V(i%, m% i, m). The state density vector can similarly be viewed as a map p: B ×E [0,1]. Despite this departure from matrix notation, it should be remembered that V describes the linear time evolution of the state density vector p. Factoring the state set into ‘biota’ and ‘environment’ parts allows a variety of conditional entropies to be defined, which leads ultimately to the definition of adaptability. Eqs. (1) and (2) can be combined and expressed in this biota × en6ironment notation as:
Conrad informally used the symbols and * to denote the biota and environment transition schemes, but these schemes are not derivable from the scheme V unless a specific state density vector p is given as well. A better notation for H( ) and H( *) might have been HB (V) and HE (V), respectively, but the former terms and their relatives are so widespread in his work that it would not be helpful to change them at this point. Third, we have the uncertainty function of the biota transition given the environment (writing functional equations now):
H(V)(p)=% % p(i, m)%%ent V(i%, m% i, m)
Finally, we have the mutual information function between the biota and environment:
i m
(4)
i% m%
where we make the space-saving abbreviation ent x = − x log x, with the usual special case that ent 0 =0. As in Eq. (2), H(V) is a linear functional, not a number. In Conrad’s terms it gives the ‘entropy of the ecosystem transition scheme’ (Conrad, 1983). More precisely, it is a function that takes a state density p and tells us the uncertainty associated with the next state change. From this definition (Eq. (4)) we can derive five related entropies. Their definitions can be streamlined with the following abbreviations, which are motivated by basic identities: V(i% i, m)=%V(i%, m% i, m)
H( *)= H(V) −H( *)
(7)
Fourth, we have the uncertainty function of the environment transition given the biota: H( * )= H(V) − H( )
H( : *)= H( )+H( *)− H(V)
(8)
(9)
All of the entropies defined here (Eqs. (4)–(9)) are functions that map probability state vectors to the non-negati6e real numbers. A value of 0 is interpreted as certainty; there is no such thing as ‘negative uncertainty’ here. Fig. 1 summarizes the relationships between the entropies. These equations and this figure must be interpreted with great care. Many information theory texts feature a similar Venn diagram-like picture, and list similar equations, but the quantities represented in those texts are numbers, show-
m%
V(m% i, m)=%V(i%, m% i, m) i%
First, we have the ‘marginal entropy for the behavior of the biota’: H( )(p)= % % p(i, m)% ent V(i% i, m) i m
(5)
i%
Second, we have the ‘marginal entropy for the behavior of the environment’: H( *)(p)=% %p(i, m)% ent V(m% i, m) i m
m%
(6)
Fig. 1. Relations between the entropy functions associated with the Markov process model of an ecosystem.
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K.G. Kirby / BioSystems 64 (2002) 33–41
ing the entropy relationships between two random variables X and Y obtained from a joint distribution (X, Y). In contrast, the H(…) quantities defined in the Conrad theory and shown in Fig. 1 are functions (of state probability vectors p), not numbers. Further, they are properties of Markov processes, i.e. they deal with random transitions and random states, not ordinary random variables. At this point, the formulation of the Conrad adaptability theory is almost complete. The final step is the definition of adaptability itself. Quoting verbatim from the Adaptability book, ‘‘the adaptability of the biota is given by: H( ˆ *) max * [H( *) such that A]
(10)
where H( *) is a maximum over all possible transition schemes subject to condition A. This condition is: the half-life of the biota is not decreased at all’’ (Conrad, 1983). A casual gloss on this is easy: the adaptability of a living system is defined as the uncertainty of the most uncertain environment in which it can survive. Yet it was not this casual gloss to which Conrad made continual reference over the years. He made use of the entropic quantities in the formalism itself. Three difficulties arise here. We will first pose them, then provide some interpretations and ‘workarounds’ in the next section. First, the maximum in Eq. (10) implies that we can make sense of inequalities across the H( *)’s. But these are functions, not numbers. Now we can certainly have function inequalities, whereby fB g means f(x)Bg(x) for all x, and so on, but one must now be careful because the negation of f Bg is no longer f] g, for example. Yet we find Conrad freely using the latter kind of inferences between these entropies. This problem does not come up in information theory (Khinchin, 1957; Cover and Thomas, 1991), where one assumes the Markov process is stationary and one converts these functions to numbers by evaluating them at the welldefined stationary density, as we did in Eq. (3). By contrast, in adaptability theory ‘no assumption about stationarity is made’ (Conrad, 1983) so this move is not open to us. (Despite this statement, the presentation in Adaptability did not identify them as functions, which added to the difficulty of the presentation).
Second, the definition of adaptability implies the definability of an entity denoted by ˆ *, ‘the most uncertain environment the biota can tolerate,’ whose entropy is given by Eq. (10). But it is not clear that such a system can exist within this formalism. The maximum of a set of entropy functions of Markov systems may not be an entropy function of a Markov system. (Compare this to a more elementary situation. If one defines k= max{ f, g} where f and g are quadratic functions, k is usually not a quadratic function itself). Third, the ‘condition’ A seems to be a predicate on the biota state set B that must be defined supplementary to the transition scheme. So this suggests the ecosystem is actually a pair (V, A). What are the constraints on this predicate? Does any predicate A on B result in a formal ecosystem? That would seem to evacuate the meaning from the entire formalism.
3. The lessons of adaptability The three questions posed above are best tackled together. The path to the answers is cleared by answering another question first: did Conrad think this Markov process scheme was a model of an ecosystem? Emphatically, no. He was the creator of some of the earliest ecosystem simulations (Conrad, 1969), simulations that might now be seen as examples of ‘artificial life’ programs (two decades ahead of their time). Until that work appeared, ecosystem simulation usually involved numerical integration of sets of differential-difference equations. Conrad’s approach in 1969, by contrast, was quite ‘modern.’ It employed algorithms operating on discrete data structures, a rough precursor of an object-oriented approach, in which objects in the program represented objects in the virtual world. For someone who had pioneered such an approach, modeling an ecosystem with a Markov process would be reactionary and unilluminating. Then what are all those omegas? In the Adaptability book we find, 120 pages after the original definitions, a tantalizingly odd explanation. We read that, despite the mathematical setup which begins with the Markov process and defines the
K.G. Kirby / BioSystems 64 (2002) 33–41
entropies from it, it is the entropies that are primary, and we are told to view H( ) as the ‘fundamental object’ and as the ‘inferred object.’ ‘‘The epistemological status of the transition scheme is thus inherently vague. From the practical and operational point of view it is not meaningful.’’ (Conrad, 1983: 173)
An ecosystem might be indeed modeled as a complex object-oriented system, say, which from a long distance away might be viewable as a Markov process. There is likely no canonical way to view it as such, however. But there would always be the entropies. The actual entropy functions would not be important, only the fact that they were there and that they were related by certain mathematically correct equations. Further, the systems that adaptability theory studies need not be ecosystems in the strict sense. Any system that can respond to disturbances could be regarded as the biota part of the system V, and thus be treatable within adaptability theory (Conrad, 1993). The value of the adaptability formalism was that it served as a guide to intuition; it was not a scientific theory in the strictest sense. Explanatory power arose not from it but outside of it, from empirical reports of biological phenomena, which were then interpreted within the formalism, in some way ‘canalized,’ to yield new insights. Let us see how this works in the actual use of the theory. Applying an identity derivable from Eqs. (7)–(9) to the ‘most uncertain environment the biota can tolerate’, we have adaptability decomposed into three terms: H( ˆ *)=H( ˆ )+H( ˆ * ˆ ) −H( ˆ ˆ *)
(11)
Here I have switched from the notation H(…), which denoted linear functionals above, to the notation H(…), which indicates numbers. The considerations cited above license us to assume that we are holding a given state probability density vector p fixed during this entire presentation, so, even though p might not be a stationary density, we take H(…) =H(…)(p). We are look-
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ing at the interrelationships between the quantities here, not comparing them as functions. All of the papers using adaptability theory make use of some form of identity (Eq. (11)). A relatively recent explication may be found in Conrad (1993) (see also Conrad, 1995). The terms in this identity are related to properties of the biota. H( ˆ ) is an indicator of biotic behavioral diversity. H( ˆ * ˆ ) increases with the magnitude fluctuations of the environment not attributable to the biota. H( ˆ ˆ *) increases with the autonomy of the biota from the vagaries of the environment, creating degrees of freedom in the biota that can enable them to possess greater computing power. However, Eq. (11) shows that this last term makes a negati6e contribution to the system’s adaptability. This served as a suggestion to Conrad that there was a tradeoff principle operating here: greater adaptability is balanced against greater computing power. The sum of the first and third terms on the right side of Eq. (11) is the mutual information between biota and environment, as defined in Eq. (9). This quantity may be held constant by allowing both the biota diversity term H( ˆ ) and the biota autonomy term H( ˆ ˆ *) to increase together or decrease together. When the biota use a great deal of redundancy to process information, so as to increase reliability, both terms increase. Conrad (1993) concludes that this connection between evolutionary adaptability and redundancybased variability provides the link between the formalism of adaptability theory and the tradeoff between (1) structural programmability, (2) evolvability, and (3) efficiency. This tradeoff may be called the Conrad Thesis, and was featured in a majority of his papers over the years. A thorough presentation of this thesis without the adaptability formalism may be found in Conrad (1988). The most recent statement is in Conrad and Zauner (2001); an alternative treatment is given by Kirby (1998). We conclude that the role of the adaptability formalism was to serve as an informal tool for exploring the nature of information processing in living systems. Conrad offered many other variants (e.g. the ‘two-time formalism’ and hierarchical entropies; see Conrad, 1983), but they are used
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K.G. Kirby / BioSystems 64 (2002) 33–41
in the same heuristic manner as the original Markov process approach presented here. Another continuing concern in the work of Conrad was the role of quantum processes in life. He never coupled this work directly to the entropy-based adaptability formalism, but as we shall see in the next section, it is a good fit, and ironically dovetails with some of the difficult ideas in the ‘fluctuon’ theory of physics he was working on intensely during the last decade of his life.
4. From classical to quantum entropy From a classical system described by an n-dimensional state density 6ector p(t), we move to quantum system described by an n× n state density matrix ª(t). Corresponding to the requirement that p(t) be nonnegative and sum to 1, we require that ª(t) be nonnegative (i.e. x*ªx ] 0 for all vectors x) and have trace 1. These matrix and vector representations coincide in the special case when ª is diagonal with entries zil =pi. In the general case, off-diagonal entries create the ‘coherences’, which give quantum mechanics its peculiar character and quantum computing its power. (Unfortunately, space does not permit a review of the quantum mechanical concepts assumed in this section; the reader may consult Vedral, 2001 or Nielsen and Chuang, 2000 for background). Instead of the time evolution p(t +1) =Mp(t) we now have z(t+ Dt)=U(Dt)ª(t)U*(Dt) where U is a unitary matrix and U* its conjugatetranspose. If we were to work in a way strictly analogous to the classical Markov process approach in Section 2 we would focus on U. In the heuristic spirit of adaptability theory, and for reasons of space, we talk about the uncertainties of the states themselves, as expressed in entropy functions of the state density matrix ª. The 6on Neumann entropy of the state density matrix ª is defined to be
S(ª)= − % u log u
(12)
u u(z)
where u(ª) is a list of all eigenvalues of ª, including repetitions. In the special case where ª is diagonal this quantity is the classical Shannon entropy. Once we move to a quantum mechanical context, the way we represent the state space Q as a product of biota and environment state spaces becomes particularly interesting. In classical systems theory the state space of a composite system is the Cartesian product of the individual state spaces: Q= B× E in our case here. In quantum systems, by contrast, state spaces of composite systems are tensor products of the individual state spaces. (Standard expositions of quantum theory take this as a basic postulate; see Fuchs, 2001 for a recent justification in terms of contextuality of measurements). In the present case, we would write this as Q= B E. If systems B and E are individually n- and m-dimensional quantum systems, respectively, when coupled together they become an nm-dimensional quantum system (not an n+ m-dimensional system as in the classical case). Physically, this multiplicative increase in dimensionality arises because superpositions of all possible state-pairings are now allowed. One consequence of this is that correlations between the two subsystems can be of a nature not found in classical systems, as we shall see. The state of the entire system can be represented as an nm× nm density matrix ª with entries ªijkl, with rows labeled by the nm pairs ij and columns labeled by the nm pairs kl. This representation allows us to define ‘partial traces’ that contract ª to smaller matrices: (ª B)ik %ªijkj and (ª E)jl %ªijil j
i
The n ×n matrix ª and the m×m matrix ª E represent the states of the subsystems B and E, respectively, averaged over all the possibilities of the entire system. Definition (Eq. (12)) allows us to define S(ª B) and S(ª E) in exactly the same way as S(ª). This in turn allows us to define conditional entropies as before: B
S(ª B ª E)= S(ª)− S(ª E)
K.G. Kirby / BioSystems 64 (2002) 33–41
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Fig. 2. The diagram on the left displays the von Neumann entropies of the toy biota/environment quantum system given by the state density matrix in Eq. (13). The negative conditional entropy of subsystem B with respect to subsystem E indicates a non-classical ‘supercorrelation’ between the two parts of the system. Once system B is observed, the system collapses into the state given by the density matrix in Eq. (14); the resulting entropies are shown on the right. The supercorrelation has now vanished, and the mutual information has been reduced.
and S(ª E ª B)= S(ª)− S(ª B) along with the mutual information S(ª B:ª E)=S(ª B)+ S(ª E) −S(ª) What is quite distinctive about quantum entropy is that quantum conditional entropy can be negati6e. For example, the spin singlet state of two entangled spin-1/2 particles (each being a 2-D system) is characterized by the density matrix Á0 Ã0 z= Ã Ã0 Ä0
0 1/2 −1/2 0
0 0Â − 1/2 0 Ã Ã 1/2 0Ã 0 0Å
Calculations reveal a total entropy S(ª) of zero. This zero value is a consequence of ª being a pure state, not a probabilistic mixture. Yet the two marginal entropies, call them S(ª B) and S(ª E), are 1, which has the strange implication that the subsystems have greater uncertainty than the entire system (entropy is additive, so one would normally expect the uncertainty of a part to be no greater than that of the whole). Further calculations with this matrix show that the conditional entropies S(ª B ª E) and S(ª E ª B)
are both −1 and, thus, the mutual information is + 2. This is a peculiar kind of quantum ‘supercorrelation.’ Cerf and Adami (1995, 1999) provide an amusing interpretation wherein this system consists of one quantum bit and one ‘anti’ quantum bit, the latter carrying negative information and equivalent to a quantum bit moving backward in time (the term ‘bit’ here is specific to 2-D quantum subsystems, and is the unit of entropy when the logarithm is taken to the base 2). There is no requirement that both conditional entropies be identical or negative. A more typical case is the state density matrix Á 19 1 Ã−9 ª= Ã 33 Ã 1 Ä 6
−9 11 −3 −2
1 −3 1 0
6 Â − 2Ã Ã 0 Ã 2 Å
(13)
which has entropies as depicted in Fig. 2. Subsystem B has negative conditional entropy with respect to E, but not vice versa. In these low-dimensional ‘toy’ examples, I have labeled the subsystems B and E to suggest that negative entropy plays a role in biological (biota× en6ironment) systems. If life draws on quantum coherence (Penrose, 1996; Matsuno, 1999) then it is certainly the case that when two
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involved subsystems interact in isolation this kind of supercorrelation will result. This conclusion has interesting ramifications when put back into adaptability theory. As pointed out in the previous section, the quantity H( ˆ ˆ *) is an indicator of computing power, yet Eq. (11) shows that it makes a negative contribution to adaptability. The best we could hope for in a classical setting is that it be zero. In quantum systems we can hope for more: H( ˆ ˆ *) can be less than zero, allowing extra adaptabilities to emerge from non-classical supercorrelations between biota and en6ironment. This would move the biota even further away from structurally programmable modes of computing. The behavior of the toy system under measurement is also worth noting. If a measurement is performed on the biota system B (for an observable whose eigenvectors form the basis in which ª is written), the state of the system reduces to one represented by the density matrix Á 19 1 Ã −9 ª% = Ã 33 Ã 0 Ä 0
−9 11 0 0
0 0Â 0 0Ã Ã 1 0Ã 0 2Å
(14)
When we calculate the entropies for this postmeasurement state, we find that the overall entropy S(ª) has increased. This is natural, as there is now new uncertainty as to which B eigenstate it collapsed into. More interestingly, the negative conditional entropy S(ª B ª E) has now become positive: the ‘supercorrelation’ has disappeared. Fig. 2 summarizes the change in entropy values. Interpreting this in the Conrad theory, we conclude that external observation of the biota has remo6ed a source of extra adaptability! The introduction of quantum entropies into the classical entropy formulation of Conrad’s adaptability theory provides a heretofore unrecognized thread connecting his two books: Adaptability, published in 1983, and Quantum Gra6ity and Life, which he was completing when he passed away. The fluctuon theory set forth in the latter book is several orders of magnitude more complex than the adaptability theory addressed in this paper, and space does not permit a review here. What we
can note at this juncture is that negative entropy (as ‘anti-entropy’) plays a role in the fluctuon theory’s explanation of life as inextricably woven into the amplification of microscopic decorrelation and recorrelation processes (Conrad, 1998).
5. Conclusion Michael Conrad’s profound insights into information processing in biological systems did not reduce biology to computer science, but elevated computer science to a higher plane that encompassed the manifold potentialities of nature. One irony was that the creator of one of the earliest artificial life programs (Conrad, 1969) argued ‘against artificial life’ (Conrad, 1994), against the claim that the essence of life could be captured in digital computer simulations. Nor, he went on, could life be captured in the scheme of classical physics. Even current versions of quantum theory might be inadequate. He once said that someone at a conference in the late 1980s had called him a ‘vitalist.’ This remark only caused him to shrug and smile and say, ‘‘I don’t know, maybe I am.’’ Yet this attribution of vitalism can only be spoken with a smile. It was his goal to have a scientific theory of life, a theory that showed the biological to be woven together with the physical, a theory that revealed nature as a plenum of possibilities. These possibilities could also be harnessed for information processing. Central to this theory was the ‘vertical’ flow between microscopic, mesoscopic and macroscopic levels, and a view of nature as constantly sliding in and out of consistency. He had his students work on small case studies of this vertical integration (e.g. Kirby and Conrad, 1986; Chen and Conrad, 1994; Zauner and Conrad, 2000), but attainment of the grand goals of his theory lies in the future. While the full theory promised to be quite complex (the fluctuon theory), a heuristic theory would serve in the interim as a guide to intuitions. This was the adaptability theory. Although the formalism of adaptability theory was not adopted by other scientists, many ideas that it generated were. The formalism’s amenability to the incorporation of quantum mechanical concepts, as shown
K.G. Kirby / BioSystems 64 (2002) 33–41
in this paper, serve to tighten the links between adaptability theory and its full-strength successor.
Acknowledgements This work was supported by a Northern Kentucky University academic sabbatical grant. The author acknowledges the Computer Science Department at Sogang University in Seoul, where this work was undertaken. The author especially wishes to thank Deborah Conrad and Klaus-Peter Zauner for their efforts in retrieving some of Michael Conrad’s difficult-to-find papers and quickly sending them across the Pacific.
References Cerf, N.J., Adami, C., 1995. Negative entropy and information in quantum mechanics. LANL eprint archive quantph/9512022. Cerf, N.J., Adami, C., 1999. Quantum extension of conditional probability. Phys. Rev. A 60, 897 – 939. Chen, J.-C., Conrad, M., 1994. Learning synergy in a multilevel neuronal architecture. BioSystems 32, 111 –142. Conrad, M., 1969. Computer experiments on the evolution of co-adaptation in a primitive ecosystem. Ph.D. Thesis, Biophysics Program, Stanford University. Conrad, M., 1972. Statistical and hierarchical aspects of biological organization. In: Waddington, C.H. (Ed.), Towards a Theoretical Biology, vol. 4. Edinburgh University Press, Edinburgh. Conrad, M., 1979. Mutation –absorption model of the enzyme. Bull. Math. Biol. 41, 387 –405. Conrad, M., 1983. Adaptability. Plenum Press, New York. Conrad, M., 1988. The price of programmability. In: Herken, R. (Ed.), The Universal Turing Machine, A Half-Century Survey. Oxford University Press, Oxford, pp. 285 –307.
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Conrad, M., 1993. Adaptability theory as a guide for interfacing computers and human society. Syst. Res. 10 (4), 3 – 23. Conrad, M., 1994. From brain structure to vacuum and back again: the great chain of being model. Nanobiology 3, 99 – 121. Conrad, M., 1995. The ecosystem as an existential computer. In: Patten, B.C., Jorgensen, S.E., Auerbach, S.I. (Eds.), Complex Ecology: the Part – Whole Relationship in Ecosystems. Prentice-Hall, Englewood Cliffs, NJ, pp. 609 – 622. Conrad, M., 1998. Quantum gravity and life. BioSystems 46, 29 – 39. Conrad. M., Zauner, K.-P., 2001. Conformation-based computing: a rationale and a recipe. In: Sienko, T., Conrad, M., Rambidi, N., Adamatzky, A. (Eds.), Molecular Computing. MIT Press, Cambridge, MA. Cover, T.M., Thomas, J.A., 1991. Elements of Information Theory. Wiley, New York. Fuchs, C.A., 2001. Quantum foundations in the light of quantum information. LANL eprint archive quant-ph/0106166. Khinchin, A.I., 1957. Mathematical Foundations of Information Theory. Dover, New York. Kirby, K., 1998. Exaptation and torsion: toward a theory of natural information processing. BioSystems 46, 81 – 88. Kirby, K., Conrad, M., 1986. Intraneuronal dynamics as a substrate for evolutionary learning. Physica D 22, 150 – 175. Matsuno, K., 1999. Cell motility as an entangled quantum coherence. BioSystems 51, 15 – 19. Nielsen, M.A., Chuang, I.L., 2000. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge. Penrose, R., 1996. Shadows of the Mind. Oxford University Press, Oxford. Shannon, C.E., 1948. The mathematical theory of communication. Bell Syst. Tech. J. 27, 379 – 423 and 623 – 656. Vedral, V., 2001. The role of relative entropy in quantum information theory. LANL eprint archive quant-ph/ 0102094. Zauner, K.-P., Conrad, M., 2000. Enzymatic pattern processing. Naturwissenschaften 87 (8), 360 – 362.
Biological adaptabilities and quantum entropies Kevin G. Kirby Department of Mathematics and Computer Science, Northern Kentucky Uni6ersity, Highland Heights, KY 41099, USA Received 7 May 2001; received in revised form 25 July 2001; accepted 14 August 2001
Abstract The entropy-based theory of adaptability set forth by Michael Conrad in the early 1970s continued to appear in his work for over two decades, and was the subject of the only book he published in his lifetime. He applied this theory to a host of subjects ranging from enzyme dynamics to sociology. This paper reviews the formalism of adaptability theory, clarifying some of its mathematical and interpretive difficulties. The theory frames the computational tradeoff principle, a thesis that was the most frequently recurring claim in his work. The formulation of adaptability theory presented here allows the introduction of quantum entropy functions into the theory, revealing an interesting relationship between adaptability and another one of Conrad’s deep preoccupations, the role of quantum processes in life. © 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Adaptability; Entropy; Quantum information; Supercorrelation; Bioinformational science
1. Introduction The development and application of an entropybased theory of adaptability was a recurring element in the work of Michael Conrad for at least 23 years, and was the subject of Adaptability, the only book he published in his lifetime (Conrad, 1983). The formalism became the lingua franca connecting his various interests, and he used it to analyze everything from mutation buffering in enzymes (Conrad, 1979) to a society’s response to computer technology (Conrad, 1993). In presentations of various and sundry topics in papers from 1972 through 1995, the ubiquitous ‘H( *)’ — the marginal entropy that became the signature of his theory —was sure to make an appearance. E-mail address: [email protected] (K.G. Kirby).
While many of Conrad’s themes and ideas had widespread influence, the adaptability formalism itself was apparently not contagious. This may be due to the complexities of the formalism, the bewildering variety of reformulations of it even within the confines of his book, and the difficulties in what he himself would call the ‘epistemological status’ of the equations. Nevertheless, he stood by the formalism for over two decades. It is the thesis of this paper that the unusual use to which he put Shannon’s entropy-based information theory can indeed be made approachable, and, moreover, it reveals links to contemporary quantum information theory. Conrad (1972, 1983) derived his scheme from Khinchin’s presentation of Shannon’s theory, an approach based on the entropy of transitions in a
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Markov process (Shannon, 1948; Khinchin, 1957; modern treatment is Cover and Thomas, 1991). In the next section, I will explain Conrad’s formalism, modernizing the notation a bit and streamlining the definitions. In Section 3, I will review the ways the formalism has been interpreted and applied, showing in particular its connection to the computational tradeoff principle (Conrad, 1988). In Section 4, I will move to a new use of adaptability theory, relating it to quantum information theory and tying it in to one of the other great themes of his work, the role of quantum processes in living systems. I conclude by discussing the relationship between adaptability theory and the fluctuon theory.
used here is chosen to anticipate the theory’s extension into quantum density matrices later in this paper). If the system is in state qj at time t, the uncertainty about what state it will be in at time t+ 1 is given by hj (M) = − %mij log mij
This is the Shannon entropy of the jth column of M, viewed as a probability density vector. If at time t we do not know for sure which state qj the system is in, but only know the state density p(t), then we can only speak of the expected uncertainty as to what happens next: H(M)(p) = %pj hj (M) h(M)Tp
2. The ecosystem entropy formalism Conrad (1983) framed adaptability theory within a formal model of ecosystems based on Markov processes. The scene for the theory is a very large but finite state space Q = {q1, q2, …, qn }. At a loss to describe the unknowably complex dynamics on Q, we turn to the use of a state density 6ector p(t), each component pj (t) representing the probability that the system is in state qj at time t. Everything we know about the ecosystem’s dynamics is encapsulated in a matrix M whose columns sum to 1 and whose entries mij represent conditional probabilities: mij =Pr[system is in state qi at time t +1 system was in state qj at time t] The state density vector evolves in discrete time by a linear transformation: p(t +1)=Mp(t) Since the columns of M sum to 1, trajectories are confined to the probability simplex
!
Dn − 1 = pDn pi ]0 and %i pi =1
"
Adopting a helpful abuse of language, we can say p(t) itself is the state of the system. If p(t) is at one of the vertices of the simplex (i.e. one component is 1 and the others are 0), it is said to be a pure state; otherwise, it is mixed (the language
(1)
i
(2)
j
Thus, associated with each Markov process M we have a linear functional H(M) = h(M)T defined on the probability simplex, which tells us the ‘uncertainty about what happens next’ for each state density vector p. In the context of information theory, p lists the probabilities of emission of n different symbols by a source. The application of one step in the dynamics (multiplication by M) would represent a ‘data processing step.’ Importantly, for the purposes of information theory it is assumed that this Markov process is stationary, meaning there is a state density vector p* that is a fixed point of M, i.e. Mp* = p*. We define the number H(M) = H(M)(p*)
(3)
This is the ‘entropy rate’ of the Markov process M. Taking this as a starting point, Conrad decomposed Q into a Cartesian product of biota and en6ironment states: Q=B× E. The term biota is more general than ‘organism’ (we could be talking about anything from an organelle to a society), yet not so all-encompassing as ‘biological system’ (which might include an environment). As the environment itself may be partly living, the term biota ultimately denotes ‘the living system of interest.’ In this scheme, each ecosystem state qi becomes a pair (i, m)B×E. Following Conrad’s lead, we
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will represent the Markov process matrix here by V. Rather than keeping track of matrix subscripts, it is easier to write V as a map V: (B × E)×(B× E)[0, 1]. It brings out the sense of this expression as a conditional probability to write V(i%, m%, i, m) as V(i%, m% i, m). The state density vector can similarly be viewed as a map p: B ×E [0,1]. Despite this departure from matrix notation, it should be remembered that V describes the linear time evolution of the state density vector p. Factoring the state set into ‘biota’ and ‘environment’ parts allows a variety of conditional entropies to be defined, which leads ultimately to the definition of adaptability. Eqs. (1) and (2) can be combined and expressed in this biota × en6ironment notation as:
Conrad informally used the symbols and * to denote the biota and environment transition schemes, but these schemes are not derivable from the scheme V unless a specific state density vector p is given as well. A better notation for H( ) and H( *) might have been HB (V) and HE (V), respectively, but the former terms and their relatives are so widespread in his work that it would not be helpful to change them at this point. Third, we have the uncertainty function of the biota transition given the environment (writing functional equations now):
H(V)(p)=% % p(i, m)%%ent V(i%, m% i, m)
Finally, we have the mutual information function between the biota and environment:
i m
(4)
i% m%
where we make the space-saving abbreviation ent x = − x log x, with the usual special case that ent 0 =0. As in Eq. (2), H(V) is a linear functional, not a number. In Conrad’s terms it gives the ‘entropy of the ecosystem transition scheme’ (Conrad, 1983). More precisely, it is a function that takes a state density p and tells us the uncertainty associated with the next state change. From this definition (Eq. (4)) we can derive five related entropies. Their definitions can be streamlined with the following abbreviations, which are motivated by basic identities: V(i% i, m)=%V(i%, m% i, m)
H( *)= H(V) −H( *)
(7)
Fourth, we have the uncertainty function of the environment transition given the biota: H( * )= H(V) − H( )
H( : *)= H( )+H( *)− H(V)
(8)
(9)
All of the entropies defined here (Eqs. (4)–(9)) are functions that map probability state vectors to the non-negati6e real numbers. A value of 0 is interpreted as certainty; there is no such thing as ‘negative uncertainty’ here. Fig. 1 summarizes the relationships between the entropies. These equations and this figure must be interpreted with great care. Many information theory texts feature a similar Venn diagram-like picture, and list similar equations, but the quantities represented in those texts are numbers, show-
m%
V(m% i, m)=%V(i%, m% i, m) i%
First, we have the ‘marginal entropy for the behavior of the biota’: H( )(p)= % % p(i, m)% ent V(i% i, m) i m
(5)
i%
Second, we have the ‘marginal entropy for the behavior of the environment’: H( *)(p)=% %p(i, m)% ent V(m% i, m) i m
m%
(6)
Fig. 1. Relations between the entropy functions associated with the Markov process model of an ecosystem.
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K.G. Kirby / BioSystems 64 (2002) 33–41
ing the entropy relationships between two random variables X and Y obtained from a joint distribution (X, Y). In contrast, the H(…) quantities defined in the Conrad theory and shown in Fig. 1 are functions (of state probability vectors p), not numbers. Further, they are properties of Markov processes, i.e. they deal with random transitions and random states, not ordinary random variables. At this point, the formulation of the Conrad adaptability theory is almost complete. The final step is the definition of adaptability itself. Quoting verbatim from the Adaptability book, ‘‘the adaptability of the biota is given by: H( ˆ *) max * [H( *) such that A]
(10)
where H( *) is a maximum over all possible transition schemes subject to condition A. This condition is: the half-life of the biota is not decreased at all’’ (Conrad, 1983). A casual gloss on this is easy: the adaptability of a living system is defined as the uncertainty of the most uncertain environment in which it can survive. Yet it was not this casual gloss to which Conrad made continual reference over the years. He made use of the entropic quantities in the formalism itself. Three difficulties arise here. We will first pose them, then provide some interpretations and ‘workarounds’ in the next section. First, the maximum in Eq. (10) implies that we can make sense of inequalities across the H( *)’s. But these are functions, not numbers. Now we can certainly have function inequalities, whereby fB g means f(x)Bg(x) for all x, and so on, but one must now be careful because the negation of f Bg is no longer f] g, for example. Yet we find Conrad freely using the latter kind of inferences between these entropies. This problem does not come up in information theory (Khinchin, 1957; Cover and Thomas, 1991), where one assumes the Markov process is stationary and one converts these functions to numbers by evaluating them at the welldefined stationary density, as we did in Eq. (3). By contrast, in adaptability theory ‘no assumption about stationarity is made’ (Conrad, 1983) so this move is not open to us. (Despite this statement, the presentation in Adaptability did not identify them as functions, which added to the difficulty of the presentation).
Second, the definition of adaptability implies the definability of an entity denoted by ˆ *, ‘the most uncertain environment the biota can tolerate,’ whose entropy is given by Eq. (10). But it is not clear that such a system can exist within this formalism. The maximum of a set of entropy functions of Markov systems may not be an entropy function of a Markov system. (Compare this to a more elementary situation. If one defines k= max{ f, g} where f and g are quadratic functions, k is usually not a quadratic function itself). Third, the ‘condition’ A seems to be a predicate on the biota state set B that must be defined supplementary to the transition scheme. So this suggests the ecosystem is actually a pair (V, A). What are the constraints on this predicate? Does any predicate A on B result in a formal ecosystem? That would seem to evacuate the meaning from the entire formalism.
3. The lessons of adaptability The three questions posed above are best tackled together. The path to the answers is cleared by answering another question first: did Conrad think this Markov process scheme was a model of an ecosystem? Emphatically, no. He was the creator of some of the earliest ecosystem simulations (Conrad, 1969), simulations that might now be seen as examples of ‘artificial life’ programs (two decades ahead of their time). Until that work appeared, ecosystem simulation usually involved numerical integration of sets of differential-difference equations. Conrad’s approach in 1969, by contrast, was quite ‘modern.’ It employed algorithms operating on discrete data structures, a rough precursor of an object-oriented approach, in which objects in the program represented objects in the virtual world. For someone who had pioneered such an approach, modeling an ecosystem with a Markov process would be reactionary and unilluminating. Then what are all those omegas? In the Adaptability book we find, 120 pages after the original definitions, a tantalizingly odd explanation. We read that, despite the mathematical setup which begins with the Markov process and defines the
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entropies from it, it is the entropies that are primary, and we are told to view H( ) as the ‘fundamental object’ and as the ‘inferred object.’ ‘‘The epistemological status of the transition scheme is thus inherently vague. From the practical and operational point of view it is not meaningful.’’ (Conrad, 1983: 173)
An ecosystem might be indeed modeled as a complex object-oriented system, say, which from a long distance away might be viewable as a Markov process. There is likely no canonical way to view it as such, however. But there would always be the entropies. The actual entropy functions would not be important, only the fact that they were there and that they were related by certain mathematically correct equations. Further, the systems that adaptability theory studies need not be ecosystems in the strict sense. Any system that can respond to disturbances could be regarded as the biota part of the system V, and thus be treatable within adaptability theory (Conrad, 1993). The value of the adaptability formalism was that it served as a guide to intuition; it was not a scientific theory in the strictest sense. Explanatory power arose not from it but outside of it, from empirical reports of biological phenomena, which were then interpreted within the formalism, in some way ‘canalized,’ to yield new insights. Let us see how this works in the actual use of the theory. Applying an identity derivable from Eqs. (7)–(9) to the ‘most uncertain environment the biota can tolerate’, we have adaptability decomposed into three terms: H( ˆ *)=H( ˆ )+H( ˆ * ˆ ) −H( ˆ ˆ *)
(11)
Here I have switched from the notation H(…), which denoted linear functionals above, to the notation H(…), which indicates numbers. The considerations cited above license us to assume that we are holding a given state probability density vector p fixed during this entire presentation, so, even though p might not be a stationary density, we take H(…) =H(…)(p). We are look-
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ing at the interrelationships between the quantities here, not comparing them as functions. All of the papers using adaptability theory make use of some form of identity (Eq. (11)). A relatively recent explication may be found in Conrad (1993) (see also Conrad, 1995). The terms in this identity are related to properties of the biota. H( ˆ ) is an indicator of biotic behavioral diversity. H( ˆ * ˆ ) increases with the magnitude fluctuations of the environment not attributable to the biota. H( ˆ ˆ *) increases with the autonomy of the biota from the vagaries of the environment, creating degrees of freedom in the biota that can enable them to possess greater computing power. However, Eq. (11) shows that this last term makes a negati6e contribution to the system’s adaptability. This served as a suggestion to Conrad that there was a tradeoff principle operating here: greater adaptability is balanced against greater computing power. The sum of the first and third terms on the right side of Eq. (11) is the mutual information between biota and environment, as defined in Eq. (9). This quantity may be held constant by allowing both the biota diversity term H( ˆ ) and the biota autonomy term H( ˆ ˆ *) to increase together or decrease together. When the biota use a great deal of redundancy to process information, so as to increase reliability, both terms increase. Conrad (1993) concludes that this connection between evolutionary adaptability and redundancybased variability provides the link between the formalism of adaptability theory and the tradeoff between (1) structural programmability, (2) evolvability, and (3) efficiency. This tradeoff may be called the Conrad Thesis, and was featured in a majority of his papers over the years. A thorough presentation of this thesis without the adaptability formalism may be found in Conrad (1988). The most recent statement is in Conrad and Zauner (2001); an alternative treatment is given by Kirby (1998). We conclude that the role of the adaptability formalism was to serve as an informal tool for exploring the nature of information processing in living systems. Conrad offered many other variants (e.g. the ‘two-time formalism’ and hierarchical entropies; see Conrad, 1983), but they are used
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K.G. Kirby / BioSystems 64 (2002) 33–41
in the same heuristic manner as the original Markov process approach presented here. Another continuing concern in the work of Conrad was the role of quantum processes in life. He never coupled this work directly to the entropy-based adaptability formalism, but as we shall see in the next section, it is a good fit, and ironically dovetails with some of the difficult ideas in the ‘fluctuon’ theory of physics he was working on intensely during the last decade of his life.
4. From classical to quantum entropy From a classical system described by an n-dimensional state density 6ector p(t), we move to quantum system described by an n× n state density matrix ª(t). Corresponding to the requirement that p(t) be nonnegative and sum to 1, we require that ª(t) be nonnegative (i.e. x*ªx ] 0 for all vectors x) and have trace 1. These matrix and vector representations coincide in the special case when ª is diagonal with entries zil =pi. In the general case, off-diagonal entries create the ‘coherences’, which give quantum mechanics its peculiar character and quantum computing its power. (Unfortunately, space does not permit a review of the quantum mechanical concepts assumed in this section; the reader may consult Vedral, 2001 or Nielsen and Chuang, 2000 for background). Instead of the time evolution p(t +1) =Mp(t) we now have z(t+ Dt)=U(Dt)ª(t)U*(Dt) where U is a unitary matrix and U* its conjugatetranspose. If we were to work in a way strictly analogous to the classical Markov process approach in Section 2 we would focus on U. In the heuristic spirit of adaptability theory, and for reasons of space, we talk about the uncertainties of the states themselves, as expressed in entropy functions of the state density matrix ª. The 6on Neumann entropy of the state density matrix ª is defined to be
S(ª)= − % u log u
(12)
u u(z)
where u(ª) is a list of all eigenvalues of ª, including repetitions. In the special case where ª is diagonal this quantity is the classical Shannon entropy. Once we move to a quantum mechanical context, the way we represent the state space Q as a product of biota and environment state spaces becomes particularly interesting. In classical systems theory the state space of a composite system is the Cartesian product of the individual state spaces: Q= B× E in our case here. In quantum systems, by contrast, state spaces of composite systems are tensor products of the individual state spaces. (Standard expositions of quantum theory take this as a basic postulate; see Fuchs, 2001 for a recent justification in terms of contextuality of measurements). In the present case, we would write this as Q= B E. If systems B and E are individually n- and m-dimensional quantum systems, respectively, when coupled together they become an nm-dimensional quantum system (not an n+ m-dimensional system as in the classical case). Physically, this multiplicative increase in dimensionality arises because superpositions of all possible state-pairings are now allowed. One consequence of this is that correlations between the two subsystems can be of a nature not found in classical systems, as we shall see. The state of the entire system can be represented as an nm× nm density matrix ª with entries ªijkl, with rows labeled by the nm pairs ij and columns labeled by the nm pairs kl. This representation allows us to define ‘partial traces’ that contract ª to smaller matrices: (ª B)ik %ªijkj and (ª E)jl %ªijil j
i
The n ×n matrix ª and the m×m matrix ª E represent the states of the subsystems B and E, respectively, averaged over all the possibilities of the entire system. Definition (Eq. (12)) allows us to define S(ª B) and S(ª E) in exactly the same way as S(ª). This in turn allows us to define conditional entropies as before: B
S(ª B ª E)= S(ª)− S(ª E)
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Fig. 2. The diagram on the left displays the von Neumann entropies of the toy biota/environment quantum system given by the state density matrix in Eq. (13). The negative conditional entropy of subsystem B with respect to subsystem E indicates a non-classical ‘supercorrelation’ between the two parts of the system. Once system B is observed, the system collapses into the state given by the density matrix in Eq. (14); the resulting entropies are shown on the right. The supercorrelation has now vanished, and the mutual information has been reduced.
and S(ª E ª B)= S(ª)− S(ª B) along with the mutual information S(ª B:ª E)=S(ª B)+ S(ª E) −S(ª) What is quite distinctive about quantum entropy is that quantum conditional entropy can be negati6e. For example, the spin singlet state of two entangled spin-1/2 particles (each being a 2-D system) is characterized by the density matrix Á0 Ã0 z= Ã Ã0 Ä0
0 1/2 −1/2 0
0 0Â − 1/2 0 Ã Ã 1/2 0Ã 0 0Å
Calculations reveal a total entropy S(ª) of zero. This zero value is a consequence of ª being a pure state, not a probabilistic mixture. Yet the two marginal entropies, call them S(ª B) and S(ª E), are 1, which has the strange implication that the subsystems have greater uncertainty than the entire system (entropy is additive, so one would normally expect the uncertainty of a part to be no greater than that of the whole). Further calculations with this matrix show that the conditional entropies S(ª B ª E) and S(ª E ª B)
are both −1 and, thus, the mutual information is + 2. This is a peculiar kind of quantum ‘supercorrelation.’ Cerf and Adami (1995, 1999) provide an amusing interpretation wherein this system consists of one quantum bit and one ‘anti’ quantum bit, the latter carrying negative information and equivalent to a quantum bit moving backward in time (the term ‘bit’ here is specific to 2-D quantum subsystems, and is the unit of entropy when the logarithm is taken to the base 2). There is no requirement that both conditional entropies be identical or negative. A more typical case is the state density matrix Á 19 1 Ã−9 ª= Ã 33 Ã 1 Ä 6
−9 11 −3 −2
1 −3 1 0
6 Â − 2Ã Ã 0 Ã 2 Å
(13)
which has entropies as depicted in Fig. 2. Subsystem B has negative conditional entropy with respect to E, but not vice versa. In these low-dimensional ‘toy’ examples, I have labeled the subsystems B and E to suggest that negative entropy plays a role in biological (biota× en6ironment) systems. If life draws on quantum coherence (Penrose, 1996; Matsuno, 1999) then it is certainly the case that when two
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involved subsystems interact in isolation this kind of supercorrelation will result. This conclusion has interesting ramifications when put back into adaptability theory. As pointed out in the previous section, the quantity H( ˆ ˆ *) is an indicator of computing power, yet Eq. (11) shows that it makes a negative contribution to adaptability. The best we could hope for in a classical setting is that it be zero. In quantum systems we can hope for more: H( ˆ ˆ *) can be less than zero, allowing extra adaptabilities to emerge from non-classical supercorrelations between biota and en6ironment. This would move the biota even further away from structurally programmable modes of computing. The behavior of the toy system under measurement is also worth noting. If a measurement is performed on the biota system B (for an observable whose eigenvectors form the basis in which ª is written), the state of the system reduces to one represented by the density matrix Á 19 1 Ã −9 ª% = Ã 33 Ã 0 Ä 0
−9 11 0 0
0 0Â 0 0Ã Ã 1 0Ã 0 2Å
(14)
When we calculate the entropies for this postmeasurement state, we find that the overall entropy S(ª) has increased. This is natural, as there is now new uncertainty as to which B eigenstate it collapsed into. More interestingly, the negative conditional entropy S(ª B ª E) has now become positive: the ‘supercorrelation’ has disappeared. Fig. 2 summarizes the change in entropy values. Interpreting this in the Conrad theory, we conclude that external observation of the biota has remo6ed a source of extra adaptability! The introduction of quantum entropies into the classical entropy formulation of Conrad’s adaptability theory provides a heretofore unrecognized thread connecting his two books: Adaptability, published in 1983, and Quantum Gra6ity and Life, which he was completing when he passed away. The fluctuon theory set forth in the latter book is several orders of magnitude more complex than the adaptability theory addressed in this paper, and space does not permit a review here. What we
can note at this juncture is that negative entropy (as ‘anti-entropy’) plays a role in the fluctuon theory’s explanation of life as inextricably woven into the amplification of microscopic decorrelation and recorrelation processes (Conrad, 1998).
5. Conclusion Michael Conrad’s profound insights into information processing in biological systems did not reduce biology to computer science, but elevated computer science to a higher plane that encompassed the manifold potentialities of nature. One irony was that the creator of one of the earliest artificial life programs (Conrad, 1969) argued ‘against artificial life’ (Conrad, 1994), against the claim that the essence of life could be captured in digital computer simulations. Nor, he went on, could life be captured in the scheme of classical physics. Even current versions of quantum theory might be inadequate. He once said that someone at a conference in the late 1980s had called him a ‘vitalist.’ This remark only caused him to shrug and smile and say, ‘‘I don’t know, maybe I am.’’ Yet this attribution of vitalism can only be spoken with a smile. It was his goal to have a scientific theory of life, a theory that showed the biological to be woven together with the physical, a theory that revealed nature as a plenum of possibilities. These possibilities could also be harnessed for information processing. Central to this theory was the ‘vertical’ flow between microscopic, mesoscopic and macroscopic levels, and a view of nature as constantly sliding in and out of consistency. He had his students work on small case studies of this vertical integration (e.g. Kirby and Conrad, 1986; Chen and Conrad, 1994; Zauner and Conrad, 2000), but attainment of the grand goals of his theory lies in the future. While the full theory promised to be quite complex (the fluctuon theory), a heuristic theory would serve in the interim as a guide to intuitions. This was the adaptability theory. Although the formalism of adaptability theory was not adopted by other scientists, many ideas that it generated were. The formalism’s amenability to the incorporation of quantum mechanical concepts, as shown
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in this paper, serve to tighten the links between adaptability theory and its full-strength successor.
Acknowledgements This work was supported by a Northern Kentucky University academic sabbatical grant. The author acknowledges the Computer Science Department at Sogang University in Seoul, where this work was undertaken. The author especially wishes to thank Deborah Conrad and Klaus-Peter Zauner for their efforts in retrieving some of Michael Conrad’s difficult-to-find papers and quickly sending them across the Pacific.
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