UNDER CONSTRUCTION - this is largely a text dump from "Hierarchy Theory as the Formal Basis of Evolutionary Theory" by Stephen Wood at the Theoretical Physics Research Unit, Birkbeck University of London. Stephen Wood's Homepage A lot more text dumping from other papers and re-writing with my own thoughts added is necessary. Hardhats are required here.
Chapters 3 and 4 explored the idea that problems can be solved by searching in a
space of states. These states can be evaluated by domain-specific heuristics and tested to
see whether they are goal states. From the point of view of the search algorithm, however,
BLACK BOX each state is a black box with no discernible internal structure. It is represented by an arbitrary
data structure that can be accessed only by the problem-specific routines—the successor
function, heuristic function, and goal test.
Multilevel Landscapes in Combinatorial Optimisation: by Chris Walshaw, Martin G. Everett
We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found and then iteratively refined at each level, coarsest to finest. Although the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid methods), it has only recently been suggested as a metaheuristic for combinatorial optimisation problems. It has been proposed that, for such problems, multilevel coarsening is equivalent to recursively filtering the solution space to create a hierarchy of increasingly coarser and smaller spaces. It is also suggested that perhaps this aids the local search algorithms used to refine the solution on each level by somehow ‘smoothing ’ the landscape of the solution spaces. In this paper, with some example problem instances drawn from graph partitioning and the travelling salesman problem, we take a detailed look at how the coarsening affects the hierarchy of solution landscapes. In particular we are interested in how the coarsening and hence filtering of the original space impacts on the maximum, minimum and average values of the cost function in the coarsened spaces. However, we also explore the manner in which the density of problem instances can moderate the effectiveness of a multilevel refinement algorithm.
Hierarchies of constraint
‘Hierarchies can be profitably viewed as systems of constraint’ (Allen and Starr, 1982: 11). Allen
and Starr (1982) discuss constraint in terms of information flow through the hierarchy. They adopt
Koestler’s metaphor of Janus (Koestler, 1967: 47-49). In the Roman mythology the god Janus had
two faces; the Latin for door ianua is from the same root. Each holon is thus a doorway through
which information enters and departs, flowing down the hierarchy from the environment and flowing
up from lower levels of the hierarchy (Allen and Starr, 1982: 9). The position of the holon in the
hierarchy is determined by the way in which the holon filters information that it receives. The
asymmetry of information exchange produces relations of constraint between holons. Constraining
holons filter out the signal that they receive from constrained holons and remain largely unaffected.
Constrained holons receive the signal from constraining holons relatively unfiltered and thus are
significantly affected. (This account of constraint and information exchange is derived from Allen
and Starr, 1982: 20; cf. Dawkins, 1976: 14).
Signals pass out from the genome and modify the environment to produce the phenotype. This is
what we call development. Development is an interaction between the genetic signals and the
environment. The dividing line between the phenotype and the environment is not precise: the
phenotype ‘is a bit of the environment locally modified by the genetic information’ (Cairns-Smith,
1982: 80). It is possible to imagine that the phenotype, the manifestation of the effects of the genetic signals, extends into the environment beyond the bounds of the body housing the corresponding genes. This is the essence of Dawkins’ idea of the ‘extended phenotype’ (Dawkins, 1982).
We can envisage the phenotype itself as a set of holons which differ in the extent to which they filter genetic signals as they pass out into the environment. Phenotypic holons that exert little constraint on the genome express the genetic signal relatively unfiltered. Phenotypic holons that exert heavy constraint on the genome express the genetic signal significantly filtered. Thus continuous genetic differences between organisms in a population may be expressed as continuous phenotypic differences, if the corresponding genetic signals are relatively unfiltered, or as discontinuities, if the genetic signals are significantly filtered. The accumulation of genetic changes will cause gradual modifications of the phenotype in the first case, but sudden shifts between stable states in the second.
In this way phenotypic holons can be said to constrain the dynamics of genetic change. These
constraints are properties of the developmental system: they are developmental constraints. Genetic
signal is filtered in such a way that across individuals, and indeed across species, qualitatively
different morphologies are produced. (An understanding of the developmental system in terms of its
inherent constraints cannot be derived from the study of a single individual; contra Dawkins [1989a:
203].)
If holons are doorways then information passes through them in two directions. If we believe that
development is the result of genetic information passing out through a hierarchy of holons into the
environment, then there must be a process in which genetic information flows back the other way. I
suggest that information flows back from the environment to the genome when organisms reproduce.
On their return journey genetic signals pass through a hierarchy of adaptive constraints. The
position of a phenotypic feature in this hierarchy is determined by the selective or adaptive advantage of that feature. As in the case of development we must think not about a single individual, but a set of individuals, in this case the population. Selective filtering occurs when the population reproduces as a whole to provide the next generation. The predominance of a gene in the next generation is proportional to the strength of the signal arriving from the environment back in the communal genome, or gene pool. Adaptive, i.e. selectively advantageous features amplify the signal, whereas maladaptive features attenuate it. A particular phenotypic feature may filter the signal of many genes. It is also possible that selective filtering of different features is correlated in some way. There is no need to conceive a simple relation between genetic signal and selective constraint. It is on genetic signals that selection acts: genetic signals are selectively filtered. This is the essence of another of Dawkins’ ideas: the ‘selfish gene’ (Dawkins, 1989b). Dawkins has been dubbed a reductionist for advocating the gene as the level at which selection acts. I have shown here that his position is exactly that expected from hierarchy theory.
Homologies are rules of interpretation that make the phenotypes of organisms meaningful.
Homologies are developmental constraints conserved among organisms. Homologies are the rules
operating at the phenotypic level that constrain the dynamics of the genetic level (cf. Allen and Starr, 1982: 42). Through descent with modification then, organisms accumulate inherited constraints on their genetic dynamics, or as Riedl (1977) would put it, on their adaptive freedom. The type is the totality of constraints inherited by the organism. The homologies, the parts of the type, are the individual constraints inherited by the parts of the organism. A general taxon or type characterises ‘a set of species sharing a common pattern of constraints and adaptive opportunities … the key event in the origin of a [general] taxon is a change in the pattern of constraints’ (Wagner, 1986: 154-155).
I have described above a ‘feedback regulatory cycle’ operating between genotype and phenotype,
similar to that envisaged by Riedl (1977). In order to explain the stability of homologues over
evolutionary time, Riedl saw the necessity of ‘feedback loops of cause and effect both from the
genome to the phenome and in the reverse direction’ (Riedl, 1977: 364). This sounds rather puzzling
but can be understood in terms of the expectations of hierarchy theory. The dynamics of gene
frequencies may be the cause of phenotypic change, but the effects are constrained by the phenotype itself. Thus information flows both ways: from genotype to phenotype in the causal relationship enshrined in the ‘central dogma’ of molecular biology, and from phenotype to genotype as constraints enshrined in the systems approach (Riedl, 1977; Wagner, 1986). I might even suggest that Riedl’s notion of burden, or systemic position, is equivalent to the position in the hierarchy of constraint.
Structures of high burden have great stability and are unlikely to be rejected or modified by natural
selection (Riedl, 1978: 239).
Dynamical systems theory
Like Dawkins (1989a), Alberch (1982) describes the idea of developmental constraints with the aid
of a thought experiment. Consider, for sake of example, that the whole diversity of a phenotype can
be expressed in terms of two variables, x and y. The distribution of forms found in nature is not
continuous. Instead, phenotypes cluster and certain regions of the xy space remain empty. Now let
us take a population of one of the natural forms and breed the population for a large number of
generations. The effect of natural selection is eliminated as far as possible, by enforcing random
mating and minimising competition. The overall genetic variability of the population can also be
increased through the use of mutagens. Score all the new phenotypes in terms of x and y, including
teratologies. We will get the same phenotype clusters as before, plus new ones, which will be
naturally lethal or non-functional phenotypes. ‘However, there will still be states that are prohibited
by developmental constraints’ (Alberch, 1982: 318). The basic effect of developmental constraints
on the apportionment of morphological variation is that ‘a continuous distribution of genotypes can
result in a discontinuous distribution of phenotypes’ (Alberch, 1982: 319).
The theoretical framework that Alberch (1982) provides for understanding developmental constraints
is dynamical systems theory: ‘Developmental systems are complex non-linear dynamical systems. It
is an intrinsic property of such systems that they will fall into a discrete number of stable states, i.e.
we should a discrete and bounded distribution of phenotypes. Furthermore, non-linear dynamical
systems will exhibit preferred transitions of form’ (Alberch, 1982: 327-328). The analysis of
development as a dynamical system enables possible stable states of morphology to be identified and
also the preferred transformations between those states. The morphogenetic process is conceived as a set of simple, locally-acting “assembly” rules (Alberch, 1982: 321). Genetic change perturbs the
parameters of the developmental system, but as long as the parameters stay within certain limits, the morphology remains unchanged. The morphology is said to be self-regulating or canalised
(Waddington, 1957). However, if a particular parameter reaches a threshold value then a sudden shift to a different stable state occurs. This effect is known in the language of dynamical systems theory as ‘bifurcation’. The parameter space for a particular dynamical system is said to have ‘bifurcation boundaries’ at which the global behaviour of the system, such as the resulting morphology, shifts from one stable state to another. Oster and Alberch (1982) describe ‘how the bifurcations in the developmental program acts as a filter, giving order to the random mutations in the genome, so as to present natural selection with a small subset of the possible phenotypes’ (Oster and Alberch, 1982: figure 11, legend; my italics). Thus developmental bifurcations ‘filter random mutations, giving them a non-random character’ (Oster and Alberch, 1982: 454).
The view of the developmental process derived from the theory of non-linear dynamical systems is
compatible with that provided by hierarchy theory. In fact, they complement each other. On the
phenomenological level, the notion of developmental constraints is left obscure by dynamical systems
theory. A phenotype can never be expressed in terms of just two variables. Hierarchy theory, as
applied to systematics, clarifies the notion. Homologies are developmental constraints and, through
descent with modification, are inherited by parts of organisms. Dynamical systems theory provides
the basis of constraint at a deeper level. Developmental bifurcations filter genetic signals, producing
variation at the morphological level which is constrained or canalised into particular stable states.
Quantity to quality
Gould (1980b) provides a lucid description of the two proposed modes of macroevolutionary change:
gradualism and punctuated equilibria. Gradualism asserts that evolution proceeds by the continuous,
gradual change at both the level of the gene and the total morphology. The theory of punctuated
equilibria (Eldredge and Gould, 1972; Gould and Eldredge, 1977) asserts that species appear rapidly
and then remain stable for the rest of their history. Gradualism is forced to explain the existence of
discontinuities in nature as gaps in the preservation of the fossil record. Punctuationism, on the other
hand, sees the gaps as real, to be expected by the theory. The theory explains discontinuity in terms
of a process of speciation which requires rapid change in both genotype and phenotype in a small
population (Gould, 1980b: 183).
The theme of Gould (1980b) is that gradualism has found favour because of the Western preference
for slow, orderly transformation. A preference for revolutionary, cataclysmic change belongs to a
different tradition, namely the tradition of dialectics derived by Engels from Hegel’s philosophy:
‘The dialectical laws are explicitly punctuational. They speak, for example, of the “transformation of
quantity into quality.” This may sound like mumbo-jumbo, but it suggests that change occurs in leaps
following a slow accumulation of stresses that a system resists until it reaches the breaking point’
(Gould, 1980b: 184-185). I argue that the concept of a “transformation of quantity into quality”
offers an explanation of the existence of discontinuities in nature which does not require reference to
a separate macroevolutionary theory. Quantitative change at the genetic level gives rise to
qualitative change at the morphological level. Change at the level of the genome may be continuous,
but discontinuous at the level of the overall form. If some phenotypic characters do change
continuously it is because the corresponding genetic signals are expressed relatively unfiltered. It is
these characters that have been studied in genetic experiments that have supposedly demonstrated
evolution to be change in gene frequencies. The properties of the developmental system are such that genetic changes, even if copious, small and undirected, can still give rise to specific, large, directed changes of form: ‘These [small, genetic] changes can have substantial impact on adult phenotypes because they operate by altering rates of development early in ontogeny, with cascading effects throughout later growth’ (Gould, 1980a: 45).
A search filter is a search query expressed in the form of a logical expression.
