Traffic bunching

In heavy traffic, such as on motorways in rush-hour, there is often oscillation in speed and there can even be mysterious ’emergent’ halts. The use of variable speed limits can result in everyone getting along a given stretch of road quicker.

Soros (worth reading) has written an article that suggests that this is all to do with the humanity and ‘thinking’ of the drivers, and that something similar is the case for economic and financial booms and busts. This might seem to indicate that ‘mathematical models’ were a part of our problems, not solutions. So I suggest the following thought experiment:

Suppose a huge number of  identical driverless cars with deterministic control functions all try to go along the same road, seeking to optimise performance in terms of ‘progress’ and fuel economy. Will they necessarily succeed, or might there be some ‘tragedy of the commons’ that can only be resolved by some overall regulation? What are the critical factors? Is the nature of the ‘brains’ one of them?

Are these problems the preserve of psychologists, or does mathematics have anything useful to say?

Dave Marsay

Hercock’s Cohesion

Robert G. Hercock Cohesion: The Making of Society 2009.

Having had Robert critique some of my work, I could hardly not comment on this think-piece. It draws on modern complexity theory and a broad view of relevant historical examples and current trends to create a credible narrative. For me, his key conclusions are:

  1. “[G]iven a sufficient degree of communication … the cooperative assembly of [a cohesive society] is inevitable.”
  2. To be cohesive, a society should be “global politically federated, yet culturally diverse”.

The nature of communication envisaged seems to be indicated by:

 “From smoke signals, and the electric telegraph, through to fibre optics, and the Internet … the manifest boom in all forms of communication is bringing immense capabilities to form new social collectives and positive cultural developments.”

 I ‘get’ that increasing communication will bring immense capabilities to support the cooperative assembly of a cohesive global society, but am not convinced the effective exploitation of the capability in this way is inevitable. In chapter 6 (‘Bridges’) Robert says:

 “The truth is we now need a new shared set of beliefs. … Unfortunately, no one appears to have the faintest idea what such a common set of beliefs should look like, or where it might arise from, or who has responsibility to make it happen, or how, etc. Basically this is the challenge of the 21st century; we stand or fall on this battle for a common cultural nexus.”  

 This is closer to my own thinking.

People have different understandings of terms like ‘federated’. My preference is for subsidiarity: the idea that one has the minimum possible governance, with reliance on the minimum possible shared beliefs and common cultures. In complex situations these minimum levels are not obvious or static, so I would see an effective federations as engaging tentatively at a number of ‘levels’, ‘veering and hauling’ between them, and with strong arrangements for ‘horizon scanning’ and debate with the maximum possible diversity of views. Thus there would be not only cultural diversity but ‘viewpoint diversity within federated debate’. What is needed seems somewhat like Holism and glocalization 

Thinking of the EU, diversity of monetary policy might make the EU as an institution more cohesive while making their economies less cohesive. To put it another way, attempts to enforce cohesion at the monetary level can threaten cohesion at the political level. So it is not clear to me that one can think of a society as simply ‘being cohesive’. Rather it should be cohesive in the sense appropriate to its current situation. Cohesion should be ‘adaptive’. Leadership and vision seem to be required to achieve this: it is not automatic.

In the mid 80s many of those involved in the development of communications technologies thought that they would promote world peace, sometimes citing the kind of works that Robert does. I had and have two reservations. Firstly, the quality of communications matters. Thus [it was thought] one probably needed digital video, mobile phones and the Internet, all integrated in way that was easy to use. [The Apple Macintosh made this credible.] Thus, if there was a clash between Soviet secret police and Jewish protestors [common at the time], the whole world could take an informed view, rather than relying on the media. [This was before the development of video faking capabilities]. Secondly, while this would destabilize autocratic regimes, it was another issue as to what would happen next. It was generally felt that the only possible ‘properly’ stable states were democratic, but views differed on whether such states would necessarily stabilize.

Subsequent experience, such as the Arab spring, support the view that YouTube and Facebook undermine oppressive regimes. But I remain unconvinced that ‘the cooperative assembly of [a cohesive society] is inevitable’ in Africa, the Middle East,Russia or South America’, or that more communications would make it so. It certainly seems that if the process is inevitable, it can be much too slow.

My own thinking in the 80s was informed by the uncertainty and complexity theory Keynes, Whitehead, Turing and Smuts, which predates that which Robert cites, and which informed the development of the United Nations as a part of ‘the cooperative assembly of a cohesive global society’. Robert seems to be arguing that according to modern theory such efforts were not necessary, but even so they may have been beneficial if all they did was speed the process up by a few generations. Moreover, the EU example seems to support my view that these theories are usefully more advanced than their contemporary counter-parts.

The financial crash of 2008 occurred part way through the writing of the book. Like any history, explanations differ, and Robert gives a credible account in terms of modern complexity theory. But logic teaches us to be cautious about such post-hoc explanations. It seems to me that Keynes’ theory explains it adequately, and having been developed before the event should be given more credence.

 Robert seems to regard the global crash of 2008 as a result of a loss of cohesion :

“When economies, states and societies lose their cohesion, people suffer; to be precise a lot of people end up paying the cost. In the recession of 2008/09 … “

But Keynes shows how it is cohesion (‘sticking together’) that causes global crashes. Firstly, in a non-globalized economy a crash in one part can be compensated for by the stability of another part, a bit like China saving the situation, but more so. Secondly, (to quote Patton) ‘if everyone is thinking alike then no-one is thinking’. Once group-think is established ‘expectations’ become ossified, and the market is disconnected from reality.

Robert’s notion of cohesion is “global politically federated, yet culturally diverse”. One can see how in 2008 and currently in the EU (and North Africa and elsewhere) de jure and de-facto regulatory structures change, consistent with Robert’s view. But according to Keynes this is a response to an actual or potential crisis, rather than a causative factor. One can have a chain of  crises in which political change leads to emergent social or economic problems, leading to political change and so-on. Robert seems to suppose that this must settle down into some stable federation. If so then perhaps only the core principles will be stable, and even these might need to be continually reinterpreted and refreshed, much as I have tried to do here.

On a more conceptual note, Robert has the qualifies the conclusion with “The evidence from all of the fields considered in this text suggests …”.  But the conclusion could only be formally sustained by an argument employing induction. Now, if improved communications is really going to change the world so much then it will undermine the basis of any induction. (In Whitehead’s terms, induction only works with an epoch but here the epoch is changed.) The best one could say would be that on current trends a move towards greater cohesion appears inevitable. This is a more fundamental problem than only considering evidence from a limited range of fields. More evidence from more fields could not overcome this problem.

Dave Marsay

The End of a Physics Worldview (Kauffman)

Thought provoking, as usual. This video goes beyond his previous work, but in the same direction. His point is that it is a mistake to think of ecologies and economies as if they resembled the typical world of Physics. A previous written version is at npr, followed by a later development.

He builds on Kant’s notion of wholes, noting (as Kant did before him) that the existence of such wholes is inconsistent with classical notions of causality.  He ties this in to biological examples. This complements Prigogine, who did a similar job for modern Physics.

Kauffman is critical of mathematics and ‘mathematization’, but seems unaware of the mathematics of Keynes and Whitehead. Kauffman’s view seems the same as that due to Bergson and Smuts, which in the late 1920s defined ‘modern science’. To me the problem behind the financial crash lies not in science or mathematics or even in economics, but in the brute fact that politicians and financiers were wedded to a pre-modern (pre-Kantian) view of economics and mathematics. Kauffman’s work may help enlighten them on the need, but not on the potential role for modern mathematics.

Kauffman notes that at any one time there are ‘adjacent possibles’ and that in the near future they may come to pass, and that – conceptually – one could associate a probability distribution with these possibilities. But as new possibilities come to pass new adjacent possibilities arise. Kauffman supposes that it is not possible to know what these are, and hence one cannot have a probability distribution, much of information theory makes no sense, and one cannot reason effectively. The challenge, then, is to discover how we do, in fact, reason.

Kauffman does not distinguish between short and long run. If we do so then we see that if we know the adjacent possible then our conventional reasoning is appropriate in the short-term, and Kauffman’s concerns are really about the long-term: beyond the point at which we can see the potential possibles that may arise. To this extent, at least, Kauffman’s post-modern vision seems little different from the modern vision of the 1920s and 30s, before it was trivialized.

Dave Marsay

GLS Shackle, imagined and deemed possible?

Background

This is a personal view of GLS Shackle’s uncertainty. Having previously used Keynes’ approach to identify possible failure modes in systems, including financial systems (in the run-up to the collapse of the tech bubble), I became concerned  in 2007 that there was another bubble with a potential for a Keynes-type  25% drop in equities, constituting a ‘crisis’. In discussions with government advisers I first came across Shackle. The differences between him and Keynes were emphasised. I tried, but failed to make sense of Shackle, so that I could form my own view, but failed. Unfinished business.

Since the crash of 2008 there have been various attempts to compare and contrast Shackle and Keynes, and others. Here I imagine a solution to the conundrum which I deem possible: unless you know different?

Imagined Shackle

Technically, Shackle seems to focus on the wickeder aspects of uncertainty, to seek to explain them and their significance to economists and politicians, and to advise on how to deal with them. Keynes provides a more academic view, covering all kinds of uncertainty, contrasting tame probabilities with wicked uncertainties, helping us to understand both in a language that is better placed to survive the passage of time and the interpretation by a wider – if more technically aware – audience.

Politically, Shackle lacks the baggage of Lord Keynes, whose image has been tarnished by the misuse of the term ‘Keynesian’. (Like Keynes, I am not a Keynesian.)

Conventional probability theory would make sense if the world was a complicated randomizing machine, so that one has ‘the law of large numbers’: that in the long run particular events will tend to occur with some characteristic, stable, frequency. Thus in principle it would be possible to learn the frequency of events, such that reasonably rare events would be about as rare as we expect them to be. Taleb has pointed out that we can never learn the frequencies of very rare events, and that this is a technical flaw in many accounts of probability theory, which fail to point this out. But Keynes and Shackle have more radical concerns.

If we think of the world as a complicated randomizing machine, then as in Whitehead, it is one which can suddenly change. Shackle’s approach, in so far as I understand it, is to be open to the possibility of a change, recognize when the evidence of a change is overwhelming, and to react to it. This is an important difference for the conventional approach, in which all inference is done on the assumptions that the machine is known. Any evidence that it may have change is simply normalised away. Shackle’s approach is clearly superior in all those situations where substantive change can occur.

Shackle terms decisions about a possibly changing world ‘critical’. He makes the point that the application of a predetermined strategy or habit is not a decision proper: all ‘real’ decisions are critical in that they make a lasting difference to the situation. Thus one has strategies for situations that one expects to repeat, and makes decisions about situations that one is trying to ‘move on’. This seems a useful distinction.

Shackle’s approach to critical decisions is to imagine potential changes to new behaviours, to assess them and then to choose between those deemed possible. This is based on preference not expected utility, because ‘probability’ does not make sense. He gives an example of  a French guard at the time of the revolution who can either give access to a key prisoner or not. He expects to lose his life if he makes the wrong decision, depending on whether the revolution succeeds or not. A conventional approach would be based on the realisation that most attempted revolutions fail. But his choice may have a big impact on whether or not the revolution succeeds. So Shackle advocates imagining the two possible outcomes and their impact on him, and then making a choice. This seems reasonable. The situation is one of choice, not probability.

Keynes can support Shackle’s reasoning. But he also supports other types of wicked uncertainty. Firstly, it is not always the case that a change is ‘out of the blue’. One may not be able to predict when the change will come, but it is sometimes possible to see that there is an economic bubble, and the French guard probably had some indications that he was living in extraordinary times. Thus Keynes goes beyond Shackle’s pragmatism.

In reality, there is no strict dualism between probabilistic behaviour and chaos, between probability and Shackle’s complete ignorance. There are regions in-between that Keynes helps explore. For example, the French guard is not faced with a strictly probabilistic situation, but could usefully think in terms of probabilities conditioned on his actions. In economics, one might usefully think of outcomes as conditioned on the survival of conventions and institutions (October 2011).

I also have a clearer view why consideration of Shackle led to the rise in behavioural economics: if one is ‘being open’ and ‘imagining’ then psychology is clearly important. On the other hand, much of behavioral economics seems to use conventional rationality as some form of ‘gold standard’ for reasoning under uncertainty, and to consider departures from it as a ‘bias’.  But then I don’t understand that either!

Addendum

(Feb 2012, after Blue’s comments.)

I have often noticed that decision-takers and their advisers have different views about how to tackle uncertainty, with decision-takers focusing on the non-probabilistic aspects while their advisers (e.g. scientists or at least scientifically trained) tend to, and may even insist on, treating the problem probabilistically, and hence have radically different approaches to problem-solving. Perhaps the situation is crucial for the decision-taker, but routine for the adviser? (‘The agency problem.’) (Econophysics seems to suffer from this.)

I can see how Shackle had much that was potentially helpful in the run-up to the financial crash. But it seems to me no surprise that the neoclassical mainstream was unmoved by it. They didn’t regard the situation as crucial, and didn’t imagine or deem possible a crash. Unless anyone knows different, there seems to be nothing in Shackle’s key ideas that provide as explicit a warning as Keynes. While Shackle was more acceptable that Keynes (lacking the ‘Keynesian’ label) he also still seems less to the point. One needs both together.

See Also

Prigogine , who provides models of systems that can suddenly change ‘become’. He also  relates to Shackle’s discussion on how making decisions relates to the notion of ‘time’.

Dave Marsay

Systemism: the alternative to individualism and holism

Mario Bunge Systemism: the alternative to individualism and holism Journal of Socio-Economics 29 (2000) 147–157

“Three radical worldviews and research approaches are salient in social studies: individualism, holism, and systemism.”

[Systemism] “is centered in the following postulates:
1. Everything, whether concrete or abstract, is a system or an actual or potential component of a system;
2. systems have systemic (emergent) features that their components lack, whence
3. all problems should be approached in a systemic rather than in a sectoral fashion;
4. all ideas should be put together into systems (theories); and
5. the testing of anything, whether idea or artifact, assumes the validity of other items, which are taken as benchmarks, at least for the time being.”

Thus systemism resembles Smuts’ Holism. Bunge uses the term ‘holism’ for what Smuts terms wholism: the notion that systems should be subservient to their ‘top’ level, the ‘whole’. This usage apart, Bunge appears to be saying something important. Like Smuts, he notes the systemic nature of mathematics is distinction to those who note the tendency to apply mathematical formulae thoughtlessly, as in some notorious financial mathematics

Much of the main body is taken up with the need for micro-macro analyses and the limitations of piece-meal approaches, something familiar to Smuts and |Keynes. On the other hand he says: “I support the systems that benefit me, and sabotage those that hurt me.” without flagging up the limitations of such an approach in complex situations. He even suggests that an interdisciplinary subject such as biochemistry is nothing but the overlap of the two disciplines. If this is the case, I find it hard to grasp their importance. I would take a Kantian view, in which bringing into communion two disciplines can be more than the sum of the parts.

In general, Bunge’s arguments in favour of what he calls systemism and Smuts called holism seem sound, but it lacks the insights into complexity and uncertainty of the original.

See also

Andy Denis’ response to Bunge adds some arguments in favour of Holism. It’s main purpose, though, is to contradict Bunge’s assertion that laissez-faire is incompatible with systemism. It is argued that a belief in Adam Smith’s invisible hand could support laissez faire. It is not clear what might constitute grounds for such a belief. (My own view is that even a government that sought to leverage the invisible hand would have a duty to monitor the workings of such and hand, and to take action should it fail, as in the economic crisis of 2007/8. It is now clear how politics might facilitate this.)

Also my complexity.

Dave Marsay

From Being to Becoming

I. Prigogine, From Being to Becoming: Time and Complexity in the Physical Sciences, WH Freeman, 1980 

 See new page.

Summary

“This book is about time.” But it has much to say about complexity, uncertainty, probability, dynamics and entropy. It builds on his Nobel lecture, re-using many of the models and arguments, but taking them further.

Being is classically modelled by a state within a landscape, subject to a fixed ‘master equation’ describing changes with time. The state may be an attribute of an object (classical dynamics) or a probability ‘wave’ (quantum mechanics). [This unification seems most fruitful.] Such change is ‘reversible’ in the sense that if one reverses the ‘arrow of time’ one still has a dynamical system.

Becoming refers to more fundamental, irreversible, change, typical of ‘complex systems’ in chemistry, biology and sociology, for example. 

The book reviews the state of the art in theories of Being and Becoming, providing the hooks for its later reconciliation. Both sets of theories are phenomenological – about behaviours. Prigogine shows that not only is there no known link between the two theories, but that they are incompatible.

Prigogine’s approach is to replace the notion of Being as being represented by a state, analogous to a point in a vector space,  by that of an ‘operator’ within something like a Hilbert Space. Stable operators can be thought of as conventional states, but operators can become unstable, which leads to non-statelike behaviours. Prigogine shows how in some cases this can give rise to ‘becoming’.

This would, in itself, seem a great and much needed subject for a book, but Prigogine goes on to consider the consequences for time. He shows how time arises from the operators. If everything is simple and stable then one has classical time. But if the operators are complex then one can have a multitude of times at different rates, which may be erratic or unstable. I haven’t got my head around this bit yet.

Some Quotes

Preface

… the main thesis …can be formulated as:

  1. Irreversible processes are as real as reversible ones …
  2. Irreversible processes play a fundamental constructive role in the physical world …
  3. Irreversibility … corresponds … to an embedding of dynamics within a vaster formalism. [Processes instead of points.] (xiii)

The classical, often called “Galilean,” view of science was to regard the world as an “object,” to try to describe the physical world as if it were being seen from the outside as an object of analysis to which we do not belong. (xv)

… in physics, as in sociology, only various possible “scenarios” can be predicted. [One cannot predict actual outcomes, only identify possibilities.] (xvii)

Introduction

… dynamics … seemed to form a closed universal system, capable of yielding the answer to any question asked. (3)

… Newtonian dynamics is replaced by quantum mechanics and by relativistic mechanics. However, these new forms of dynamics … have inherited the idea of Newtonian physics: a static universe, a universe of being without becoming. (4)

The Physics of Becoming

The interplay between function, structure and fluctuations leads to the most unexpected phenomena, including order through fluctuations … . (101)

… chemical instabilities involve long-range order through which the system acts as a whole. (104)

… the system obeys deterministic laws [as in classical dynamics] between two bifurcation points, but in the neighbourhood of the bifurcation points fluctuations play an essential role and determine the “branch” that the system will follow. (106) [This is termed ‘structurally unstable”]

.. a cyclic network of reactions [is] called a hypercycle. When such networks compete with one another, they display the ability the ability to evolve through mutation and replication into greater complexity. …
The concept of structural stability seems to express in the most compact way the idea of innovation, the appearance of a new mechanism and a new species, … . (109)

… the origin of life may be related to successive instabilities somewhat analogous to the successive bifurcations that have led to a state of matter of increasing coherence. (123)

As an example, … consider the problem of urban evolution … (124) … such a model offers a new basis for the understanding of “structure” resulting from the actions (choices) of the many agents in a system, having in part at least mutually dependent criteria of action. (126)

… there are no limits to structural instability. Every system may present instabilities when suitable perturbations are introduced. Therefore, there can be no end to history. [DJM emphasis.] … we have … the constant generation of “new types” and “new ideas” that may be incorporated into the structure of the system, causing its continual evolution. (128)

… near bifurcations the law of large numbers essentially breaks down.
In general, fluctuations play a minor role … . However, near bifurcations they play a critical role because there the fluctuation drives the average. This is the very meaning of the concept of order through fluctuations .. . (132)

… near a bifurcation point, nature always finds some clever way to avoid the consequences of the law of large numbers through an appropriate nucleation process. (134)

… For small-scale fluctuations, boundary effects will dominate and fluctuations will regress. … for large-scale fluctuations, boundary effects become negligible. Between these limiting cases lies the actual size of nucleation. (146)

… We may expect that in systems that are very complex, in the sense that there are many interacting species or components, [the degree of coupling between the system and its surroundings] will be very large, as will be the size of the fluctuation which could start the instability. Therefore … a sufficiently complex system is generally in a metastable state. (147) [But see Comments below.]

… Near instabilities, there are large fluctuations that lead to a breakdown of the usual laws of probability theory. (150)

The Bridge from Being to Becoming

[As foreshadowed by Bohr] we have a new form of complimentarity – one between the dynamical and thermodynamic descriptions. (174)

… Irreversibility is the manifestation on a macroscopic scale of “randomness” on a microscopic scale. (178)

Contrary to what Boltzmann attempted to show there is no “deduction” of irreversibility from randomness – they are only cousins! (177)

The Microscopic Theory of Irreversible Processes

The step made … is quite crucial. We go from the dynamical system in terms of trajectories or wave packets to a description in terms of processes. (186)

… Various mechanisms may be involved, the important element being that they lead to a complexity on the microscopic level such that the basic concepts involved in the trajectory or wave function must be superseded by a statistical ensemble. (194)

The classical order was: particles first, the second law later – being before becoming! It is possible that this is no longer so when we come to the level of elementary particles and that here we must first introduce the second law before being able to define the entities. (199)

The Laws of Change

… Of special interest is the close relation between fluctuations and bifurcations which leads to deep alterations in the classical results of probability theory. The law of large numbers is no longer valid near bifurcations and the unicity of the solution of … equations for the probability distribution is lost. (204)

This mathematization leads us to a new concept of time and irreversibility … . (206)

… the classical description in terms of trajectories has to be given up either because of instability and randomness on the microscopic level or because of quantum “correlations”. (207)

… the new concept implies that age depends on the distribution itself and is therefore no longer an external parameter, a simple label as in the conventional formula.
We see how deeply the new approach modifies our traditional view of time, which now emerges as a kind of average over “individual times” of the ensemble. (210)

For a long time, the absolute predictability of classical mechanics, or the physics of being, was considered to be an essential element of the scientific picture of the physical world. … the scientific picture has shifted toward a new, more subtle conception in which both deterministic features and stochastic features play an essential role. (210)

The basis of classical physics was the conviction that the future is determined by the present, and therefore a careful study of the present permits the unveiling of the future. At no time, however, was this more than a theoretical possibility. Yet in some sense this unlimited predictability was an essential element of the scientific picture of the physical world. We may perhaps even call this the founding myth of classical science.
The situation is greatly changed today. … The incorporation of the limitation of our ways of acting on nature has been an essential element of progress. (214)

Have we lost essential elements of classical science in this recent evolution [of thought]? The increased limitation of deterministic laws means that we go from a universe that is closed to one that is open to fluctuations. to innovations.

… perhaps there is a more subtle form of reality that involves both laws and games, time and eternity. (215) 

Comments

Relationship to previous work

This book can be seen as a development of the work of Kant, Whitehead and Smuts on emergence, although – curiously – it makes little reference to them [pg xvii]. In their terms, reality cannot logically be described in terms of point-like states within spaces with fixed ‘master equations’ that govern their dynamics. Instead, it needs to be described in terms of ‘processes’. Prigogine goes beyond this by developing explicit mathematical models as examples of emergence (from being to becoming) within physics and chemistry.

Metastability

According to the quote above, sufficiently complex systems are inherently metastable. Some have supposed that globalisation inevitably leads to an inter-connected and hence complex and hence stable world. But globalisation could lead to homogenization or fungibility, a reduction in complexity and hence an increased vulnerability to fluctuations. As ever, details matter.

See Also

I. Prigogine and I. Strengers Order out of Chaos Heinemann 1984.
This is an update of a popular work on Prigogine’s theory of dissipative systems. He provides an unsympathetic account of Kant’s Critique of Pure Reason, supposing Kant to hold that there are “a unique set of principles on which science is based” without making reference to Kants’ concept of emergence, or of the role of communities. But he does set his work within the framework of Whitehead’s Process and Reality. Smuts’ Holism and Evolution, which draws on Kant and mirrors Whitehead is also relevant, as a popular and influential account of the 1920s, helping to define the then ‘modern science’.

Dave Marsay

Composability

State of the art – software engineering

Composability is a system design principle that deals with the inter-relationships of components. A highly composable system provides recombinant components that can be selected and assembled in various combinations … .”For information systems, from a software engineering perspective,  the essential features are regarded as modularity and statelessness. Current inhibitors include:  

“Lack of clear composition semantics that describe the intention of the composition and allow to manage change propagation.”

Broader context

Composability has a natural interpretation as readiness to be composed with others, and has broader applicability. For example, one suspects that if some people met their own clone, they would not be able to collaborate. Quite generally, composability would seem necessary but perhaps not sufficient to ‘good’ behaviour. Thus each culture tends to develop ways for people to work effectively together, but some sub-cultures seem parasitic, in that they couldn’t sustain themselves on their own.

Cultures tend to evolve, but technical interventions tend to be designed. How can we be sure that the resultant systems are viable under evolutionary pressure? Composability would seem to be an important element, as it allows elements to be re-used and recombined, with the aspiration of supporting change propagation.

Analysis

Composability is particularly evident, and important, in algorithms in statistics and data fusion.  If modularity and statelessness are important for the implementation of the algorithms, it is clear that there are also characteristics of the algorithms as functions (ignoring internal details) that are also important.

If we partition a given data set, apply a function to the parts and the combine the result, we want to get the same result no matter how the data is partitioned. That is, we want the result to depend on the data, not the partitioning.

In elections for example, it is not necessarily true that a party who gets a majority of the votes overall will get the most candidates elected. This lack of composability can lead to a loss of confidence in the electoral process. Similarly, media coverage is often an editor’s precis of the precis by different reporters. One would hope that a similar story would emerge if one reporter had covered the whole. 

More technically, averages over parts cannot, in general, be combined to give a true overall average, whereas counting and summing are composable. Desired functions can often be computed composably by using a preparation function, then composable function, then a projection or interpretation function. Thus an average can be computed by finding the number of terms averaged, reporting the sum and count, summing over parts to give an overall sum and count, then projecting to get the average. If a given function can be implented via two or more composable functions, then those functions must be ‘conjugate’: the same up to some change of basis. (For example, multiplication is composable, but one could prepare using logs and project using exponentiation to calculate a product using a sum.)

In any domain, then, it is natural to look for composable functions and to implement algorithms in terms of them. This seems to have been widespread practice until the late 1980s, when it became more common to implement algorithms directly and then to worry about how to distribute them.

Iterative Composability

In some cases it is not possible to determine composable functions in advance, or perhaps at all. For example, where innovation can take place, or one is otherwise ignorant of what may be. Here one may look for a form of ‘iterative composability’ in which one hopes tha the results is normally adequate, there will be signs if it is not, and that one will be able to improve the situation. What matters is that this process should converge, so that one can get as close as one likes to the results one would get from using all the data.

Elections under FPTP (first past the post) are not composable, and one cannot tell if the party who is most voter’s first preference has failed to get in. AV (alternative vote) is also not composable, but one has more information (voters give rankings) and so can sometimes tell that there cannot have been a party who was most voters first preference who failed to get in. If there can have been, one could have a second round with only the top parties’ candidates. This is a partial step towards general iterative composability, which might often be iteratively composable for the given situation, much more so than fptp.

Parametric estimation is generally composable when one has a fixed number of entities whose parameters are being estimated. Otherwise one has an ‘association’ problem, which might be tackled differently for the different parts. If so, this needs to be detected and remedied, perhaps iteratively. This is effectively a form of hypothesis testing. Here the problem is that the testing of hypotheses using likelihood ratios is not composable. But, again, if hypotheses are compared differences can be detected and remedial action taken. It is less obvious that this process will converge, but for constrained hypothesis spaces it does.

Innovation, transformation, freedom and rationality

It is common to suppose that people acting in their environment should characterise their situation within a context in enough detail to removes all but (numeric) probabilistic uncertainty, so that they can optimize. Acting sub-optimally, it is supposed, would not be rational. But if innovation is about transformation then a supposedly rational act may undermine the context of another, leading to a loss of performance and possibly crisis or chaos.

Simultaneous innovation could be managed by having an over-arching policy or plan, but this would clearly constrain freedom and hence genuine innovation. To much innovation and one has chaos, too little and there is too little progress.

A composable approach is to seek innovations that respect each other’s contexts, and to make clear to other’s what one’s essential context is. This supports only very timid innovation if the innovation is rational (in the above sense), since no true (Knightian) uncertainty can be accepted. A more composable approach is to seek to minimise dependencies and to innovate in a way that accepts – possibly embraces – true uncertainty. This necessitates a deep understanding of the situation and its potentialities.  

Conclusion

Composability is an important concept that can be applied quite generally. The structure of activity shouldn’t impact on the outcome of the activity (other than resource usage). This can mean developing core components that provide a sound infrastructure, and then adapting it to perform the desired tasks, rather than seeking to implement the desired functionality directly.

Dave Marsay