Non-Standard Computation : an overview

Modification history

This is a highly compressed whistle-stop tour of non-standard computation -- what it is, and what interesting, valuable and deep research can be done in this area.

Non-Standard computation is in contrast to "standard" or "classical" computation, by which I mean our classical Turing / von Neumann perspective. Over the last decade or so we have begun to understand just how narrow a perspective this is.

Our appreciation of the physical embodiment of computation, our development of biologically-inspired computational models, and our understanding of emergent systems has reached exciting heights. Breakthroughs in all these areas are occurring, or are on the horizon.

Firstly, they have begun to reach sufficient maturity that common themes can be observed within and between these areas, that interesting and deep research questions can be asked. Yet they are not so well developed that all their novelty has been mined -- there are a lot of exciting fundamental new discoveries still to be made. The time is right to commit to Research Programmes in non-standard computation.

Secondly, there is now sufficient computational power and capacity available to implement or simulate large, interesting models -- getting qualitatively different results impossible to obtain any other way. For example, the UK's development of e-science and the GRID could provide such power. Additionally, non-von Neumann style devices such as Field Programmable Gate Arrays (FPGAs) offer exciting new possibilities for large scale simulations.

Embodied in physics : non-Turing

There is a categorical difference between a computational model and the computational device that might implement it. A model is a mathematical abstract description, say of an algorithm. A computer is a physical device existing in the real world, subject to errors, wear and tear, finiteness, and the actual laws of physics themselves, that progresses through a series of steps to execute the algorithm.

The model tends to get most of the study in computer science. But we need to make explicit just how the choice of model is driven by the physical reality, and just how the view of what is computable is driven by the choice of model.

It is well-known that analogue computational models describe computations that cannot be achieved in discrete digital computational models -- they can achieve what to a digital model looks like an infinite amount of computation in a finite time. Since the physical world is itself digital, these analogue models cannot be effectively implemented. (However, the world is digital only on very small scales. How much computation can an analogue implementation actually do?)

Quantum computational models also describe computations that cannot be achieved effectively with classical digital models. But the physical world is quantum mechanical, so these models potentially can be realised. Turing machines are not an adequate description of computability. The real world routinely performs quantum-mechanical "computations" that cannot be effectively implemented on a Turing machine. Since the Turing machine is the basis of all our theory of computation, algorithms, programming languages, specification models, refinement calculi, and so on, a lot of work needs to be done to upgrade these to include quantum models. We might have to wait a while for commercial quantum computers, but when they arrive, Moore's law suggests they will grow in power very quickly. Doubling a classical computer's register length (roughly) doubles its computing speed, but adding just one bit to a quantum computer's register doubles its speed. We need to be ready to exploit them once they arppear.

When we perform a real computation, we choose to interpret certain observations of the executing physical device as corresponding to aspects of the mathematical model. But the concrete physical device includes features not in the abstract computational model. (Consider protein folding -- if DNA is modelled abstractly just as a string of bases, then it is puzzling that greatly separated positions are correlated. The solution is clear once we consider the physical realisation in the concrete real world: these correspond to spatially nearby positions when the corresponding protein is folded in 3D.) We can choose to observe other features, such as the time taken by a computation, or the power drain, which might correspond to different computations. Known to the security world as "side channels", these have recently come to the fore in Differential Power Analysis (DPA), a way of cracking crypto algorithms. Focussing on the abstract model to the detriment of the physical implementation leads to oversights.

Cellular automata are inspired by certain physical models (crystals, spin networks, etc.) Ironically for a so-called "non-Von" paradigm, they were developed by von Neumann. They perform their computations as an array of simple cells, each a state machine that evolve through time in a way dependent on its own current state and the states of neighbouring cells. Conway's "Game of Life" is the most well-known CA. CAs can act as universal computers, and have emergent structures (such as the famous "gliders"). Wolfram has studied 1D CAs in detail, and shown their close connections with chaos theory. Unsurprisingly, due to some of their inspiration, they can be used for physics simulations, from study of condensed matter phase transitions to self-organising quantum gravity spin networks.

Inspired by biology : non-Von, non-symbolic

Our computing artefacts are not the only things that can be considered to perform computations. Computation is in the eye of the beholder, and we can choose to perceive many "natural" devices as performing computations. And these do not fall into the classical von Neuman model. Since much of the natural world can be perceived as doing some form of search, optimisation or pattern recognition, and since many problems of interest can be cast in these terms, these computations, as well as being interesting in their own right, hold the promise of inspiring novel problem-solving techniques.

These approaches tend to be "non-symbolic" computations. They explore a staggeringly vast phase space, yet can necessarily consider only a vanishingly small part of it. Obviously they can provide no guarantees of optimal results. However, they are astonishingly good at providing "satisficing" results. (But we get too hung up on theoretical best case analyses: "good enough" really is good enough.) They also tend to be robust in the presence of noise, error, and change -- in other words, in the presence of the real world.

Neural networks are inspired by the workings of the brain. A network of weighted nodes can be used as a robust pattern matcher, and pattern categoriser. The "learning algorithms" that decide on the weights are a variety of optimisation algorithms, not as biologically inspired as the network itself.

Darwinian evolution provides another inspiration. Genetic algorithms, genetic programming, and much Artificial Life research uses concepts such as mating, variation and selection of the fittest from biological evolution.

Evolution in the wild involves many species. Ecology studies heterogeneous agents coevolving, involved in cooperation and parasitism, in arms races, and forming dependency webs.

Darwinian evolution is slow. Nothing the agents learn during their lives is passed on to their offspring, except at second order. Lamarckian evolution is faster -- and can be considered the route taken by intelligent, learning agents. Economics, trading, and so on can be thought of in Lamarckian evolutionary terms. Swarmware -- flocking or schooling behaviour -- is inspired by the behaviour of large swarms of similar agents copying their neighbours' behaviours, and thereby potentially improving their own.

The immune system is another biological pattern recognition and classification system -- it learns to distinguish self from other, and destroy the other. Additionally, the immune system performs important maintenance and repair functions. Whereas evolutionary paradigms are interested in improving the performance of best of a large population of essentially similar agents, immune systems' behaviour is an emergent property of the entire population of diverse agents, and improve by in weeding out the weakest players, replacing them by agents as different as possible from them. The immune system is probably currently one of the computationally least understood of all the biological paradigms.

Biologically-inspired models are often used where the problem is too difficult to solve by more conventional means, and part of that difficulty lies in determining an appropriate initial state for a large complex system. Appealing again to biology, one solution is to start from a relatively simple initial state, and let it "grow" to a suitable maturity, rather than attempting to "switch on" a large arbitrary, or hand-crafted, configuration that is almost certainly non-saisficing. So one area of biology that should have important lessons for computing is embryology, the study of development from a single cell to an entire organism. For example, nanotechnology assemblers might have to grow their own numbers before they start to build the artefact of interest. In a multi-agent network, the initial population might start small, and decide for itself how to grow to fulfil its purpose.

Intelligence via embodiment and bottom-up layering of functionality is promulgated as an alternative to top-down symbolic processing. Brooks' subsumption architecture, inspired partly by insect models, has made great strides in intelligent robotics. Collision avoidance, path finding, mapping, and so on can arise naturally from layered architectures. Embodiment ensures that problems of noisy sensors and faulty actuators are addressed, and robust solutions found.

These techniques can be fruitfully combined, too. Evolutionary algorithms have been used to find good neural network topologies. CAs can be used to implement swarm algorithms. Small neural nets can be used for a layer in a subsumption architecture. And so on. This areas is in its infancy. There is vast potential for using biologically-inspired algorithms to optimise and classify a vast range of problem -- and of meta-problems (finding not only good values of problem-specific parameters, but second order parameters describing the shape of the problem space itself, then third order parameters, and so on).

We can choose to study these natural computational models for two separate, independently valuable reasons. • We can try to understand how they work in the real world, getting a deeper understanding of biology, or economics. • And we can let them inspire ways to solve our own, different problems -- emphasising the parts that help, de-emphasising the parts that do not.

Emergent systems : more is different

Chaos theory, complex adaptive self-organising systems, artificial life: all these areas of study show that systems become qualitatively different when they have many, rather than a few, components. Such systems have emergent properties -- properties of the entire system that are not properties of the individual components. As our own artefacts grow in complexity, they too exhibit emergent properties, some desirable, some rather less so.

Currently the term "emergent" is often used as a synonym for "unpredictable" or just plain"incomprehensible". However, if we are ever to advance beyond simple computational artefacts, we need a science of emergence. It may be true that the precise prediction of a complex system is impossible: it may be computationally irreducible. But that does not mean that no progress can be made (any more than the existence of the Halting Problem prevents us from proving termination of interesting programs). We can better understand the process of emergence: we can classify, we can discover structures and processes, we can develop laws governing general properties, we can find areas of greater predictability, and so on.

The role of contingency is crucial. Given the vastness of the phase space explored by complex growing systems, they can explore even in principle only a vanishingly small portion of it: they are highly non-ergodic. (For example, the number of proteins actually expressed is vanishingly small compared to the number that could be expressed in the age of the universe, which itself is even more vanishingly small compared to the total number combinatorically possible.) In nature, precisely which portion is explored is controlled as much by chance as by necessity. The role that chance plays on engineered emergence needs to be explored further.

Complex systems should be "grown" rather than "switched on". The form of growth is a kind of emergent property, too. For example, an entire individual grows, emergently, from a single cell. We are used to thinking of dynamical systems having intial transients that decay soon after being switched on. When a system instead grows, the picture is very different. The "transient" never decays: intial conditions crucially affect future development. We are also used to thinking in equilibrium terms: an evolving, growing, in some sense living, system is a far-from-equilibrium system; we need new laws to describe and new intuitions to think about such systems.

Change is an important property of emergent systems. As a system grows, as its scope increases, the way it works needs to change. As a system's environment changes (possibly in reaction to the system's own actions, possibly for other external reasons) the system needs to change the way it interacts. Static structures, processes and rules are not a good model when change is needed. Not only do we need to engineer the initially desired emergent properties, we need to engineer ability to change in desired, if unanticipated, ways.

The behaviour of a complex emergent system is as strongly influenced by its environment as by its intrinsic structure. Evolved systems can be considered to encode interesting aspects of their environments that they need to respond to. Move a system into a different environment, and it ceases to be such an effective solution to the problem (of survival in the natural world, of its engineered purpose if it is a designed artefact). The same system may solve different problems in different environments; different systems may solve the same problem in different environments. The effect of environment on emergence needs to be studied.

An emergent property is at a higher level than its component parts, for example, cells emerging from molecules, organisms emerging from cells. As the example shows, systems can have many layers of emergence, each providing components for the next. Current experiments in generating emergent properties seem to "stick" at around two levels of emergent hierarchy. This may be because the assumed combinators, or laws, are fixed. Real emergent systems have hierarchical emergent laws, too. The emergence of hierarchical combinators needs to be studied.

If Kauffman's arguments about quantum decoherence are right, that the universe is the way it is because it has to be that way, and if Rees and Smolin are right, that the parameters are very finely tuned to allow complexity -- then maybe we can't build in arbitrary physical laws to our simulations and hope to get realistic complex outputs -- maybe we have to build in the real physical laws?

This may not be a fundamental problem, however. As opposed to natural emergent systems, there are also artificial ones -- such as global economics emerging from local trading rules, and other social systems. They have key differences from natural systems: the system's existence is not so dependent on precise parameter tuning (as evidenced by the existence of different kinds of such systems); the components (now agents) may break the "laws"; and the "laws" may differ at different places and times, even to the extent of contradicting other laws. Such artificial systems are potentially of even more interest than natural emergence, since we want to engineer emergence in our artefacts, since we want to achieve a purpose.

Non-standard computational Research programme possibilities

The realisation that physical embodiment affects computational models, and that biological systems can be considered as computational models, should change the way we look at computer science. Common themes run through these areas: massive parallelism, emergent properties, and simulation. These are enormous areas, and several research projects spring to mind. A few of these are outlined below, and in the accompanying FPGA page.

1. A science of emergence

We would like to be able to design complex multi-agent systems to have desirable, and not to have undesirable, emergent properties. There is currently little or no theory about how to do so.

1.1 Understanding emergence

We need to understand the current status of emergence research, to build up a language and classification of emergent systems.

1.2 An emergence laboratory

We need to be able to experiment with emergent systems, to increase understanding, to answer questions that require further empirical evidence, and as a route to eventual design. (Computational resources for investigating emergent properties are likely to be large. An FPGA simulator might be appropriate.)

1.3 Engineering emergence

We need to be able to design and build systems with desired emergent properties.

2. Artificial biology

I present the two areas I am most interested in, since they fit well with my main interest of Engineering Emergence. The many other areas of artifical biology have been relegated to section 4, and left as a series of interesting questions.

2.1 Artificial Immunology

2.2 Artificial Embryology

3. Quantum computing

Much effort is going into building actual quantum computers: that is not my interest here. Rather, we should be ready to exploit these devices once they appear. Current descriptions of quantum algorithms talk in terms of manipulating unitary matrices, which is equivalent to describing classical algorithms in terms of logic gates. (Much) higher level languages are needed, and all the support paraphernalia that goes with them for them both to have have a solid foundation, and to be usable in software development.

3.1 Quantum computer science

3.2 Quantum software engineering

Computational resources for simulating quantum algorithms are likely to be exponentially large. An FPGA simulator might be appropriate.

4. Other biologically inspired computational models

Neural networks. The "learning algorithms" that decide on the weights are a variety of optimisation algorithms, not as biologically inspired as the network itself. • Can analogues of the chemical communication in the brain be usefully added? • Can more biologically-inspired learning algorithms be developed?

Darwinian evolution. • Can different biologically-inspired mating algorithms be developed? Asexual, "symbiotic" (as in mitochondria in cells), polyploidal, "gene-hopping", ... • Can more "natural" chromosomal encodings be found? • Can better cost functions be developed?

Ecology. • What is the best way to use coevolution to breed more robust genetic algorithm solutions?

Subsumption architecture. • How can subsumption devices be designed to perform desired tasks? • How to get many small devices to cooperate to perform larger tasks.

Cellular automata. • What is the effect of dimensionality, even fractal dimensions? • When to use synchronous v. asynchronous updates? • What are the trade-off with size of neighbourhood, shape of neighbourhood (square, hexagonal, quasi-crystal Penrose tilings, irregular), and the number of states? • What can non-deterministic CAs do? • What can quantum CAs do? • What would be a sensible high level CA programming language?

Physical devices. • What other kinds of computational models can a physical device present? • How many different models can a single device present? • How can we deduce what models are possible in advance? • How can we design in, and design out, suitable and unsuitable observational models from our devices?


I would like to thank John Clark for several stimulating discussions in this area, and for commenting on these drafts.