AI Expert Newsletter

AI - The art and science of making computers do interesting things that are not in their nature.

June 2005

• Introduction
• Neural Gases
  • Running the DemoGNG applet
  • The neural gas
  • The growing neural gas
  • Applications
  • Links
• There's a Hole in my Budget - qualitative reasoning, economics, and Flanders and Swann
  • Flanders and Swann in Prolog
  • Infinite loops, control-structure bugs, and the twist in the tale
  • Qualitative reasoning
  • From novice economist to expert
  • Cognitive models and teaching
  • How to improve your research output
  • Links and other references
• "Lisp has all the visual appeal of oatmeal with fingernail clippings mixed in"
  • Links

Introduction

In our March issue, I explored some of the neural-net applets around the Web. One, intriguing because of the slowly growing network of lime green dots which writhes around inside as if struggling to escape, was the DemoGNG applet for competitive learning, written by Hartmut Loos and Bernd Fritzke at Bochum University. The applet says it implements "neural gases" and "growing neural gases"; since I'd not come across these before, I decided to write about them for this issue. There are also a few items vaguely connected with Lisp, some fun, some serious; and - inspired by a British comic song - a meander through qualitative reasoning, cognitive models in education, and economics. As ever, we welcome news, views, and comments.

Neural Gases

So what's a neural gas? The DemoGNG applet page mentioned above links to the applet source code and documentation, to a user guide, and to the authors' report Some Competitive Learning Methods. This explains that a neural gas is a particular kind of competitive learner. In general, competitive learning networks learn to cluster or classify their inputs, with different output nodes representing different clusters or classes. The outputs compete for inputs to represent. Kohonen self-organising feature maps are a well-known example. (As the comp.ai.neural-nets FAQ explains in How many kinds of Kohonen networks exist?, there are several different kinds, with subtleties that are often ignored.)

Sometimes when using competitive learners for clustering, we want the clusters for their own sake, to tell us about the structure of the data. Sometimes we want to reduce each cluster to a representative element, as in signal quantisation. This first trains the learner on a sample of typical input signals, giving a set of clusters. Hopefully, the number of clusters is much less than the number of possible signals. We then allocate each cluster a number, and compress future data by mapping each signal to the number of the cluster it falls in, transmitting the number in place of the signal. And sometimes, we want information about the connectivity between clusters. At the end of this article, there's a nifty example, skeletonising images of bacterial colonies to line segments. For more information on competitive learners, see the Some Competitive Learning Methods report: it's a nice comparison of competitive learners, using uniform terminology throughout, and not limited to neural gases.

Let's now turn from competitive learners in general to neural gases in particular, and a demonstration.

Running the DemoGNG applet

Start by loading DemoGNG. It has a lot of controls, of which I'll use probability distribution (just above bottom left); max nodes (to its right); start, stop, and reset; and the network model menu (at the top). Note that after a run, you need to press stop, and then reset to discard the current network and make a new. There's a speed control on the right, which you may want to change to suit your browser and computer.

Change the probability distribution to Rectangular. This draws a pale blue rectangle depicting the space of possible signals. On starting a run, the applet will create a network, and then train it by repeatedly feeding it two-dimensional input vectors which have an equal chance of lying anywhere within the rectangle, i.e. a uniform probability distribution.

Check the network model menu is set to its default of "growing neural net". Then press start, and see the network grow to the maximum number of nodes. These will be roughly evenly distributed within the rectangle.

Next, start another run in the rectangle, but before the run ends, change probability distribution to Ring. The network will flow and deform to redistribute its nodes evenly within the ring - a good example because it's highly symmetric. It's fun to see how nodes just inside the hole move toward and link to those in the body of the ring, breaking connections with their old neighbours. It reminds me of how new friendships form and old ones dissolve as one moves from one city to another.

You can move nodes yourself by left-clicking and dragging them. You can also move the entire signal distribution: Change probability distribution to Right Mouse. This will draw a small rectangle for the input space. Pressing the right mouse button will jump it to a new position; you can then release the button to leave it there, or drag and then release. This is a nice way to see how the network adapts to fast and slow changes in its inputs.

The neural gas

Now we've demonstrated growing neural gases, let's see how they work. We'll start with neural gases, because growing neural gases combine these with other components. You can get a feel for what a neural gas does and how fast it does it from the applet. From the network model menu, select Neural Gas. Change probability distribution to Rectangular, and max nodes to 10. If you just ended a run, press reset to generate a new network. Press start, and watch the nodes spread out evenly. Then try with 50 nodes.

Here's the essence of the algorithm:

  Initialize the network to contain N randomly distributed nodes.



  Then

    Take the next input, i.e. a vector, which in this

    demo is two-dimensional.

  

    Calculate each node's distance from the input.

  

    Move the node nearest to the input - the winning node -

    still nearer it.



    Move the second-nearest node nearer the input too,

    but not so far.



    Repeat with all the other nodes. As with the nearest

    and second-nearest, they are moved less far as their

    distance increases.

  And repeat the loop

The complete algorithm as written in the Neural Gas page of Some Competitive Learning Methods contains calculations not shown above. These determine how far each node moves towards the input, and how the movement depends on time. The distance moved is a product of three factors. The first, the ε term, decreases with time, and gives a kind of annealing, so that movements are large when finding an initial solution, but get smaller as the net refines it. The second, h_λ, term, decreases with time and also with how far a node is from winning. (A quick qualitative-reasoning check on whether factors increase or decrease with their arguments helps when reading these algorithms, to avoid getting lost in details.) The third factor is simply the vector distance between the node and the input. Note that the report sometimes uses the subscript _i to mean "initial", as well as an index.

The algorithm doesn't come with much explanation, but there's a readable introduction in Chapter 2 of Growing Neural Gas. Experiments with GNG, GNG with Utility and Supervised GNG, a Master's thesis by Jim Holmström, Uppsala. Some notation, such as the symbol for an input vector, differs between this and Some Competitive Learning Methods. It's worth exploring how the parameters in the bottom line of menus affect learning; the course notes on Classification and Clustering Using Artificial Neural Networks by Hamed Muhammed at Uppsala suggest some experiments.

The growing neural gas

The growing neural gas algorithm is described on the Growing Neural Gas page of Some Competitive Learning Methods. It starts with two nodes and grows its network a node at a time, using links generated by competitive Hebbian learning to guide their placement. Since these weren't in the neural gas, let's look at them next.

Competitive Hebbian learning, in this sense, trains a set of nodes on a sequence of inputs. Unlike in the neural gas, the nodes don't move; the goal is for them to grow links to one another as learning proceeds. A link between two nodes means the nodes are close together - neighbours - relative to the average spacing of the inputs. Links only grow between nodes in regions where inputs occur, so reflect the distribution of inputs. The mathematics behind this, cited in the Competitive Hebbian Learning page, and described in Topology representing networks by Thomas Martinetz and Klaus Schulten, is specialised; but you can get a feel for where links place themselves by using the applet. Set the network model to Competitive Hebbian Learning, and then run it on various probability distributions, remembering that inputs are only generated in the pale blue areas.

As well as inserting new nodes, we move existing ones each time the network receives an input; but instead of moving all of them as in a plain neural gas, we move only two. One of these is the winner, i.e. the one nearest the input; the other is the one of its neighbours (i.e. the nodes linked to it) closest to the input.

When nodes move, they drag their links with them. The growing neural gas has an aging mechanism to remove old links and nodes that no longer reflect the distribution of inputs.

Growing neural gases are an elegant idea, and tempting because they don't need to know the input distribution. However, as ever with machine learning, it's worth testing learning performance, comparing with alternative methods, and looking at the statistical basics. Let's now turn to applications.

Applications

If you're interested in neural gases, Google will turn up an assortment of applications. One possibility, suggested in the Weblog of Semantic Web researcher Leo Sauermann, is analysing large semantic graphs. This would be an interesting experiment, as would others using non-numeric inputs. You'd need a measure of distance between inputs, but once that's done, the algorithm would probably carry across without trouble. The Neuromimetic intelligence team at INRIA are working on a possibly related project, using neural gases (which they find better than growing neural gases) for document classification.

I'll finish with an Indian project, which used growing neural gases to classify filamentous bacteria in sewage, by skeletonising images of them into line segments. As the authors say in Shape Extraction of Volumetric Images of Filamentous Bacteria Using a Topology Preserving Feature Map, the line segments were obtained in a very direct way, as just the links between nodes.

Filamentous bacteria live in colonies, joining together into long thin filaments of cells. When sewage is treated by mixing with air and letting microorganisms clean it by feeding on organic waste, amongst its inhabitants are assorted species of filamentous bacteria. In small concentrations, these are useful because they bind together aggregates of other bacteria living in the mix, meaning, for instance, that these precipitate out readily, and so are easy to remove. But large concentrations gum up the system; so it's important to know what species, and in what numbers, are living in your sewage tanks. This involves classifying the filaments' species by looking at properties such as their branching ratio, something almost impossible to do manually because they are so complex.

The authors first obtained images of the filaments using confocal laser scanning microscopy. This enables the 3D object to be scanned as a sequence of 2D sections; all done optically, so the filaments don't need to be physically sliced. After some preliminary cleaning up and separation into individual filaments, the images were converted into lists of voxels (the 3D equivalent of pixels). These voxel coordinates were then fed one at a time to a modified growing neural gas algorithm; each voxel caused a change in node positions, and possibly a new node to be formed. (The paper calls them processors rather than nodes.) The links amongst the nodes in the final state of the growing neural gas were the skeleton of the image, a collection of line segments. It was much easier to search these for properties indicating their species than it would have been the original image.

Compared with conventional skeleton-finding methods, say the authors, theirs was much less sensitive to noise, as well as being efficient in invariance to image rotation. The paper gives details of the algorithm, and shows some of the images before and after neural gassing.

Neural gases are a nice example of local behaviour generating global structure. They make me think of how a pack of hunting or foraging animals might be organised, if the inputs were food: to benefit the pack as a whole, it makes sense to have a "specialist" covering the region of most abundant food, and for its neighbours to pay attention to that area too; but this has to be balanced against the risk of neglecting the rest of the territory. Or perhaps the nodes could represent research groups, and the inputs scientific ideas. I'll leave it to the cynic to decide where research funding might enter this interpretation.

Links

www.neuroinformatik.ruhr-uni-bochum.de/ini/VDM/research/gsn/DemoGNG/GNG.html - The DemoGNG (Version 1.5) applet, with links to source code, applet manual, and Some Competitive Learning Methods. The latter is at www.neuroinformatik.ruhr-uni-bochum.de/ini/VDM/research/gsn/JavaPaper/.

www.booru.net/download/MasterThesisProj.pdf - Growing Neural Gas. Experiments with GNG, GNG with Utility and Supervised GNG, Master's Thesis by Jim Holmström, Uppsala, 2002. Chapter 2 explains the growing neural gas algorithm. See also the screen shots and explanations in Unsupervised ontogenetic networks by Bernd Fritzke, at www.ki.inf.tu-dresden.de/~fritzke/papers/ncc2_4.pdf.

www.itee.uq.edu.au/~cogs2010/cmc/chapters/SOM/ - The Self Organizing Map: Unsupervised Competitive Learning, by Simon Dennis, Queensland, 1997. A nice intro to competitive learning, with applets and exercises for them.

www.arch.usyd.edu.au/~rob/java/applets/neuro/LearningVectorQuantisationDemo.html - Learning Vector Quantisation by Rob Saunders. An applet demonstrating the Learning Vector Quantisation algorithm, with Voronoi cells shown. The source can be downloaded. See also Simple Competitive Learning by Jenny Orr, Williamette, www.willamette.edu/~gorr/classes/cs449/Unsupervised/competitive.html. This explains vector quantisation, with diagrams.

hw001.gate01.com/frog/kaeru/FrogCells.html - "FROG CELLS" by Maeda Mameo. Saunders's page links to a "Frog Cells" page. That link is broken, but this one is probably the same. It displays dynamically updated Voronoi and Delaunay diagrams - a good way to appreciate them - and the code was used by Saunders in his applet.

www.ks.uiuc.edu/Publications/Papers/paper.cgi?tbcode=MART94 - Online version of Topology representing networks by Thomas Martinetz and Klaus Schulten, from Neural Networks, volume 7, 1994. Introduces and proves the properties of competitive Hebbian learning.

www.cb.uu.se/~hamed/data_mining/lab3.pdf - Classification and Clustering Using Artificial Neural Networks, by Hamed Muhammed, Uppsala, 2001. Part of a data-mining course; introduces DemoGNG and sets exercises on the effect of changing its parameters.

mdp-toolkit.sourceforge.net/ - Modular toolkit for Data Processing. An open-source Python toolkit which views neural nets and other learning algorithms as processing elements through which data is piped. Processing elements can be linked, combining them into "flows". Includes a growing neural gas, as well as principal component analysis, independent component analysis, and slow feature analysis. Last updated at the end of 2004.

www.gsf.de/ILIAD/reports/ICIAP99.pdf - Shape Extraction of Volumetric Images of Filamentous Bacteria Using a Topology Preserving Feature Map, by U. Bhattacharya, V. Liebscher, A. Datta S. K. Parui, K. Rodenacker, and B. B. Chaudhuri, Indian Statistical Institute Calcutta and Neuherberg Germany, 1999. For beautiful microphotographs of filamentous bacteria, see www.environmentalleverage.com/Filamentous%20Bacteria%20Photomicrographs.htm at Environmental Leverage: Turning Liabilities to Leverage!!!!

www.ks.uiuc.edu/Research/Neural/neural_gas.html - Robot Control with the "Neural Gas" Algorithm. Summary of research by Stanislav Berkovich at the Theoretical and Computational Biophysics Group, Urbana-Champaign. Uses a neural gas to divide a robot's visual input space into smaller patches which are more easily modelled by local maps.

leobard.twoday.net/stories/659872/ - Leo Sauermann's blog. Sauermann works on the Semantic Web at DFKI and maintains gnowsis (at www.gnowsis.org/), DFKI's open-source Semantic Desktop environment. In his blog entry for 29 April 2005, he proposes DemoGNG for analysing large semantic nets.

www.inria.fr/rapportsactivite/RA2004/cortex/uid36.html - Project page of the Neuromimetic intelligence team at INRIA, who are using neural gases for document classification.

humanresources.web.cern.ch/humanresources/external/training/tech/special/DISP2003/DISP-2003_L21A_30Apr03.ppt - PowerPoint slides on intelligent signal processing, by Léonard Studer, CERN. A presentation of neural nets (and fuzzy logic and evolutionary computing) for signal processing, with mention of Kohonen maps and brief mention of growing neural gases.

www.faqs.org/faqs/ai-faq/neural-nets/part1/section-11.html - How many kinds of Kohonen networks exist?, in the comp.ai.neural-nets FAQ.

www.neural-forecasting.com/source_code.htm - Neural Network Software Resources for Developers, from the Portal on forecasting with artificial neural networks. Links to DemoGNG and other neural net software, in C, C++, Java, Lisp, and Scheme.

There's a Hole in my Budget - qualitative reasoning, economics, and Flanders and Swann

On the penultimate day of 2001, I took a train to Maastricht to see a historic event, the introduction of the Euro. The inauguration ceremony in Maastricht's lovely city square made an excellent and lavishly EU-funded New Year's party; and after it, I continued with a trip around Europe. I read articles in the Dutch papers about coin collectors with no guilders to collect; in the Belgian, about the ink rubbing off forged Euro notes; in the Spanish, allergies due to nickel in the Euro coins; in the Portuguese, the lady who got her conversion factor wrong and charged an entire month's salary into her mobile phone. But these were small difficulties: all the hastily-cobbled conversion software ran as it should, and Europe avoided total monetary collapse.

I ended up in Frankfurt at the Bundesbank's Money Museum. Amongst its exhibits were a display about the benefits of monetary union; a Federal Reserve comic strip proclaiming the virtues of global free markets; and a Virtual Economy gaming console where you could be Finance Minister, balancing the economy against the demands of workers and businesses whilst being careful not to hurl it into a massive inflationary spiral. In fact, the whole museum is devoted to monetary stability and the avoidance of inflation. Which, given what happened during the 1920s, is not surprising. In 1920, a dollar was worth 40 Mark; by November 1923, it was 4.2×10¹² Mark.

Inflation in Britain never soared to such levels, but in 1951, it must have been sufficient to inspire the comic-song duo Flanders and Swann to write There's a Hole in my Budget. The lyrics go like this:

DS: There's a hole in my budget, dear Harold, dear Harold,
There's a hole in my budget, dear Harold, my dear.

MF: Then mend it, dear Healy, dear Dennis, dear Dennis,
Then mend it dear chancellor, dear Dennis, my dear.

DS: But how shall I mend it, dear Wilson, dear Wilson,
But how shall I mend it, dear Wilson, my dear?

MS: By building up exports, dear Dennis, dear Dennis,
By increased production, dear eyebrows, my dear.

DS: But that means working harder, dear Harold, dear Harold,
And the workers must have more incentives, my dear.

MF: Then decrease taxation, dear Healy, dear Healy,
And raise all their wages, dear Healy, my dear.

DS: But where is the money to come from, dear Premier,
But where is the money to come from, my dear?

MF: Why, out of your budget, dear Healy, dear Healy,
Why, out of your budget, dear Dennis, my dear.

DS: But there's a hole in my budget, dear Harold, my dear!

Flanders and Swann in Prolog

It's fun to translate this into Prolog. There are many possibilities, depending whether you're writing a planner, economic simulator, or whatever; how general you want it to be; and how much you want it to explain its conclusions. Here's a simple version, written thinking of the lines in the song as goals and subgoals:

  can_get_money :- can_raise_exports.

  can_raise_exports :- can_raise_production.

  can_raise_production :- can_raise_workers_incentives.

  can_raise_workers_incentives :- can_lower_taxes.

  can_raise_workers_incentives :- can_raise_wages.

  can_lower_taxes :- can_get_money.

  can_raise_wages :- can_get_money.

  1 ?- consult('d:/ainews/flanders_and_swann.pl').

  % d:/ainews/flanders_and_swann.pl compiled 0.00 sec, 48 bytes



  Yes

  2 ?- can_get_money.

  ERROR: Out of local stack

  Exception: (2,794,796) can_raise_workers_incentives ? 

Here's a different translation:

  direct_subgoal( get_money, raise_exports ).

  direct_subgoal( raise_exports, raise_production ).

  direct_subgoal( raise_production, raise_workers_incentives ).

  direct_subgoal( raise_workers_incentives, lower_taxes ).

  direct_subgoal( raise_workers_incentives, raise_wages ).

  direct_subgoal( lower_taxes, get_money ).

  direct_subgoal( raise_wages, get_money ).



  subgoal( A, B ) :- direct_subgoal( A, B ).

  subgoal( A, B ) :- direct_subgoal( A, Z ), subgoal( Z, B ).

  8 ?- subgoal( get_money, SG ).



  SG = raise_exports ;

  etc...



  SG = raise_wages ;



  SG = get_money ;

As any good Prolog textbook will explain, the subgoal predicate is easily extended to find the path between two actions. The causal network can also be enhanced, for example by adding probabilities and Bayesian reasoning.

Infinite loops, control-structure bugs, and the twist in the tail

The Budget song is funny because it describes an infinite loop. There's no solution, because you need the resource you're fixing in order to fix it. This reminded me of an exercise in the first edition of Patrick Winston's Artificial Intelligence. (Make sure to look for the first edition - later ones don't have this excercise.)

The exercise, written by Richard Brown, is problem 16-6 in Chapter 17. He conjectures a correspondence between story twists and program bugs, and gives an example of a story involving recursion with no stopping case. This is Larry Niven's Convergent Series, in which a man escapes from a demon by drawing a pentacle on its stomach. The laws of demonology constrain demons to remain inside pentacles, and the demon senses the pentacle and shrinks itself to fit. The pentacle shrinks too; the demon shrinks more; and so on ad infinitum. The Budget song mirrors the same control-structure bug.

Brown identifies other bugs: the deadly embrace of parallel programming is mirrored in O. Henry's short story The Gift of the Magi; the Lisp FUNARG problem, by the Rosencrantz and Guildenstern business. Other kinds of bug, Brown says, can be summarised by giving the morals of fairy tales. The first moral to come to my mind was "Don't count your chickens before they're hatched". Could this be an admonition not to dimension dynamic arrays until you know how big they have to be? Brown ends by setting a problem to choose a bug, characterise it, and then write a short story with the bug as its twist.

Qualitative reasoning

Returning to Flanders and Swann, their song demonstrates a simple form of qualitative reasoning. Quantitative reasoning calculates with numerical values; qualitative reasoning abstracts these away into symbolic values. Thus, the Budget song is only interested in whether its economic variables go up or down, not by how much or how fast. Sometimes, a qualitative answer is necessary because we don't have precise numerical values; sometimes, because it's all we need in order to understand how something works - e.g. the direction each wheel turns in a gear train, or whether your Christmas tree lights will go out when a bulb blows in the circuit.

One kind of qualitative reasoning is what we do when we say "I want a beer. OK, I'm out of cash; got to go to a cash dispenser and get out £20; oh blow it, I was overdrawn, and I'll be even more overdrawn now". In other words, negative plus negative is negative. We can easily code such rules in a symbolic language such as Lisp or Prolog:

  sum(  neg, neg, neg ).

  sum(  pos, pos, pos ).

  mult( pos, pos, pos ).

  mult( pos, neg, neg ).

  mult( pos, zer, zer ).

  mult( neg, neg, pos ).

One problem with qualitative reasoning is that there isn't always enough information to determine a unique answer. If I tell you I was overdrawn but have now paid money into my account, you can't tell - unless I say how much I paid in - whether I am still overdrawn. It's easy enough to program around this, by making rules return sets of answers, saying for example that the sum of positive and negative can be any of negative, zero, or positive:

  sum( neg, pos, [neg,zer,pos] ).

In spite of combinatorial explosion, qualitative reasoning is extremely useful, and has attracted a lot of research. You can experiment with it for yourself: the third edition of Ivan Bratko's book PROLOG Programming for Artificial Intelligence has an excellent chapter on it. (Make sure to look for the third edition - earlier ones don't have all the material.) Prolog code for the book's examples can be downloaded from the book's Web site, and includes qualitative models of simple circuits, a bath filling with water, and a block on a spring.

From novice economist to expert

On the Budget song page, just after the lyrics, Flanders tells us:

Yes, I'm rather proud of that, actually, how in 1951 how we explained this involved economic truth in simple revue terms. Alas, all too true today.

Why do I say that? Riesbeck was interested in how novices reorganise their memory structures as they learn. He compared how novice and expert economists solve problems, and developed models of each type of reasoning and a theory for how novices reorganise their memory as they gain expertise.

Compare two of his examples:

Q: If the federal budget deficit went down, what would happen to the rate of inflation?
A: I think inflation would go down, because people would have less money taken out of their weekly income, they would hopefully do some more saving, have more money to live on, not be pushing for higher wages, and that wouldn't be pushing inflation up.

With the resulting structure of taxes and expenditures, the President is not going to be balancing the Federal budget in 1984 or any other year. With high growth choked off by high interest rates, budget deficits are going to be bigger, not smaller. The result: more demands for credit and higher interest rates.

Lester Thurow, Newsweek, 21 September 1981, p. 38

The first transcript is from a novice who, Riesbeck claims, viewed the economy in terms of the goals of economic actors - workers, businesses, Government - and the subgoals needed to achieve these goals. For example, a business's goal is to raise income. One way to do so - a subgoal - is to raise sales. Other goals are to raise prices and to lower costs. One of the Government's goals is to lower the budget deficit (even though, in the real world, its actions might make one think otherwise).

In total, Riesbeck interprets the transcript in terms of these three actors, each having one top-level goal and two or three subgoals:

  GOVERNMENT GOAL: lower the budget deficit

                   SUBGOAL: raise taxes

                   SUBGOAL: lower consumer spending



  WORKERS' GOAL:   raise money

                   SUBGOAL: lower taxes

                   SUBGOAL: raise wages

                   SUBGOAL: lower consumer spending 



  BUSINESS GOAL:   raise money

                   SUBGOAL: raise sales

                   SUBGOAL: lower costs

                   SUBGOAL: raise prices

Now look at Riesbeck's second example, from a Newsweek finance columnist. This is more abstract, focussing directly on the economic variables and how they influence one another. Riesbeck claims such reasoning can be modelled as searching a graph whose nodes are economic variables, connected by links indicating whether one variable varies directly or inversely with the other. To answer a question such as "If the federal budget deficit went down, what would happen to the rate of inflation?" would involve searching the graph for a chain of influence links from budget deficit to inflation. This is more efficient than novice reasoning, because the data structure being searched is simpler, being a single graph instead of an intertangled forest of different actors' goal trees.

Cognitive models and teaching

Models like Riesbeck's are useful in education, by helping teachers understand what goes on in their pupils' heads. One of the earliest and most famous is Young and O'Shea's production system model of children's subtraction. They say that in standard educational theory, children's faulty arithmetic is due to misremembering "number facts", aggravated by problems such as bad setting-out, fatigue, carelessness, and poor concentration. So standard educational theory - this was written in 1981 - had concentrated on models for recall and use of correct number facts.

However, the children's errors may actually be due to failures in execution. We should regard the child as faithfully following wrong algorithms, not wrongly executing the correct one. Of course, this is a hypothesis; but it's worth taking seriously, because if true, it gives us more control over teaching.

Young and O'Shea went on to build a production system model of how children might acquire wrong algorithms rather than correct ones. They started with production system rules which modelled correct subtraction, and then showed that by slight changes to some of the rules, their model could account for more than 2/3 of the errors observed in their subjects' subtractions. Some of the rules were faulty because of wrong actions; others, because of wrong conditions. Consider this subtraction:

  50

  46

  --

  16

  --

Young and O'Shea concluded, amongst other things, that teachers tend to regard a a new skill as something that must be "consolidated" by repeated practice, as in language books that introduce past tenses and then set translations with all past tenses and no present tenses. But the rule model shows we must also train the student to discriminate when to apply rules, i.e. to get the conditions right. Such training may actually be hindered by consolidation. In the language example, old present tense rules may be lost or changed in favour of past tense ones.

Researchers have developed such cognitive models in many areas of education. I mentioned Riesbeck's model of economic learning; a more recent one, focussing on how experts switch easily between textual information and X-Y graphs, and why novices have difficulty doing the same, is How Does an Expert Use a Graph? A Model of Visual and Verbal Inferencing in Economics by Herbert Simon and colleagues at Carnegie Mellon.

Such work might help teachers explain to students how macroeconomics (the aggregate behaviour of economies in terms of variables such as inflation, Gross National Product, and unemployment) is related to microeconomics (the behaviour of individual actors). Could it also help us build artificially-intelligent economists? You wouldn't want to experiment on an actual economy, and the Money Museum gaming consoles I mentioned at the start of this article are too heavy to move; but if anybody wants to try out a computerised Chancellor of the Exchequer or Finance Minister program, I can point them at a testbed. It's a Web-based virtual UK economy which I and colleagues built a few years ago for teaching introductory economics, and where you play the role of Chancellor, set taxes and other variables, and then watch how your changes affect the economy over the next ten years.

How to improve your research output

If, when reading this article, you hadn't heard of Flanders and Swann, it may help to think of them as the English equivalent to Tom Lehrer. I'll end this article with some advice from his song Lobachevsky. Written for mathematicians, it applies just as well to AI:

Plagiarize,
Let no one else's work evade your eyes,
Remember why the good Lord made your eyes,
So don't shade your eyes,
But plagiarize, plagiarize, plagiarize...
Only be sure always to call it please, "research".

Links and other references

www.nyanko.pwp.blueyonder.co.uk/fas/and_hole.html - There's a Hole in my Budget, from Flanders and Swann Online.

www.ecscoutnet.co.uk/information/campfire/Hole.html - The song is based on the traditional There's a Hole in my Bucket. Here's one of many listings, from the Erith and Crayford Scouts songs site. (For people from Maastricht, there's a version in local dialect at www.limburgzingt.nl/maast-l.htm; search for "ummer".)

Problem 16-6 (by Richard Brown) in Chapter 17, Problems to think about, from Artificial Intelligence (1st edition) by Patrick Winston, 1977.

www.online-literature.com/o_henry/1014/ - The Gift of the Magi by O. Henry, at The Literature Network.

xarch.tu-graz.ac.at/autocad/lisp/FAQ-link/msg00653.html - Posting explaining the Funarg problem on comp.lang.lisp by Barry Margolin, 27 April 2000.

Chapter on Qualitative Reasoning, from the book PROLOG Programming for Artificial Intelligence (3rd edition) by Ivan Bratko, 2001.

cwx.prenhall.com/bookbind/pubbooks/bratko3_ema/chapter1/deluxe.html - Student resources page for PROLOG Programming for Artificial Intelligence, with source code for each chapter. The menu at the top selects the chapter: qualitative reasoning is near the end.

www.upc.es/web/QR2002/Papers/QR2002%20-%20Bandelj.pdf - Qualitative Simulation with CLP, by Aleksander Bandelj, Ivan Bratko and Dorian Šuc Faculty of Computer and Information Science University of Ljubljana, Slovenia.

www.qrg.northwestern.edu/papers/files/qpolitics_v4_KDF.pdf - Towards a qualitative model of everyday political reasoning, by Kenneth Forbus and Sven Kuehne, Northwestern University.

Knowledge reorganization and reasoning style by Chris Riesbeck, in Developments in expert systems, edited by Mike Coombs, Academic Press, 1984.

Errors in children's subtraction, by Richard Young and Tim O'Shea, from Cognitive Science, volume 5, issue 2, 1981.

www.cogs.susx.ac.uk/users/bend/cogmod/compex2.html - A Production System Model of Subtraction, by Benedict du Boulay and Steve Torrance, from the Cognitive Modelling course at Sussex University, 2002. Shows output from a production system written in Pop-11 and based on the Young and O'Shea rules.

www-2.cs.cmu.edu/afs/andrew.cmu.edu/usr13/al28/camera/TLS_94_1/TLS_94_1.html - How Does an Expert Use a Graph? A Model of Visual and Verbal Inferencing in Economics by Hermina Tabachneck, Anthony Leonardo, and Herbert Simon, Carnegie Mellon. See also their CaMeRa: A Computational Model of Multiple Representations, at www-2.cs.cmu.edu/afs/andrew.cmu.edu/usr0/psycam/public/camera/TLS_95_2/TLS_95_2.html.

www.cmpe.boun.edu.tr/courses/cmpe560/spring2004/dalk.ppt#8 - Qualitative Modeling in Education by Forbus and Bredeweg. See also Bredeweg's thesis at staff.science.uva.nl/~bredeweg/pdf/thesis/03Groen.pdf.

www.bized.ac.uk/virtual/economy/ - Virtual Economy, by Andy Beharrell, Keith Church, Jocelyn Paine, and Graham Stark. Our be-your-own-Chancellor educational simulation of the British economy. See also Graham Stark's Virtual Chancellor at www.virtual-worlds.biz/vwc/.

www.inc.com/magazine/19950915/2624.html - When Money Flowed Like Water, by Doron Swade, from Inc. Magazine, September 1995. About the Phillips machine, a water-based analogue computer economic model, built from old Lancaster bomber parts.

www.j-paine.org/NarrativeTechnology.html - The Lives Behind the Numbers On the Screen: Illustrating the Social Consequences of Economic Change By Telling Stories On the Web. Some experiments in generating, from the output of Virtual Economy, stories about the effects of economic changes on individual people.

nostoc.stanford.edu/jeff/personal/diary/diary.html - Diary of an Insane Cell Mechanic - A psychologist's descent into molecular biology, by Jeff Shrager. Shrager worked on a model of multiple-representation reasoning in gas laser physics, which partly inspired the model described in Simon et. al.'s How Does an Expert Use a Graph?. He also wrote this Diary, and - amongst many other things - is principal investigator on the Web-based programmable biological knowledge-base BioLingua, built on top of BioLisp.

nostoc.stanford.edu/jeff/precog/ - Brain parts sort of involved in PreCognition: An fMRI study. Shrager's parody of the electronic journal Cognitive Science Online.

en.wikipedia.org/wiki/Tom_Lehrer - Wikipedia entry for Tom Lehrer.

"Lisp has all the visual appeal of oatmeal with fingernail clippings mixed in"

I found that quote, by Perl creator Larry Wall, in an interview with Mark Jason Dominus about his book Higher Order Perl, which tells how to program with higher-order functions in Perl. It looks like an insult to Lisp, but it's really a compliment: Dominus has taken the notion of higer-order functional programming, popular in Lisp, for use in real-world Perl tasks such as typesetting, diagram generation, and HTML hacking. Higer-order functional programming passes functions as arguments to other functions, making it easy to iterate over structures without coding your own loops. In Lisp, calling

  (mapcar #'+ '(1 2 3 4) '(5 6 7 8)) 

  (mapcar #'* '(1 2 3 4) '(5 6 7 8)) 

Lisp is regarded by Paul Graham, author of On Lisp and ANSI Common Lisp, as the basis for his new language Arc, as he explains in Being Popular and The Hundred-Year Language. Arc will be succinct, because "succinctness is power"; but will not be object oriented, because "there are five reasons people like object-oriented programming, and three and a half of them are bad". Other essays on Graham's site include A Unified Theory of VC Suckage, Why Nerds are Unpopular, and an essay on essay-writing. Did you know the word "meander" comes from a river in Turkey? I do now.

But perhaps I should forget Lisp. In his collection More About Real Programmers, Adrian Hilton tells us "Real programmers don't use LISP. Only effeminate programmers use more parentheses than actual code". The collection is one of many on the theme "Real programmers don't eat quiche", possibly inspired by Real Programmers Don't Use Pascal. The article is a companion to the heart-warming story of Mel, from the days when real computers were made from vacuum tubes and drums, and to optimise your code, you timed how fast the drum rotated.

Hilton's collection ends with the real history of Unix:

In 1969, AT & T had just terminated their work with the GE/Honeywell/ AT & T Multics project. Brian and I had just started working with an early release of Pascal from Professor Nichlaus Wirth's ETH labs in Switzerland and we were impressed with its elegant simplicity and power. Dennis had just finished reading 'Bored of the Rings', a hilarious National Lampoon parody of the great Tolkien 'Lord of the Rings' trilogy. As a lark, we decided to do parodies of the Multics environment and Pascal. Dennis and I were responsible for the operating environment. We looked at Multics and designed the new system to be as complex and cryptic as possible to maximize casual users' frustration levels, calling it Unix as a parody of Multics, as well as other more risque allusions. Then Dennis and Brian worked on a truly warped version of Pascal, called 'A'. When we found others were actually trying to create real programs with A, we quickly added additional cryptic features and evolved into B, BCPL and finally C. We stopped when we got a clean compile on the following syntax:

   for(;P("\n"),R-;P("|"))for(e=C;e-;

   P("_"+(*u++/8)%2))P("| "+(*u/4)%2);

Links

www.theperlreview.com/Interviews/mjd-hop-20050407.html - Interview with Mark Jason Dominus, from the Perl Review.

www.xoltar.org/2005/mar/17/higher-order-perl.html - Review of Higher Order Perl by Damian Conway, explaining how it will transform your programming.

www.paulgraham.com - Paul Graham. His essays are linked from the left-hand navigation bar. See also en.wikipedia.org/wiki/Paul_Graham.

www.pbm.com/~lindahl/real.programmers.html - Real Programmers Don't Use Pascal. See www.pbm.com/~lindahl/mel.html for The story of Mel.

www.suslik.org/Humour/Computer/Langs/real_prog2.html - More About Real Programmers, jokes collected by Adrian Hilton, ending with UNIX and C a Hoax.

www.topsail.org/tao.html - The Tao Of Programming.

www.dooyoo.co.uk/printed-books/bored-of-the-rings-the-harvard-lampoon/#sd - Review of Bored of the Rings.

Past newsletters are available at either www.ddj.com or www.ainewsletter.com. As ever, interesting links and ideas for future issues are very welcome. Feel free to contact either myself (below) or Jocelyn <popx@j-paine.org> with comments, thoughts and suggestions.

Until next month,

Dennis Merritt

Copyright ©2005 Amzi! inc., CMP, and Jocelyn Paine. All Rights Reserved