When humans analyse a situation, they often do so in a seemingly unstructured and roundabout manner, with much halting and change of direction. This is as much noticed in areas of seeming precision, such as mathematics, as it is in areas of imprecision, such as politics. The proof of a mathematical theorem is but a final trace over what has gone before. It may look logical and orderly, but that will not usually be a good description of the mental activity which led to it. If we choose to implement directed theorem proving techniques which are "neat and orderly", we should expect them to fail on all but the most trivial proofs. We might instead choose to implement methods which do not have the specificity of the direct approach, but may handle a larger class of problems.

Someone scratching around "on the back of an envelope" may be plunging into many different areas and be using many different methods of analysis together. A person can combine these methods at a single focus, and smoothly switch among methods as needed.

Most of the tools developed for machine analysis are highly directional, and usually singly talented. Examples are spreadsheets for financial analysis, the critical path method of planning, expert systems, neural networks. There are often weak forms of undirectedness introduced into these tools, the backward solving of spreadsheets, the forward and backward passes of CPM, back propagation in neural networks. The extra information these weak forms of undirectedness can provide is welcome, but limited. The other major flaw with these methods is the necessity to code the particular problem into the very limited representation available from the algorithm, resulting in complex problems being watered down or dismembered and destroyed.

The reason for directionality in algorithmic methods is easy to perceive - the underlying machine method is directional in its operation. The machine jumps to a new instruction, and it stores data in locations, overwriting the old. The storage capacity of the location is limited, and the structure of the stored information must be known in advance, not change dynamically. As long as the purpose can be directly described in jump-store terms with simple phasing that can be decided beforehand, an algorithm is a powerful tool. As we move towards higher level analysis, the jump-store approach becomes less appropriate.

The basic element of methods such as Expert Systems and Neural Networks is the directed node, as shown in Figure 1. It has inputs and an output. All the information available for processing comes in on its input connections. A typical Expert System rule might be

IF A > B THEN C = 6

Here the logic seems to demand direction, in that A and B must be known before an output is produced for C. Large and complex structures can be built from these rules, but all have essentially the same structure (Figure 2), an artifact of the base form. A search may be needed to assemble the structure, but it too is a directed search and may not be incremental in that the search is performed before values are known.

If we are confident that such a structure can represent the particular analysis being attempted, then directed nodes should be adequate to support the analysis. If the analysis is in any way complex, a different structure may be needed, one which is not so specifically directed towards solution of the problem. When machines were severely limited in computing power and memory, the notion of undirected activity would have seemed very wasteful. It would already be known with some certainty what the output should be, and the machine was programmed to that end.

A structure which represents the approach of high level analysis might look like Figure 3, where there are many possible outputs. It is not that the analysis is not goal directed, just that the goal is at a higher level than any direct output, and the goal can only be reached by satisfying the interactions among the lower level, explicit goals. Some explicit goals may become inputs to establish what other goals would be, there being insufficient structure to resolve all goals at once. We need to differentiate here between multiple goals with a swirl of interactions among them, and a specific goal having many facets.

The lack of specificity at a high level could be handled either by having a multitude of directed analyses, with some need for communication among them, or by allowing directionality to be transient. This approach forces a change to the base component, such that now it has the potential for bi-directionality on each of its connections. The bi-directionality confers several benefits - if the node finds an input unimpressive, it can choose to change it, drive it out and see what returns, the node may have information about C to guide it when combining A and B, and it obtained the initial direction of working from what prompted the processing, transmission in a pathway to it. The node can choose to send an output to C, or to A or B, or any combination or none. The pathways have existence (memory resource), as well as the node, so can transmit more complex information.

We should emphasise here that there need to be many more nodes in an undirected network than there would be rules in an Expert System, because each rule would be made out of nodes. Once the inputs are known for the Expert System rule, it changes to jump-store, whereas the undirected nodes use structure for the logic and structure for the information storage. Information storage can take the form of logic and data, so the logic can be used to later interpret the data. See [1] for an example of indexing a list, where logical structure is generated to interpret and store data. [1] also describes the different methods of information transmission that may be occurring in the structure to support different "modes of reasoning".

If we return to that Expert System rule

IF A > B THEN C = 6

We can implement the elements of this statement in an undirected way, including the logical implication. The entire statement is undirected, in that logical control may be asserted, or the statement may provide a logical output. Once implemented and asserted, if we happen to know that C is not 6, then we may know something useful about A and B, in that A <= B. By removing the undirectedness of the base element, we have combined the different methods of reasoning employed in Expert Systems and Constraint Reasoning. If we wish to make use of this information, we are obliged to increase the complexity of the information being transmitted. If A and B are integers, and we already know B to be 12, then we need to transmit 0..11 (say) for A. The logical states required to transmit or receive information about statements are, unsurprisingly, not those of Boolean logic, premised as it is on direction.

Critical path planning (CPM) was originally developed to plan and track simple linear projects such as erecting a high rise building. There has been some effort to graft the rules of an Expert System onto the basic CPM algorithm implemented on a computer, as resource allocation in particular can be very primitive in comparison with a skilled Project Manager. The forward and backward passes are strongly directed, and employ an extremely simple algorithm, incapable of adding two numbers together to modify its behaviour. For example, one might wish to say that

Duration_C = 10 + StartDate_B - StartDate_A

The algorithm has no means of handling such variation during its operation. While some breakout could be proposed for specific cases, there is no general method because the algorithm has the flow of control.

A simple project plan might be viewed as shown in Figure 5. The elements of the project are combined in a directed way to produce a finish date. Backward analysis is performed to produce a measure of float. This may be adequate for simple projects, but many projects have an entirely different structure, where there are many influences and requirements, many changing dynamically as the project proceeds. Complex projects will include alternative ways of doing things, contingencies, termination criteria, invalidating the simple directional analysis.

A more realistic view might be as shown in Figure 6. The influences of cost and outcome extend deep into the project,. Of course, it also becomes multi-dimensional with risk analysis and topologically variable as the influences interact with each other. A description of the area showing the application of an undirected network in greater detail is given in [2]. The point being made here is that directed methods of analysis are unsuited to relatively common planning problems, and often lead to an undesirable mindset on the part of the tool users.

Data mining has been described as "extracting useful patterns from data". There are several problems here, the first being where does one start. Any system finding patterns will often have to inference across empty space because a great deal of "obvious" data is left out, and may have no idea of cause and effect, nor of temporal irreversibility. A reasonable starting strategy is to look for difference or activity. An advantage of undirected analysis is that islands of activity, when found, can grow and coalesce precisely because the analysis at any point is undirected and can be turned around to work with new connections. Accreting islands of activity, then coalescing them into archipelagos and then into continents is a much more powerful approach than attempting to strike out from one point, no matter how good a starting point it appears to be. Undirected operators can tolerate the starting point turning out to be the middle of a logical arc, whereas directed operators cannot.

If interesting rules are found, it is simple to connect the confidence level to values in the rule, such that

Confidence level 0.8-1.0

IF X IN {3..7}

AND Y IN {2..9}

THEN Z > 20

One method used to find patterns in data mining is to convert the data into a form amenable to analysis with a neural network. [3] has described the difficulty of conversion, particularly of data that is not smoothly numeric, and the limited operations available from the method. A more significant problem is the directionality of the method, springing from a clear Input/Output structure.

One might look at the human brain, observe that it is made from directional cells, and conclude that a mechanised form of directional cell would be adequate to carry out analysis. The cellular operations in our brains are many levels below our reasoning, the cells can transmit complex messages through repeated firings, and there are many loops, allowing patterns of firing to "ring" within the structures. How the human manages undirected analysis is not the subject of this paper. What is clear is that the human is capable of it, and can bring many seemingly different forms of analysis to a focus on any point in the structure. A similar conceptual error would arise if one looked at the signal processing in a television set, saw the many resistors there, and concluded that the signal processing of a television set, or something like it, could be built only using resistors, without amplification or feedback or switching.

If we take as an example a diversified corporation with many divisions, and a profit which is stable over time, the simple Input/Output structure reveals little because the profit output doesn't change, whereas there may be very large changes occurring the next level down, crucial to survival of the organisation. One could respond that the wrong output layer is being analysed, but if there is sufficient human analytic capacity brought to bear to recognise which are the important inputs and outputs in the structure, then there is little need for automated mining of the data. As the economy changes over time, the correlation among the inputs may change, leading to catastrophic failure, as shown in Figure 8, where the swings become wild and then lose phase.

 It may not be enough to observe summations, but also the derivatives of those summations. If a particular technique is selected, a choice has been made on the effects that will be capable of being mined in the data. Undirected methods offer a wider choice of operations, and can analyse in any direction.

As an analogue of a nervous system with receptors and effectors (a neural network with amplification and feedback and switching), we should expect short reflex arcs and higher and higher level clumps, in the manner of Figure 9. If the data being mined is reasonably extended, we should expect operators requiring the manipulation of objects and logic and numbers. Undirected operators can provide the richness and variety required to both evaluate the effects and implement the connections by patching operators into the structure. With any node being potentially an input or an output, the analysis can switch around to predict a receptor input, control an effector output or generate a structure to guard against failure of a high level output.

Directed methods of analysis have more to do with initial development on machines with limited resources than with "fitness of purpose" for high level analysis. The "jump-store" cycle needs to be many levels below such analysis.

A general method of analysis using nodes interacting in an undirected network would seem to provide much greater scope for successfully automating the analysis of complex problems.

1. Brander, J. Multi Modal Methods of Information Transmission - AAAI Spring Symposium 1998

2. Brander, J and Dawe, M . Use of Constraint Reasoning to Integrate Risk Analysis with Project Planning - The International Journal of Project and Business Risk Management - Winter 1997