# Unknown Title
## On reasoning in networks with qualitative uncertainty
## Simon Parsons*and E. H. Mamdani
Department of Electronic Engineering,
Queen Mary and Westfield College, Mile End Road, London, E1 4NS, UK.
## Abstract
In this paper some initial work towards a new approach to qualitative reasoning under un certainty is presented. This method is not only applicable to qualitative probabilistic reasoning, as is the case with other methods, but also allows the qualitative propagation within networks of values based upon possi bility theory and Dempster-Shafer evidence theory. The method is applied to two simple networks from which a large class of directed graphs may be constructed. The results of this analysis are used to compare the quali tative behaviour of the three major quantita tive uncertainty handling formalisms, and to demonstrate that the qualitative integration of the formalisms is possible under certain as sumptions.
## 1 INTRODUCTION
In the past few years, the use of reasoning about qual itative changes in probability to deal with uncertainty has become widely accepted, being applied to domains such as planning [Wellman 1990b] and generating plau sible explanations [Henrion and Druzdzel 1990]. Such a qualitative approach has certain advantages over quantitative methods, not least among which is the ability to model domains in which the relation between variables is uncertain as a result of incomplete knowl edge, and domains in which numerical representations are inappropriate.
The existence of the latter, as Wellman [1990a] points out, is often due to the precision of numerical methods which can, in certain circumstances, lead to knowledge bases being applicable only in very narrow areas be cause of the interaction between values at a fine level of detail. Since they view the world at a higher level of abstraction, qualitative methods are immune to such
*Current address: Advanced Computation Laboratory, Imperial Cancer Research Fund, P. 0. Box 123, Lincoln's Inn Fields, London, WC2A 3PX, UK
problems; the small complications such interactions cause simply have no effect at the coarse level of detail with which qualitative methods are concerned.
The focus of the qualitative approach of Wellman and Henrion and Druzdzel is assessing the impact of evi dence. That is assessing how the change in probabil ity of one event due to some piece of evidence affects the probability of other events. For instance, taking a patient's temperature and finding that it is 38C is evidence that increases the probability that she has a fever, which in turn increases the probability that she has measles.
Now, when using the qualitative method we reason with a restricted set of values. Instead of using the full range of real numbers we are only interested in whether values are positive [+], negative [-], zero [0], or any of the three [?]. Thus we can determine that since the probability of fever increases, the change in prob ability is [+], and use this to decide that the change in probability of measles is also [+]. This is clearly weaker information than that obtained by traditional methods but may still be useful [Wellman 1990a], in particular since qualitative results may be obtained in situations where no numerical information may be de duced.
## 2 A NEW QUALITATIVE APPROACH
This paper presents a new approach to reasoning about qualitative changes. This work is drawn from the first author's thesis [Parsons 1993] in which may be found a number of extensions to the work described here. The motivation behind this work was to integrate dif ferent approaches to reasoning under uncertainty, in particular probability, possibility [Zadeh 1978] [Dubois and Prade 1988a], and evidence [Shafer 1976] [Smets 1988) theories. Thus, our qualitative approach differs from that described above in that it is concerned with changes in possibility values [Parsons 1992a] and belief values [Parsons 1992b) as well as probability values. As a result we need a general way of referring to values that may be probabilities, possibilities or beliefs.
Definition 2.1: The certainty value of a variable X taking value x, val(x), is either the probability of X, p( x), the possibility of X, II( x), or the belief in X, bel(x).
We set our work in the framework of singly connected networks in which the nodes represent variables of in terest, and the edges represent explicit dependencies between the variables. When the edges, of such graphs are quantified with conditional probability values they are similar to those studied by Pearl [1988], when pos sibility values are used the graphs are similar to those of Fonck and Straszecka [1991J and when belief values are used the graphs are those studied by Smets [ 1991 J .
Each node in a graph represents a binary valued vari able. The probability values associated with a vari able X which has possible values x and ·X are p(x) and p( ·x), and the possibility values associated with X are II(x) and II(·x). Belief values may be assigned to any subset of the values of X, so it is possible to have up to three beliefs associated with X; bel({x}), bel({·x}) and bel({x,·x}). For simplicity these will be written as bel(x), bel(·x) and bel(x U ·x). This rather restrictive framework is loosened in [Parsons 1993J where non-binary values and multiply connected are considered.
Wellman [ 1990a, 1990bJ and Henrion and Druzdzel [ 1990J base their work upon the premise that a suitable interpretation of "a positively influences c" is that:
$$( 1 ) ( 2 ) ( 3 )$$
This seems reasonable, but it is a premise; there are other ways of encoding the information that seem equally intuitively acceptable, for instance p( c I a) > p(c) and p(c I a) > p(·c I a) [Dubois and Prade 1991J. Since our aim was to provide a method that was suit able for integrating formalisms we wanted to start from first principles thus minimising the number of neces sary assumptions. As a result, a different approach was adopted as described below.
Given a link joining nodes A and C as in Figure 1, we are interested in the way in which a change in the value of a, say, expressed in a particular formalism, in fluences the value of c. Note that the arrow between A and C does not necessarily indicate a causal relation ship between them, rather it suggests that propagation of qualitative changes will be from A to C.
Figure 1: A simple network
<details>
<summary>Image 1 Details</summary>
### Visual Description
## Diagram: Simple Directed Relationship
### Overview
The image displays a minimal, abstract diagram consisting of two circular nodes connected by a single, straight directional arrow. The diagram represents a direct, one-way relationship or transition from a starting point to an endpoint.
### Components/Axes
* **Node A:** A white circle with a black outline, positioned on the left side of the diagram. The label "A" is placed directly below it.
* **Node C:** An identical white circle with a black outline, positioned on the right side of the diagram. The label "C" is placed directly below it.
* **Arrow:** A solid black line with a filled arrowhead at its right terminus. It originates from the right edge of Node A and points directly to the left edge of Node C, indicating a left-to-right flow.
### Detailed Analysis
The diagram is purely structural and contains no quantitative data, scales, or axes. The analysis is therefore qualitative:
* **Spatial Relationship:** Node A is the origin, and Node C is the destination. The arrow creates a clear vector from left to right.
* **Visual Style:** The elements are rendered in a simple, high-contrast black-and-white line art style. There is no color, shading, or texture.
* **Labeling:** The labels "A" and "C" are in a serif typeface, positioned centrally beneath their respective nodes.
### Key Observations
1. **Directionality:** The arrow is unambiguous, establishing a one-way path from A to C. There is no indication of a return path or bidirectional relationship.
2. **Simplicity:** The diagram is stripped of all extraneous detail. There are no intermediate nodes, branching paths, or descriptive text beyond the single-letter labels.
3. **Label Gap:** The sequence jumps from "A" to "C," omitting "B." This could imply that an intermediate step is being abstracted away or that the relationship is being defined between two specific, non-sequential entities.
### Interpretation
This diagram is a fundamental representation of a **directed relationship**. It serves as a visual primitive that can be used to model countless concepts:
* **Process Flow:** A → C could represent a simplified process where step A leads directly to outcome C, with intermediate steps implied or irrelevant for the current context.
* **Causation:** It can illustrate that A is a direct cause or precursor to effect C.
* **State Transition:** In systems theory, it could show a system moving from state A to state C.
* **Abstract Mapping:** It may simply denote a mapping or function where input A produces output C.
The omission of "B" is the most notable feature, prompting the viewer to question what is being intentionally excluded from the model. The diagram's power lies in its abstraction; it conveys the core idea of a direct, singular connection without specifying the nature of the entities or the mechanism of the relationship. It is a template awaiting context.
</details>
We can then model the impact of evidence that affects the value of A in terms of the change in certainty value of a and ·a, relative to their value before the evidence was known, and use knowledge about the way that a change in, say, val(a) affects val(c) to propagate the effect of the evidence. We define the following relationships that describe how the value of a variable X changes when the value of a variable Y is altered by new evidence:
Definition 2.2: The certainty value of a variable X taking value x is said to follow the certainty value of variable Y taking value y, val(x) follows val(y), if val(x) increases when val(y) increases, and val(x) decreases when val(y) decreases.
Definition 2.3: The certainty value of a variable X taking value x is said to vary inversely with the cer tainty value of variable Y taking value y, val(x) varies inversely with val(y), if val(x) decreases when val(y) increases, and val(x) increases when val(y) decreases.
Definition 2.4: The certainty value of a variable X taking value x is said to be independent of the cer tainty value of variable Y taking value y, val(x) is independent of val(y), if val(x) does not change as val(y) increases and decreases.
The way in which the variation of val(x) with val(y) is determined is by establishing the qualitative value of the derivative 8val(x)\8val(y) that relates them. If the derivative is known, it is a simple matter to cal culate the change in val(x) from the change in val(y). Thus to determine the change at C in Figure 1 we have:
$$\Delta \val(c) = \Delta \val(a) \otimes \left[ \frac{\partial \val(c)}{\partial \val(a)} \right]_{b=0}
\quad \text{and } \quad \Delta \val(-c) = \Delta \val(a) \otimes \left[ \frac{\partial \val(-c)}{\partial \val(a)} \right]_{b=0}
\quad \text{if } x \in [0,1], \quad x \neq 0, \quad x \neq -1
\end{equation}$$
$$[ \frac { \partial v ( c ) } { \partial v ( a ) } ]$$
where [ x J is [+ J if x is positive, [ O J if x is zero and [-J if x is negative, and EB and 0 denote qualitative addition and multiplication respectively:
| EB | [+J | [OJ | [-J | [?J |
|------|-------|-------|-------|-------|
| [+J | [+J | [+J | [?J | [?J |
| [OJ | [+J | [OJ | [-J | [?J |
| [-J | [?J | [-J | [-J | [?J |
| [?J | [?J | [?J | [?J | [?J |
| 0 | [+J | [OJ | [-J | [?J |
|-----|-------|-------|-------|-------|
| [+J | [+J | [OJ | [-J | [?J |
| [OJ | [OJ | [OJ | [OJ | [OJ |
| [-J | [-J | [OJ | [+J | [?J |
| (?J | (?J | [OJ | [?J | (?J |
We can express this as a matrix calculation (after Far-
reny and Prade [ 1989 ]) :
$$\begin{array}{ll}
\left[ \Delta v _ { a } ( c ) \right] = \left[ \frac{\partial v _ { a } ( c )}{\partial t} \right] \cdot \left[ \frac{\partial v _ { a } ( - c )}{\partial t} \right] & = \left[ \frac{\partial v _ { a } ( c )}{\partial t} \right] \cdot \left[ \frac{\partial v _ { a } ( - c )}{\partial t} \right] \\
\end{array}$$
Clearly val(c) follows val(a) when 8val(c)\8val(a) = [ +], val(c) varies inversely with val(a) when 8val(c)\8val(a) = [-] and is independent of val(a) when 8val(c)\8val(a) = [ 0].
## 3 QUALITATIVE CHANGES IN SIMPLE NETWORKS
Applying probability theory to the example of Figure 1, so that val(x) becomes p(x) in (2) and ( 3 ) , and re ferring to the directed link joining A and C as A--+ C, we have the following simple result which agrees with the assumption ( 1 ) made by Wellman as a basis for his qualitative probabilistic networks when conditional values are taken as constant, as they are throughout this work:
Theorem 3.1: The relation between p(x) and p(y) for the link A --+ C is such that p( x) follows p(y) iff p( x I y) > p( x I -.y), p( x) varies inversely with p(y) iff p(x I y) < p(x I -.y) and p(x) is independent of p(y) iff p(x I y) = p(x I -.y) for all for all x E {c, -.c}, yE{a,-.a}.
Proof: Probability theory tells us that p( c) = p( a) p(c I a) + p(-.a)p(c I -.a) and p(a) = 1p(-.a) so that p(c) = p(a)[p(c I a) - p(c I -.a)] + p(cl- .a) and [ 8val(c)\8val(a) J = [ p(cla)- p(cl-.a) J . Similar rea soning about the way that p(c) varies with p(-.a) and p(-.c) varies with p(a) and p(-.a) gives the result. D
By convention [ Pearl1988 J two nodes A and C are not connected in a probabilistic network if p( a I c) = p( a I -.c ) . In addition, since p( a) and p( -.a) are related by p(a) = 1p(-.a), we can say that if p(c) follows p(a), then p( -.c) varies inversely with p( a) and follows p( -.a). The assumption that conditional probabilities are con stant does not seem to cause problems when propagat ing changes in singly connected networks as discussed here. However, the assumption does become problem atic when handling multiply connected networks [ Par sons 1993 ] .
Applying possibility theory to the network of Figure 1, and writing II(x) for val(x) in (2) and ( 3 ) , we can establish a relationship between II(a) and II(c). Un fortunately, unlike the analogous expression for prob ability theory, this involves the non-conditional value II( a). This complicates the situation since the exact form of the qualitative relationship between II( a) and II(c) depends upon whether II(a) is increasing or de creasing. '�'e have:
Theorem 3.2: The relation between II(x) and II(y), for all x E {c,-.c}, y E {a,-.a}, for the link A--+ C is such that II(x) follows II(y) if min(II(x I y),II(y)) > min(II(x I -.y),II(-.y)) and II(y) < II(x I y). If min(II(x I y), II(y)) ::; min(II(x I -.y),II(-.y)) and II(y) < II(x I y) then II(x) may follow II(y) up if II(y) is increasing, and if min(II(x I y), II(y)) >min (II(x I -.y), II(-.y)) and II(y) � IT(x I y) then II(x) may follow II(y) down if II(y) is decreasing. Other wise II(x) is independent of II(y).
Proof: Possibility theory gives II( c)= sup[min(II(c I a),II(a)),min('�r(c I -.a),II(-.a))]. This may not be differentiated, but consider how a small change in II( a) will affect II(c). If II(a) is the value that determines II( c), any change in II( a) will be reflected in II( c). This must happen when min(II(c I a),II(a)) > min(II(c I -.a),II(-.a)) and II(a) < II(c I a). If II(a) is increas ing and the second condition does not hold, it may become true at some point, and so the increase may be reflected in II (c). Similar reasoning may be applied when II( a) is decreasing and the first condition is ini tially false. Thus we can write down the conditions relating II(c) and II(a), while those relating II(c) and II(-.a) as well as those relating II(-.c) and II(a) and II( -.a) may be obtained the same way. D
To formalise this we can say that [ 8val(c)\8val(a)] = [iJ if ii(x) may follow II(y) up and [ 8val(c)\8val(a) J = [!J if ii(x) may follow II(y) down while extending 0 to give:
| 0 | [+J [OJ [-J [?J [iJ [!] |
|-----|-----------------------------|
| [+J | [+J (OJ [-J [?J [+,OJ [OJ |
| [OJ | [OJ [OJ [OJ [OJ [OJ [OJ |
| [-J | [-J [OJ [+J [?J [OJ [-, OJ |
| [?J | [?J [OJ [?] [?J [+,0] [-,0] |
where [ + , OJ indicates a value that is either zero or positive. Normalisation, the possibilistic equivalent of p(a) = 1-p(-.a), ensures that max(II(a), II(-.a)) = 1. Thus at least one of II( a) and II(-.a) is 1, and at most one of II(a) and II(-.a) may change, so II(x) changes when either II(y) or II(-.y) changes.
Writing bel( x) for val( x) in (2) and ( 3 ) , and using Dempster's rule of combination [ Shafer 1976 J to com bine beliefs in the network of Figure 1, we have:
Theorem 3.3: The relation between bel(x) and bel(y) for the link A --+ C is such that bel(x) follows bel(y) iff bel(x I y) > bel(x I y U -.y), bel(x) varies inversely with bel(y) iff bel(x I y) < bel(x I y U -.y) and bel(x) is independent of bel(y) iff bel(x I y) = bel(x I y u-.y) for all x E {c,c}, y E {a, a}.
Proof: By Dempster's rule bel( c)= La �{ a , -,a } m(a) bel(c I a). Now, from Shafer [ 1976 J m(a) = bel(a), m(-.a) = bel(-.a) and m(au-.a) = 1-bel(a)-bel(-.a). Thus 8bel(c)\8bel(a) = bel(c I a)-bel(c I aU-.a). Sim-
ilar reasoning about the way that bel(c) varies with bel(·a) and bel(-.c) varies with bel(a) and bel(·a) gives the result. 0
Note that bel( c) is the belief in hypothesis c given all the available evidence, while bel( c I aU ·a) is the belief induced on c by the marginalisation on { c U -.c} of the joint belief on the space { c U ·c} x {a U ·a}. Thus bel( c ) follows bel( a) if cis more likely to occur given a than given the whole frame. Other results are possible when alternative rules of combination, such as Smets' disjunctive rule [ Smets 1991], are used.
The results presented in this section allow the propa gation of changes in value from A to C given condi tionals of the form val(c I a). It is possible to derive similar results for propagation from C to A [Parsons 1993] which say, for instance, that if p(c) follows p(a), then p( a) follows p( c).
## 4 A COMPARISON OF THE THREE FORMALISMS
It is instructive to compare the qualitative behaviours of the simple link of Figure 1 when the conditional values that determine its behaviour are expressed us ing probability, possibility and evidence theories. This comparison exposes the differences in approach taken by the qualitative formalisms, providing some basis for choosing between them as methods of knowledge rep resentation.
One way of representing the possible behaviours that a link may encode, is to specify the possible values of �val(·a), �val(c) and �val(·c) for given values of �val( a). Thus for probability theory we have:
$$p ( a ) = 1 If Δ p ( a ) = [0] If Δ p ( a ) = [-] p ( a ) ≠ 1 If Δ p ( a ) = [+] If Δ p ( a ) = [0] If Δ p ( a ) = [-]$$
For any value of p(a), either [8p(c)\8p(a)J = [+J, or [8p(c)\8p(a)] =[-],and:
$$p ( c ) = 1 If \Delta p ( c ) = [0] Then \Delta p ( -c ) = [0] If \Delta p ( c ) = [-] Then \Delta p ( -c ) = [-] If \Delta p ( c ) = [+] Then \Delta p ( -c ) = [-]$$
The criterion on which the choice of probability theory is most likely to depend, is whether or not it is ap propriate that [8val(x)\8val(·x)] = [ -] in every case since it is possible to model this in other formalisms, and impossible to avoid it in probability theory.
In possibility theory we have:
$$\begin{array}{ll}
\overline { \Delta I ( - \sigma ) } = 1 & \text { If } \Delta I ( \sigma ) = [0] \\
& \text { Then } \Delta I ( - \sigma ) = [-]
\end{array}
Then \overline { \Delta I ( - \sigma ) } = [?], be co$$
$$\begin{array}{ll}
\overline{\rho ( a ) } & = 1 & \text { If } \Delta I I ( a ) = [ + , with } \\
& = [0, -a) & \text { If } \Delta I I ( a ) = [0] \\
& = [-, \overline{\rho ( a ) } ] & \text { If } \Delta I I ( a ) = [-]
\end{array}$$
For any II(a), either [8II(c)\8II(a)J [0] or [8II(c)\8II(a)] =[+]while:
$$\begin{array}{ll}
I(c) = 1 & \text{if } \Delta I(c) = [0] \\
I(c) \neq 1 & \text{if } \Delta I(c) = [+1, -1] \\
I(c) \neq 1 & \text{if } \Delta I(c) = [-1, +1] \\
I(c) \neq 1 & \text{if } \Delta I(c) = [-1, -1]
\end{array}$$
where �II(x) =[?]is taken to mean �II(x) = [+], [0] or [-]. Thus possibility theory can represent a wider range of behaviours than probability theory.
However, possibility theory has one major limitation that is not shared by probability theory, and that is the fact that it does not have an inverting link. If val( a) increases, it is only possible to have val( c ) decreasing if val(·a) decreases and val(c) follows it. This restricts the representation to the situation in which val( a) :ft 1 and increases to 1 and this may be inappropriate.
Evidence theory is the least restricted of the three. Here we have:
$$\begin{array}{ll}
\text{ours} & \text{bel}(a) = 1 & If } \Delta_{bel}(a) = [0] & Then } \Delta_{bel}(a) = [?] \\
\text{onal} & \text{If } \Delta_{bel}(a) = [-] & Then } \Delta_{bel}(a) = [?] \\
\text{us-} & \text{bel}(a) \neq 1 & If } \Delta_{bel}(a) = [+] & Then } \Delta_{bel}(a) = [?] \\
\text{This} & \text{If } \Delta_{bel}(a) = [-] & Then } \Delta_{bel}(a) = [?] \\
\text{aken} & \text{If } \Delta_{bel}(a) = [+] & Then } \Delta_{bel}(a) = [?] \\
\text{s for} & \text{If } \Delta_{bel}(a) = [-] & Then } \Delta_{bel}(a) = [?] \\
\end{array}$$
For any bel(a), [8bel(c)\8bel(a)J = [+], [0], or [-], while:
$$\begin{array}{ll}
bel(c) = 1 & If \Delta bel(c) = [?] \\
& If \Delta bel(c) = [?] \\
bel(c) \# 1 & If \Delta bel(c) = [?] \\
& If \Delta bel(c) = [?] \\
\end{array}$$
so that there are no restrictions on the changes.
The purpose of this comparison is not to suggest that one formalism is the best in every situation. Instead, it is intended as some indication of which formalism is best for a particular situation. If a permissive formal ism is required, then evidence theory may be the best choice, while probability might be better when a more restrictive formalism is needed.
## 5 QUALITATIVE CHANGES IN MORE COMPLEX NETWORKS
The analysis carried out in Section 3 allows us to pre dict how qualitative changes in certainty value will be propagated in a simple link between two nodes. Now, the change at C depends only on the change at A, and differential calculus tells us that 8z\8x = 8z\8y · oy\ox so the behaviours of such links may be composed. Thus we can predict how qualitative changes are propagated in any network, quantified by
probabilities, possibilities or beliefs where every node has a single parent.
Figure 2: A more complex network
<details>
<summary>Image 2 Details</summary>
### Visual Description
## Directed Graph Diagram: Convergent Flow Structure
### Overview
The image displays a simple directed graph (or flowchart) consisting of three circular nodes connected by two directed edges (arrows). The diagram illustrates a convergent flow where two separate source nodes point to a single destination node. The entire diagram is monochrome, rendered in black lines on a white background.
### Components/Axes
* **Nodes:** Three circular nodes, each labeled with a single uppercase letter.
* **Node B:** Positioned in the top-left quadrant of the image.
* **Node C:** Positioned in the top-right quadrant of the image.
* **Node D:** Positioned in the bottom-center of the image, below and between nodes B and C.
* **Edges (Arrows):** Two straight lines with arrowheads indicating direction.
* **Edge from B to D:** A straight line originating from the bottom of Node B and terminating with an arrowhead pointing to the top-left side of Node D.
* **Edge from C to D:** A straight line originating from the bottom of Node C and terminating with an arrowhead pointing to the top-right side of Node D.
* **Labels:** The only text present are the labels for the nodes: "B", "C", and "D". No axis titles, legends, or numerical data are present.
### Detailed Analysis
The diagram's structure is symmetrical and hierarchical.
* **Spatial Layout:** The two source nodes (B and C) are placed at the same vertical level (top), separated horizontally. The target node (D) is placed centrally below them, creating a triangular or V-shaped composition.
* **Flow Direction:** The arrows establish a clear, unidirectional flow of information, dependency, or process from the top nodes (B and C) to the bottom node (D). This represents a many-to-one or convergent relationship.
* **Visual Elements:** All nodes are identical in size and shape (simple circles). The connecting lines are of uniform thickness. There is no variation in color, line style, or node shape to indicate different types of entities or relationships.
### Key Observations
1. **Symmetry:** The diagram exhibits bilateral symmetry along a vertical axis running through Node D. The positions of B and C are mirror images, as are the angles and lengths of the connecting arrows.
2. **Convergence:** The primary structural feature is the convergence of two distinct paths into a single point (Node D). This is the central visual and conceptual message.
3. **Simplicity and Abstraction:** The diagram uses minimal, abstract symbols (circles, lines, letters). It contains no specific context, labels for the relationships, or quantitative data. Its meaning is entirely dependent on the interpretation of the labels B, C, and D within a larger system.
### Interpretation
This diagram is a foundational representation of a **convergent system** or a **dependency structure**.
* **What it Suggests:** It models a scenario where two independent entities, processes, or data sources (B and C) both contribute to, influence, or feed into a single downstream entity, process, or result (D). This is a common pattern in:
* **Causal Models:** Where B and C are causes and D is an effect.
* **Data Flow:** Where B and C are inputs to a function or process D.
* **Organizational Charts:** Where B and C are subordinates or departments reporting to a manager D.
* **Logical Arguments:** Where B and C are premises supporting a conclusion D.
* **Relationship Between Elements:** The relationship is strictly directional and non-cyclical. There is no feedback loop from D back to B or C. The diagram implies that D's state or existence is dependent on B and C, but not necessarily vice-versa.
* **Notable Absence:** The lack of labels on the arrows is significant. It abstracts away the *nature* of the relationship (e.g., "causes," "provides data to," "reports to"), focusing solely on the *existence* and *direction* of the connection. This makes the diagram a versatile template applicable to many fields.
* **Peircean Investigative Reading:** As a sign, this diagram is an **icon** (it resembles the structure it represents) and a **symbol** (the letters B, C, D stand for arbitrary concepts). The arrows are **indexical signs**, pointing directly from cause to effect. The viewer is invited to interpret the underlying system by assigning meaning to the symbols B, C, and D. The diagram's power lies in its abstraction, allowing it to model the deep structure of countless real-world phenomena where multiple factors combine to produce a single outcome.
</details>
We now extend these results to enable us to cope with networks in which nodes may have more than one par ent. To do this we consider the qualitative effect of two converging links such as those in Figure 2. Since we are only dealing with singly connected networks, B and C are independent and the overall effect at D is determined by:
There are two ways of tackling the network of Figure 2 in probability theory. We can either base our calcula tion on probabilities of the form p(d I b) which implies the simplifying assumption that the effect of B on D is independent of the effect of C (and vice versa), or we can use the proper joint probabilities of the three events B, C, and D, using values of the form p( d I b, c).
$$\begin{array}{c}
\left[ \Delta v _ { a } ( d ) \right] = \left[ \frac{\partial u _ { a } ( b )}{\partial v _ { a } ( c )}\right] \cdot \left[ \frac{\partial u _ { a } ( c )}{\partial v _ { a } ( c )}\right] \cdot \left[ \frac{\partial u _ { a } ( c )}{\partial v _ { a } ( c )}\right]
\end{array}
\begin{array}{c}
\left[ \Delta v _ { b } ( d ) \right] = \left[ \frac{\partial u _ { b } ( c )}{\partial v _ { b } ( c )}\right] \cdot \left[ \frac{\partial u _ { b } ( c )}{\partial v _ { b } ( c )}\right] \cdot \left[ \frac{\partial u _ { b } ( c )}{\partial v _ { b } ( c )}\right]
\end{array}
\begin{array}{c}
\left[ \Delta v _ { c } ( d ) \right] = \left[ \frac{\partial u _ { c } ( c )}{\partial v _ { c } ( c )}\right] \cdot \left[ \frac{\partial u _ { c } ( c )}{\partial v _ { c } ( c )}\right] \cdot \left[ \frac{\partial u _ { c } ( c )}{\partial v _ { c } ( c )}\right]
\end{array}$$
In the first case we assume that the effects of B and C on D are completely independent of one another so that the variation of D with B (and D with C) is just as described by Theorem 3.1, the joint effect being established by using ( 4) to obtain:
which gives the same results as the expression given by Wellman [1990a] for evaluating the same situa tion. vVith the other approach, writing the network as B&C .... D, we have:
$$established by using (4) to obtain:$$
Theorem 5.1: The relation between p(z) and p(x) for the link B&C .... D is determined by:
$$\left[ \frac { \partial p ( z ) } { \partial p ( x ) } \right] = [ p ( z | x , y ) + p ( z - w , y )$$
$$- p ( z | x , y ) - p ( z | - w , y )$$
for all x E {b, -,b}, y{c, -,c} and z E {d, -,d}.
Proof: We have p(d) = LbE{b,-,b}cE{c,-,c} p(b, c, d) = Lbe{b,-,b}cE{c,..,c} p(b, c)p(d I b, c). Since B and C are independent, p(b, c) = p(b)p(c). Using p(x) = 1 p(-,x) and differentiating we find that [ 8p(d)\8p(b)] = p(c) [ p(d I b, c)-p(d I -,b, c)] + p( -, c) [p(d I b, -, c) - p(d I -,b,-,c)] = p(c){[p(d I b,c) + p(d I -,b,-,c)] - [(p(d I b, -, c) + p(d I -,b, c) ]} + [ (p(d I b, -,c)- p(d I ..,b, -,c) ] . From this, and similar results for the variation of p( d) with p(-,b), p(c) and p(-,c), and the way p(-,d) changes with p( b), p( -,b), p( c) and p( -,c), the result follows. D
Thus the way in which for instance p( d) is dependent upon p(b) is itself dependent upon a term just like the synergy condition introduced by Wellman, and apply ing ( 4) we get an expression which has a similar be haviour to that given by Wellman for a synergetic re lation. In possibility theory we have a similar result to that for the simple link:
Theorem 5.2: The relation between II(x), IT(y) and II(z), for all x E {b,-,b}, y E {c,..,c}, z E {d,-,d} for the link B&C .... D is such that:
- (1) II(z) follows II( x) iff II( x, y, z) > sup[II( ..,x, y, z), II(x,-.y,z),II(..,x,-,y,z)] and II(x) < min(II(z I x,y), II(y)), or II(x, -,y, z) > sup[II(x, y, z), II(..,x, y, z), II(-,x,..,y,z)] and II(x) < min(II(z I x,-,y),II(-,y)).
- (2) II(z) may follow II(x) up iff II(x, y, z) ::::; sup [ IT(-.x,y,z),IT(x,-.y,z),II(..,x,..,y,z)] and II(x) < min(II(z I x, y), II(y)), or II(x, -.y, z) ::::; sup[II(x, y, z), II( -,x, y, z), II(-,x, -,y, z) ] and II(x) < min(II(z I x, ..,y), II(-,y)).
- (3) II(z) may follow II(x) down iff II(x, y, z) > sup[II(-.x, y, z),II(x, -,y, z), II(-,x,-,y, z)] and II(x);::: min(IT(z I x, y), II(y)), or II(x, -,y, z) > sup[IT(x, y, z), II( -,x, y, z), II(-,x, -,y, z) ] and II(x) 2': min(IT(z I x,-,y),II(-,y)).
- (4) Otherwise II(z) is independent of II(x).
Proof: As for Theorem 3.2, the result may be de termined directly from IT(d) = SUPxe{x,..,x},YE{y,-,y} II(x,y,z) and II(x,y,z) = II(z I x,y)IT(x)II(y).D
When we use belief values we may take the relationship between B, C and D to be determined by one set of conditional beliefs of the form bel(d \ b, c), or by two sets of conditional beliefs of the form bel( dlb). For conditionals of the form bel(d I b, c) we have:
Theorem 5.3: The relation between bel(z) and bel(x) for the link B &C .... D is determined by:
$$\begin{aligned}
\{ \frac { | b ( z ) | } { | b ( x ) | } \} & = [ b ( z | x , y ) - b ( z | x - u , y ) ] \\
& = [ b ( z | x , - y ) - b ( z | x - u , - y ) ]
\end{aligned}$$
$$\begin{aligned}
& \textcircled { 1 } [ b ( z | x , y U - y ) ] - \textcircled { 2 } [ b ( z | x U - x , y U - y ) ] \\
& = \textcircled { 3 } [ b ( z | x U - x , y U - y ) ] + \textcircled { 4 } [ b ( z | x U - x , y U - y ) ]
\end{aligned}$$
For all x E {b,--,b}, y E {c,--,c}, z E {d,--,d}. Proof: By Dempster's rule of combination, bel(d) = Lb�{b,-.b},c�{c,-.c} m(b)m(c)bel(dib,c). Now, m(x) = bel(x), m(--,x) = bel(--,x) and m(x U --,x) = 1 -bel(x) -bel(--,x), so that [8bel(d)\8bel(b)] = [(bel(d I b, c) bel( d I b U --,b, c)] Ef) [(bel( d I b, -,c)-bel( d I b U --,b, -,c)] EB [ (bel(d I b, cu--,c)-bel(d I bu--,b, cu--,c). From this, and similar results for the variation of bel( d) with bel(--,b), bel(c) and bel(--,c), and the way bel(--,d) changes with bel(b), bel( --,b), bel( c) and bel( --,c) the result follows. D
Thus bel( d) follows bel(b) iff bel(d I b, c) > bel(d I bu--,b,c), bel(d I b,--,c) > bel(d I bu--,b,--,c), and bel(d I b, c U -,c) > bel(d I b U --,b, c U -,c). For conditionals of the form bel( d I b) we obtain:
Theorem 5.4: For the link B&C--+ D, bel(z) follows bel(x) if bel(z I x) � bel(zixU--,x) and is indeterminate otherwise for all x E {b,--,b}, y E {c,--,c}, z E {d,--,d}.
Proof : Dempster's rule of combination tells us that bel(d) = Lb�{b,-.b},c�{ c,-.c},bvc:Jd bel(dib)m(b) bel( dic)m( c). As a result, [8bel( d)\8bel(b )] = [(bel( d I b)-bel( d I bu--,b )]{bel( di--,c) [ 1 +bel( c )+bel( --,c) -m( c)] +bel( di--,c) [ 1 +bel( c)+ bel( -,c)-m( -,c)] +bel( die) [ 1 + bel(c) + bel(--,c)- m(c U -,c)]}+ bel(dl--,b) [m(c)bel(d I c)+ m(--,c)bel(d I -,c)+ m(c U --,c)bel(dic U -,c)). Since m(x) :::; 1 for all x, [8bel(d)\8bel(b)] = [+] if bel(d I b) � bel(dlb U -,b) and [?) otherwise. From this, and similar results for the variation of bel( d) with bel(--,b), bel(c) and bel(--,c), and the way bel(--,d) changes with bel(b), bel(--,b), bel(c) and bel(--,c) the result follows. D
Thus the formalisms again exhibit differences in be haviour across the same network.
The expressions derived in this section are those ob tained by using the precise theory of each formalism. This is important since it ensures the correctness of the integration introduced in Section 6. However, for reasoning using single formalisms, it may provde ad vantageous to extend the simpler apporach adopted by Wellman [1990a) to possibility and evidence theories.
Finally a word on the scope of the reasoning that we can perform as a result of our analysis. The differential calculus tells us that .6-z = Ax· [8z\8x] + .6-y · [8z\8y], provided that x is not a function of y. Thus we can clearly use the results derived above to propagate qual itative changes in probability, possibility and belief functions through any singly connected network.
## 6 INTEGRATION THROUGH QUALITATIVE CHANGE
The work described in this paper so far has extended qualitative reasoning about uncertainty handling for malisms to cover possibility and belief values as well as probability values. Not only is this useful in it self in providing a means of reasoning according to the precise rules of probability, possibility and Dempster Shafer theory when there is incomplete numerical in formation, but it can also provide a way of integrating the different formalisms.
Consider the following medical example. The network of Figure 3 encodes the medical information that joint trauma (T) leads to loose knee bodies (K), and that these and arthritis (A) cause pain (P). The incidence of arthritis is influenced by dislocation (D) of the joint in question and by the patient suffering from Sjorgen 's syndrome (S) . Sjorgen's syndrome affects the inci dence of vasculitis (V), and vasculitis leads to vas culitic lesions (L).
<details>
<summary>Image 3 Details</summary>
### Visual Description
## Directed Acyclic Graph (DAG): Dependency Flow Diagram
### Overview
The image displays a directed acyclic graph (DAG) consisting of eight nodes, each represented by a circle containing a single uppercase letter. The nodes are arranged in three hierarchical levels, with directed edges (arrows) indicating a one-way flow or dependency relationship from parent nodes to child nodes. The graph has a clean, technical diagram style with black outlines and arrows on a white background.
### Components/Axes
* **Nodes:** Eight circular nodes, each labeled with a unique letter: `T`, `D`, `S`, `K`, `A`, `V`, `P`, `L`.
* **Edges:** Directed arrows connecting the nodes, indicating the direction of flow or dependency.
* **Spatial Layout:** The nodes are organized into three distinct horizontal rows or levels.
* **Top Row (Level 1):** Contains nodes `T`, `D`, and `S` from left to right.
* **Middle Row (Level 2):** Contains nodes `K`, `A`, and `V` from left to right. `K` is positioned below and between `T` and `D`. `A` is positioned below and between `D` and `S`. `V` is positioned below and to the right of `S`.
* **Bottom Row (Level 3):** Contains nodes `P` and `L`. `P` is positioned below and between `K` and `A`. `L` is positioned below and to the right of `V`.
### Detailed Analysis
**Node Connections and Flow:**
The graph defines the following specific relationships via directed edges:
1. **From Top Row:**
* Node `T` (top-left) has a single outgoing edge pointing to node `K` (middle-left).
* Node `D` (top-center) has a single outgoing edge pointing to node `A` (middle-center).
* Node `S` (top-right) has two outgoing edges: one pointing to node `A` (middle-center) and another pointing to node `V` (middle-right).
2. **From Middle Row:**
* Node `K` (middle-left) has a single outgoing edge pointing to node `P` (bottom-left).
* Node `A` (middle-center) has a single outgoing edge pointing to node `P` (bottom-left).
* Node `V` (middle-right) has a single outgoing edge pointing to node `L` (bottom-right).
**Summary of Paths:**
* Path 1: `T` → `K` → `P`
* Path 2: `D` → `A` → `P`
* Path 3: `S` → `A` → `P`
* Path 4: `S` → `V` → `L`
### Key Observations
* **Convergence Points:** Node `P` is a convergence point, receiving input from two distinct paths (via `K` and `A`). Node `A` is also a convergence point, receiving input from both `D` and `S`.
* **Divergence Point:** Node `S` is a divergence point, influencing two separate downstream branches (`A` and `V`).
* **Terminal Nodes:** Nodes `P` and `L` are terminal nodes (sinks) with no outgoing edges.
* **Source Nodes:** Nodes `T`, `D`, and `S` are source nodes (roots) with no incoming edges.
* **Symmetry:** The graph exhibits partial symmetry. The left branch (`T`→`K`→`P`) and the central branch (`D`→`A`→`P`) both terminate at `P`. The right branch (`S`→`V`→`L`) is separate and terminates at `L`.
### Interpretation
This diagram is a classic representation of a dependency graph, workflow, or causal model. It visually encodes the following information:
* **Sequential Dependencies:** The arrows define a strict order of operations or influence. For example, `P` cannot occur or be computed until both `K` and `A` are complete. `A` depends on both `D` and `S`.
* **Parallel Processes:** The branches originating from `T`, `D`, and `S` can potentially proceed in parallel until they converge. The `S`→`V`→`L` branch operates independently of the processes leading to `P`.
* **System Structure:** The graph suggests a system where multiple initial conditions (`T`, `D`, `S`) feed into intermediate processes (`K`, `A`, `V`), which then produce final outcomes (`P`, `L`). The fact that `P` integrates information from two major branches (`T`-based and `D`/`S`-based) implies it is a primary or composite result, while `L` is a secondary result derived solely from `S`.
* **Potential Applications:** Such a diagram could model a software build process (where letters represent modules), a project management plan (tasks and dependencies), a Bayesian network (probabilistic dependencies), or a data transformation pipeline. The lack of specific labels means the interpretation is abstract, but the structural relationships are explicitly defined.
</details>
Figure 3: A network representing medical knowledge The strengths of these influences are given as proba bilities:
| p(k I p(k I p(a I p(a I | p(v Is) | | 0.1 | | | |
|---------------------------|-----------|----------|---------|-----|-------------|-----|
| | | t) --,t) | 0.6 0.2 | p(v | I--,s) | 0.3 |
| | d,s) | | 0.9 | p(a | 1--,d,s) | 0.6 |
| | d,--,s) | | 0.6 | p(a | I--,d,--,s) | 0.4 |
beliefs:
k, bel(p I
a)
bel(p I
k,--,a)
--,k, bel(p I
a)
k
bel(p I
U
--,k, k,
bel(p I
a)
a U
-,a)
--,k,
--,p I
-,a)
--,k, bel (
bel(
--,p I
0.9
0.7
0.7
0.6
0.7
0.5
=
=
=
=
a U
-,a)
0.4
All other conditional beliefs are zero and possibilities:
$$\begin{cases} \pi ( 1 - v ) = 1 \\ \pi ( - l | v ) = 0 . 1 \end{cases}$$
We can integrate this information allowing us to say how our belief in the patient in question being in pain,
and the possibility that the patient has vasculitic le sions, vary when we have new evidence that she is suffering from Sjorgen's syndrome. From the new ev idence we have �p(s) = [+], �p(-.s) = [-] , �p(t) = [0], �p( -.t) = [0], �p( d) = [0] and �p( -.d) = [0]. Since a change of [0] can never become a change of [ +] or [ -] we can ignore the latter changes. Now, from Theorem 3.1 and Theorem 5. 1 we know that:
$$\left[ \frac { \partial p ( v ) } { \partial p ( s ) } \right] = - 1 \quad \text{and} \quad \left[ \frac { \partial p ( v ) } { \partial p ( - s ) } \right] = + 1$$
so that �p(a) =[+], and �p(v) =[-] from which we can deduce that �p(-.a) =[-] and �p(-.v) = [+].
To continue our reasoning we need to establish the change in belief of a and the change in possibility of l. To do this we make the monotonicity assumption [Parsons 1993] that if the probability of a hypothesis increases then both the possibility of that hypothesis and the belief in it do not decrease. As well as being intuitively acceptable, this assumption is the weakest sensible relation between values expressed in different formalisms, and is compatible both with the principle of consistency between probability and possibility val ues laid down by Zadeh [1978] and the natural exten sion of this principle to belief, necessity [Dubois and Prade 1988b], and plausibility values.
The assumption also says that if the probability de creases then the possibility and belief do not increase, and so we can say that �bel( a)= [+, OJ, �bel(-.a) = [ - , 0], �II(v) = [ -,0] and �II(-.v) = [+, 0]. Now we apply Theorem 5.3 to find that:
$$\begin{array}{ll}
\left[ \frac{\partial b e l ( p )}{\partial b e l ( a ) } \right] = [ + ] \\
\left[ \frac{\partial b e l ( - p )}{\partial b e l ( a ) } \right] = [ - ]
\end{array}$$
Since we are initially ignorant about the possibility of vasculitis, we have II(v) = II(-.v) = 1, so that Theo rem 3.2 gives:
$$\begin{aligned}
\left[ \frac { \partial p ( t ) } { \partial p ( v ) } \right] = [ 0 ] \\
\left[ \frac { \partial p ( - t ) } { \partial p ( - v ) } \right] = [ 0 ]
\end{aligned}$$
Hence we can tell that �bel(p) = [+, 0], bel(-.p) [ -, OJ and �II( v) = �II( -.v) = [OJ. The result of the new evidence is that belief in the patient's pain may in crease, while the possibility of the patient having vas culitic lesions is unaffected. Thus we can use numeri cal values and qualitative relationships from different uncertainty handling formalisms to reason about the change in the belief of some event given information about the probability of a second event, and can in fer whether the possibility of a third event also varies. As a result reasoning about qualitative change allows some integration between formalisms.
## 7 DISCUSSION
There is an important difference between the approach to qualitative reasoning under uncertainty described here, and that of Wellman [1990a, bJ. Despite their name, Wellman's Qualitative Probabilistic Networks do not describe the qualitative behaviour of probabilis tic networks exactly. In particular, some dependencies between variables are ignored in favour of simplicity, and synergy relations are sometimes introduced to rep resent them where it is considered to be important.
In our approach, since it is based directly upon the various formalisms, the qualitative changes predicted are exactly those of the quantitative methods. This has been demonstrated in [Parsons and Saffiotti 1993J which analyses the representation of a real problem in a number of different qualitative and quantitative for malisms. In this analysis we make qualitative predic tions about the impact of evidence of faults in an elec tricity distribution network and compare these with the real quantitative changes. In every ca<;e, for proba bility, possibility and belief values, the qualitative pre dictions were correct. This verification is a good in dication of the validity of the approach, and suggests that it will be useful in situations where incomplete information prevents the application of quantitative methods.
Our qualitative method also provides a means of in tegrating uncertainty handling formalisms on a purely syntactic basis. For any hypothesis x about which we have uncertain information expressed, say in proba bility theory and possibility theory, we can make the intuitively reasonable assumption that if p( x ) increases II( x ) does not decrease, and thus translate from proba bility to possibility without worrying what probability or possibility actually mean.
As a result any desired semantics may be attached to the values, a feature which finesses the problem of the acceptability of the semantics which must be faced by other, semantically based, schemes for integration (eg. [Baldwin 1991]). The only problem with switch ing semantics would be that some combination rules might no longer apply, Dempster's rule in the case of Baldwin's voting model semantics, which would entail a re-derivation of the appropriate propagation condi tions. Since the qualitative approach does not a priori, rule out any combination scheme, this is not a major difficulty.
Finally, there is one important way that this method might be improved. The main disadvantage of any qualitative system is that there is no distinction be tween small values and large values, so 0.001 is qualita tively the same as 100, 000. As a result we cannot dis tinguish between evidence that induces small changes in the certainty of a hypothesis and evidence that in duces large changes. This problem has been recognised for some time, and there is now a large body of work on order of magnitude reasoning (for example [Raiman
1986], [Parsons and Dohnal 1992]) which attempts to automate reasoning of the form it "If A is bigger than B and B is bigger than C then A is bigger than C". The applications to our system are obvious, and we intend to do some work on this in the near future.
## 8 SUMMARY
This paper has introduced a new method for qualita tive reasoning under uncertainty which is equally ap plicable to all uncertainty handling techniques. All that need be done to find the qualitative relation be tween two values is to write down the analytical ex pression relating them and take the derivative of this expression with respect to one of the values. This fact was illustrated by results from the qualitative analysis of the simplest possible reasoning networks in each of the three most widely used formalisms.
Having established the qualitative behaviours of prob ability, possibility and evidence theories, the differ ences between these behaviours were discussed at some length, before knowledge of this behaviour was used to establish a form of qualitative integration between for malisms. In this integration numerical and qualitative data expressed in all three formalisms was used to help derive the change in belief of one node in a directed graph and the possibility of another from knowledge of a change in the probability of a third, related, node.
## Acknowledgements
The work of the first author was partially supported by a grant from ESPRIT Basic Research Action 3085 DRUMS, and he is endebted to all of his colleagues on the project for their help and advice.
Special thanks are due to Mirko Dohnal, Didier Dubois, John Fox, Frank Klawonn, Paul Krause, Rudolf Kruse, Henri Prade, Alessandro Saffiotti and Philippe Smets for uncomplaining help and construc tive criticism. The anonymous referees also made a number of useful comments.
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