(ab,in, de)duction in Hierarchical Fuzzy Representation.
An example

(abridged version)

Maurice Clerc 1995

Introduction

An important principle in HFR (Hierarchical Fuzzy Representation) is that each "object of memory" or "memobject" is defined only by its (fuzzy)relationships with all the others, in a finite closed world. Also one of its purposes is to give results which are "psychologically valid", in words of comparison, classification, reasoning. That is why, as far as possible, some parameters are coming from real tests with human beings. As some other models, HFR makes no difference between the classical inference forms (e.g.abduction, induction, deduction), and, in fact, between any logical figure. I show here how it can run against a simple example, coming from a paper of Pei Wang about his Non-axiomatic Reasoning System (NARS) [10] , but with another formalization.

Phase 1: Formalization (human and computer)

The activation/inhibition or evocation relation

is used to formalize all assertions like "is", "is a", "has", "is not" .... It can be understand as follows:

"if x is evoked (in memory) then y too (with strength s)"

The evocation strength s is a fuzzy value whose support is either in [-1,0[ (inhibition) or in [0,1] (activation). With this notations,the example of Pei Wang can be rewritten as follow:

                       J1: bird   flyer                         J2: dove  bird                        J3: dove   swimmer                       J4: swan  bird                         J5: swan  flyer                         J6: swan  swimmer                          J7: penguin  bird                         J8: penguin  flyer                       J9: penguin  swimmer

Data are not logically consistent, as often in the real world (See J1, J7, J8). Here, there is no fuzzyness at all. Nevertheless, all what follows is still valid in general case, for one knows how to compute sum and product of unimodal fuzzy values (e.g. [4]).

Phase 2: Building the network (computer)

Except for the reverse links, the building is obvious. I choose here very simple (and very classical !) activation and propagation rules (weighted-sum computation, e.g. also Ron Sun [9]).

Reverse links

Let's suppose the information "bird" is completely evoked (activated) in memory. In the finite closed world defined by the initial knowledge, it impliesthat we must also have, more or less, either "dove", or "swan", or "penguin". That is why the systems creates links from "bird" to this three nodes.

Figure 1. Network of the closed world given by J1-J8

Phase 3: Fuzzy representations of concepts by spreading activation (computer)

In HFR, generally speaking, a concept at level n is a constellation of activations of nodes at level n-1. In particular, the constellation obtained by activating to 1 just one node, is called the significance of this node [3]. It is interesting to note that there is a kind of circularity, just as in a dictionary, that is to say the "meaning" of each object depends of the context(all "co-evoked" objects). That is why I call sometimes the worlds generated by HFR "circular worlds".

Table 1. Fuzzy representation (significance) of concept bird

		bird	flyer	dove	swimmer	swan	penguin
bird	time 1	1.00	0.80	0.42		0.42	0.42
	time 2	1.00	0,80	0.42	0.30	0.42	0.42

Phase4: Evocations of concepts (computer)

There are many ways to calculate the similarity between two fuzzy sets (See e.g. [7]),and then the "strength" of evocation from a concept by another, but a good choice (from a psychological viewpoint) seems to be non-linear functions of the angle of the significances, seen as vectors [1, 2, 8].

Table 3. Evocations after stabilization (time step 2) . Read from left to right. Ex.: "bird evokes flyer (0.69)".

	bird	flyer	dove	swimmer	swan	penguin
bird	1.00	0.69	0.54	0.44	0.75	0.32
flyer	0.57	1.00	0.43	0.04	0.47	-0.25
dove	0.59	0.56	1.00	-0.45	0.17	-0.17
swimmer	0.37	0.04	-0.35	1.00	0.54	0.54
swan	0.86	0.65	0.18	0.73	1.00	0.43
penguin	0.37	-0.34	-0.18	0.73	0.43	1;00

Phase 5: Fuzzy representations of stereotypes (computer)

After Phase 4, it is now easy to built some classes by fuzzy clustering of the concepts (See e.g. [5, 6]),and their "representatives", called stereotypes (and not prototypes. as often said). For example. the class of bird is the fuzzy set:

{bird/1, flyer/0.69, dove/0.59, swimmer/0.44, swan/0.86, penguin/0.37}

For each class, the stereotype is calculated as "weighted average"., and then normalized in order to become a significance..

Table 4. Significance of stereotype _bird

	bird	flyer	dove	swimmer	swan	penguin
_bird	1.00	0.74	0.32	0.46	0.61	0.37

Phase6: Evocations of stereotypes (computer)

Exactly in the same way as for concepts, we can calculate "how much"' a stereotype evokes another. In fact, stereotypes are (new) concepts.

Table 5. Evocations of a stereotype by another. Ex.: "_bird evokes _flyer(0.94)".

	_bird	_flyer	_dove	_swimmer	_swan	_penguin
_bird	1.00	0.94	0.91	0.51	0.97	0.43
_flyer	0.82	1.00	0.96	0.24	0.75	0.14
_dove	0.81	0.97	1.00	0.11	0.71	0.08
_swimmer	0.50	0.27	0.12	1.00	0.63	0.80
_swan	0.98	0.88	0.82	0.66	1.00	0.54
_penguin	0.28	0.11	0.06	0.54	0.28	1.00

Maybe there is still some inconsistency between (penguin, bird, flyer), but not as obvious as in the initial knowledge.

Phase7: Interpretations (human and computer)

Now. we can answer some questions. In principle, all requests to HFR are like"I have the information X (with the possibility/plausibility/confidence/strength x); What about Y ?" . Of course, it can be said sometimes in a shorter way.

Table 6. Some requests, and some answers. The level "descriptors" indicates just the given pieces of evidence.

LEVEL
	descriptors	concepts/significances	stereotypes
Is a bird a flyer ?	0.80	0.69	0.94
What is a "typical" bird ?	dove/0.80 swan/0.80 penguin/0.80	swan/0.86 dove/0.59 penguin/0.37	swan/0.98 dove/0.81 penguin/0.28
Is a bird a swimmer ?	0.00	0.44	0.51
Is a dove a flyer ?	0.00	0.56	0.97

Comments of Table 6

Numerical values has to be interpreted in a linguistic way, e.g. 0.8="Very probably". It is interesting to note that there is no monotonicity.

Is a bird a flyer ?

At the very beginning, there is only a direct link bird => flyer, So the answer is "Very probably" (0.80). But by working more on this question HFRhas to cope with the inconsistency between (penguin, bird, flyer). In particular, penguin is more or less a bird and not a flyer. So the"confidence" is decreasing: "Probably" (0.69). By construction, stereotypes maybe less consistent as initial concepts, and there is indeed the case: penguin is almost entirely ignored in the stereotype "bird", so the confidence is much higher:"Pretty sure" (0.94), even if "bird" is still in the stereotype"penguin" (0.28). Even if we do not have seen the initial inconsistency, the important increase, from 0.69 to 0.94, points out there may be something wrong in the data.

What is a "typical" bird?

The swan "wins": "Almost sure" (0.86 and, after, 0.98). Usually, this could be an abduction (from J1: a bird is a flyer and J5: a swan is a flyer)

Is a bird a swimmer ?

Finally "Probably" (0.51). Again, the great difference with the value at level concepts "Maybe" (0.44) may indicate an inconsistency in the data. Usually, this is called an induction (from J4: a swan is a bird and J6: a swan is a swimmer)

Is a dove a flyer ?

At level concepts "Maybe" (0.56). This value could seem small. After all, one could say the "naive" deduction

J2: a dove is a bird (0.8) and J1: a bird is a flyer( 0.8). so dove is a flyer (0.64=0.8*0.8)

But in fact dove is not a very "good" bird. So 0.56 seems a good compromise.This "subtleties" disappear at level stereotype, and the answer becomes"Surely" (0.97), and this time, the value is too high.

Conclusion

For the moment, I do not have enough results with human beings to really evaluate the psychological validity of the above conclusions, but intuitively,they seem reasonable.

And by the way, what would be YOUR answers ?

References

[1]X. Chanet, Décompositions floues, ressemblances, catégorisations,1992, France Télécom: Annecy, France.

[2] M. Clerc, Validité psychologique des représentations floues. (Info. In Cognito, 1, Décembre 1995, 3-5) (English version available)

[3] M. Clerc, F. Guérin, et al. Représentations floues dans un mémoriel. in JIOSC (Journées Internationales d'Orsay sur les Sciences Cognitives). 1994. Orsay, France: CNRS.

[4] D. Dubois, H. Prade, La théorie des possibilités (Masson,Paris, 1985).

[5] T. Gu, B. Dubuisson, Similarity of classes and fuzzy clustering, Fuzzy Sets& Systems 34 (1990) 213-221.

[6] K. Hattori, Y. Torri, Effective algorithms for the nearest neighbor method in the clustering problem, Pattern recognition 26 (1993) 741-746.

[7] C.P. Pappis, A comparative assessment of measures of similarity of fuzzy values, Fuzzy Sets & Systems 56 (1993) 171-174.

[8] S.A. Sloman, Feature-based induction, Cognitive psychology 25 (1993)231-280.

[9] R. Sun, A neural network model of causality, IEEE transactions on neural networks 5 (1994) 604-611.

[10] P. Wang, From Inheritance Relation to Non-Axiomatic Logic, International Journal of Approximate Reasoning 11 (1994) 281-319.