Incomplete knowledge

Incomplete knowledge will be categorized into three types:

Knowledge having a sort of label that shows that it may be wrong
Theorist [Poole 1987], which involves hypothetical reasoning, divides logical knowledge into hypotheses and facts. Once again, hypotheses involve incomplete knowledge, because among them inconsistencies can occur. In this case, however, a hypothesis is knowledge that is to be abduced.
Theorist labels a knowledge set including the knowledge that may be abduced as a hypothesis set. In logic based hypothetical reasoning systems like Theorist, the justification of abduced knowledge (hypothesis) is guaranteed by a logical process.
Knowledge having a sort of measure
In probabilistic reasoning [Pearl 1988], knowledge has a sort of measure for plausibility. Knowledge whose plausibility is not 1.0, is incomplete knowledge. However, knowledge is not always abduced; it is something that appears with a degree of possibility.
The ε-semantics gives a strange connection between logic and probability. Some abduction systems like Cost-based abduction [Charniak 1994] or Probabilistic Horn abduction [Poole 1993a, 1993b] introduce possibility. In addition, Poole has shown that propositional probabilistic Horn abduction is equal to a Bayesian network. In these cases, however, possibility works as a sort of weight (cost) for selecting good hypotheses.
Some knowledge that is missing
Every knowledge base at present lacks the knowledge that is needed to make inferences. Therefore, missing knowledge must be supplemented. From the viewpoint that the knowledge needed for an inference is missing, we can say that the knowledge base is incomplete. Though supplemented knowledge is equal to hypotheses in (1) above, no candidate hypotheses are prepared. At times, therefore, it is hard to justify those hypotheses that are abduced. Hypotheses generated by CMS can be regarded as knowledge of this type.

The incomplete knowledge mentioned in this research belongs to the type of incomplete knowledge in (3). In this paper, inference with incomplete knowledge means to abduce missing knowledge, and to create such knowledge from the knowledge in the knowledge base by abduction and analogical mapping. By using these strategies, an inference system can create new knowledge, but the number of candidate hypotheses is very large and it is hard to justify these hypotheses. Therefore, slight changes are made to the definition for the incompleteness of knowledge as follows;

Some knowledge that is missing, but substitute knowledge exists
In this case, a certain piece of necessary knowledge is missing from the knowledge base; however, some other knowledge analogous to the missing knowledge exists. Therefore, even if a set of hypotheses is not given explicitly, a certain hypothesis can be mapped to a certain piece of implicit knowledge in the knowledge base. As a result, the mapped knowledge can be substitute knowledge for an inference.
Here, knowledge in the knowledge base is used to make the inference, and it is a justified hypothesis, not a new hypothesis. However, the combination of hypotheses is new. In this situation, the analogical mapping plays a significant role; it provides a guarantee to the combination of hypotheses.
Furthermore, if the mapped knowledge is mapped to knowledge that must be abduced to explain an observation, this knowledge is new and equal to a hypothesis categorized into type (3). The justification of the hypothesis and the explanatory coherence can be guaranteed by the analogical mapping.

The fact that a hypothesis can be mapped to knowledge in the knowledge base means that a hypothesis categorized into type (3) can be abduced.

References:

[Charniak 1994] Charniak E. and Shimony S. E.: Cost-based abduction and MAP explanation, rtif. Intell., vol. 66, pp. 345--374 (1994).
[Pearl 1988] Pearl J.: Probablistic Reasoning in Intelligent Systems: Networks of Plausible Inference (2nd Ed.), Morgan Kaufmann (1988).
[Poole 1987] Poole D., Aleliunas R., Goebel R.: Theorist: A Logical Reasoning System for Defaults and Diagnosis, The Knowledge Frontier: Essays in the Representation of Knowledge (N.J.Cercone, G.MaCalla Eds.), Springer-Verlag (1987).
[Poole 1993a] Poole D.: Probablistic Horn Abduction and Bayesian networks, Artif. Intell., vol. 64, pp. 81--129 (1993).
[Poole 1993b] Poole D.: Logic Programming, Abduction and Probability: a top-down anytime algorithm for estimating prior and posterior probabilities, New Generation Computing, vol. 11(3-4), pp. 377--400 (1993).