First of all, here is what the PLN book has to say about default inference (in the chapter on Inference Control):
Inference control contains a great number of subtleties, only a small minority of which has been considered here. Before leaving the topic we will consider one additional aspect, lying at the borderline between inference control, PLN inference proper, and cognitive science. This is the notion of “default inheritance,” which plays a role in computational linguistics within the Word Grammar framework (Hudson 1984), and also plays a central role in various species of nonmonotonic “default logic” (Reiter, 1980; Delgrande and Schaub, 2003). The treatment of default inference in PLN exemplifies how, in the PLN framework, judicious inference control and intelligent integration of PLN with other structures may be used to achieve things that, in other logic frameworks, need to be handled via explicit extension of the logic. To exemplify the notion of default inheritance, consider the case of penguins, which do not fly, although they are a subclass of birds, which do fly. When one discovers a new type of penguin, say an Emperor penguin, one reasons initially that they do not fly – i.e., one reasons by reference to the new type’s immediate parent in the ontological hierarchy, rather than its grandparent. In some logical inference frameworks, the notion of hierarchy is primary and default inheritance of this nature is wired in at the inference rule level. But this is not the case with PLN – in PLN, correct treatment of default inheritance must come indirectly out of other mechanisms. Fortunately, this can be achieved in a fairly simple and natural way. Consider the two inferences (expressed informally, as we are presenting a conceptual discussion not yet formalized in PLN terms) A) penguin --> fly <0> bird --> penguin <.02> |- bird --> fly B) penguin --> bird <1> bird --> fly <.9> |- penguin --> fly The correct behavior according to the default inheritance idea is that, in a system that already knows at least a moderate amount about the flight behavior of birds and penguins, inference A should be accepted but inference B should not. That is, evidence about penguins should be included in determining whether birds can fly – even if there is already some general knowledge about the flight behavior of birds in the system. But evidence about birds in general should not be included in estimating whether penguins can fly, if there is already at least a moderate level of knowledge that in fact penguins are atypical birds in regard to flight. But how can the choice of A over B be motivated in terms of PLN theory? The essence of the answer is simple: in case B the independence assumption at the heart of the deduction rule is a bad one. Within the scope of birds, being a penguin and being a flier are not at all independent. On the other hand, looking at A, we see that within the scope of penguins, being a bird and being a flier are independent. So the reason B is ruled out is that if there is even a moderate amount of knowledge about the truth-value of (penguin --> fly), this gives a hint that applying the deduction rule’s independence assumption in this case is badly wrong. On the other hand, what if a mistake is made and the inference B is done anyway? In this case the outcome could be that the system erroneously increases its estimate of the strength of the statement that penguins can fly. On the other hand, the revision rule may come to the rescue here. If the prior strength of (penguin --> fly) is 0, and inference B yields a strength of .9 for the same proposition, then the special case of the revision rule that handles wildly different truth-value estimates may be triggered. If the 0 strength has much more confidence attached to it than the .9, then they won’t be merged together, because it will be assumed that the .9 is an observational or inference error. Either the .9 will be thrown out, or it will be provisionally held as an alternate, non-merged, low-confidence hypothesis, awaiting further validation or refutation. What is more interesting, however, is to consider the implications of the default inference notion for inference control. It seems that the following may be a valuable inference control heuristic: 1. Arrange terms in a hierarchy; e.g., by finding a spanning DAG of the terms in a knowledge base, satisfying certain criteria (e.g., maximizing total strength*confidence within a fixed limitation on the number of links). 2. When reasoning about a term, first do deductive reasoning involving the term’s immediate parents in the hierarchy, and then ascend the hierarchy, looking at each hierarchical level only at terms that were not visited at lower hierarchical levels. This is precisely the “default reasoning” idea – but the key point is that in PLN it lives at the level of inference control, not inference rules or formulas. In PLN, default reasoning is a timesaving heuristic, not an elementary aspect of the logic itself. Rather, the practical viability of the default-reasoning inference-control heuristic is a consequence of various other elementary aspects of the logic, such as the ability to detect dependencies rendering the deduction rule inapplicable, and the way the revision rule deals with wildly disparate estimates.
What occurred to me a couple days ago is that one could perhaps formalize the above notion of “default inference by inference control” more rigorously ... and go a little further by articulating an explicit default inference rule based on the hierarchy mentioned there.
This could be done in a number of ways. For instance, one could define a notion of OntologicalInheritance, based on the spanning dag mentioned above.
One can build a default logic based on the spanning dag G, as follows. Suppose B lies below A in the dag G. And, suppose that, for predicate F, we have
meaning that F applies to A with truth value t. Then, we may say that “If ~F(B) is not known, then F(B)”. In other words, we may assume by default that B possesses the properties of the terms above it in the hierarchy.
More formally, we might propose a “default inference rule” such as:
IMPLICATION AND OntologicalInheritance B A Evaluation F A NOT Evaluation Known( NOT (Evaluation F B ) ) Evaluation F B
There is no reason that a rule like this can’t be implemented within PLN. Note that implementing this rule within PLN gives you something nice, which is that all the relationships involved (OntologicalInheritance, Known, NOT, etc.) may be probabilistically quantified, so that the outcome of the inference rule may be probabilistically quantified.
The “Known” predicate is basically the K predicate from standard epistemic logic (K_a, where a is the reasoning system itself in this case.)
The hierarchy G needs to be periodically rebuilt as it is based on abstracting a dag from a graph of probabilistic logical relations. And, the results of the above default inference rule may be probabilistic and may be merged with the results of other inference rules.
The results from using this rule, in principle, should not be so different from if the rule were not present. However, the use of an ontological hierarchy in this way leads to a completely different dynamics of inference control, which in practice WILL lead to different results....
And, note finally that for this sort of ontological approach can also be used in the context of association spreading. In this case, it may take the form (for example) of ontology-based pruning of Assocation relationships. The Association between pigeon and flyer may be pruned because it is essentially redundant with a set of two Association relationships that are implicit in the hierarchy:
Association pigeon bird Association pigeon flyer
Using an ontology-pruned set of Association links to guide inference that is partially based on ontology-driven default inference rules, provides an alternative approach to solving the frame problem, that doesn’t require introducing a notion of hierarchy into the underlying logic.