Robust logics Original Research Article

Suppose that we wish to learn from examples and counter-examples a criterion for recognizing whether an assembly of wooden blocks constitutes an arch. Suppose also that we have preprogrammed recognizers for various relationships, e.g., on-top-of(x,y), above(x,y), etc. and believe that some possibly complex expression in terms of these base relationships should suffice to approximate the desired notion of an arch. How can we formulate such a relational learning problem so as to exploit the benefits that are demonstrably available in propositional learning, such as attribute-efficient learning by linear separators, and error-resilient learning? We believe that learning in a general setting that allows for multiple objects and relations in this way is a fundamental key to resolving the following dilemma that arises in the design of intelligent systems: Mathematical logic is an attractive language of description because it has clear semantics and sound proof procedures. However, as a basis for large programmed systems it leads to brittleness because, in practice, consistent usage of the various predicate names throughout a system cannot be guaranteed, except in application areas such as mathematics where the viability of the axiomatic method has been demonstrated independently. In this paper we develop the following approach to circumventing this dilemma. We suggest that brittleness can be overcome by using a new kind of logic in which each statement is learnable. By allowing the system to learn rules empirically from the environment, relative to any particular programs it may have for recognizing some base predicates, we enable the system to acquire a set of statements approximately consistent with each other and with the world, without the need for a globally knowledgeable and consistent programmer. We illustrate this approach by describing a simple logic that has a sound and efficient proof procedure for reasoning about instances, and that is rendered robust by having the rules learnable. The complexity and accuracy of both learning and deduction are provably polynomial bounded.