Learning a Plan in the Limit

Induction is one of the major forms of reasoning; it is therefore essential to include it in a commonsense reasoning framework. This paper examines a class of induction-based planning problems. These problems can be solved, but not in the classical sense, where one and only one output—a correct solution—is to be provided. Here a sequence of outputs, or hypothetical solutions, is output, with finitely of them being possibly incorrect, up to a point where the sequence becomes constant, having converged to a correct solution. Still it is usually not possible to ever discover when convergence has taken place, hence it is usually not possible to definitely know that the problem has been solved, a behavior known in the literature on inductive inference, or formal learning theory, as identification in the limit. We present a semantics for iterative (looping) planning based on identification in the limit, with the planner learning from positive examples (as opposed for instance to negative examples, or answers to queries). Potential plans are repeatedly hypothesized, and a correct plan will be converged upon iff such a plan exists. We show that this convergence happens precisely when a particular logical entailment relation is satisfied. The existing system KPLANNER, which generates iterative plans by generalizing over two examples, is analogous to a fragment of this procedure. We describe an optimizing version, which attempts to satisfy a goal in the best possible way by defining preferences on plans, in the form of an order relation. We also discuss potential extensions to planning which cannot be solved by induction (and a fortiori, cannot be solved in the classical sense).