A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

Automatic Recognition of Tractability in Inference Relations

A procedure is given for recognizing sets of inference rules that generate polynomial time decidable inference relations. The procedure can automatically recognize the tractability of the inference rules underlying congruence closure. The recognition of tractability for that particular rule set constitutes mechanical verification of a theorem originally proved independently by Kozen and Shostak. The procedure is algorithmic, rather than heuristic, and the class of automatically recognizable tractable rule sets can be precisely characterized. A series of examples of rule sets whose tractability is non-trivial, yet machine recognizable, is also given. The technical framework developed here is viewed as a first step toward a general theory of tractable inference relations.

Toward a Theory of Representation Design

This research is concerned with designing representations for analytical reasoning problems (of the sort found on the GRE and LSAT). These problems test the ability to draw logical conclusions. A computer program was developed that takes as input a straightforward predicate calculus translation of a problem, requests additional information if necessary, decides what to represent and how, designs representations capturing the constraints of the problem, and creates and executes a LISP program that uses those representations to produce a solution. Even though these problems are typically difficult for theorem provers to solve, the LISP program that uses the designed representations is very efficient.

Reasoning from Incomplete Knowledge in a Procedural Deduction System

One very useful idea in AI research has been the notion of an explicit model of a problem situation. Procedural deduction languages, such as PLANNER, have been valuable tools for building these models. But PLANNER and its relatives are very limited in their ability to describe situations which are only partially specified. This thesis explores methods of increasing the ability of procedural deduction systems to deal with incomplete knowledge. The thesis examines in detail, problems involving negation, implication, disjunction, quantification, and equality. Control structure issues and the problem of modelling change under incomplete knowledge are also considered. Extensive comparisons are also made with systems for mechanica theorem proving.

The Definition and Implementation of a Computer Programming Language Based on Constraints

The constraint paradigm is a model of computation in which values are deduced whenever possible, under the limitation that deductions be local in a certain sense. One may visualize a constraint 'program' as a network of devices connected by wires. Data values may flow along the wires, and computation is performed by the devices. A device computes using only locally available information (with a few exceptions), and places newly derived values on other, locally attached wires. In this way computed values are propagated. An advantage of the constraint paradigm (not unique to it) is that a single relationship can be used in more than one direction. The connections to a device are not labelled as inputs and outputs; a device will compute with whatever values are available, and produce as many new values as it can. General theorem provers are capable of such behavior, but tend to suffer from combinatorial explosion; it is not usually useful to derive all the possible consequences of a set of hypotheses. The constraint paradigm places a certain kind of limitation on the deduction process. The limitations imposed by the constraint paradigm are not the only one possible. It is argued, however, that they are restrictive enough to forestall combinatorial explosion in many interesting computational situations, yet permissive enough to allow useful computations in practical situations. Moreover, the paradigm is intuitive: It is easy to visualize the computational effects of these particular limitations, and the paradigm is a natural way of expressing programs for certain applications, in particular relationships arising in computer-aided design. A number of implementations of constraint-based programming languages are presented. A progression of ever more powerful languages is described, complete implementations are presented and design difficulties and alternatives are discussed. The goal approached, though not quite reached, is a complete programming system which will implicitly support the constraint paradigm to the same extent that LISP, say, supports automatic storage management.