A significant problem in the design and construction of an artificial neural network for function approximation is limiting the magnitude and the variance of errors when the network is used in the field. Network errors can occur when the training data does not faithfully represent the required function due to noise or low sampling rates, when the network's flexibility does not match the variability of the data, or when the input data to the resultant network is noisy. This paper reports on several experiments whose purpose was to rank the relative significance of these error sources and thereby find neural network design principles for limiting the magnitude and variance of network errors.

Courses in Computer Science have undergone significant improvement during the last few years. This improvement has resulted from the introduction of guiding principles such as structured programming and top-down design. Unfortunately, even with the use of these principles, buggy programs are commonplace. More guiding principles are needed.

It is proved that an automatic theorem proving system consisting of resolution, paramodulation, factoring, equality reversal, simplification, and subsumption removal is complete in first-order logic with equality. When restricted to equality units, the system is similar to the Knuth-Bendix procedure for deriving consequences from equalities. However, our proofs of completeness are restricted to the case in which the ordering on words (terms or atoms) that is required in this type of process is order-isomorphic to the positive integers. The completeness of resolution and paramodulation without the functionally reflexive axioms is a simple corollary of our result. The methods used are based upon the familiar ideas associated with semantic trees, and should be helpful in showing that other theorem proving systems with equality are complete.

An extenston of the Knuth-Bendix algorithm for finding complete sets of reductions is described.The extension is intended to handle equational theories which can be split into two parts, R and T, such that each equation m R can be construed as a reduction and T represents an equational theory for which a finite, complete umficat~on algorithm ~s known.The extension ts demonstrated in the case when T is the theory of a fimte number of associaUve and commutatwe functions and to which the extension is presently restricted An tmplementatlon of the extended Knuth-Bendtx algorithm has produced complete sets of reductions for free commutattve groups, commutative rings wtth umt, and distributive lattices.

The concern here is with proof procedures which are generalizations of input or unit deduction. The author's generalizations of input deduction involve lemmas, whereas those of unit deduction involve longer clauses and are akin to Robinson's P1 deduction. Chang's theorem, which establishes the equivalence of input and unit refutation, is extended to these generalizations. Completeness results of Henschen, Wos, and Kuehner for input or unit deduction applied to Horn sets are generalized to apply also to non-Horn sets. A key result is that any unsatisfiable set can be refuted by a lock linear resolution procedure in which the only lemmas are positive clauses composed entirely of instances of a small set of literals which can be specified in advance. In an implementation such lemmas would be generated only infrequently, thus allowing one to periodically gather the lemmas, discard other generated clauses, and restart the proof procedure.