Fixed Point vs. First-Order Logic on Finite Ordered Structures with Unary Relations

We prove that first order logic is strictly weaker than fixed point logic over every infinite classes of finite ordered structures with unary relations: Over these classes there is always an inductive unary relation which cannot be defined by a first-order formula, even when every inductive sentence (i.e., closed formula) can be expressed in first-order over this particular class. Our proof first establishes a property valid for every unary relation definable by first-order logic over these classes which is peculiar to classes of ordered structures with unary relations. In a second step we show that this property itself can be expressed in fixed point logic and can be used to construct a non-elementary unary relation.

Self-Supervised ARTMAP

Computational models of learning typically train on labeled input patterns (supervised learning), unlabeled input patterns (unsupervised learning), or a combination of the two (semisupervised learning). In each case input patterns have a fixed number of features throughout training and testing. Human and machine learning contexts present additional opportunities for expanding incomplete knowledge from formal training, via self-directed learning that incorporates features not previously experienced. This article defines a new self-supervised learning paradigm to address these richer learning contexts, introducing a neural network called self-supervised ARTMAP. Self-supervised learning integrates knowledge from a teacher (labeled patterns with some features), knowledge from the environment (unlabeled patterns with more features), and knowledge from internal model activation (self-labeled patterns). Self-supervised ARTMAP learns about novel features from unlabeled patterns without destroying partial knowledge previously acquired from labeled patterns. A category selection function bases system predictions on known features, and distributed network activation scales unlabeled learning to prediction confidence. Slow distributed learning on unlabeled patterns focuses on novel features and confident predictions, defining classification boundaries that were ambiguous in the labeled patterns. Self-supervised ARTMAP improves test accuracy on illustrative lowdimensional problems and on high-dimensional benchmarks. Model code and benchmark data are available from: http://techlab.bu.edu/SSART/.