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.

Cerebellar Learning in an Opponent Motor Controller for Adaptive Load Compensation and Synergy Formation

This paper shows how a minimal neural network model of the cerebellum may be embedded within a sensory-neuro-muscular control system that mimics known anatomy and physiology. With this embedding, cerebellar learning promotes load compensation while also allowing both coactivation and reciprocal inhibition of sets of antagonist muscles. In particular, we show how synaptic long term depression guided by feedback from muscle stretch receptors can lead to trans-cerebellar gain changes that are load-compensating. It is argued that the same processes help to adaptively discover multi-joint synergies. Simulations of rapid single joint rotations under load illustrates design feasibility and stability.