We resolve an old problem, namely to design a ‘natural’ machine model for accepting the complements of recursively enumerable languages. The model we present is based on non-deterministic Turing machines with ‘one-sided’ advice. We prove that these machines precisely accept the co-RE languages, without restriction on the advice functions that are used. We show that for accepting a co-RE language, one-sided advice is as powerful as arbitrary advice, but also that linearly bounded one-sided advice is sufficient. However, ‘long’ sublinear advice can make Turing machines with one-sided advice accept more co-RE languages than ‘short’ sublinear advice. We prove that infinite proper hierarchies of language classes inside co-RE can be devised, using suitable increasing sequences of bounding functions for the allowed advice. The machine model and the results dualize for the family of recursively enumerable languages.

The neuroidal tabula rasa (NTR) as a hypothetical device that is capable of performing tasks related to cognitive processes in the brain was introduced by L. G. Valiant in 1994. Neuroidal nets represent a computational model of the NTR. Their basic computational element is a kind of a programmable neuron called neuroid. Essentially, it is a combination of a standard threshold element with a mechanism that allows modification of the neuroid's computational behaviour. This is done by changing its state and the settings of its weights and of threshold in the course of computation. The computational power of an NTR crucially depends both on the functional properties of the underlying update mechanism that allows changing of neuroidal parameters and on the universe of allowable weights. We will define instances of neuroids for which the computational power of the respective finite-size NTR ranges from that of finite automata, through Turing machines, up to that of a certain restricted type of BSS machines that possess super-Turing computational power. The latter two results are surprising since similar results were known to hold only for certain kinds of analog neural networks.

A finite automaton—the so-called neuromaton, realized by a finite discrete recurrent neural network, working in parallel computation mode, is considered. Both the size of neuromata (i.e., the number of neurons) and their descriptional complexity (i.e., the number of bits in the neuromaton representation) are studied. It is proved that a constraint time delay of the neuromaton output does not play a role within a polynomial descriptional complexity. It is shown that any regular language given by a regular expression of length n is recognized by a neuromaton with Θ( n ) neurons. Further, it is proved that this network size is, in the worst case, optimal. On the other hand, generally there is not an equivalent polynomial length regular expression for a given neuromaton. Then, two specialized constructions of neural acceptors of the optimal descriptional complexity Θ( n ) for a single n -bit string recognition are described. They both require O(n 1/2 ) neurons and either O(n) connections with constant weights or O(n 1/2 ) edges with weights of the O(2 √n ) size. Furthermore, the concept of Hopfield languages is introduced by means of so-called Hopfield neuromata (i.e., of neural networks with symmetric weights). It is proved that the class of Hopfield languages is strictly contained in the class of regular languages. The necessary and sufficient so-called Hopfield condition stating when a regular language is a Hopfield language, is formulated. A construction of a Hopfield neuromaton is presented for a regular language satisfying the Hopfield condition. The class of Hopfield languages is shown to be closed under union, intersection, concatenation and complement, and it is not closed under iteration. Finally, the problem whether a regular language given by a neuromaton (or by a Hopfield acceptor) is nonempty, is proved to be PSPACE-complete. As a consequence, the same result for a neuromaton equivalence problem is achieved.