Pseudorecursive varieties 457) exhibit a lack of recursive uniformity, expressing the failure of universal and existential quantifiers to reverse. Several examples are given of personal encounters with infeasible or errant quantifier reversal. Results strengthening and applying pseudorecursiveness are followed by the study of a property of spectra that is not uniform. These foreshadow an abstraction of this notion and its integration with the algebraic and computational studies—steps that may eventually help explicate Tarski's claim that recursively enumerable, nonrecursive but pseudorecursive equational (...) theories are nonetheless decidable. (shrink)

The Church-Turing Thesis (CTT) is often paraphrased as ``every computable function is computable by means of a Turing machine.'' The author has constructed a family of equational theories that are not Turing-decidable, that is, given one of the theories, no Turing machine can recognize whether an arbitrary equation is in the theory or not. But the theory is called pseudorecursive because it has the additional property that when attention is limited to equations with a bounded number of variables, one obtains, (...) for each number of variables, a fragment of the theory that is indeed Turing-decidable. In a 1982 conversation, Alfred Tarski announced that he believed the theory to be decidable, despite this contradicting CTT. The article gives the background for this proclamation, considers alternate interpretations, and sets the stage for further research. (shrink)