While it goes without saying that computing devices need space and time (spacetime), nothing in Turing’s theory of idealized computation prepares one for the rich nature of a computer’s dependence on the underlying spacetime geometry. Nor does that theory prepare one for the vast array of non-Turing computers (and one computer à la Turing) obtained by painting ordinary Turing machine hardware o...

A new computing model, called the active element machine (AEM), is presented that demonstrates Turing incomputable computation using quantum random input. The AEM deterministically executes a universal Turing machine (UTM) program η with random active element firing patterns. These firing patterns are Turing incomputable when the AEM executes a UTM having an unbounded number of computable steps...

A configuration of a Turing machine is given by a tape content together with a particular state of the machine. Petr Kůrka has conjectured that every Turing machine – when seen as a dynamical system on the space of its configurations – has at least one periodic orbit. In this paper, we provide an explicit counter-example to this conjecture. We also consider counter machines and prove that, in t...

Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms.

Frequency computation was introduced in [16]. Trakhtenbrot [17] proved the existence of a continuum of functions computable by frequency Turing machines with frequency 1 2 . In contrast, every function computable by a frequency Turing machine with frequency exceeding

The time separation results concerning classes of languages over a single-1etteK alphabet accepted by multi-tape nondeterministic Turing machines. well-known from Seiferas, Fischer and Meyer (1978), are supplemented. Moreover, via a universal machine a modified time complexity measure UTIME of Turing machines computations which is sensitive to multiplication by constants (i.e., UTIME(t) s UTIME...