Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at least for functions f .. R → N, the same class of computable functions. Nevertheless, there are infinite time computable functions f : R → R that are not one-...

The main current issue about hypercomputation concern the following thesis: it is physically possible to build a hypercomputation model. To prove this thesis, one possible strategy could be to physically build an oracle Turing machine. More precisely, it is about finding a physical theory where a hypercomputation model will be able to use some external information from nature. Such an informati...

We demonstrate how the DSD programming language can be used to design a DNA stack machine and to analyse its behaviour. Stack machines are of interest because they can efficiently simulate a Turing machine. We extend the semantics of the DSD language to support operations on DNA polymers and use our stack machine design to implement a non-trivial example: a ripple carry adder which can sum two ...

We provide two complexity measures that can be used to measure the running time of algorithms to compute multiplications of long integers. The random access machine with unit or logarithmic cost is not adequate for measuring the complexity of a task like multiplication of long integers. The Turing machine is more useful here, but fails to take into account the multiplication instruction for sho...

Church’s thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing’s work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church’s thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing’s work involve a subtle but p...

A Turing machine that computes Fibonacci numbers is de-

Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at least for functions f .. R → N, the same class of computable functions. Nevertheless, there are infinite time computable functions f : R → R that are not one-...

Relativized forms of deterministic and nondeterministic time complexity classes are considered. It is shown, at variance with recently published observations, that one basic step by step simulation technique in complexity theory does not relativize (i.e., does not generalize to computations with oracles), using the current definitions for time and space complexity of oracle computations. A new ...

We show that, for any ε > 0, there exists a language accepted in strong ε·log n space by a 2-way deterministic Turing machine working with a single binary worktape, that cannot be accepted in sublogarithmicweak space by any pebble machine (i.e., a 2-way nondeterministic Turing machine with one pebble that can be moved onto the input cells). Moreover, we provide optimal unary lower bounds on the...

In the previous lectures, we have introduced the concept of deterministic oracle Turing machines. The concept of non-deterministic oracle Turing machines is just a direct extension of that to the non-deterministic case. To be precise, a non-deterministic oracle Turing machine (sometimes abbreviated to “oracle NTM”) is just like a deterministic oracle TM, but with a non-deterministic transition ...