We investigate the Church-Kalmar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be dierent.) We also look at various "obstacles" to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the "design" of our future computer). We also ask ourselves, how all this reects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.

We show that the halting times of infinite time Turing Machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of "supertasks" or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Godel's constructible hierarchy: namely that of L where is the supremum of halting times. A number of other open questions are thereby answered. 1 Introduction: Infinite Time Turing Machines Hamkins and Lewis in [4] give an account of the construction of these machines (first developed by Kidder and Hamkins in 1989) and develop the basic theory of this notion of computability. The reader should refer to this paper for a clear and full account of their basic properties, from which all the results and definitions of this introduction are taken. But to summarise: an infinite time Turing machine has...

A version of the Church-Turing Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, non-well-founded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard Church-Turing Thesis.

This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.

A new model of parallel computation - a so called Parallel Turing Machine (PTM) - is proposed. It is shown that the PTM does not belong to the two machine classes suggested recently by van Emde Boas, i.e., the PTM belongs neither to the first machine class consisting of the machines which are polynomial-time and linear-space equivalent to the sequential Turing Machine, nor to the second machine class which consists of the machines which satisfy the parallel computation thesis. Further the notion of a pipelined PTM is introduced and the "period" is defied as a complexity measure suitable for evaluating the efficiency of pipelined computations. It is shown that to within...

to a practical application. A dozen years later the first stored-program electronic digital computers began to spring into existence. All were modelled on the universal Turing machine. Today's digital computers also are in essence universal Turing machines. 2. Is There a Known Upper Bound to Computability? Many textbooks on the fundamentals of computer science offer examples of informationprocessing tasks that are, it is claimed, absolutely uncomputable, in the sense that no machine can be specified to carry out these tasks. For example, it is said that no machine can repond to any given (finite) string of binary digits in accordance with the following rules: 3 (1) Answer '1' if the string is a program that will cause a universal Turing machine on whose tape it is inscribed to execute only a finite number of operations (such programs are called 'terminating'). (2) Answer '0' if the string is not a terminating program; i.e. if the st

We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla over 0. We show that we can apply an argument going back to that of H. Friedman to prove that the set 1-degrees of certain \Sigma 1 2 -correct KP-models of the form L oe (oe ! ! L 1 ) have minimal upper bounds. 1 Introduction Obtaining minimality results in degree theory has a long history: the methods go back to those of Spector when he constructed minimal Turing degrees, and to Gandy-Sacks, [2], for minimal hyperdegrees. The perfect set construction is the common thread to these proofs. A further feature, which is shared, either directly or indirectly, by such arguments, is the use of a selection principle in order to typically, directly shrink a perfect set T ` ! ! to a T 0 so that a particular function is either continuous one-to-one, or constant on the branches of T 0 . For ex...

We characterise explicitly i, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the L i -stables. It also implies that the machines devised are "\Sigma 2 Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. 1 Introduction: Hamkins and Lewis in [4] give a notion of computability degree based on Infinite Time Turing Machines. They define: Definition 1.1 For f; g 2 ! ! write f 1 g /! 9p 2 ! OE g p (0)#f . The right hand side here is to be interpreted in the usual manner for Turing computability, except that computations are allowe...

Submitted to Nonlinearity We introduce a new type of shift dynamics as an extended model of symbolic dynamics, and investigate the characteristics of shift spaces from the viewpoints of both dynamics and computation. This shift dynamics is called a functional shift that is defined by a set of bi-infinite sequences of some functions on a set of symbols. To analyze the complexity of functional shifts, we measure them in terms of topological entropy, and locate their languages in the Chomsky hierarchy. Through this study, we argue that considering functional shifts from the viewpoints of both dynamics and computation give us opposite results about the complexity of systems. We also describe a new class of shift spaces whose languages are not recursively enumerable. 1

Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences over the ground model of the standard natural numbers:   ¢¡-Comprehension Axiom scheme) is insufficient. £-¤¦¥¨ § (second order number theory with a ©��