Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with super-Turing power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or super-Turing. 1

Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper-) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the self-contradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.

We discuss the question of how to operationally validate whether or not a “hypercomputer ” performs better than the known discrete computational models. 1.

The accelerated Turing machine (ATM) is the work-horse of hypercomputation. In certain cases, a machine having run through a countably infinite number of steps is supposed to have decided some interesting question such as the Twin Prime conjecture. One is, however, careful to avoid unnecessary discussion of either the possible actual use by such a machine of an infinite amount of space, or the difficulty (even if only a finite amount of space is used) of defining an outcome for machines acting like Thomson’s lamp. It is the authors ’ impression that insufficient attention has been paid to introducing a clearly defined counterpart for ATMs of the halting/non-halting dichotomy for classical Turing computation. This paper tackles the problem of defining the output, or final message, of a machine which has run for a countably infinite number of steps. Non-standard integers appear quite useful in this regard and we describe several models of computation using filters.

The accelerated Turing machine (ATM) is the work-horse of hypercomputation. In certain cases, a machine having run through a countably infinite number of steps is supposed to have decided some interesting question such as the Twin Prime conjecture. One is, however, careful to avoid unnecessary discussion of either the possible actual use by such a machine of an infinite amount of space, or the difficulty (even if only a finite amount of space is used) of defining an outcome for machines acting like Thomson’s lamp. It is the authors’ impression that insufficient attention has been paid to introducing a clearly defined counterpart for ATMs of the halting/non-halting dichotomy for classical Turing computation. This paper tackles the problem of defining the output, or final message, of a machine which has run for a countably infinite number of steps. Non-standard integers appear quite useful in this regard and we describe several models of computation using filters.

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We discuss the question of how to operationally validate whether or not a “hypercomputer ” performs better than the known discrete computational models. 1