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In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To… (More)

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

- Gualtiero Piccinini, Darren Abramson, Dave Chalmers, Jack Copeland, Martin Davis, John Earman +7 others
- 2011

This paper defends a modest version of the Physical Church-Turing thesis (CT). Following an established recent trend, I distinguish between what I call Mathematical CT—the thesis supported by the original arguments for CT— and Physical CT. I then distinguish between bold formulations of Physical CT, according to which any physical process—anything doable by… (More)

We show how to determine the k-th bit of Chaitin's algorithmically random real number Ω by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N , has solutions for an odd or an even number of values of N. We also… (More)

- Toby Ord, Tien D. Kieu
- 2005

The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical models of computation which claim to be able to solve the halting problem for Turing machines. We show that such… (More)

- Toby Ord, Alan Blair, Melvin Dresher
- 2002

– We present a new paradigm extending the Iterated Prisoner's Dilemma to multiple players. Our model is unique in granting players information about past interactions between all pairs of players – allowing for much more sophisticated social behaviour. We provide an overview of preliminary results and discuss the implications in terms of the evolutionary… (More)

We show how to determine the ¢-th bit of Chaitin's algorithmically random real number £ by solving ¢ instances of the halting problem. From this we then reduce the problem of determining the ¢-th bit of £ to determining whether a certain Diophantine equation with two parameters, ¢ and ¤ , has solutions for an odd or an even number of values of ¤. We also… (More)

It is often claimed that from the moment of conception embryos have the same moral status as adult humans. This claim plays a central role in many arguments against abortion, in vitro fertilization, and stem cell research. In what follows, I show that this claim leads directly to an unexpected and unwelcome conclusion: that natural embryo loss is one of the… (More)

- Toby Ord, Tien D. Kieu
- IJUC
- 2009

While it is well known that a Turing machine equipped with the ability to ip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set X may be coded as a probability p X such that if a Turing machine is given a coin which lands heads with probability p X it can compute any… (More)