We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy’s thesis.

Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectively-computable functions on string representations of numbers.

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...

Abstract. A theory of one-tape linear-time Turing machines is quite different from its polynomial-time counterpart since one-tape linear-time Turing machines are closely related to finite state automata. This paper discusses structural-complexity issues of one-tape Turing machines of various types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in linear time, where the running time of a machine is defined as the height of its computation tree. We clarify how the machine types affect the computational patterns of one-tape linear-time Turing machines.

The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turing-computable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.

A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development

Acknowledgements Thanks to Leslie Smith, the rest of the Computing Science dept, and the ex-members of the CCCN. The University and the Computing Science department provided funding for this project. M. Elton deserves thanks for showing me how difficult it is to go head- to-head with contemporary philosophers who have abandonded `old fashioned ' notions of empirical validity, epistemological justification, and ontological grounding. Thanks most of all to friends who aren't in any way associated with the university, the department, or what I did here. They provided a nicely separate world away from the invented world of academia to which I could escape. M S and R deserve special mention, as does L who showed me something real at a point where I thought it wasn't. I ought to point out that this is all Darragh Smyths fault. And I'd like to thank my parents, our parents, those beings that crawled, in the spirit of scientific enterprise and discovery, from the sea into the trees (and later that descended from the trees) all those years ago. Without you, I wouldn't be here, I'd be in the sea, like the dolphins, having a whale of a time. i

This paper is about the problem of sense extension. "Sense" can be understood as its non-technical manifestation --- the sense or meaning of a word. Specifically, this paper deals with sense extension as applied to non-literal meaning, i.e. the generation of metaphor.

In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty| arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...

In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty| arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...