A Natural Axiomatization of Church's Thesis. Nachum Dershowitz and Yuri Gurevich. July 2007.



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 GÃ¶del and others suggested may be possible.

While not directly dealing with programming languages, I still think this paper might be of interest, since our field (and our discussions) are often concerned with computability (or effective computation, if you prefer).

The idea the Church's Thesis can be proven is not new, and indeed there seems to be a cottage industry devoted to this issue (for a quick glance search the paper for references to Gandy and work related to his).

Even if the discussion of ASMs is not your cup of tea, it seems like a good idea to keep in mind the distinctions elaborated in the conclusions section between "Thesis M" (the "Physical C-T Thesis"), the "AI Thesis", and the "standard" meaning of the C-T thesis.

Another quote (from the same section) that may prove useful in future LtU discussions: We should point out that, nowadays, one deals daily with more flexible notions of algorithm, such as interactive and distributed computations. To capture such non-sequential processes and non-classical algorithms, additional postulates are required.