In the computational literature the term
“Church-Turing thesis” is applied to a
variety of different propositions usually not equivalent to the original thesis—CTT-O; some even go far beyond
anything either Church or Turing wrote.
Several but not all are fundamental assumptions of computer science. Others
(such as the various physical computability theses we have discussed) are important in the philosophy of computing
and the philosophy of physics but are
highly contentious; indeed, the label
“Church-Turing thesis” should not mislead computer scientists or anyone else
into thinking they are established fact
or even that Church or Turing endorsed
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B. Jack Copeland ( firstname.lastname@example.org) is
Distinguished Professor of Philosophy at the University of
Canterbury in Christchurch, New Zealand, and Director of
the Turing Archive for the History of Computing, also at
the University of Canterbury.
Oron Shagrir ( email@example.com) is Schulman
Professor of Philosophy and Cognitive Science at the
Hebrew University of Jerusalem, Jerusalem, Israel.
Copyright held by the authors.
Publication rights licensed to ACM. $15.00
classical concept, saying, “[E]quilibrat-
ing can be so easily, reproducibly, and
mindlessly accomplished” that we may
“take the operation of equilibrating as
an effective one,” even if “the functions
computable in principle given Turing’s
operations and equilibrating include
Over the years, there have been sever-
al departures from Turing’s 1936 analy-
sis, as the needs of computer science
led to a broadening of the algorithm
concept. For example, Turing’s fourth
axiom, which bounds the number of
parts of a system that can be changed
simultaneously, became irrelevant
when the algorithm concept broadened
to cover parallel computations. The fu-
ture computational landscape might
conceivably include more extensive re-
visions of the concept, if, for example,
physicists were to discover that hard-
ware effective in Doyle’s extended sense
is a realistic possibility.
If such hardware were to be developed—hardware in which operations
are effective in the sense of being “
easily, reproducibly, and mindlessly accomplished” but not bounded by Turing
computability—then would the appropriate response by computer scientists
be to free the algorithm concept from
CTT-A? Or should CTT-A remain as a
constraint on algorithms, with instead
two different species of computation being recognized, called, say, algorithmic
computation and non-algorithmic computation? Not much rides on a word, but
we note we prefer “effective computation” for computation that is bounded
by Turing computability and “
neo-effective computation” for computation
that is effective in Doyle’s sense and not
bounded by Turing computability, with
“neo” indicating a new concept related
to an older one.
The numerous examples of notional
“hypercomputers” (see Copeland9 for
a review) prompt similar questions. Interestingly, a study of the expanding literature about the concept of an infinite-time Turing machine, introduced by
Joel Hamkins and Andy Lewis in 2000,
shows that a number of computer scientists are prepared to describe the in-finite-time machine as computing the
halting function. Perhaps this indicates
the concept of computation is already
in the process of bifurcating into “
effective” and “neo-effective” computation.