A function f is computable =def there is an algorithm that computes f;
i.e., there is an algorithm A such that for all input i, A(i)=f(i)
and A specifies how f's input and output are related.
algorithm for P =def a finite procedure (i.e., a finite set of
instructions) for solving P that is:
"effective" or unambiguous for the computer or human who will execute it;
i.e., all steps of the procedure must be clear and well-defined for the
executor, and
it must eventually halt,
and it must output a correct solution to P.
Turing Machines
history of term "computable"
Turing's Thesis: a function is computable iff it is TM-computable.
Church's Thesis: a function is computable iff it is lambda-definable (= recursive)
What is a computer? Philosophy
Searle, "Is the Brain a Digital Computer?"
Yes; because everything is a digital computer
Hayes et al. symposium
computer = machine that can take as I/P
patterns that describe changes to themselves
& other patterns, and that causes the
described changes to occur
Is the universe a computer? (Lloyd, Wolfram)
What is an algorithm? Philosophy
Cleland:
Mundane procedures (causal processes,
including recipes) are effective procedures
that are not TM-computable
Effectiveness of mundane procedures depends on external
world
Preston on difference between recipes and algorithms
recipes are more like specifications
What is hypercomputation?
=def computation of functions that can't be TM-computed
Turing's "oracle" machines
Putnam's & Gold's "trial & error" machines (TM, but last answer counts)