Can differential calculus (limits, integrals, derivatives) be encoded in lambda calculus?

Question

I am wondering, if the Church-Turing thesis holds (all effectively calculable functions are computable by Turing machines/lambda calculus) and I can compute the limit of a function by hand, what is the encoding of e.g. the derivative $\lim_{h\to 0} \frac{df(x)}{dh} = \frac{f(x+h)-f(x)}{h}$.

I know the encoding for the divide and plus sequences, but how would one encode the limit?

this sounds like turing machines and our models of computability have problems dealing with real numbers in general — qrzx, Dec 16 '14 at 00:28
Yes, and real numbers are extremely important to our understanding of limits, and hence derivatives. — Ben Grossmann, Dec 16 '14 at 00:30
Are there any models of computation that allow for computation with reals ( or a representation of reals )? I found this Wikipedia entry stating it is probably impossible to realize reals in our universe, but surely there must be approaches to deal with them in an abstract form? — qrzx, Dec 16 '14 at 00:36
Huh, ok, there exist proof that you cannot map the reals to the natural numbers, so I guess that makes it very difficult — qrzx, Dec 16 '14 at 00:38
Well, by some definition, you could probably just call the axioms of mathematics (ZFC, perhaps) a "model of computation". — Ben Grossmann, Dec 16 '14 at 00:38
But this does not allow us to build machines that compute for us... — qrzx, Dec 16 '14 at 00:43
Not in the robust mathematical sense, anyway. However, there are certainly machines that "calculate" certain derivatives, either formally or via numerical approximation. — Ben Grossmann, Dec 16 '14 at 00:49
Limit is just second order logic formula you can encode it using lambda calculus. — Trismegistos, Jan 12 '15 at 13:08

score 1 · Answer 1 · answered Dec 16 '14 at 01:04

First of all, we have to be careful here. Just because you and I can solve particular limits by hand, do you have an algorithmic method (by hand or otherwise) to solve limits that'll work for EVERY limit problem?

Also, it would be impossible to represent the result of each limit problem finitely since the cardinality of the reals outnumber the cardinality of finite length strings over any finite alphabet. Because of this, I would say computing limits in general (and thus derivatives) is uncomputable.

Of course however, there are numerical methods that give us good approximations for these kind of things.

Can differential calculus (limits, integrals, derivatives) be encoded in lambda calculus?

1 Answers1