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If I understand it correctly, an algorithm that computes the value of a real function $f$ has computational complexity $O(g(n))$ if the following holds: When we compute $f$ to precision $\delta$ requires on the order of $g(n)$ steps.

However, what if we have an algorithm that first "finds a more efficient algorithm to compute $f$", and then computes $f$?

In other words, what if we have an algorithm $A$ that does the following:

  1. Find an efficient algorithm $B$ for computing $f$.

  2. use $B$ to compute $f$.

In that case, we can no longer speak of the computational time it would take to compute $f(5)$ for example, because it fully depends on whether Algorithm $A$ has already found algorithm $B$. In other words, the computing time required to compute $f(5)$ if $5$ is the first comoputed number is far greater than the computational time required to compute $f(5)$ after $f(3)$ is already computed.

My question is, is there a concept/theory about this kind of algorithm that first finds another algorithm before computing a function? Specifically I am wondering about analysis of the computational complexity of such algorithms.

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    $\begingroup$ Would you say Mathematica basically does what you are asking? You give it equations to solve and it automatically figures out which algorithm to use to solve those equations, then solves them. $\endgroup$ – Mehrdad Dec 31 '17 at 21:08
  • $\begingroup$ Check out itu.dk/people/sestoft/pebook, it's relevant. $\endgroup$ – Nathan Ringo Dec 31 '17 at 22:47
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There is a well-known algorithm, Levin's universal search algorithm, whose mode of operation is identical. Consider for example the problem of finding a satisfying assignment for a formula which is guaranteed to be satisfiable. Levin's universal search runs all potential algorithms in parallel, and if any algorithm outputs a satisfying assignment, stops and outputs this assignment. If the optimal algorithm for the problem runs in time $f(n)$, then Levin's algorithm runs in time $O(f(n))$ (with a possibly huge constant) if implemented correctly.

While Levin's algorithm is impractical (due to the huge constants involved), it is very interesting theoretically. See the Scholarpedia article for more on universal search.

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Suppose we have a function f which takes an argument x of type A, and outputs another function which takes an argument y of type B and returns a result of type C. In your words, f takes an argument x and returns an "algorithm" which takes inputs of type B and outputs results of type C.

The function f has the type

A → (B → C)

Indeed, it takes x : A and returns a function of type B → C. But such an f is equivalent to a function g : A × B → C which takes both x and y at once and gives you the final result. Indeed, there is an isomorphism between the types

A → (B → C)

and

A × B → C

because we can define g in terms of f as

g(x, y) := f(x)(y)

and we can define f in terms of g as

f(x) := (y ↦ g(x,y))

The operation of passing from g to f is called currying and functional programmers use it all the time. In computability theory the idea of taking one input and outputing a function (algorithm) is emboddied in the s-m-n theorem.

The answer to your question is "yes, people do this all the time". But there is also a moral: an algorithm which finds an algorithm is still just an algorithm.

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    $\begingroup$ +1 for that final sentence. Well said. $\endgroup$ – John Coleman Dec 31 '17 at 15:04
  • $\begingroup$ "an algorithm which finds an algorithm is still just an algorithm". Yes that is true, but there is a fundamental difference. If a "direct" algorithm computes $f(5)$ twice, it will cost him $c+c$ steps to do it, where $c$ is the steps required to compute $f(5)$ once. On the other hand, if an algorithm that first searches for another algorithm computes $f(5)$ it costs him $c_1+c_2$ steps, where $c_1$ is the cost of the first computation, and $c_2$ of the second, and $c_1>c_2$, likely vastly bigger. This is the phenomenon I'm interested in. $\endgroup$ – user56834 Dec 31 '17 at 15:13
  • $\begingroup$ @Programmer2134 would compiler optimizations be the concept you're interested in? I'm unsure of the theory behind this at all (especially its interactions with complexity theory), but this could be a potential example $\endgroup$ – Mark Jan 1 '18 at 1:53
  • $\begingroup$ The buzzword to look for is "partial evaluation". $\endgroup$ – Andrej Bauer Jan 1 '18 at 12:43

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