Is effective solvability a coherent and/or useful concept?
I am aware of <PERSON>'s proof of the undecidability of the halting problem (and I think I understand it). What I'm asking is quite different. I shall define what I mean by "effectively solvable":
A problem $P$ is effectively solvable if there exists an agent $A_P$, such that for any specific instance of $P \, (P_i)$, $A$ can solve that instance of the problem $(P_i)$ in a finite time.
Imagine $A_P$ as a superintelligence with unbounded (but finite) computational power, and unbounded (but finite) memory. $P$ is effectively solvable if $A_P$ would eventually (in finite time) produce a correct solution for any given instance of $P$.
It may be the case that there does not exist any algorithm that "solves" a certain class of problems say sigma $$ problems (suppose that sigma problems are a certain kind of mathematical problem). However, this does not mean the problem is not effectively solvable. For any given sigma problem, a sufficiently intelligent agent (or sufficiently competent mathematician) may be able to eventually find its solution(s) (or determine that no such solutions exist) using an (unbounded*) number of mathematical tricks. The mathematical methods applied to sigma problems is limited only by the creativity of the agent concerned. There are essentially an infinite number of strategies, from which a finite subset may be drawn from in sequence (forming an algorithm) to apply to a particular sigma problem.
Effective solvability (or effective decidability) is different from the traditional notion of decidability in that problem that are effectively solvable do not require a finite number of steps which always guarantee a solution. In fact, for each instance of an effectively solvable problem, a different algorithm may be used. There may be no general algorithm that solves all instances of a problem, but for each instance of the problem, a tailor made algorithm may exist.
Going entirely off my intuition here, I think the following is a definition of "effectively solvable":
A problem $P$ is effectively solvable for $A_P$ if $\not\exists$ an instance $P_i$ of $P | A_P$ does not solve $P_i$ in finite sequence of steps.
OR:
A problem $P$ is effectively solvable if $\forall$ instances $P_i$ of $P$, $A_P$ can correctly solve $P_i$ in finite time.
If there is a tailor made algorithm for every instance $P_i$ of a problem $P$, then a sufficiently intelligent agent can in practice solve every instance of the problem (by searching solution space for said algorithm when confronted with that instance of the problem). 603
Thus even if in theory no general solution to the problem exists, in practice, specific solutions to each $P_i$ always exist (provided the above criteria is satisfied).
While some of such problems may be "undecidable", they are in actuality (for all practical purposes) "effectively solvable".
The set of effectively solvable problems is a superset (and I believe that it is a proper superset) of the set of decidable problems.
I do not know whether the halting problem is effectively solvable (I'm starting to believe that it's not). That's not my major interest though (I am curious to the answer) What I am truly interested in is effective solvability.
I am interested in effective solvability, because the range of problems that can be solved by an AI is not (necessarily) the range of problems that can be solved by a Turing machine, but the range of problems that are effectively solvable. I feel that the traditional (<PERSON>) notion of decidability is insufficient when the computer is a sufficiently (pun intended) intelligent agent. Thus when we ask the question:
"What problems can a computer solve?"
I feel the set of effectively solvable problems and not (necessarily) the set of decidable problems is the answer.
What are some problems that are not effectively solvable by your definition?
I do not know of any, and I suspected there may be none. To hazard a guess however:
1. Problems which may involve randomness may not be effectively solvable. In particular if the computer gets the right answer, but doesn't know it got the right answer, then it can never solve that instance of the problem. Such problems are thus not effectively solvable.
As a concrete example of such a problem, imagine a machine delta. Delta is non deterministic. Delta (uses its input) to randomly select/construct an arbitrary Turing machine $M$. 603