Yes, Glen’s final para (and the one before) were the only interpretation I meant.
I wasn’t trying to be imaginative at all in the short thing I wrote about costs in relation to reversibility. Just the plain-vanilla stuff, for the sake of maybe articulating a theorem within the usual assumptions. So, for a classical computer, if I used the expression “well-formed”, I just meant “computable” in the usual Church-Turing sense, and written in a representation that does execute on whatever computer one is working with. So, generates an answer and stops. For quantum computers, also deterministic, but in the sense of unitary evolution of a state. I haven’t spent time learning how this works, so I just guess at what it must be. If I recall, for Schor’s algorithm to factor products of primes, one starts with some representation for the product, then evolves for a while, and at some definite time later, probability concentrates on a subset of the qubits that indicate what the factors are, with some probability of ambiguity. Since all this is unitary, there must be a conserved volume of state space, so I have supposed that this unitary evolution shouldn’t asymptotically converge forever on a best-estimator for the factors; rather it should pass through some best estimator at a finite time and then diverge again into things that are not easy to interpret. Something like a Poincare cycle. If I am not wrong in my imagination of how this works, then the thing that would take the place of halting in the classical computer would be whatever tells you how long you should evolve before trying to interpret the output, or else a criterion that the output is as well-concentrated (locally in time) as it is going to get. Since unitary evolution is reversible (in idealization), I suppose one could even look for a minimum of wave-function support, and if one has passed that, back up the computation to get close to the minimizer state. Whereas the notion of “all the computable programs” is believed (as I understand it) to be defined for a classical Turing machine (even if uncomputable to enumerate), I have never heard anyone talk about, and haven’t tried to imagine myself, what all the interpretable inputs to a quantum computer might be, as some kind of cellularization of the multi-qubit input state. So I tried to stay away from more than mentioning quantum computing. To try to use these formal models as metaphors for reasoning with ambiguous methods is of course interesting to do. But I didn’t mean to do that. To say anything that might have content would require arguing for some method for using metaphors from the formal world, to the unavailable world of “reasoning”, and I don’t claim to have anything to offer toward that. Eric > On Jan 14, 2025, at 14:10, glen <geprope...@gmail.com> wrote: > > Thanks. I get your answer to (1). I'm still unclear on (2). Yours is more > useful than Frank's because his leaves open what it means in relation to > computation. Both your clause "answer becomes obvious/self-evident" and > Eric's original sense of tautology (with excess) give me some hints at what's > required beyond Frank's. That sense has something to do with the successive > iteration ... like "effectiveness" I guess. If you have a state and there's > no ambiguity in the *next* state, then that sentence has an inevitable, > deterministic successor. I guess it need not even be deterministic. But a > sampling strategy has to be also be well-formed such that the computation can > move along without supervision by a daemon like us or some outside agent. > I.e. well-formed means has a unique inference. I don't see how it can be a > computation without this inevitable chunking along. > > In whatever my poor conception of normal logic is, whatever > transformations/inferences you make on a sentence takes you to another (true) > sentence. But your choice of transformation can take it further from whatever > final form you want, as well as closer to that final form. But there's a > sense of the word "computation" that implies it's completely mechanical. No > logician is needed for the transformations to occur. > > Do y'all intend to say such things? I mean, for reversibility to obtain, it > kinda has to be that way, right? You can't have arbitrary branches in the > inferences if you want to restore the initial state from the final state. > Unless, perhaps, you can ensure that the logic, itself, is somehow convex ... > so that every sentence is derivable from every other sentence. Is that what > we're talking about? Sorry for my confusion and thanks for having the > conversation in public! > > On 1/13/25 16:49, steve smith wrote: >> glen asked >>> Question 2: What does "well-formed" mean in this concept of computation? >> 2) I suppose this is EricS's question, but here is my answer. I think of >> "well formed questions" being the province of "science" more than >> "engineering" or "computation" but am not prepared to say that either are >> fundamentally different than "science". In Science I intuit a maxim that >> "if the question is well enough crafted, the answer becomes >> obvious/self-evident"... some kind of Willem du Occam corollary? 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