*If the number of bits is constant, then any polynomial time based on it is constant.*
Big-O statements (like an algorithm taking constant or O(1) time) are statements about asymptotic behavior, i.e, how the function behaves in the limit (usually, as input size tends to infinity). If you don't look at them in the limit, then big-O-ish language (like constant time) is meaningless.
How meaningless? Even undecidable languages have a constant time "algorithm" if you consider the input size to be held to a constant. So without viewing things in the limit, *all* problems become computationally equivalent in the asymptotic language.
**Update:** added citation from parent node
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You're right, blokhead. I came to conclusions too fast after reading Q::S documentation. I think the documentation is badly formulated about this point, too.
Thanks for correcting.
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I really don't want to beat this thread into the ground, but...
*It is reasonable to assume that some inputs will vary from 1 to very large. I do not agree with you that it is reasonable to assume that input numbers will in this case. There will be some interesting size of number, and not much variation around it.*
My point is not that one *must always* consider input sizes from 1 to infinity. It's completely reasonable to only be interested in finding the most efficient algorithm for a specific range of input sizes. But you originally said that in such a case, the running time of a polynomial algorithm would be constant, as if it were a meaningful statement. With inputs restricted to a fixed interval, it is the case that (with very few exceptions) *any function is big-O of any other function*, so big-O language is meaningless. (Although you didn't explicitly mention big-O, it is implied when you say that the running time is "constant." If this isn't what you meant, then fine.)
If one is really only interested in a specific range of inputs (which is fine), one should use actual running time numbers for comparison (which are comparable and therefore meaningful), not asymptotics (which are all equivalent when not being considered in the limit).
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