[mmarx]

> That's comparing apples to oranges, because the two input types have vastly different lengths relative to their value.

It really is not, because complexity is fundamentally a function of _the length of the input_ (specifically: it measures how the runtime (or space usage) grows with growing input). If you have longer input, your Turing machine can spend more time to compute its answer. Also note that this assumes that _the input_ is encoded in unary, if you get binary input and need to spend time and space to convert it to unary representation, sure, that will lead to an exponential blow-up.

Edit: > Just think about the rough number of steps you would need to add or even multiply two very large numbers, e.g. in a Turing machine. It would be obviously vastly more if the numbers are given in unary rather than in binary.

First, note that those are not actually pseudo-polynomial, as the number of steps needed for addition (or multiplication) depend on the number of digits, not the numeric values involved. Yet, even here unary encoding _does not have worse complexity_, since you still take polynomially many steps in the length of the input (i.e., the length of the unary encodings of the input values). Yes, all the inputs will be exponentially larger, so in practice it's certainly not the better algorithm, but _the complexity_ is not worse, since that's only concerned with the asymptotic behaviour.

[cubefox]

When we deal with algorithms involving numbers, we are interested in actual numbers, not in their numeral representation. It would be absurd to compare unary and binary addition relative to the same numeral length. We are ultimately interested in numbers, not numerals. Adding one million to one million is simply much faster for binary addition than for unary addition, even if unary addition has better complexity relative to its own encoding than binary addition has to its encoding. We are interested in complexity relative to the numerical value, not in the length of their respective encodings.

Edit: By the way, as the Wikipedia piece notices, the binary addition algorithm is O(log(n)) in time _relative to the value_, while unary addition is presumably O(n) relative to the value (just writing the numeral out alone takes O(n) steps). So the time complexity is in fact better. Probably the same holds for space complexity. Moreover, only in this case makes a comparison even sense, since we are comparing the same values in both cases, while for the numeral case we would be comparing the lengths of two different types of numerals, apples to oranges.

[mmarx]

> It would be absurd to compare unary and binary addition relative to the same numeral length.

Why? The Turing machine has no concept of numeric values, it only knows about the length of whatever the input is.

> Adding one million to one million is simply much faster for binary addition than for unary addition, even if unary addition has better complexity relative to its own encoding than binary addition has to its encoding.

From a complexity standpoint, adding one million to one million is O(1), irregardless of the encoding.

> We are interested in complexity relative to the numerical value, not in the length of their respective encodings.

But the complexity relative to the numerical value is not even well-defined, since it _depends_ on the choice of the encoding.

> By the way, as the Wikipedia piece notices, the binary addition algorithm is O(log(n)) in time _relative to the value_, while unary addition is presumably O(n) relative to the value (just writing the numeral out alone takes O(n) steps). So the time complexity is in fact better.

It really is not better. Time complexity is _always_ measured relative to the length of the input, and for binary encoding, the length of the input is O(log(n)), so, taking O(log(n)) steps for the addition is linear, same as the linear time needed for adding numbers in unary encoding (which is basically just copying the input to the output).

> Moreover, only in this case makes a comparison even sense, since we are comparing the same values in both cases, while for the numeral case we would be comparing the lengths of two different types of numerals, apples to oranges.

But _the numeral is not the input_, its _encoding_ is. This is really the whole point, and it is _precisely_ because you get different complexities for different encodings.

[cubefox]

> Why? The Turing machine has no concept of numeric values, it only knows about the length of whatever the input is.

That's irrelevant. We are interested in addition, or multiplication, or whatever, which is an operation between numbers (values), and it can be faster or slower depending on which numerals (encoding) are used.

> But the complexity relative to the numerical value is not even well-defined, since it _depends_ on the choice of the encoding.

Yes it depends on the encoding, no it is of course well-defined. You can measure two different algorithms which use two different encodings, relative to the same thing, the value.

> complexity is _always_ measured relative to the length of the input

That's simply not true. There is even a Wikipedia article about it. Even if it calls it "pseudo".

> But _the numeral is not the input_, its _encoding_ is

A numeral is an encoding of a number.